Hydrogen Bonding: A Theoretical Perspective (Topics in Physical Chemistry)

Hydrogen Bonding: A Theoretical Perspective

STEVE SCHEINER

Oxford University Press

HYDROGEN BONDING

TOPICS IN PHYSICAL CHEMISTRY

A Series of Advanced Textbooks and Monographs Series Editor, Donald G. Truhlar

F. Iachello and R. D. Levine, Algebraic Theory of Molecules P. Bernath, Spectra of Atoms and Molecules J. Cioslowski, Electronic Structure Calculations on Fullerenes and Their Derivatives E. R. Bernstein, Chemical Reactions in Clusters J. Simons and J. Nichols, Quantum Mechanics in Chemistry G. A. Jeffrey, An Introduction to Hydrogen Bonding S. Scheiner, Hydrogen Bonding: A Theoretical Perspective

HYDROGEN BONDING A Theoretical Perspective

STEVE SCHEINER

New York Oxford Oxford University Press 1997

Oxford University Press Oxford New York Athens Auckland Bangkok Bogota Bombay Buenos Aires Calcutta Cape Town Dar es Salaam Delhi Florence Hong Kong Istanbul Karachi Kuala Lumpur Madras Madrid Melbourne Mexico City Nairobi Paris Singapore Taipei Tokyo Toronto Warsaw and associated companies in Berlin Ibadan

Copyright © 1997 by Oxford University Press, Inc. Published by Oxford University Press, Inc., 198 Madison Avenue, New York, New York 10016 Oxford is a registered trademark of Oxford University Press. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data Scheiner, Steve. Hydrogen bonding : a theoretical perspective / Steve Scheiner. p. cm. — (Topics in physical chemistry) Includes bibliographical references and index. ISBN 0-19-509011-X 1. Hydrogen bonding. 2. Quantum chemistry. I. Title. II. Series. QD461.S36 1997 541.2'26—dc21 96-48698

9 8 7 6 5 4 3 2 1 Printed in the United States of America on acid-free paper

To my mother: You will not be forgotten

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Preface

hy a theoretical perspective of hydrogen bonding? After all, there have been numerW ous texts, monographs, and compilations written about hydrogen bonds over the years . Much of this literature has taken the viewpoint of the crystallographer or spec1-7

troscopist, with the emphasis placed on structural aspects of the H-bonded complexes in their equilibrium geometries or their modes of internal vibration. Quantum chemical calculations offer a rich source of supplementary information concerning this phenomenon. First, much of the literature that has accumulated over the years concerning H-bonds has been gathered in solvents of various types. One is then presented with the problem of separating the intrinsic properties of the complex under investigation from the perturbations incurred by interactions with the solvent medium. In contrast, in vacuo investigation of systems, in isolation from surroundings, is a definite strength of computational methods, as they are free of complicating solvent effects. While spectroscopic data can provide details of the equilibrium geometries of H-bonded complexes, it has proven difficult to extract the energetics of the interaction. Growing sophistication of computer hardware and more efficient algorithms have dramatically enhanced the level of accuracy that can be expected from the calculations, which can address the energetics directly. Theoretical approaches offer an additional dividend: dissection of the interaction energy into various physically meaningful components, which can provide insights into the fundamental nature of the interaction. Whereas experimental data are largely relevant to the global minimum in the potential energy surface, computational methods can map entire domains of this surface. One can locate secondary minima and stationary points of higher order in addition to the minima. It is also feasible to identify interconversion pathways from one minimum to another, along with magnitudes and shapes of energy barriers along these paths. Quantum chemical methods have capabilities that are not limited to energetics. It is a straightforward matter to examine in detail the electronic redistributions that accompany the formation of the H-bond. The precise nature of the displacements of the nuclei that com-

viii

Preface

prise the normal vibrational modes can be elucidated with an accuracy that eludes analysis of purely experimental data. This information presents the possibility of genuine understanding of the perturbations in vibrational spectra that accompany formation of a H-bond. Thus, quantum chemistry has a great deal to offer the field of hydrogen bonding. And indeed, the body of pertinent calculated data has been growing at a rapid rate. This book is intended to digest this vast amount of information and organize it into a form that is understandable to a general reader who is interested in hydrogen bonding but has little in the way of formal training in theoretical calculations. A brief introduction is presented so that the reader might obtain some appreciation of the basic ideas behind the computation of various properties, and to prepare for the jargon one is likely to encounter in the literature. This explanation is intentionally brief and simplified: the reader is referred to many fine reviews, monographs, and texts if interested in more detail about these methods or the underlying theory. A glossary of common abbreviations is included to aid the reader in recalling the definition of each term as it appears. While quantum chemical studies of H-bonded systems date back to the 1960s, this book concentrates on calculations made principally since 1980 or so. These results are more reliable in a number of respects, particularly from the standpoint of higher levels of theory and the characterization of stationary points on the potential energy surface. (In fact, an earlier text on the subject of H-bonding3 had compiled a listing of theoretical studies in the 1960-73 timeframe for those interested in some of the earlier work.) The group of articles selected is not an exhaustive one; rather, I have culled those that were considered best able to illustrate a given point, taking into account also their level of accuracy. The emphasis in this book is on ab initio calculations. Semiempirical methods are not designed or parametrized to treat intermolecular interactions well. Indeed, the original formulations of some semiempirical methods did not predict H-bonds to exist at all. There have been attempts to patch them up over the years so as to permit a modicum of attractive potential where a H-bond is expected. However, one cannot depend on the reliability of such approaches. Another important distinction between ab initio and semiempirical methods is that the former will, at least in principle, approach reality as the basis set is enlarged and as the treatment of electron correlation is made more complete. It is thus possible to estimate how close one is to this asymptote, even if enormous basis sets with high orders of correlation are not feasible for a given chemical system, by monitoring the results of the calculations as the level of ab initio theory is improved, one step at a time. If the data are stable to additional improvements of the method, a certain measure of confidence may be attached to the results. Such is not the case with semiempirical methods wherein a single result is obtained, with no real way to improve upon it or test its reliability against a higher level of comparable theory. A new type of method has been gaining popularity very rapidly. Density functional theory (DFT) bypasses the conventional concept of individual molecular orbitals and instead optimizes the total electron density, including all electrons. Based originally on some concepts from solid-state physics, the method scales to a lower order with respect to the number of atoms or electrons, as compared to conventional ab initio theory where the computational effort rises roughly as the fourth or higher power of N. Consequently, this new approach has enormous potential to treat systems that are far too large for ab initio methods to examine. The DFT approach is maturing quickly; it seems that major new developments appear in the literature on a monthly basis. Along with these enhancements in the methodology have come comparisons with data generated by the older and more reliable techniques. Results for hydrogen bonds have been mixed so far; there is still some question

Preface ix

as to which kinds of functionals are most appropriate. The next few years will likely witness improvements to the point where DFT calculations become competitive with conventional ab initio methods in terms of accuracy. However, because this method is relatively new, and has received only limited testing to this point, I have for the most part avoided any extensive discussion of the DFT results here. It is not unlikely that shortly after this book appears, the situation will have changed and the time will be right for an entire text devoted to application of density functional methods to H-bonded systems. The organization of this text is as follows. Chapter 1 presents the reader with a capsule summary of quantum chemical methods. The intent is not to make the reader an expert in various theoretical approaches but rather to provide a basic understanding of the techniques, their strengths, and limitations. This chapter introduces and explains much of the jargon and includes a glossary of abbreviations that a reader is likely to encounter in the original literature. Chapter 1 also provides a tentative definition of a hydrogen bond, and how quantum chemical calculations can address the various contributing factors. A simple example is provided to illustrate the central points for later reference with more complicated systems. This chapter also delves into some detail on the most common sources of error encountered in computations of this sort. Chapter 2 surveys the field of H-bonds that have been studied to date, focusing on smaller molecules for which the calculations are most definitive. These small molecules serve as models of the functional groups that occur in larger systems as well. This chapter focuses on the energetics of various combinations of partners, and the details of their equilibrium geometries. Emphasis is placed on systematic relationships between the properties of the constituent molecules and the nature of each H-bond. Also discussed are the perturbations that occur in each molecule as the H-bond is formed. One way in which the fundamental forces responsible for the formation of a H-bond can be probed is by examining of the force field that restores the equilibrium geometry after small geometrical distortions. This field is directly manifested by the normal vibrational modes that exhibit themselves in the vibrational spectrum of the complex. Chapter 3 is hence devoted to a discussion of the vibrational spectra of H-bonded complexes, and what can be learned from their calculation by quantum chemical methods. While the vibrational frequencies are directly related to the forces on the various atoms, the intensities offer a window into the electronic redistributions that accompany the displacement of each atom away from its equilibrium position, so vibrational intensities are also examined in some detail. Of particular interest are relationships between the vibrational spectra and the energetic and geometric properties of these complexes. The vibrational spectrum provides information about the potential energy surface in the immediate vicinity of the minimum. Chapter 4 broadens the scope by considering wider swaths of the surface. Large deviations from the equilibrium geometry are examined, some of which take the system to a secondary minimum on the surface. This chapter discusses paths between various minima that pass through stationary points of higher order, and provides a broad picture of the general topology of the entire surface. Hydrogen bonding is particularly important in condensed phases where it can significantly affect such properties as boiling point or crystal structure. H-bonds seldom occur in isolation in condensed phases but are commonly part of a chain of molecules held together by such interactions. The effect of one H-bond on another is the subject of Chapter 5, dealing with cooperativity phenomena. The source of this effect is probed, in terms of electronic redistributions and various contributors to the full energetic interaction. The magnitude of cooperativity is considered, as a function of the number of contiguous H-bonds, and the

x

Preface

asymptotic limit of an infinitely long chain of H-bonds. Another central question is whether it is worth the energetic expense of bending the H-bonds in a chain, so as to enable the two ends to approach one another to form an additional H-bond and a cyclic structure of the entire chain. The classification of any given interaction as a H-bond is not always a trivial matter. There are many situations which have certain characteristics in common with a H-bond but others are lacking. Chapter 6 considers a number of interactions whose designation as an "official" H-bond could be called into question. Some of these situations include the possibility of the C—H group acting as a proton donor or an electronegative atom in a covalent bond of only small polarity serving as a suitable proton acceptor. Also examined is the question as to whether an interaction in which one or both of the partners bears an electric charge should be considered as a true H-bond. Interactions of this type comprise some of the strongest H-bonds known. The characteristics of the proton transfer potentials in these ionic H-bonds are particularly interesting, sometimes containing two wells while only a single minimum is present in other cases. The chapter explores the relationships between the strength of the H-bond and the intrinsic acidity and basicity of the subunits. A detour is taken to explore the intriguing question as to whether there is a catalytic advantage to one of the two lone pairs of an oxygen atom in the carboxylate group. References 1. G. C. Pimentel and A. L. McClellan. The Hydrogen Bond. Freeman: San Francisco, 1960. 2. S. N. Vinogradov and R. H. Linnell. Hydrogen Bonding. Van Nostrand-Reinhold: New York, 1971. 3. M. D. Joesten and L. J. Schaad. Hydrogen Bonding. Marcel Dekker: New York, 1974. 4. P. Schuster, G. Zundel, and C. Sandorfy, Eds. The Hydrogen Bond: Recent Developments in Theory and Experiments. North-Holland Publishing Co.: Amsterdam, 1976. 5. P. Schuster, Ed. Hydrogen Bonds. Vol. 120. Springer-Verlag: Berlin, 1984. 6. G. A. Jeffrey and W. Saenger. Hydrogen Bonding in Biological Structures. Springer-Verlag: Berlin, 1991. 7. D. A. Smith, Ed. Modeling the Hydrogen Bond. Vol. 569. American Chemical Society: Washington, D.C., 1994.

Contents

Abbreviations

xvii

I QUANTUM CHEMICAL FRAMEWORK 1.1 Quantum Chemical Techniques 3 1.1.1 Basis Sets 4 1.1.2 Electron Correlation 7 1.1.3 Geometries 10 1.2 Definition of a Hydrogen Bond 11 1.2.1 Geometry 12 1.2.2 Energetics 13 1.2.3 Electronic Redistributions 13 1.2.4 Spectroscopic Observations 13 1.3 Quantum Chemical Characterization of Hydrogen Bonds 1.3.1 H-bond Geometries 15 1.3.2 Thermodynamic Quantities 15 1.3.3 Electronic Redistributions 18 1.3.4 Spectroscopic Observations 18 1.4 A Simple Example 19 1.5 Sources of Error 22 1.6 Basis Set Superposition 23 1.6.1 Secondary Superposition 24 1.6.2 Important Properties of Superposition Error 25 1.6.3 Historical Perspective 25 1.7 Energy Decomposition 28 1.7.1 Kitaura-Morokuma Scheme 32

14

xii Contents 1.7.2 Alternate Schemes 34 1.7.3 Perturbation Schemes 37 2

GEOMETRIES AND ENERGETICS 2.1 XH ZH3 53 2.1.1 BSSE 56 2.1.2 Substituent Effects 60

2.2 XH YH2 61 2.2.1 Comparative Aspects 62 2.2.2 Angular Features 64 2.2.3 Alternate Complexes and Geometries 66 2.2.4 Energy Components 67 2.3 HYH ZH3 69 2.3.1 Substituents 71 2.4 XH XH 71 2.5 HYH YH2 77 2.5.1 Binding Energy of Water Dimer 78 2.5.2 Complexes Containing H2S 79 2.5.3 Substituent Effects 81 2.6 (ZH3)2 84 2.7 Carbonyl Group 89 2.7.1 Substituent Effects 93 2.8 Carboxylic Acid 94 2.8.1 Carboxylic Acid Dimers 99 2.9 Nitrile 101 2.10 Imine 103 2.11 Amide 105 2.11.1 Interaction with Carboxylic Acid and Ester 2.12 Nucleic Acid Base Pairs 113 2.13 H-Bonds versus D-Bonds 118 2.13.1 Water Molecules 120 2.14 Summary 121 3

112

VIBRATIONAL SPECTRA 3.1 Method of Calculation 139 3.2 Accuracy Considerations 141 3.3 (HX)2 143 3.4 H3Z HX 148 3.4.1 Analysis of Intensities 150 3.4.2 Anharmonicity 152 3.4.3 Other Properties 154 3.4.4 Relationship between H-Bond Strength and Spectra 3.5 H2Y...HX 156 3.5.1 Alkyl Substituents 159 3.5.2 Other Properties 159

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3.6 H2Y...HYH 160 3.6.1 Polarizability 162 3.6.2 Comparison between (H2O)2 and (H2S)2 163 3.6.3 Effects of Electron Correlation and Matrices 166 3.6.4 Substituent Effects 168 3.6.5 NMR spectra 171 3.7 Expected Accuracies 171 3.7.1 HF Dimer 171 3.7.2 Water Dimer 173 3.8 HYH ... NH 3 175 3.9 (NH3)2 177 3.10 Carbonyl Oxygen 179 3.10.1 Relationship between E and v 180 3.10.2 Formaldehyde + Water 181 3.10.3 Formaldehyde + HX 182 3.11 Imine 184 3.12 Nitrile 185 3.12.1 Correlation and Anharmonicity 186 3.12.2 HCN as Proton Donor 191 3.12.3 HCN Dimer 192 3.13 Amide 195 3.14 Summary 197 4

EXTENDED REGIONS OF POTENTIAL ENERGY SURFACE 4.1 Ammonia Dimer 208 4.2 H 2 O ... HX 209 4.3 (HX)2 209 4.3.1 Anisotropies of Energy Components 210 4.3.2 Interconversion Pathways 212 4.3.3 HC1 Dimer 213 4.4 Water Dimer 215 4.4.1 Characterization of Possible Minima and Stationary Points 215 4.4.2 Components of the Interaction Energy 220 4.5 Carbonyl Group 223 4.6 Amines 225 4.7 Summary 226

5

COOPERATIVE PHENOMENA 5.1 HCN Chains 232 5.1.1 Geometries 232 5.1.2 Energetics 234 5.1.3 Dipole Moments 235 5.1.4 Vibrational Spectra 235 5.1.5 Quadrupole Coupling Constants 5.1.6 Cyclic Chains 240

239

xiii

xiv

Contents

5.2 HCCH Aggregates 240 5.2.1 Trimers 241 5.2.2 Tetramers and Pentramers 242 5.3 Hydrogen Halides 245 5.3.1 Open versus Cyclic Trimers 245 5.3.2 Three-Body Interaction Energies 246 5.3.3 Larger Oligomers 248 5.4 Water 252 5.4.1 Extended Open Chains 253 5.4.2 Branching Clusters 257 5.4.3 Cyclic Oligomers 257 5.4.4 Identification of True Minima 262 5.4.5 Substituent Effects 270 5.5 Mixed Systems 272 5.5.1 Geometries 273 5.5.2 Energetics 274 5.5.3 Vibrational Spectra 275 5.5.4 Effects of Electron Correlation 278 5.5.5 Other Mixed Trimers 280 5.6 Summary 282 6

WEAK INTERACTIONS, IONIC H-BONDS, AND ION PAIRS 6.1 Weak Acceptors 292 6.1.1 Dihalogens 292 6.1.2 CO 294 6.1.3 CO2 295 6.1.4 NNO 296 6.1.5 SO2 297 6.1.6 CC12 298 6.2 C—H as Proton Donor 298 6.2.1 Alkynes 299 6.2.2 Alkanes 302 6.2.3 Metal Atoms as Acceptors 306 6.2.4 Hydride as Proton Acceptor 307 6.3 Symmetric Ionic Hydrogen Bonds 308 6.3.1 Hydrogen Bihalides 308 6.3.2 Comparison with Other Anionic H-bonds 310 6.3.3 Cationic H-bonds 316 6.3.4 Comparisons between Cations and Anions 318 6.3.5 Alkyl Substituents 319 6.3.6 Other Considerations 320 6.4 Asymmetric Ionic Systems 321 6.4.1 General Principles 321 6.4.2 Test of Quantitative Relationships 322 6.5 Syn-Anti Competition in Carboxylate 326 6.5.1 Ab Initio Calculations 326

Contents

6.5.2 Experimental Findings 328 6.5.3 Carboxylic Group 328 6.5.4 Solvent Effects 328 6.5.5 Resolution of the Question 329 6.6 Neutral versus Ion Pairs 330 6.6.1 Amine-Hydrogen Halide 330 6.6.2 Carboxyl/Carboxylate Equilibrium 6.6.3 Experimental Confirmation 337 6.6.4 Long Chains 339 6.7 Summary 341 6.7.1 Low Polarity of Acceptor 341 6.7.2 C-H Donors 342 6.7.3 Ionic H-Bonds 343 6.7.4 Neutral Versus Ion Pairs 345 Index of Complexes Subject Index

371

365

335

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Abbreviations

ACPF ANO

APT BSSE

CASSCF

CC CC cc CCD CCSD CEPA CF CPF CI

Approximate Coupled-Pair Functional approach to compute electron correlation Atomic Natural Orbitals. A basis set constructed from maximization of the occupancy numbers of the natural orbitals of a given atom from a CI calculation. Atomic Polar Tensor. An analytic means of considering the effect of atomic motion upon the dipole moment of a given system. Basis Set Superposition Error. The error incurred in a computation of the interaction energy when the basis functions of one monomer artificially improve the basis set of its partner (and vice versa), thereby lowering the energy by the variation principle. Complete Active Space Self-Consistent Field. An MCSCF calculation which includes all excitations of a given type from a chosen set of reference molecular orbitals. Coupled Cluster. A means of including dynamic electron correlation that includes higher-order excitations. Counterpoise Correction. A means of correcting BSSE. Correlation Consistent, referring to a specific class of basis sets. Coupled Cluster including Double excitations. Coupled Cluster including Single and Double excitations. Coupled Electron Pair Approximation for including dynamic correlations. Charge Flux. Loss or gain of electron density as an atom is displaced. Coupled Pair Functional procedure for including dynamic electron correlation. Configuration Interaction. A means of including electron correlation by mixing in to the Hartree-Fock wave function, configurations generated by excitations from occupied to virtual MOs.

xviii Abbreviations

CISD CT DISP DZ DZP ECP

EFG

ES EX

GIAO GTO HF

IEPA

IGLO KM LCAO

LCCM MBPT MBS

MCSCF

MINI MO MPn

Configuration Interaction using Single and Double excitations. Charge Transfer. The component of the interaction energy resulting from excitation of the electrons of one subunit into the vacant MOs of its partner. Dispersion energy. An interaction resembling London forces, present only in post-SCF calculations. Double-Zeta basis set. Similar to minimal basis set except that each orbital consists of a pair: an inner and outer function. Double-Zeta Polarized basis set. Like DZ but also including polarization functions. Effective Core Potential. Also known as pseudopotentials. A procedure for considering only the valence electrons explicitly; used mainly with large atoms. Electric Field Gradient. The rapidity with which the electric field generated by a given molecular system is changing, usually evaluated at the position of a nucleus. Electrostatic energy. The Coulombic interaction between the static charge clouds on two molecular entities. Exchange energy. Part of the interaction energy between static charge clouds of two subunits, resulting from Pauli exchange between them. Similar to steric repulsion for molecular interactions. Gauge-Including Atomic Orbitals. A class of orbitals which are designed to permit computation of chemical shift tensors in NMR spectra. Gaussian-type orbital. Functions which differ from hydrogen-like orbitals in that the r dependence is exp(— r2). Hartree-Fock. Calculations based on the Hartree-Fock approximation of each electron moving in the time-averaged field of the others. No dynamic electron correlation is included. Independent Electron Pair Approximation. A means of including dynamic electron correlation where the total correlation energy is partitioned into a sum of contributions from each occupied pair of spin orbitals. Individual Gauge for Localized Orbitals Kitaura-Morokuma means of partitioning the total interaction energy of a given complex. Linear Combination of Atomic Oribtals. Usually refers to the practice of constructing each molecular orbital in terms of functions centered on each atom. Linearized Coupled Cluster Technique Many Body Perturbation Theory. A means of including electron correlation, similar to MP. Minimal Basis Set. One orbital is used to represent each of the orbitals of each shell that is full or partially filled. Examples: Is for H or He; 1s, 2s, 2px, 2py, and 2pz for Li-Ne. Multi-Configuration Self-Consistent Field. A means of variationally minimizing the energy of several electron configurations of a given system simultaneously, so as to provide a better description of its electronic structure. A type of minimal basis set, the most common being MINI-1. Molecular Orbital. nth-order M011er-Plesset theory. Means of including electron correlation.

Abbreviations

NBO

NEDA NPAD

NQCC PES POL

QCISD SAPT SCEP SCF STO ZPVE + //

xix

Natural Bond Orbital. Orbitals resulting from a sort of localization scheme that resembles the traditional concepts of 2-center bonds and lone electron pairs. Natural Energy Decomposition Analysis. A means of decomposing the total energy using natural bond orbitals. Normalized Proton Affinity Difference. A measure of the relative proton affinities of the two partners in a H-bond, indicating the likelihood of observing an ion pair. Nuclear Quadrupole Coupling Constant Potential Energy Surface. Polarization energy. The component of the interaction energy that results when the electric field of one subunit perturbs the electron density on its partner. Also abbreviated as PL. Quadratic Configuration Interaction using Single and Double excitations. Symmetry-Adapted Perturbation Theory. A means of partitioning the interaction energy into various components. Self-Consistent Electron Pair. A correlation technique that considers electrons two at a time. Self-Consistent Field. Commonly used synonymously with Hartree-Fock (HF). No dynamic correlation included. Slater-Type Orbital. Functions which loosely resemble hydrogen-like orbitals, especially insofar as the dependence is exp(— r). Zero-Point Vibrational Energy. The vibrational energy contained by a molecule or complex at 0° K. Indication that the basis set includes very diffuse functions. Double-slash indicates a distinction between the level of theory at which a geometry was optimized and the level for which the energy was computed. For example, MP2/6-31G(2d,2p)//SCF/6-31G* indicates a MP2/631G(2d,2p) calculation of a structure optimized at the SCF/6-31G* level.

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HYDROGEN BONDING

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I Quantum Chemical Framework

I. I Quantum Chemical Techniques

The following represents a capsule summary of the nature of ab initio quantum chemical calculations, intended to provide the reader with the minimal background necessary to comprehend the theoretical literature of H-bonds and to evaluate the quality and reliability of a given calculation. For more details about quantum chemistry in general or the specific methods, the reader is referred to any of a number of fine texts and review articles that have been written on the subject1-8. Quantum chemical methods are based on the time-independent Schrodinger equation

where 9€ represents the Hamiltonian of the system. The Hamiltonian is a quantum mechanical description, in terms of operators, of the kinetic energy of the particles, as well as the interactions between all electrons and nuclei. is a wave function and represents the "trajectories" of the particles. Due to the quantum nature of electrons, and the Heisenberg Uncertainty Principle, classical trajectories are inappropriate, so the paths are described instead in terms of probabilities of the particles being at any given point in space. Equation (1.1) is an eigenvalue problem, with E representing the energy of the system. Most quantum chemical calculations invoke the Born-Oppenheimer approximation which distinguishes between the electrons and the much heavier nuclei. Consequently, it is a good approximation to treat the nuclei as fixed in space, with the electrons moving in the electric field generated by them. Equation (1.1) cannot be solved exactly because each electron repels all of the others, leading to a many-body problem. The usual method adopted to circumvent this difficulty has been the Hartree-Fock approximation, which in essence reduces the problem to a single particle by time-averaging the motion of all electrons other than the one in question. In other words, electron 1 is considered to move in the field of the

3

4

Hydrogen Bonding

electron cloud associated with the probability distribution of all other electrons. The same idea is applied to electron 2 which moves in the time-averaged field of electron 1 plus all the others, and so on. Of course, solution of the 1 -electron Hartree-Fock equation for each electron changes its probability density, thereby altering the field it sets up for the other electrons. Consequently, the equations are solved iteratively, until the 1-particle wave functions, and the fields generated therefrom, no longer change appreciably from one cycle to the next. Because of the generation of this "self-consistent field," the SCF abbreviation is sometimes used synonymously with HF, that is, Hartree-Fock. This Hartree-Fock approximation6,7 neglects a very important phenomenon. Since the electrons are constantly aware of each others' presence, via their electrostatic repulsion, they would tend to correlate their motions so as to avoid one another as much as possible. One can take as an analogy a pair of quarreling roommates that wander through their apartment from one room to the next in such a way as to avoid contact with each other. When one is in the kitchen, the other will be in the bedroom, and so on. This phenomenon will lower the energy of the system by reducing the time that the electrons of like charge spend close to each other. Not only does correlation lower the energy of the system, but it also affects the overall electron density of the system. 1.1.1 Basis Sets Most quantum chemical treatments describe each molecular orbital as a linear combination of atomic orbitals9,10. In this so-called LCAO approximation, each atom has assigned to it certain functions that resemble the standard s, p, d, and so forth atomic orbitals that are centered about the nucleus. There are certain important differences, however. Whereas the hydrogen-like orbitals die off as exp(— r), where r is the distance from the nucleus and a constant, the integrals that must be incorporated into the Hartree-Fock matrix using this form of the orbital are difficult to evaluate. These "Slater-type orbitals" (STOs) are usually replaced by a small number of Gaussian functions, where exp(— r) is replaced by exp(— r2). The quadratic dependence of r in the exponent greatly simplifies the form of the integrals, particularly those that involve several atomic centers simultaneously. So much simpler, in fact, that it is computationally more efficient to evaluate a large number of integrals involving Gaussians than a much smaller number of STO integrals. Moreover, a series of Gaussians with progressively larger values of orbital exponent a can fairly closely reproduce a Slater-type function. As a result, most modem quantum chemical calculations are performed using basis sets composed exclusively of Gaussian functions. The collections of orbitals that are applied to calculations are referred to as "basis sets" and typically fall into one of several categories. The smallest employs one orbital to represent each of the orbitals of each shell that is full or partially filled. A minimal basis set for a first-row atom like Li or F would thus contain 1 s, 2s,2p x, 2p , and 2pz orbitals, whereas H or He would be described by a single 1 s orbital, as indicated in the first row of Table 1.1. Perhaps the most commonly used basis set of minimal type is STO-3G11. The name refers to the fact that each Slater-type orbital of the minimal basis set is replaced by a triad of Gaussian functions. (This triad is called a "contraction," and the three functions are referred to as "primitives.") Other minimal basis sets of this type, albeit less widely used, are STO4G and STO-6G. There are a number of ways that a minimal basis set can be improved. One approach is to provide more flexibility by doubling the number of functions. A "double- " basis set is similar to minimal, except that each atomic orbital is "split" into two. The flexibility of such

Quantum Chemical Framework

5

Table 1 . 1 Some of the most common types of basis sets, and the orbitals contained therein. Common name Unpolarized minimal split valence double-zeta triple valence Polarized split valence, polarized double-zeta, polarized double-zeta, doubly polarized containing diffuse functions

Abbreviation

Common example

H

1st Row atoms

MBS

STO-nG 3-21G,4-31G

1s

6-311G

i, ls o ls i ,ls m ,ls o

ls,2s,2pa ls,2si,2pi,2so,2po Isi,lso,2si,2pi,2so,2po ls,2si,2pi,2sm,2pm,2so,2po

6-31G**

ls i , ls o ,P

1s,2si,2pi,2so,2po,da

DZP

lsi,lso,p

lsi,lso,2si,2pi,2so,2po,d

DZ2P

1s

1si,lso,2si,2pi,2so,2po,di,do

sv

DZ

SVP

+

1si,

1s0

ls

1s

i,

6-31 + G**

o,Pi,Po

1s i ,ls o

ls,2si,2pi,2so,2po,d,sp

a b

Each p-set consists of 3 separate functions: p x ,p y ,p z : similarly, d refers to a set of 5 functions. Diffuse set of s,px,py,pz, all with small exponent.

a "DZ" basis permits each orbital to expand or contract in size to conform to the environment in which the atom finds itself. Even greater flexibility is provided by a triple- or TZ basis set. One line of reasoning has been that while splitting the valence shell is certainly worthwhile, there is little to be gained by doing the same thing to the inner shell, since these electrons will be little affected by changes in the bonding environment around the atom. Basis sets have evolved that are similar in spirit to DZ or TZ but split only the valence shells. Such "split-valence" basis sets are exemplified by 3-21G12, 6-31G13, or 6-311G14. The 6 in the latter case refers to the number of Gaussian primitives used to describe the inner shell, 1s orbital. The 3 and 1 of 6-31G indicate the number of primitives that model the inner and outer valence orbitals: 3 Gaussians are used for the inner and 1 for the outer. 6-311G is similar except that a third set of functions are added, by a single Gaussian, to split the valence shell three ways, as indicated in Table 1.1. The inner and outer s orbitals of an atom in a double- basis set are both spherical, that is, isotropic. So while the presence of two of them permits the orbital to expand, it can do so isotropically only, with no stretching in any one direction over another. Such "polarization" in a given direction would be useful in many situations. Consider for example the O—H bond in water. The direction of the O—H bond is clearly unique; a stretching of the H 1s orbital in this direction would permit a better description of the bond. Mixing the s orbital with even a small amount of a px orbital, collinear with the O—H bond axis, would enable the former to stretch in the x direction, polarizing the orbital along the bond. This stretching is illustrated graphically in Fig. l.la where the s-orbital is indicated by the circle and the px as a smaller orbital. For this reason, when added to hydrogen, p-orbitals are referred to as "polarization functions". In an analogous manner, the p-orbitals of C or O, for example, can be polarized by a small amount of a d-orbital of appropriate symmetry, as indicated in Fig. 1.1b. Polarization functions on such atoms hence correspond to orbitals of d symmetry. Rather than adding simply a px function, a full set of all three p-orbitals are used to polarize the hydrogen basis set. A full set of five d-functions are likewise used, al-

6

Hydrogen Bonding

Figure I.I Schematic representation of the manner in which (a) a p-orbital can polarize one of s-type and how (b) a p-function can be polarized by a d-function. The position of the nucleus is indicated by the dark dot.

though in some cases it is convenient to use a set of six even though the sixth corresponds roughly to an orbital of s-type symmetry. There are various conventions used to indicate when polarization functions have been added to a basis set. The addition of a "P" is commonly used, as for example when DZP refers to a polarized double- basis set. Of course, this single letter does not clarify whether all atoms have had polarization functions added, or only a subset. In most cases, the P designation should be taken to indicate polarization functions on all atoms. Another indication that has been used over the years is an asterisk. A single asterisk, as in 6-31G* indicates polarization functions (d-type) have been added to non-hydrogen atoms; a second asterisk would inform the reader of p-functions on hydrogen as well. Whereas the P designation carries no information as to the values of the exponents used for the polarization functions, the asterisks refer to specific values, in the context of well defined basis sets, for example, 6-31G**. It has become more common with faster computers to use multiple sets of polarization functions. For example, one might wish to include both a "tight" and "diffuse" set of d-functions, with large and small orbital exponents, respectively, on oxygen atoms. This double polarization set can be indicated in various ways, for example, as DZ2P. Since the asterisk convention cannot indicate this easily, it has become increasingly common to abandon these asterisks altogether and to indicate the numbers of polarization functions in parentheses. As an example, 6-31G** could be equally described as 6-31G(d,p) where the d and p polarization functions on non-hydrogen and hydrogen atoms come respectively before and after the comma. Doubling the d-functions, but leaving as a single set the p-functions on hydrogen, would then be simply indicated as 6-31G(2d,p). This sort of notation readily lends itself to the representation of orbitals of even higher angular momentum, as for example 6-311 G(3df,2pd). Another sort of function which has found a good deal of use in basis sets is of the same symmetry as those mentioned above, for example, s or p. However, it is given a particularly small orbital exponent, imparting to it a large and diffuse nature. Such diffuse orbitals are particularly useful for describing anions, as they permit the overload of electrons to better avoid one another as they take advantage of the large expanse of space over which this orbital extends. It has become common to indicate the presence of such functions by a + symbol. For example, 6-31 +G* includes a diffuse sp-shell on non-hydrogen atoms; a second + as in 6-31 + +G* indicates diffuse functions on H as well15. Test calculations have suggested that such functions on hydrogen are less important than on heavy atoms, at least for ground states16. While the vast majority of calculations make use of basis functions centered on the atomic nuclei, it is sometimes worthwhile to add other functions which have their origin along the bond axis between a given pair of nuclei. Such "bond functions" 1 7 - 2 1 can provide

Quantum Chemical Framework

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rapid convergence of various properties with respect to number of functions in the basis set. There are, on the other hand, certain hidden dangers that accompany the use of such functions and of which the user must be wary22. The choice of orbital exponents can be an important one. It is common for the exponents of the s and orbitals to be established by optimization of the energy of the atom. One may impose certain restrictions on the various exponents. For example, even-tempered basis sets are those in which the progression of orbital exponents follows a geometric sequence from one to the next, that is, .23 This prescription calls for only two parameters in an arbitrarily long list of primitive Gaussian functions. A generalization which extends the optimization to four parameters has been designated as "well-tempered"24,25. The polarization function exponents (d, f, etc.) can also be optimized, or alternately chosen so as to yield good reproduction of properties other than the total energy. With the increasingly common use of electron correlation, there has been some rethinking of these prescriptions to select orbital exponents. "Correlation-consistent" basis sets have been designed26, sometimes abbreviated with the prefix "cc", specifically to be amenable to calculations involving electron correlation. The smallest of these is cc-pVDZ (correlation-consistent, polarized valence double zeta), and is capable of yielding well over half of the total correlation energy in atoms such as Ne and B; cc-pVQZ can attain 95-99%. These sets can be augmented (indicated by the prefix "aug") by additional functions optimized for atomic anions, so as to describe diffuse electronic distributions27. Testing shows these basis sets can be superior to more standard types in certain applications28. Also designed to incorporate correlation effects into the construction of orbitals is the atomic natural orbital (ANO) basis29 which maximizes the occupation numbers from a CI calculation of the individual atom, starting from an uncontracted basis. As a general notation, it is common to enclose within square brackets the number of functions of various types. For example, [6s5p2d/4s2p] indicates that first-row atoms are represented by 6s functions, 5 sets of 3 (px, py, and pz) p-functions, and two separate sets of five d-functions. The basis set for hydrogen is indicated after the slash: 4s and 2 sets of p-functions. Because the order is always the same; s, followed by p, d, etc. it is not uncommon to leave out the letters entirely: [652/42]. In this lexicon, the 6-31G** basis set could be represented as [321/21]. If the system were to contain atoms beyond the first row of the periodic table, they would be indicated to the left of the first set of numbers, as in [second-row/first-row/H]. As the atom becomes larger, the number of basis functions needed to describe it increases as well. However, since one is most interested in the valence shell where most of the "action" occurs, the increasingly larger number of "inactive" or core functions become more and more of a nuisance. One cannot simply omit them as the valence orbitals would then collapse into smaller core orbitals (which are of much lower energy). One solution is development of "core pseudopotentials" or effective core potentials (ECP) which eliminate the need to include core functions explicitly, yet keep the valence functions from optimizing themselves into core orbitals9,30-32. Such pseudopotentials are commonly used in elements of the lower rows of the periodic table, like Br or I. 1.1.2 Electron Correlation There are a number of ways in which one may begin to correct the Hartree-Fock wave function so as to include electron correlation6,7,33,34. The simplest in concept is configuration interaction (CI) which takes the Hartree-Fock solution as a starting point, or reference con-

8

Hydrogen Bonding

figuration35,36. Other configurations are generated by permitting the excitation of one electron from the subset of occupied molecular orbitals to the subset of unoccupied or "virtual" MOs. The complete list of "singly excited configurations" is generated by considering all such possible excitations, subject to the restriction of preservation of the spin state of the ground state under study. The list is then extended to double excitations, again accounting for all possible combinations of excitations of two electrons from the occupied to the virtual MOs. A full-CI list is generated by progressing in this manner to include triple, quadruple, and higher excitations, until all N electrons have been excited. The correlated wave function is then expressed as a linear combination of the reference, Hartree-Fock configuration, plus small amounts of all of the possible excitations. Variational treatment of this trial wave function leads to the correlation energy by adjusting the relative amount that each particular configuration contributes to the final correlated wave function. Unfortunately, the number of configurations generated by all possible excitations of even relatively small systems quickly becomes astronomical and beyond the reach of any computers. For this reason, the list is commonly terminated at some point. One of the more common points of termination is after the inclusion of all single and double excitations. This CISD treatment typically captures the bulk of the correlation energy. One notable problem with termination of the full CI expansion has been referred to as the size-consistency problem37. What this means is that the same treatment of a complex is fundamentally different than that of the subunits of which it is composed. Consider for example the CISD level. Such a treatment of the first subunit would permit double excitations within it; similarly for the other subunit. A consistent theory should hence permit quadruple excitations within the complex, as this would account for simultaneous double excitations in each of the subunits. But CISD terminates the excitation list at doubles in the complex. Truncated CI treatments are therefore fundamentally poorly disposed to handle molecular interactions such as hydrogen bonds. There have been a number of correction algorithms formulated over the years to help improve this size consistency problem but they do not entirely resolve it38-40. Another means of introducing size consistency is by a quadratic approximation, QCISD41. The approach achieves this size consistency by sacrificing its variational character. It can be considered as a simplified approximate form of CCSD (see below); the method may cease to remain size consistent on going to higher levels of substitution42. Certain other types of procedures are size consistent. Coupled pair theories43-45 suffer from a different shortcoming: they are not variational. What this means is that it is possible in principle to obtain an energy lower than the true energy of the system in question. In the independent electron pair approximation (IEPA), the total correlation energy is partitioned into a sum of contributions from each occupied pair of spin orbitals. A different correlation wave function is constructed for each pair, letting their electrons be excited into the virtual MOs of the reference configuration. The total correlation energy then corresponds to the sum of all pair energies. When the IEPA approach is extended to incorporate coupling between different pairs, it becomes a coupled-pair theory. In terms of excitations from the original Hartree-Fock determinant, the correlation energy depends directly upon the double excitations, but their contributions involve quadruple excitations in an indirect way, and the latter are linked to hextuple excitations, and so on. The coupled-cluster approximation affords a means of expressing these relationships in a closed set of equations46,47. Most applications of coupledcluster theory include only double excitations, and are designated CCD48,49. More general versions of the theory that include also single and higher excitations are commonly abbre-

Quantum Chemical Framework

9

viated as CCSD50. Coupled-cluster is highly demanding of computer resources so various approximations to it have been suggested. The linear coupled-cluster approximation (LCCA) sets certain products equal to zero, and is equivalent to doubly-excited many-body perturbation theory, carried to infinite order (see below). If instead of ignoring all the product terms set equal to zero in L-CCA, certain of them are retained, one arrives at the coupled electron pair approximation (CEPA). The type of correlated method that has enjoyed the most widespread application to Hborided systems is many-body perturbation theory34, also commonly referred to as M011erPlesset (MP) perturbation theory51-53. This approach considers the true Hamiltonian as a sum of its Hartree-Fock part plus an operator corresponding to electron correlation. In other words, the unperturbed Hamiltonian consists of the interaction of the electrons with the nuclei, plus their kinetic energy, to which is added the Hartree-Fock potential: the interaction of each electron with the "time-averaged" field generated by the others. The perturbation thus becomes the difference between the correct interelectronic repulsion operator, with its instantaneous correlation between electrons, and the latter Hartree-Fock potential. In this formalism, the Hartree-Fock energy is equal to the sum of the zeroth and first-order perturbation energy corrections. The first correction to the Hartree-Fock energy appears as the second-order perturbation energy. This term can be represented as a sum over double excitations from the reference configuration:

where i refers to the orbital energy of molecular orbital i. The sum extends over all a and b which are occupied MOs and r and s which are vacant. The integral in the numerator makes use of physics notation and indicates a combination of Coulomb and exchange integrals over these particular MOs6. Similar, albeit increasingly more complex, equations can be derived for higher orders of perturbation theory. The energy, after correction by Eq (1.2), is commonly referred to as MP2. The MP3 level involves additional terms, but remains restricted to double substitutions from the reference configuration. At the fourth order, there are contributions from single, triple, and quadruple excitations, as well as doubles. Since the expressions involving the triple excitations are the most difficult, the full MP4 is sometimes simplified by neglecting them, leading to what is denoted MP4SDQ. Due to the computational efficiency of MP theory, correlation calculations have come within reach of many computational chemists. For example, the time necessary to carry out a full MP3 calculation is comparable to that of a single iteration of CID. Another strong advantage is that MP theory is size consistent, making it a good choice for molecular interactions of various sorts. Moreover, as will be illustrated in greater detail later, a large number of calculations over the years have indicated that MP2 provides results in excellent agreement with the much more computationally demanding MP4. For these reasons, the literature of correlated calculations of hydrogen bonds is largely dominated by M011er-Plesset theory. In certain cases, a single determinant does not offer an adequate representation of the electronic structure. In such cases, it is useful to perform a Multi-Configuration SCF calculation (MCSCF) wherein a number of different configurations are chosen as important, and their adjustable parameters (orbital coefficients, etc.) are variationally optimized54,55.

10

Hydrogen Bonding

This procedure suffers from a high degree of arbitrariness in the choice of just which configurations are deemed important. The calculation can be made somewhat more objective by including all excitations between a subset of occupied MOs and a subset of vacant orbitals. (These excitations are subject to certain restrictions as to multiplicity or order of excitation.) The orbitals chosen for the excitations are referred to as the "active space", and the method is dubbed Complete Active Space Self Consistent Field (CASSCF)56,57. Both MCSCF and CASSCF provide a certain fraction of the correlation energy, relative to a single configuration, Hartree-Fock, calculation. 1.1.3 Geometries Before 1980, geometry optimizations were not very well automated. Locating the minimum on the potential energy surface of even a fairly simple molecule could be a tedious chore. One geometrical parameter was usually optimized at a time, independent of the others. Since the value used for one parameter affects the optimized values of the others, the entire set of geometrical parameters had to be cycled through a second and sometimes a third time, before the geometry could be considered converged. Many of these optimizations were only partial in the sense that it was common to make certain assumptions about the geometry. For example, one might decide in advance that all C—H bonds of a methyl group would be of the same length or the geometry of an ethyl group might not be optimized at all, with certain standard bondlengths or angles being assumed. The development in the early 1980s of the means to evaluate derivatives of the energy with respect to nuclear motion, and implementation of gradient algorithms, changed the face of ab initio calculations58-60. It became possible to optimize all parameters simultaneously, searching for the minimum on a multidimensional potential energy surface. From that point in time, complete optimizations became the norm in the literature. Since the optimizations make use of a Hessian matrix consisting of the second derivative of the energy with respect to the various geometrical parameters, it became straightforward to determine if the optimization path had proceeded to a true minimum on the potential energy surface, or to a higher-order stationary point such as a transition state. (A true minimum is distinguished by the Hessian having all positive eigenvalues.) It should be understood, however, that the optimization procedures take one to a minimum on the surface, not necessarily to the minimum. In other words, there is no guarantee that the minimum one locates is the global minimum; it might equally well be a secondary minimum. A shorthand has developed by which quantum chemists communicate the nature of their calculation. The level of computation is indicated on the left of a slash, with the basis set to the right of the slash. Thus, MP2/6-31G* would indicate a MP2 calculation with a 6-31G* basis set. It is common to optimize the geometry at one level of theory, and then to apply a higher level to compute the energy at that particular molecular structure. Such calculations are indicated by a double-slash. As an example, if one were to optimize the geometry at the SCF level with a 6-31G* basis set, and then to compute the energy of this structure at the MP2 level with the larger 6-31G(2d,2p) set, this might be indicated as: MP2/631G(2d,2p)//SCF/6-31G*. These geometry optimization procedures paid an additional dividend. With a true minimum in hand, it becomes possible to compute the vibrational spectrum of a given system. A straightforward formulation allows one to extract the normal vibrational modes of the system, complete with the corresponding frequencies, but with the important proviso that

Quantum Chemical Framework

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these calculations be made from the standpoint of a minimum on the surface. Attempts to compute vibrational spectral features at a nonstationary point are meaningless. Hence, the precise determination of minima opened the door to important comparisons with experimental spectral data, comparisons that were not possible prior to that time. Whereas optimization of the geometry of a single molecule is relatively straightforward, the same procedure can encounter difficulties for a molecular complex, particularly if weakly bound. The first problem here is that the force constants for motions of one molecule relative to the other are quite a bit smaller than those for stretches or bends wholly within one molecule. One tactic to circumvent this difference is to perform a geometry optimization using frozen subunits. That is, the internal geometry of each partner can be taken as fixed, and the optimization carried out over the intermolecular parameters. The final result is thus not fully optimized but the earlier restraint can then be released and the geometry now optimized over all parameters, intramolecular as well as intermolecular. A second issue concerns the landscape of the potential energy surface. As will be detailed in a number of examples, the surface of many H-bonded complexes is rather flat. However, the surface is far from featureless, containing a number of different local minima, connected by stationary points of various orders. Depending on the particular optimization algorithm or step size chosen, it is possible for the optimization procedure to jump over a local minimum, missing it entirely. One must also be careful that the minimum identified is indeed the global minimum by scanning large regions of the surface for others that might be deeper. And due to the flatness of the potential, it is not uncommon for what appears at first sight to be a minimum to be seen on closer inspection to be a stationary point of higher order. In summary, then, ab initio methods represent a powerful means of extracting information about chemical systems. When flexible polarized basis sets are used, in conjunction with electron correlation, results of high accuracy are attainable. It is possible to locate minima on the potential energy surface as well as transient entities, such as transition states. One limitation of these methods is the size of the system that may be studied. The amount of computational resources required rises quickly with the number of atoms or electrons. Another caveat is that ab initio methods typically investigate a static situation: that is, the energy, energy derivatives, and electronic structure of one given arrangement of nuclei are calculated at a time. The ab initio methods can be supplemented by other theoretical approaches in order to simulate dynamic processes.

1.2 Definition of a Hydrogen Bond

Since its first suggestion many years ago61-65, the hydrogen bond has continued to fascinate chemists. This interaction is intimately involved in the structure and properties of water in its various phases, and of molecules in aqueous solution. In addition to the traditional role of the H-bond as a structural element in large molecules such as proteins and nucleic acids66-70, a series of such bonds appear to be vital to the functioning of a number of enzymes71-73. There are some indications that H-bonds play an even more important role in biological electron transfer across long distances than much stronger covalent bonds74. The principles of H-bonding have been taken as a means to design new materials capable of selfassembly into well-ordered crystal structures75- 77, for molecular recognition of organic molecules 78-80 , organic analogs of zeolites with supramolecular cavities and continuous

12

Hydrogen Bonding

channels81-84, for self-assembly of spherical, helical and cylindrical structures85-90. Hbonds offer an avenue for stereocontrol of certain reactions91 and for understanding the structure of monolayers92,93. A Lewis structure of a H-bond violates the octet principle of striving toward two electrons around each hydrogen. Much weaker than a conventional covalent bond, the H-bond is stronger than the van der Waals forces that bind together nonpolar molecules. While Coulombic attraction between polar molecules certainly contributes toward the interactive force, the H-bond is nonetheless considered to be more than a simple electrostatic interaction. The classic picture of a H-bond begins with a pair of molecules, both in their ground electronic state and both with a closed shell62,94. One molecule, AH, is designated the proton donor with the pertinent hydrogen covalently bound to an electronegative atom like O or F. The acceptor molecule, B, contains an electronegative atom with at least one lone pair of electrons. As the two molecules approach, the hydrogen atom forms a sort of "bridge" between them. The lone pair of the acceptor atom is pulled toward the bridging proton to form a weak bond, designated by the dotted line in the simple AH ... B notation. It is not necessary for AH and B to be separate molecules; intramolecular H-bonds are also acceptable, so long as it is possible for the proton-donating and accepting groups to attain the proper positioning. There are several consequences of this interaction that are commonly taken as criteria for formation of a H-bond95-97. 1.2.1 Geometry The attractive interaction generated by the formation of the H-bond draws the two groups closer together than would be the case in the absence of such a bond. For this reason, the distance separating the nonhydrogen atoms, R(A ... B), is typically shorter than the sum of van der Waals radii of the A and B atoms. Indeed, it is typically assumed that there is a strong correlation between the shortness of this interatomic separation and the strength of the Hbond98. Concomitant with the formation of the bond is a certain amount of stretching of the covalent A—H bond; the amount of this stretch, r, is usually closely correlated with the strength, as well as the length R(A ... B) of the H-bond94,99-101. There is also an expectation of certain directionality to the H-bond. The bridging hydrogen will tend toward the line connecting the A and B atoms. The same is presumed for the lone pair of the Y atom. Taking FH ... NH 3 as a first and simple example, the bridging proton lies directly along the F...N axis. The C3 symmetry axis of NH3, coincident with the single lone pair of the N atom, also lies along this line. The situation becomes a little more complicated when the acceptor has more than one lone pair. A simple description of the electronic structure of the carbonyl oxygen of, say H2CO, places two sp2-hybridized lone pairs at 120° angles from each other and from the C=O bond. Formation of a H-bond with HF should therefore occupy one of these two lone pairs, situating both the H and F atoms along a line approximately 120° from the C=O bond. Another source of ambiguity would arise if there were two hydrogens on the donor molecule that were each capable of participating in a H-bond. In most cases, one would expect a standard linear H-bond to form, utilizing one of these protons. Another possibility, and one to be discussed in greater detail below, would have the two protons both participating, and both oriented toward the acceptor, but neither of them lying directly along the A...B axis. More complicated situations can be envisioned when the donor has two protons and the acceptor has more than one lone pair.

Quantum Chemical Framework

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1.2.2 Energetics The strength of the H-bond is typically measured in terms of the interaction energy between the two molecules involved. Such an interaction energy is more difficult to define in the case of an intramolecular bond. In such a case, one can consider the difference in energy between the H-bonded geometry and another configuration in which the bridging hydrogen is not permitted to come within H-bonding proximity of the proton acceptor. For example, a simple 180° rotation around another bond axis can swing the proton away from the acceptor. In the gas phase, which most quantum calculations mimic, the H-bond energy is typically in the range of 2 to 15 kcal/mol. This interaction is weaker than most covalent bonds by about one order of magnitude, but stronger than nondirectional "noncovalent" forces which tend to be less than 2 kcal/mol. Nonetheless, this range is meant only as a general rule of thumb rather than as a strict threshold. One should not consider the range as exclusively the province of H-bonds, nor should interactions be discounted as H-bonds merely for lying outside this range. Another energetic aspect of H-bonds is not only the total interaction energy, but its origin as well. For example, certain means of partitioning the interaction energy attribute the bulk of the stabilization to electrostatic forces between the charge distributions of the two subunits. Within this context, an interaction of the proper magnitude, but with minimal Coulombic contribution, might not be categorized as a H-bond. 1.2.3 Electronic Redistributions Unlike the formation of covalent bonds which involve massive shifts of electron density, the rearrangements that occur as a consequence of a H-bonding interaction are more subtle. The electron distributions within each subunit remain largely intact as the H-bond is formed. Nonetheless, there are indeed shifts of electron density that do occur. While relatively small in magnitude, these shifts tend to be characteristic of H-bonds, and can be taken as a sort of fingerprint for formation of such a bond. For example, there is an overall shift of electron density from the proton acceptor molecule to the donor. (It is for this reason that the proton donor is sometimes referred to as the electron acceptor, and vice versa.) This density is drawn not only from the lone pair participating in the H-bond but from the entire molecule. Rather than residing on the bridging proton, the density bypasses this center and becomes distributed throughout the donor molecule. Indeed, the total density associated with the central hydrogen undergoes a decrease as the bond is formed. The above patterns are drawn from objective analysis of the spatial distribution of electron density maps. In contrast, any attempt to quantify the amount of charge transferred from one molecule to the other relies upon some arbitrary partitioning of the region between them as belonging to a specific molecule, and is consequently subject to some degree of arbitrariness. There is also a good deal of sensitivity to basis set in assigning charge to various atoms. Bearing in mind these caveats, typical results indicate that some 0.01 to 0.03 electrons are transferred from the proton acceptor to the donor molecule upon H-bond formation. 1.2.4 Spcctroscopic Observations One of the more striking consequences of the formation of a H-bond appears in the vibrational spectrum. The band which corresponds to the stretch of the A—H bond shifts to lower

14

Hydrogen Bonding

frequency, is intensified, and undergoes a concomitant broadening. In a quantitative sense, it has been possible to establish a nice correlation between the amount of the red shift and the strength of the H-bond94,102,103. It would also appear that the magnitude of the intermolecular H-bond stretching frequency is directly related to the strength of the H-bond. NMR spectra also have diagnostic utility 104 . The electron density shifts which arise from the H-bonding result in perturbations of the proton shielding tensor, deshielding the bridging hydrogen105-107. The isotropic shielding and anisotropies tend to correlate with the length of the H-bond 108-1l0 as do the peak volumes in the solid state heteronuclear correlation spectra111. Indeed, chemical shifts can help ascertain the secondary or tertiary structure of proteins, discriminating between -helices and -sheets112; this relationship has been tentatively attributed to the different H-bond lengths in the two structures and the aforementioned relationship113. It has been suggested that the single or double-well character of a proton-transfer potential is signaled by whether the chemical shift of a deuterium is larger or smaller than that of protium 114,115 . The 13C anisotropic chemical shift tensors of the carboxyl carbon are sensitive to H-bonding, and have been seen to correlate with vibrational frequencies116. Proton magnetic resonance measurements have been used to probe the pK of catalytic residues and proton positions upon binding an inhibitor to an enzyme117. NMR measurements of the solid state can lead to surprisingly detailed characterization of molecular geometry, including the nature of the proton transfer potential101. 15N chemical shifts have been interpreted to question a view of the catalytic mechanism of serine proteases derived from X-ray diffraction data118 and to study model systems pertinent to charge relay chains119,120. The 1 JNC, nuclear spin-spin coupling constant appears to have diagnostic value for the Hbonding of amide groups in that it increases when H-bonding occurs through the O and is lowered when this interaction involves the NH group121,122. In conjunction with the proton chemical shift, this parameter can also be used to estimate the relative strengths of H-bonds within a macromolecule such as a protein, or to distinguish between various secondary structure motifs123. NMR measurements can be used also to monitor the dynamics of H-bonds. An example is a 2H and 13C study of cooperative rearrangements among the four OH groups that form a ring in a crystalline sample124. Quadrupolar coupling constants can be of use as well. For example, a study of polypeptides by solid-state 17O NMR spectroscopy125,126 revealed a correlation between this parameter and the length of hydrogen bonds. It is also possible to use electron resonance techniques to determine characteristics of a H-bond in proteins with good accuracy, as in a recent ENDOR examination of the heme binding pocket of fluorometmyoglobin127. Another promising technique is electron spectroscopy for chemical analysis (ESCA) which can monitor hydrogen bonds near the surface of a material128. The difference in binding energies of atoms in the proton donor and acceptor offers a measure of the strength of the H-bond. While there have been few experimental measurements of the dynamics of association involved in formation of a H-bond, or the opposite dissociation, in the gas phase, there are modern approaches that show promise in this regard. For example, resonant photoacoustic spectroscopy has been used to examine possible dissociation pathways of carboxylic acid dimers129-131.

1.3 Quantum Chemical Characterization of Hydrogen Bonds Theoretical methods have been used to obtain insights into the nature of the hydrogen bond from their very inception. While this volume focuses on work published since 1980, the in-

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terested reader might gain a historical perspective from inspection of a number of review articles which summarized earlier understanding of the H-bond phenomenon132-141. In this section we discuss the manner in which quantum chemical calculations evaluate the various properties that are important to H-bonds. 1.3.1 H-bond Geometries The equilibrium geometry of any given molecular entity represents the bottom of the global minimum in the potential energy surface. Quantum chemical methods can efficiently evaluate this set of nuclear coordinates. It must be understood however, that the equilibrium structure at one level of theory will typically differ from that at another level. There is a substantial literature that deals, for example, with the effect of basis set upon molecular geometry. The various facets of the equilibrium geometry can be conveniently divided into intermolecular properties, for example the distance separating the two molecules, and intramolecular parameters, representing the bond lengths and angles within each subunit. Since the overall interaction energy in a H-bond is typically less than 15 kcal/mol, it is not surprising that the energy of the system is usually not very sensitive to the former intermolecular geometric parameters. Due to the flatness of the energy profile, the minimum in this potential can be shifted a good deal by minor changes such as a change in basis set. It is therefore common to find intermolecular aspects of H-bonds to be sensitive to the particular theoretical approach. Bending and stretching potentials for intramolecular geometrical parameters, on the other hand, are much sharper so there is less sensitivity to level of theory. Of greatest interest with regard to these intramolecular geometries are the perturbations induced in each subunit by the formation of the H-bond. Whereas the quantum chemical calculations provide a straightforward picture of the geometry at the bottom of the minimum, experimental observations pertain instead to a dynamic average. A diatomic molecule furnishes the simplest example of this difference where Re refers to the interatomic distance at the bottom of the well and the vibrational average of the ground level is denoted by Ro. Due to the weak nature of H-bonds, equilibrium and dynamically averaged quantities can differ by significant amounts. A recent study142, for example, predicts that the third-order anharmonicity within the water dimer might alter the interoxygen distance by as much as 0.13 A. 1.3.2 Thermodynamic Quantities There are several different energetic quantities that are relevant to quantum chemical evaluation of H-bond strength. The interaction of a pair of molecules, A and B, with one another to form an A...B complex can be represented by the reaction

where the connecting dot notation is used to indicate the specific interaction within the complex. The total energy change of the process in Equation (1.3) is commonly taken as E and is defined as:

For most complexation reactions, the complex is more stable than the isolated species so the process is exothermic and E is negative. All species are normally taken in their fully optimized geometries. It is important to note that the process of combining with one another

16

Hydrogen Bonding

typically causes certain changes in the internal geometries of both A and B. For this reason, the complexation energy E has folded into it the energetic consequences of such internal geometry changes, sometimes referred to as "nuclear relaxation energy." The latter term is conceptually distinct from electronic redistribution energetics that accompany the combination of A and B. A typical calculation of a molecular interaction thus involves the geometry optimization of three entities: the reactants A and B, and the complex A ... B. The subtraction of electronic energies in Equation (1.4) yields Eelec, the electronic contribution to the interaction energy. This term includes the internuclear repulsive energy within the molecule. Other contributions arise from translational, rotational, and vibrational motions of the nuclei143. Making the usual assumption of ideal gas behavior, the translational partition function of a given molecule in a container of volume V at temperature T is144

where m is the mass of the molecule, h is Planck's constant, and k is the Boltzmann constant. Hence, each of the three species involved in Reaction (1.3) contains 3/2 RT of translational energy per mole so

The rotational partition function is dependent upon the equilibrium geometry. Assuming separation of rotational and vibrational motions,

holds for temperatures significantly above absolute zero. Ia, Ib, and Ic represent the three principal moments of inertia of the molecule or complex, determined by the nuclear masses and their positions in space. The symmetry number a refers to the number of indistinguishable orientations one can obtain by rotating the molecule in space; is larger for more symmetric molecules. Equation (1.7) is simplified somewhat for a linear molecule

since there is only a single moment of inertia, I. The rotational energy is different for the nonlinear and linear cases:

A given molecule has 3n-6 normal modes of vibration (3n-5 if linear), where n is the number of atoms. Each mode i has a characteristic vibrational frequency vi and a residual energy even at absolute zero temperature. The total zero-point vibrational energy is thus:

As the temperature rises above 0° K. higher vibrational levels begin to be populated and the additional vibrational energy is:

Quantum Chemical Framework

17

The full thermodynamic interaction energy E is equal to the sum of terms:

where all E terms on the right refer to the differences between the product A...B and reactants, A plus B. The reader is cautioned that " E" is very commonly used in the computational literature when what is actually meant is Eelec. That is, many articles refer to the electronic contribution to the interaction energy as though it were the full thermodynamic E. In this book, we will attempt to clearly differentiate between E and Eelec. Perhaps a more proper designation for the electronic portion of the interaction would be De, the dissociation energy from the equilibrium geometry. Do would refer to this same quantity, after correction for zero-point vibrational energies. Another proviso concerns the signs of these quantities. For a stable complex, E is negative, signifying its formation to be exothermic, while De is taken as positive since it refers to the energy required to dissociate the complex. In the literature, H-bond energies are usually discussed as positive quantities. Other terms for this quantity are: complexation, dissociation, and interaction energy. The enthalpy of formation of A...B from its constituent molecules is equal to E with a pV correction that yields

since two molecules are combining to produce a single complex in Reaction (1.3). The statistical expressions for entropy may be used to derive equations most useful in studying molecular interactions of the type in Reaction (1.3). As there is typically a large energy separation between the ground and excited electronic states of adducts and their complex, it is valid to take the electronic entropy as zero. The translational entropy can be written as:

where m refers to the mass of the particular system and P is the pressure. Substituting in the values of the physical constants, the molar entropy can be written as

where the mass is expressed in amu, temperature in °K, and pressure in atmospheres. Applying Equation (1.15) to reaction (1.3) at 1 atrn pressure yields a molar entropy change of

The rotational entropy of each species is

18

Hydrogen Bonding

where qrot is defined in Equation (1.7). The change in molar rotational entropy can be derived from Equation ( 1 . 17) to depend upon the principal moments of inertia of reactants and products, along with the temperature. The entropy associated with each vibrational normal mode is a function of both temperature and vibrational frequency vi.

It is only necessary to sum the contributions made by each mode to the entropy of the various reactants and products to obtain the vibrational contribution to the total entropy change of the reaction. The latter total is then

The Gibbs free energy of Reaction (1.3), G, then follows simply from

1.3.3 Electronic Redistributions There are a number of ways of monitoring the distribution of electron density in any molecular entity. The total density can be computed at a number of points in space and presented as a contour map or some three-dimensional representation. Shifts are easily examined by density difference maps which plot the difference in density between two different configurations. For example, the density shifts caused by H-bond formation can be taken as the difference between the complex on one hand, and the sum of the densities of the two noninteracting subunits on the other, with the two species placed in identical positions in either case. Comparisons with x-ray diffraction data have verified the validity of this approach145. Also, the total density of the complex itself can be examined for the presence of critical points that indicate H-bonding interactions146-148. It is also feasible to examine individual molecular orbitals, again via plots over interesting regions of space. The data may be compressed into tabular form by assigning density to various atomic centers. This sort of treatment conforms to notions of atomic charges. While any sort of partitioning of the density of this sort suffers from arbitrariness, it can offer useful insights as long as the treatment is consistent from one configuration to the next. In other words, while the atomic charges themselves may not have much meaning, the changes undergone during the H-bond formation have more validity. Rather than examine the electronic distributions directly, another approach focuses upon the electrostatic potential in the region surrounding the molecule, which is a direct result of the charge arrangement149. Correlations can be drawn between H-bonding abilities of molecular entities and the potential at certain selected points in space150,151. 1.3.4 Spectroscopic Observations It is entirely feasible to compute the force field for nuclear displacements from equilibrium by quantum chemical means, leading directly to evaluation of vibrational spectra. In fact, the frequencies of the normal modes are required in order to evaluate the zero-point vibrational energies mentioned earlier so as to compute the enthalpy of formation of a H-bond. These theoretical spectra can be compared to available segments of experimental spectra152.

Quantum Chemical Framework

19

It must be remembered however that the bulk of experimental data are gathered in solvent or other condensed media whereas the theoretical results pertain to the gas phase. It is possible to evaluate elements of NMR spectra by theoretical means as well. GaugeIncluding Atomic Orbital (GIAO) procedures have been developed153 and applied to a variety of H-bonded systems154-158. An alternate technique has been dubbed Individual Gauge for Localized Orbitals (IGLO)159,160. While these permit investigation at the SCF level, correlated work is possible as well, as in the GIAO-MP2 method 161-163 , a coupledcluster GIAO-CCSD(T) procedure164, or by a multiconfiguration generalization165. Ab initio calculations have reached a stage of maturity where quadrupolar coupling constants can be computed with surprising accuracy166. For example, comparison of experimental measurements with calculated coupling parameters have enabled an elucidation of the manner in which aggregates of formamide change their basic structure as the temperature of the liquid is changed167.

1.4 A Simple Example The former analysis may perhaps be best understood by way of an example. Let us consider first the water dimer in its most stable orientation. The geometries were optimized with a modest basis set at the SCF level, but the results are illustrative nonetheless, and more accurate calculations would not influence the example in any important respect. Figure 1.2a illustrates the geometry of the monomer and the directions of the three principal moments of inertia; the dimer is depicted in Fig. 1.2b. Table 1.2 reports the total mass of each species in the first row, followed by the translational energy at 25° C, 3/2 RT. Since there are two reactants and only one product, and all with the same translational energy, Etrans for the reaction is —3/2 RT as listed in the second row of Table 1.3. The moments of inertia are strongly affected by dimerization. The greater distances of the atoms from the center of mass of the dimer, as compared to the smaller monomer, account for the much increased values of Ib and Ic. Nonetheless, the total rotational energies of the monomer and dimer are identical, according to Eq (1.9). Hence, Erot amounts to —3/2 RT as was true for Etrans. The water monomer has three normal modes of vibration. The first three rows of the vibrational frequency entries in Table 1.2 report the frequency of the monomer, followed by the corresponding frequencies of the two molecules as they occur within the dimer. One may note the perturbations caused by the interaction to the internal modes of each water molecule as they differ by up to 100 cm - 1 . The next three lines list the "new" intramolecular vibrational modes that are present in the dimer but not in the monomer. These typically

Figure 1.2 Structures of the water (a) monomer and (b) dimer, illustrating orientations of the three principal moments of inertia. The origin is at the center of mass in each case.

20

Hydrogen Bonding

Table 1.2 Properties of water monomer and dimer at 298° K and 1 atm pressure, calculated with 4-31G basis set. Monomer M, amu Etrans, kcal/mol Ia,au Ib, au Ic, au a Erot, kcal/mol v i ,cm -1

Dimer

18 0.89 1.842 4.429 6.271 2 0.89 1743 3958 4109

EZPVE, kcal/mol Evib,them , kcal/mol vib, therm' Strans, calmor-1 deg-1 S rot,cal mol-1 deg-1 Svib, cal mol-1 deg-1

14.02 0.001 34.61

10.34 0.004

36 0.89 7.492 269.28 270.73 1 0.89 1751,1789 3870,3970 4073,4116 130, 167 174,210 416,761 30.63 1.78 36.68

20.93 11.02

correspond to intermolecular stretching, wagging, and so forth. Perhaps more important, they are also of much lower frequency than the intramolecular modes. For this reason, they add only a relatively small amount to the zero-point vibrational energy. That is, the value of 30.6 kcal/mol is only slightly larger than twice the EZPVE of the water monomer. Consequently, EZPVE is only 2.6 kcal/mol for the dimerization reaction, as seen in the fourth row of Table 1.3. On the other hand, it is the low-frequency vibrations which can more easily be populated as the temperature begins to climb. This population of the low-frequency

Table 1.3 Thermodynamic parameters of water dimerization. E, H amd G in kcal m o l - 1 ; S in cal mol-1 deg - 1 . Eelec

Etrans Erot EZPVE E vib,therm

E AH Strans Srot S vib S G

-8.23

-0.89 -0.89 +2.59 +1.76

-5.66 -6.25 -32.54 0.26 11.01 -21.28 0.09

Quantum Chemical Framework

21

modes of the dimer is manifested by the larger value of E vib,therm in the next line of the table and leads to the value of 1.76 kcal/mol for Evib in Table 1.3. The exchange of six librational modes in the pair of isolated monomers for the same number of low-frequency vibrational modes in the dimer is characteristic of the H-bonding process. The system loses 3/2 RT of translational energy by combining two molecules into a single complex, and the same amount of rotational energy as a pair of rotating species form the dimer. This loss is compensated by the gain in zero-point energy from the new vibrational modes in the complex. In the water dimer in Table 1.2 for example, the six new intermolecular vibrations account for 2.65 kcal/mol of zero-point energy, as compared to the 3RT lost in librations (1.78 kcal/mol at 298° K). In addition, the low-frequency modes are rapidly populated as the temperature climbs, adding more energy to the complex and making the value of AE less negative. Taking the water dimer as an example again, the presence of these low-frequency modes adds some 1.8 kcal/mol to the complex in Evib,therm. The various contributions to the entropy of the monomer and dimer are listed in the last three rows of Table 1.2. The translational entropies may be seen in Eq (1.15) to vary as the logarithm of the mass so the monomer and dimer values are similar. The loss of translational degrees of freedom upon dimerization is hence responsible for the negative value of Strans in Table 1.3. The much larger moments of inertia of the dimer lead to a higher rotational entropy, by Eqs (1.7) and (1.17). This effect is due to the closer spacing of the rotational energy levels and their greater accessibility to thermal population. Although three degrees of rotational freedom are lost upon dimerization, a compensation occurs due to the greater rotational entropy of the dimer arising from its higher moments of inertia; Srot is therefore close to zero in this case. The high frequencies of the vibrational modes in the water monomer make occupation of any levels other than the ground state quite small. The dominance of the single complexion, all molecules in their ground state, is associated with the near-zero vibrational entropy. In contrast, just as the presence of low-frequency vibrations in the dimer permits Evib,therm to climb with temperature, the population of these levels also provides for more ways of rearranging the quanta of vibrational energy and hence to a much larger vibrational entropy. The net result of these three contributing factors is that the full AS is negative by some 21 eu. The overwhelming factor in this loss of entropy arises from the translational component. Comparison of the SCF energies of the water dimer with the pair of monomers yields an electronic contribution to the H-bond energy of — 8.23 kcal/mol, as indicated in the first row of Table 1.3. (Note that negative values indicate greater stability of the dimer and hence more binding energy.) This value is lessened by 2.59 kcal/mol as a result of the greater zeropoint vibrational energy in the dimer than the monomers. Translation and rotation each add 0.9 kcal/mol to the binding energy, but the extra energy resulting from the population of the intermolecular vibrational modes drops the interaction energy by 1.76 kcal/mol. The calculated value of E is thus —5.66 kcal/mol. The enthalpy of binding is more negative by RT. When combined with the negative value of AS, the Gibbs free energy change accompanying dimerization is close to zero. That is, the small dimerization energy is approximately canceled by the loss of entropy which accompanies the complexation. There has been some discussion in the literature as to whether a H-bond is stronger than a D-bond. That is, how does the isotopic substitution of a protium nucleus by a twice-asrnassive deuterium affect the energetics of binding. Since the electronic part of the interaction energy is based upon the Born-Oppenheimcr approximation which places the nuclei at rest, Eelec is unaffected by any isotopic substitution, including this one. Indeed, the po-

22

Hydrogen Bonding

tential energy surface upon which the nuclei move is independent of the atomic masses. The translational and rotational terms would be affected to a small extent. The largest change occurs in the vibrational terms. Doubling the mass of an atom would significantly perturb the effective mass of any vibration involving that atom and so would change the associated frequency. The effects would first be seen in the zero-point vibrational energies, EZPVE, and then in Evib,therm as the temperature climbs. This effect is described in detail below.

1.5 Sources of Error

There are a number of sources of error in the various contributors to the thermodynamics of the formation of H-bonds. First are the assumptions of ideal gas behavior which permit one to write the simple expressions in Eqs (1.5)—(1.17). These assumptions also include the full separability of vibrational and rotational motion. The vibrational terms are commonly calculated within the framework of the harmonic approximation which precludes coupling between various modes and third or higher order terms in the dependence of the energy of the molecule upon the nuclear deformations. The translational and rotational energies are probably computed rather accurately. The total mass of any system is simple and stands apart from any quantum calculations. While the rotational partition functions are sensitive to the moments of inertia, the rotational energy at room temperature is virtually independent of these properties. Moreover, equilibrium geometries can be calculated to relatively good precision with even moderate levels of theory, certainly accurate enough to obtain excellent approximations to the correct moments of inertia. More sensitive to the level of theory is the vibrational component of the interaction energy. In the first place, the harmonic frequencies typically require rather high levels of theory for accurate evaluation. It has become part of conventional wisdom, for example, that these frequencies are routinely overestimated by 10% or so at the Hartree-Fock level, even with excellent basis sets. A second consideration arises from the weak nature of the H-bonding interaction itself. Whereas the harmonic approximation may be quite reasonable for the individual monomers, the high-amplitude intermolecular modes are subject to significant anharmonic effects. On the other hand, some of the errors made in the computation of vibrational frequencies in the separate monomers are likely to be canceled by errors of like magnitude in the complex. Errors of up to 1 kcal/mol might be expected in the combination of zero-point vibrational and thermal population energies under normal circumstances. The most effective means to reduce this error would be a more detailed analysis of the vibration-rotational motion of the complex that includes anharmonicity. By far the largest source of error in calculating the energetics of hydrogen bonding arises in the electronic term. As exemplified by Eq (1.4), each contribution is computed as the difference in energy between the complex on one hand and the sum of monomers on the other. One can note in Table 1.2 that E tends to be of similar magnitude to its two constituent terms for translational, rotational, and vibrational energies. Such is not the case, however, for electronic energies. These quantities represent the energy released upon forming a given molecule from an assortment of isolated nuclei and individual electrons and are hence very large in magnitude. Nearly the same energy is released whether these components are assembled into a pair of isolated monomers or into a H-bonded complex; the difference between these two options (representing Eelec) is many times smaller than the energy of assembly in either case. Taking the water dimer as an example again, the total

Quantum Chemical Framework

23

electronic energy of this dimer is on the order of —10 5 kcal/mol. To be more precise, assembly into the pair of water monomers releases 95,266.86 kcal/mol and into the complex 95,275.08 kcal/mol. The difference, 8.23 kcal/mol, represents only 0.009% of the total. As a consequence, even very small errors in either of the large quantities, errors as small as several thousandths of a percent, will produce very large errors in their difference, Eelec. It is understood that there are few means of calculation of the electronic energy of any system that are capable of obtaining the true result within 0.01%. Indeed, many ab initio calculations, especially those that ignore correlation, will be in error by many, many times this amount. Calculation of even remotely reasonable values for interaction energies are thus dependent upon large-scale cancellation between very large errors. For example, if one calculates the energies at the Hartree-Fock level, it is implicitly assumed that the correlation energy in the dimer, typically several hundred kcal/mol, will be nearly identical to the total correlation energy of the pair of isolated monomers. Similar types of cancellation are the presumption for calculating interaction energies with less than complete basis sets.

1.6 Basis Set Superposition

With regard to basis set, there is another and more subtle hazard to the computation of interaction energy by the supermolecular approach. It is obvious that one must use the same basis set in calculating the energy of the pair of isolated monomers as for the complex. For example, the properties of the monomers and dimer of water were computed using the 4-31G basis throughout in the foregoing analysis (Tables 1.2 and 1.3). Each contributor to the interaction energy was obtained by

where the subscript refers to the specific basis set. If one were to apply a smaller basis set like 3-21G or STO-3G to the individual monomers, but retain 4-31G for the complex, it would introduce an obvious inconsistency, wiping out any possibility of the aforementioned cancellation necessary for reasonable results. So it is agreed that the same basis set must be used for the complex as for its constituent subunits. But the situation is not as simple as it sounds, as noted by Kestner in 1968168. The basis set for each monomer consists of functions centered on each of its atoms. The basis set of the dimer is larger in the sense that there are present functions centered on all atoms of both monomers. One may represent this fact by using a subscript to indicate the atoms covered by the basis set:

The larger basis set of the dimer provides additional flexibility. The electrons of monomer A are free to partially occupy the orbitals provided along with molecule B, and vice versa. Such freedom is not provided to these same electrons when the monomer is calculated in its own basis set, with B completely absent. The availability of the extra orbitals will lower the energy of each monomer, within the context of the dimer, by the variation principle which states that each additional degree of flexibility provided to the electrons permits a lowering of the energy. As a result, the complex undergoes an artificial stabilization due solely to its larger basis set, in comparison to the smaller sets of the monomers. This spurious stabilization of the complex, in excess of any genuine interaction energy, is commonly referred to as basis set superposition error (BSSE).

24

Hydrogen Bonding

How may this error be avoided? The simplest means is to calculate the energy of all species within the same set of basis functions. Since any computation of A...B must surely place functions on both A and B, the same must be true for each monomer.

The difference between this formulation and Eq (1.22) is that the basis set used to calculate the energy of A includes not only its own functions but those of B as well (and similarly for B in the same large basis set). The calculation of E(A)A...B is much like that of the full complex except that the nuclei and electrons of B are deleted. The extra functions of B included in the calculation of A are sometimes referred to as "ghost" orbitals and the procedure outlined in Eq (1.23) is commonly denoted "functional counterpoise" after the originators of the suggestion169. The m and d superscripts on E in Eqs (1.22) and (1.23), respectively, refer to whether the energies of the monomers are computed within the monomer or dimercentered basis set. The difference between these two means of calculating the interaction energy is one measure of the superposition error.

1.6.1 Secondary Superposition While the computation of the energy of each monomer must be performed within the context of the basis set of the entire complex for the sake of consistency in order to avoid BSSE, there is another consideration engendered by this approach. This issue was first raised by a number of research groups in the late 1970s and early 1980s170~174 and is related to the change in properties of each monomer associated with the ghost orbitals of its partner. Consider as a simple example a spherically symmetric atom like Ar. An atom-centered basis set would correctly reflect that Ar has no dipole (or higher) moment. Suppose now that an additional species is added to the system, another Ar atom for example. Within the context of the basis set of the pair, the spherical symmetry of the first Ar atom is lost; consequently each atom has associated with it a nonzero permanent dipole moment. The interaction energy hence contains a dipole-dipole interaction that is not present in the real dimer. Similar arguments can be extended to higher multipole moments or to elements of the polarizability tensor. These properties are different in the original basis set of a single atom as compared to that of the dimer. This line of reasoning is valid also for more complicated systems like H-bonded complexes of molecules like water. Even though each HOH molecule does indeed possess a nonzero moment, addition of the basis functions of its partner would introduce a change in this, or higher, moments. The above changes in the calculated properties of each monomer, caused by the addition of the partner functions to the basis set, together with the perturbations in the interaction energy associated with them, are frequently referred to as secondary BSSE or as basis set extension effects. Not only are these effects more difficult to remove than primary BSSE, but it is not entirely clear whether they should in fact be corrected. For example, early on, Karlstrom and Sadlej170 argued that these effects can be beneficial in that the properties of each monomer are improved by the enlargement of the basis set, as would occur if the additional functions were centered on the molecule itself rather than its partner. However, later work indicated that secondary BSSE typically represents another artifact that deteriorates the quality of the calculation. Latajka and Scheiner17S took as a model the interaction between a Li ' cation and a neutral molecule of water and showed that

Quantum Chemical Framework

25

the secondary BSSE can be quite large, comparable in magnitude to the primary effect. They suggested a crude means of correcting the extension effect and found no ensuing overcorrection. 1.6.2 Important Properties of Superposition Error There are several salient facts to bear in mind: 1. In general, BSSE is reduced as the basis set becomes larger and more flexible. However, there is no strict correspondence, and the BSSE can in fact become larger with certain additions to the basis set. Minimal basis sets are particularly prone to large BSSE, as are those like 3-21G with a poor description of the inner shells. 2. The BSSE rises rapidly as the two molecules approach one another; angular dependence is more complex. 3. The origin of BSSE makes it difficult to incorporate counterpoise corrections directly into gradients of the potential energy. 4. Whereas SCF BSSE can be reduced to negligible proportions with large basis sets, the superposition error at correlated levels goes down much more slowly, persisting at large values, even with very flexible bases. 1.6.3 Historical Perspective Understanding of the issues involved in superposition errors has evolved slowly. Consequently, the reader is liable to encounter a number of different attitudes and means of handling the problem over the years in the literature. This section is intended to provide some perspective on the problem so that the reader will be able to critically assess the impact that superposition error might have on a given set of calculations in the literature. 1.6.3.1 Early Attitudes Although many researchers ackowledged that there was some inconsistency in using a larger basis set for the complex than for the monomers, there was initial reluctance to accept the counterpoise procedure as a valid means to correct the problem when it was first introduced. The chief source for this skepticism lay in the numerical results. Many of Ihe early calculations of H-bonded complexes relied on fairly small and inflexible basis sets. It is now known that bases of this type tend toward potentials that are much less attractive than the true potential. Limitations of the era also prohibited application of correlation in most cases, eliminating a major attractive component. As a result, the H-bond attractions corresponding to these treatments are much too weak. It was only the superposition errors that were hidden in the calculations that permitted the final results to be even slightly attractive. In other words, if one does not analyze the results carefully, one can easily be misled since the spurious attractive nature of the BSSE can compensate in some sense for the unsatisfactory character of the calculations themselves. For example, an STO-3G calculation of the water dimer, if left uncorrected for basis set superposition, can yield an interaction energy not very different from experimental expectations. But when the superposition error is removed, the remaining potential, the true HF/STO-3G potential is not attractive at all, but indicates the water molecules would repel one another and not form a H-bonded complex176. Rather than recognize this observation as a legitimate failing as an error due to the STO-3G basis set or to the absence of correla-

26

Hydrogen Bonding

tion, it was tempting to attribute the repulsive potential to the counterpoise correction. And if one takes the experimental data point as the ultimate goal of a calculation, then retaining the superposition error appeared to offer a superior means of achieving that end. But such an approach is shortsighted. One must understand that precise reproduction of an experimental result is coincidental at best when using a crude method with known weaknesses. After all, a minimal basis set without correlation does not offer a realistic version of a molecular system, so why should one take an experimental H-bond energy as a criterion as to whether counterpoise is an appropriate correction? By removing the BSSE, one approaches the true picture of a given theoretical model, in this case the HF/STO-3G version of the water dimer. Even with larger and more flexible basis sets, one should not necessarily expect the calculations to duplicate experiment. Such an expectation might lead one to erroneously conclude that counterpoise corrections do not improve the accuracy of the calculations177. Nonetheless, the early disagreement of counterpoise corrected H-bond potentials with experiment spawned a number of variants of the technique which reduced the BSSE correction and left the potential more attractive than if the full error were removed. Some of these methods justified themselves on the grounds that the electrons of one molecule should not expand into the orbital space of the partner molecule that is already occupied by electrons 178-181 . Hence, damping factors were introduced or more formal means of permitting the electrons of molecule A to partially occupy only the vacant MOs of molecule B, and vice versa176,178,182. Another technique proposed employing a perturbing charge field generated by the partner183,184. An alternate treatment has been proposed to evaluate the BSSE which is based on an exact perturbation solution185. More recently, it has been demonstrated on formal and numerical grounds that the Pauli exchange principle itself prevents the electrons of A from expanding into the occupied space of B 186-191 . Consequently, it is the full counterpoise correction, as originally proposed, that should be applied to the problem of molecular interactions. It is now widely accepted that the counterpoise correction should be applied to the Hartree-Fock part of the potential182,190,193-195. (Indeed, recent work has suggested that Slater basis functions might provide a realistic alternative to the more standard Gaussians in certain cases, provided counterpoise corrections are made195.) Computer technology has made this acceptance an easy pill to swallow since the BSSE can be made negligibly small by application of large and flexible basis sets which can be handled by modern workstations. It is hence not even necessary in many cases to do the actual correction. The smallness of this error is particularly fortunate when optimizing geometries since the gradient procedures that search potential energy surfaces for stationary points do not incorporate BSSE corrections directly into their algorithms. Nevertheless, even with large basis sets, superposition errors can introduce noticeable errors into equilibrium geometries of many molecules196,197. There are alternatives to the most commonly used Boys-Bernardi counterpoise scheme. One approach that shows promise, for example, is a chemical Hamiltonian approach (CHA), pioneered by Mayer 198-201 , which attempts to isolate the superposition error directly in the Hamiltonian operator. The Schrodinger equation that is solved is hence a modified one, which yields a wave function that is hopefully free of superposition error. In the case of (HF)2, it was found that this approach mimics rather closely the results of the standard counterpoise scheme for a scries of small to moderate sized basis sets200. Later calculations202 extended these tests to other small H-bonded systems as well, again limiting their testing to basis sets no larger than 6-31G**. A recent test203 has extended the method's use-

Quantum Chemical Framework

27

fulness. The authors find for a series of H-bonded complexes that the difference between the Boys-Bernardi and CHA-corrected interaction energies diminishes as the core, that is, sp-part, of the basis set improves. They recommend the use of either correction scheme as superior to simply trying to eliminate the superposition error by improving the basis set. However, this approach has been criticized on the grounds of its inconsistency with the results of symmetry-adapted perturbation theory204 and because this Hamiltonian is nonHermitian185. A similar idea has been proposed to construct a projection operator to remove the BSSE when applied to a wave function, with results comparable to the standard counterpoise procedure when tested on He and H2205. It might be noted, however, that this approach also has its detractors who identify inconsistencies with symmetry-adapted perturbation theory204. 1.6.3.2 Correlated Levels The problem is less tractable at correlated levels where the BSSE persists at uncomfortably large magnitudes even with very large basis sets206. Most workers now agree that the full counterpoise error must be removed from correlated calculations of molecular interaction potentials190,197,204,207-210. For one thing, the counterpoise-corrected interaction energy equates nicely to the sum of perturbation theory terms, each of which is formally free of superposition error211. Nonetheless, there remains some lingering controversy as to the appropriateness of invoking this correction181,185,192,212. As an example, calculations examined the fluctuations that occur in the interaction energy of the water dimer as small perturbations are introduced into the basis set213. These perturbations included addition of a second set of d-functions on O, minor adjustments in the polarization function exponent, or the number of primitive gaussians in the contraction. The raw interaction energies for the water dimer with R(OO) = 3.0 A are illustrated by the broken curves in Fig. 1.3 as the basis set was altered. Note the large fluctuations at the MP2 as well as SCF levels. For example, the MP2 interaction energies vary between —5.5 and —8.5 kcal/mol. After counterpoise correction, on the other hand, the data is much more consistent from one basis set to the next, as illustrated by the solid curves. More recent work has confirmed these conclusions: The counterpoise method leads to an accurate description of the correlated interaction energies in the HF dimer, with the proviso that the basis set is capable of properly describing the physical forces involved209. Even with an insufficiently flexible set, the counterpoise-corrected results are more stable with respect to basis set than uncorrected data. Davidson and Chakravorty214 have recently compiled an exhaustive listing of earlier means of treating superposition error and proposed an alternative means of looking at the problem, referred to as a complete basis set. They suggest that counterpoise corrections should perhaps be supplemented by what they refer to as monomer and dimer nonadditivity corrections in order to obtain the correct interaction energy at any level. These new terms would contain within them secondary BSSE. In the water and HF dimer cases they considered, they found that the latter nonadditivity corrections are of the same sign as counterpoise corrections at the SCF level so that they move the interaction energy in the correct direction. On the other hand, the dimer nonadditivity correction is of opposite sign to counterpoise at the MP2 level, which might explain why inclusion of counterpoise corrections at this level can move the calculated interaction energy away from an experimental result. A reexamination of this analysis led others to suggest that large nonadditivity corrections result from a poor choice of basis set and do not indicate any conceptual weakness in

28

Hydrogen Bonding

Figure 1.3 Interaction energies computed for the water dirtier213 with R(OO) equal to 3.0 A. Uncorrected values are connected by broken lines and counterpoise corrected (cc) interaction energies illustrated by solid curves.

the counterpoise procedure itself211. The authors present the notion that basis sets that yield small counterpoise corrections are not necessarily best adapted to investigate molecular interactions. A basis set designed to best reproduce the important components of the interaction energy would be a better choice. While sets of the latter type may lead to a nonnegligible BSSE, counterpoise correction would yield superior results. The original authors215 argue that the difference between the true dissociation energy and its counterpoise-corrected equivalent is a nonadditive correction for basis set incompleteness. In terms of analyzing the contemporary literature, it is probably advisable to consider as overly attractive any interaction energies that have not been corrected for basis set superposition error. It is emphasized that "overly attractive" refers in this context to the correct result with a given theoretical model, not to the experimental value as a reference. Results computed at the SCF level with a counterpoise correction can probably be taken as the most accuracy one is likely to achieve with a particular basis set without correlation. There remains some difference of opinion concerning correlated results, but counterpoise corrections should probably be taken as more valid than those with no such corrections at all. There is also some lingering question as to the precise details of properly correcting BSSE when there are more than two identifiable units involved in the interaction 201,216-218 .

1.7 Energy Decomposition One of the underlying questions about hydrogen bonds is just what physical forces hold the two partner molecules together. That is, what is the fundamental nature of the H-bond? As

Quantum Chemical Framework

29

the two molecules approach one another, a number of physical phenomena can be imagined that might be involved in the forces between them 219-221 . From a strictly electrostatic point of view, the two molecules that eventually form a Hbond each have associated with them an electronic distribution which produces an electric field in the surrounding space. The interaction of the static fields of the two molecules corresponds to a purely Coulombic force which may be attractive or repulsive, depending upon the orientations of the two molecules. The energetic consequence of this interaction is commonly denoted as the electrostatic energy. This electrostatic interaction corresponds, then, to the classical Coulomb force between two charge distributions at long separations between the two subunits. At such distances, it is also useful to write the full electrostatic interaction energy as a multipole series. That is, one may construct the charge distribution of each neutral molecule as the sum of a dipole moment vector, quadrupole moment tensor, and so on to higher and higher orders. The interaction between the dipoles of the two molecules behaves as R-3 where R is the distance between centers of charge of the two. Dipole-quadrupole interactions die off as R - 4 , quadrupole-quadrupole as R - 5 , and so on, up to infinite orders. At long distances, the higher order terms are anticipated to become quite small as this series converges. Indeed, for very long separations, it is the dipole-dipole interaction which dictates the preferred angular orientation of the approach of one molecule toward the other. Should one of the molecules be charged, the ion-dipole term is of lowest order and the series will die off as R - 2 . The situation becomes less clearcut as the two molecules approach within H-bonding distance of one another. In the first place, the multipole series loses its utility since the higher order terms become progressively larger and the series does not converge properly. It may hence be misleading to consider the dipole-dipole term as dominant or indeed, as even an important contributor. Even ignoring the multipole analysis, the full electrostatic interaction becomes more difficult to define unambiguously. That is, when far apart, it is a simple matter to assign any "piece" of electron density to one molecule or the other, based simply on whether the region of interest is close to A or close to B. But as the two molecules approach one another, this dividing border becomes more vague. There is significant density in the region midway between the two molecules and it is not at all clear whether the electrons here are part of the distribution pattern of A or of B. The energetic consequence of this density overlap is commonly lumped under the rubric of "penetration" terms in the electrostatic energy, defined properly as the difference between the full electrostatic interaction energy and the infinite summation of the multipole expansion. It should also be recalled at this point that the electrostatic interaction assumes the charge density patterns of either molecule are completely unaffected by the presence of the field generated by its partner. That is, the density is "frozen" in the configuration adopted when the two molecules are fully isolated. The contributions of the various terms in the multipole expression to the full electrostatic interaction energy may be illustrated for the water dimer in Fig. 1.4222. Each curve in Fig. 1.4 represents the cumulative sum up to the indicated term. For example, the R-5 curve represents the sum of the R - 3 , R - 4 , and R-5 terms. For intermolecular separations exceeding 5 A, the first term in the series, R - 3 , which contains primarily the dipole-dipole interaction, nicely mimics the full coulombic energy. As the two molecules approach closer together, higher-order terms become necessary. At the van der Waals minimum of 3 A, even taking the series up through sixth or seventh order significantly underestimates the full term. It is hence apparent that penetration effects arc important for this uncharged H-bond in its equilibrium configuration. A second factor in the interaction between the two molecules is not classical in origin but arises instead from the requirement that the full wave function of any system, includ-

30

Hydrogen Bonding

Figure 1.4 Distance dependence of the multipole series of the electrostatic interaction energy, truncated at various orders, for the water dimer. Data from222. The values for the series truncated at R-7 differ only very slighty from the R-5 series and so are not shown explicitly.

ing a H-bonded complex, be antisymmetric with respect to interchange of any two electrons. The interchange between pairs of electrons, both of which are contained within one molecule, has already been taken into account in generating the wave function of that molecule. What is of interest here is the interchange of one electron from molecule A with one from B. Because of its mathematical description, the energetic consequence of permitting the latter exchanges into the supermolecular wave function is termed "exchange" energy. In the case of interactions between closed shell subunits, as is the situation in H-bonded complexes, this term is repulsive in nature (hence the descriptive as "exchange repulsion") and can be linked in some sense with the classic picture of "steric repulsion" between charge clouds. It must nevertheless be remembered that this force is quantum mechanical in origin and, like electrostatic energy, is calculated within the context of electron densities of each subunit that are unaltered by the presence of the partner. The sum of the aforementioned electrostatic and exchange energies is sometimes classified as the Heitler-London energy. Of course, an integral ingredient in H-bonding arises from the ability of each molecule to perturb the electron density of the other. For example, as A approaches B, its electric field induces redistributions of electron density within B and vice versa. Such redistributions are driven by the energetics of the situation and are of course stabilizing. The net stabilization achieved by the system as a result of the density redistributions can be classified as "induction" or "deformation" energy, owing to its origin. There have been attempts in the literature to further partition this deformation energy into smaller pieces. One means of breaking down this term rests on the ability to divide elec-

Quantum Chemical Framework

3I

trons and space into that belonging to A and that of B. The electronic redistribution of A can now be accomplished in one of two ways. The A electrons can move around from their original locations around A to other parts of space, previously unoccupied, but still defined as "A space." This term is commonly referred to as "polarization" energy as it fits with the usual definition of polarization of a given molecule. The other possibility is for the A electrons to invade "B space" which leads to the concept of "charge transfer" from A to B and its energetic consequence. However, the distinction between polarization and charge transfer rests on the division of space into that belonging to A or B, which is arbitrary at best, and becomes even more so as the two subunits merge into a single complex. For that reason, there are many that argue against any separation of the induction energy into these two components as they claim it leads to spurious judgments as to the fundamental nature of the interaction under study223,224. The previous components of the interaction energy can be derived in the independent particle approximation and so appear within the context of Hartree-Fock level calculations. Nevertheless, inclusion of instantaneous correlation will affect these properties. Taking the electrostatic interaction as an example, the magnitude of this term, when computed at the SCF level, will of course be dependent on the SCF electron distributions. The correlated density will be different in certain respects, accounting for a different correlated electrostatic energy. The difference between the latter two quantities can be denoted by the correlation correction to the electrostatic energy. There is another physical phenomenon which appears at the correlated level which is completely absent in Hartree-Fock calculations. The transient fluctuations in electron density of one molecule which cause a momentary polarization of the other are typically referred to as London forces. Such forces can be associated with the excitation of one or more electrons in molecule A from occupied to vacant molecular orbitals (polarization of A), coupled with a like excitation of electrons in B within the B MOs. Such multiple excitations appear in correlated calculations; their energetic consequence is typically labeled as "dispersion" energy. Dispersion first appears in double excitations where one electron is excited within A and one within B, but higher order excitations are also possible. As a result, all the dispersion is not encompassed by correlated calculations which terminate with double excitations, but there are higher-order pieces of dispersion present at all levels of excitation. Although dispersion is not necessarily a dominating contributor to H-bonds, this force must be considered to achieve quantitative accuracy. Moreover, dispersion can be particularly important to geometries that are of competitive stability to H-bonds, for example in the case of stacked versus H-bonded DNA base pairs225. Having listed the above components of the interaction energy, it is worthwhile to underscore their arbitrariness. The total energy of a system corresponds to a real physical observable, so the interaction energy, defined as the difference between energies, is likewise real. But the various components do not correspond to a quantum mechanical operator and are only as real as the arbitrary definition associated with them. As an example, the electrostatic energy arising within the Hartree-Fock formalism is similarly limited by the independent particle approximation. It is necessary to apply correlation corrections to high order to even approach the physical picture of this phenomenon. It is also questionable whether one should consider "exchange" as a separate entity since its existence is intimately connected with the quantum mechanical mandate of antisymmetry of the wave function. The notion of first precluding and then "permitting" electron exchange between subunits, so as to extract exchange energy, is no more real than is the ability to turn off and on this antisymmetry principle.

32

Hydrogen Bonding

Despite the arbitrariness of definition, the decompositions of the interaction energies of H-bonds have provided some intriguing and useful insights into the fundamental aspects of this phenomenon. Some of these will be detailed in the ensuing chapters. Nonetheless, the reader is cautioned that this arbitrariness has also spawned a number of different schemes of decomposing the energy into various segments in the literature. One must be particularly careful in comparing these sets of data since it is not uncommon for two different schemes to use the same name for components that are derived in different ways. Most schemes fall into one of two general categories. The various components can be computed directly via a perturbational scheme219,221,226 or a supermolecule approach can be taken wherein the terms are calculated as a difference between large quantities220. 1.7.1 Kitaura-Morokuma Scheme Let us take as an example what is probably the most frequently used means of decomposing the interaction energy of various complexes, including H-bonds. In the KM scheme, Kitaura and Morokuma220,227 first compute the wave functions of the two isolated subunits of the complex, A° and B°. They take this pair of wave functions as the starting point for a Hartree-Fock calculation of the complex.

Note that the electrons of A are antisymmetrized within A°, and similarly for B electrons in B°. However, no exchange of A and B electrons is permitted in A° B°. The zeroth iteration of the SCF procedure yields an energy which differs from the total energy of the pair of isolated subunits by an amount taken to be EES since it permits the field of each monomer to interact with the electron density, i°, of the partner, without perturbing that density. The exchange energy is extracted by again beginning the SCF procedure but this time permitting interchange of electrons between A and B, indicated by the AB operator below.

The energy associated with 2 differs from that of 1 by an amount defined here as the exchange energy, EEX, owing to the electron exchange within the 2 wave function. Starting with 1r and enforcing the restriction of no exchange between electrons in A and B, convergence of the iterative SCF procedure permits the electrons in each subunit to relax in the presence of the field of the partner. The extra stabilization gained as a result of this relaxation is associated with the polarization, and is labeled EPL. A similar SCF relaxation, but now allowing the full antisymmelrization of all electrons, yields an energy which is lower than the latter one by an amount which is taken to correspond to charge transfer between A and B, and is hence denoted ECT. The latter four terms do not encompass all of the effects associated with complexation. Any remaining effects, in addition to those that result from the artificial separation described above, are collected into a last "mixing" term, E MIX . Moreover, there have been attempts to further partition some of the above terms into smaller pieces as well as other mixing terms. Note that the Kitaura-Morokuma scheme limits itself to SCF calculations so includes no dispersion, nor any correlation corrections to the aforementioned terms. One of the problems of this technique lies in the mixing term which provides no physical insights. Furthermore, this term grows to uncomfortably large proportions when the interaction strengthens223. It was also found that the various components of the interaction energy are even more sensitive to basis set choice than is the total interaction energy. The

Quantum Chemical Framework

33

separation of deformation energy into charge transfer and polarization is purely artificial; indeed, the latter two effects are indistinguishable in the limit of a complete basis set. Some of these points can be illustrated with simple examples. Table 1.4 lists the components of the interaction energy of the water dinner computed by the Morokuma-Kitaura scheme for the water dimer, taken in its experimental geometry, with an interoxygen separation of 2.98 A227. The electrostatic term is clearly highly sensitive to the basis set chosen. Going from minimal STO-3G to split-valence 4-31G more than doubles this attractive force; adding d-functions reduces EES by 1.4 kcal/mol. The exchange energy is more stable but the other terms are again rather erratic. The charge transfer energy is notable in that it is quite large for the minimal basis set. Much of this contribution can be attributed to basis set superposition error which is expected to contaminate this term the most. (Efforts have been made more recently to correct the various components for BSSE194,228-232.) If one of the two species is charged, the ion can be expected to be much more effective at polarizing its partner. Furthermore, the presence of ion-dipole, ion-quadrupole, and so forth terms will enhance the electrostatic interaction. These presumptions are confirmed by the comparison of the data in Table 1.5 for the ionic complex between (NH4)+ and NH3224, with the neutral dimer in Table 1.4. The data were computed using a very large basis set, including d and functions on nitrogen. A range of different intermolecular separations is considered to illustrate the distance-dependence of the various components. Note from the bottom lines of the two tables that the ionic (H 3 NH ... NH 3 ) + complex is bound more strongly by several-fold. Whereas the electrostatic component in the neutral dimer is less than 10 kcal/mol, this term exceeds 20 kcal/mol at the same separation (3.0 A) and is greater than 30 kcal/mol at the equilibrium separation of the ionic complex (2.75 A). One can also see the rapid growth of the exchange repulsion as the two subunits approach one another, as it is the principal factor preventing their collapse into one another. While the polarization energy in the neutral water dimer is less than 1 kcal/mol, the presence of the ion in (H 3 NH ... NH 3 ) + greatly enlarges this component, even at distances as far as 3.25 A. The charge transfer term is similarly enhanced in the ionic complex. But the data in Table 1.5 also point out one of the prime weaknesses of decomposition methods like Kitaura-Morokuma. Note that as the two subunits come closer than their equilibrium separation, the polarization energy blows up beyond all reasonable proportions. This very large negative value is due to the "merging together" of the basis sets of the two molecules, particularly for a basis as extended as the one being used here. The problem arises because the polarization energy, as defined in such a scheme, fails to observe the Pauli exclusion principle and electrons from one subunit begin to occupy space already taken by

Table 1.4 Morokuma-Kitaura components of SCF interaction energy of water dimer.a

EES EEX EPL ECT EMIX

AE a

All values in kcal/mol

STO-3G

4-31G

-4.2 4.0 -0.1 -4.8 0.1 -5.1

-8.9 4.2 -0.5 -2.1 -0.3

227

-7.7

6-31G* -7.5 4.3

-0.5 -1.8 -0.1 -5.6

34

Hydrogen Bonding

Table 1.5 Morokuma-Kitaura components of SCF interaction energy of H 3 NH +... NH 3 . a R(A) 3.25 EES EEX EPL ECT EMIX

E

-18.2

5.1 -8.2 -3.4

5.2 -19.5

3.0

-23.5 11.5 -14.6 -6.0 10.8 -21.7

2.75 -31.6 25.8 -28.5 -12.3 24.8 -21.9

2.50 -44.8 57.3 -180.9

-30.1 183.7 -16.2

a

All values in kcal/mol. Data224 calculated using large basis set (S + f).

the electrons of the partner. Another symptom of the problem is the very large magnitude of the mixing term, whose positive values seem to compensate for the overly attractive polarization (and charge transfer energies). There have been numerous schemes proposed in the literature to circumvent some of these difficulties. One worth mention sets up a "Pauli blockade" which prevents any of the intermediate wave functions from violating the Pauli principle233. It effectively combines into a single term the KM polarization, charge transfer, and mix terms. There is also the problem that basis set superposition error contaminates each of the components, making a physical interpretation difficult. Various means have been devised over the years to circumvent this problem193,194,234. 1.7.2 Alternate Schemes A reduced variational space method235, related to the KM procedure, has been developed in which the orbitals of one fragment are optimized in the field of the frozen orbitals of its partner. Truncation of the variational space by deletion of unoccupied orbitals of one partner or the other is the pathway to evaluation of polarization, charge-transfer, and BSSE terms. When applied to the water dimer235, the Coulomb and exchange sum dominates the interaction but charge transfer and polarization terms are needed for proper angular dependence. A quite different scheme has been proposed by Weinhold et al.236-238 which is based on natural bond orbitals. After an initial transformation of the atomic orbital basis set into natural atomic orbitals which optimize the occupancy, one obtains a set of core plus valence orbitals with high occupancy, and another set of residuals with low occupancy. The natural bond orbitals are derived from the formation of an optimal orthonormal set of directed hybrids which translate to a set of localized orbitals that correspond roughly to the traditional concepts of chemical bonds and lone pairs. In this framework, the total energy of the dimer is partitioned into two principal components. E arises when all unoccupied bond orbitals are deleted and is associated with the electrostatic interactions, plus dominant effects of the Pauli principle (that is, steric repulsions). The other component is denoted E and is representative of charge-transfer delocalization. While E * is many times smaller than E the contributions of these two terms to the interaction energy, indicated by the A, are much more comparable in magnitude:

Quantum Chemical Framework

35

E can be related to the Heitler-London energy, the sum of electrostatic and exchange interactions, while the SCF deformation energy, containing both intramolecular polarization and intermolecular charge transfer, corresponds roughly to E Analysis of their wave functions for the water dimer revealed that the bulk of the charge transfer consisted of density shifting from the lone pair of the proton acceptor to the antibond between the oxygen and bridging hydrogen of the donor molecule. This work emphasized the importance of delocalization of electron density into the unoccupied orbitals in stabilizing the water dimer. The authors estimate that 5.4 kcal/mol arises from the specific donation from the aforementioned lone pair to O— H antibond. Their version of the Heitler-London energy (electrostatic plus exchange) is repulsive, in contrast to most other decomposition treatments wherein the attractive Coulombic force is stronger than the exchange repulsion. One of the more intriguing findings of the Reed-Weinhold treatment is that relatively large energetic stabilizations result from only very small amounts of charge being transferred, generally less than 0.01 electrons. Reed et al.236 compared their results for a number of systems with the KM scheme which attributes much of the H-bond attraction to electrostatic energy. The authors attribute the distinction to the operational definition of charge transfer. They claim that the KM electrostatic term contains contributions from determinants which are better described as donoracceptor in nature, leaving to the charge transfer energy only that portion of the acceptor orbital which is Schmidt orthogonalized to the partner's donor orbital. On the other hand, Olszewski et al. echo the KM contention of the dominance of electrostatics in their studies of H-bonding of various pairs of simple hydrides233. One may conclude that the distinction between the two treatments is largely a semantic one which underscores the arbitrary nature of any means of partitioning the interaction energy. A more modern version of this general scheme based on natural bond orbitals involves a decomposition of the SCF part of the interaction energy into electrostatic, charge transfer, and deformation terms239. While these terms are similar in name to the KM components, there are significant differences in formulation. The electrostatic term, for example, contains an exchange contribution as it enforces antisymmetry of the appropriate wave function. It, moreover, is evaluated from fragment wave functions deformed by the interaction, so contains induction/polarization energy. The deformation energy is repulsive as it comprises the distortions of the monomer electron clouds so as to maintain orthogonality of their wave functions. Their analysis of the water dimer led Glendening and Streitwieser239 to attribute strong contributions to the binding from both electrostatics and charge transfer, with sizable repulsive forces arising from the deformation. The results are exhibited in Table 1.6 where they may be compared with the more commonly used MorokumaKitaura procedures. Figure 1 .5 illustrates the dependence of each of these components upon the particulars of the basis set. Their behavior can be compared to that of the total interaction energy, indicated by the solid line. Most of these terms are reasonably basis set insensitive, that is no more sensitive than the total E itself. Still another form of decomposition is based on first transforming the canonical molecular orbitals into localized MOs of a different sort. The so-called localized charge distributions240 also contain contributions, not necessarily integral, from the various nuclei. The total energy of a H-bonded complex like the water dimer is partitioned into kinetic and potential energy terms. The drive for a given charge distribution to spread out so as to lower its kinetic energy is balanced against the "suction" of the charge into a region of low potential energy.

Table 1.6 Comparison between Morokuma-Kitaura and natural energy decomposition analysis (NEDA)239. All values in kcal/mol, calculated with 4-31G basis set. KM total E CT ES POL EX MIX DEF

BSSE

-7.8 -2.4 - 10.5 -0.6 6.2 -0.5

NEDA -7.8 -13.3 -17.8

24.8 -1.1

Figure 1.5 Values of natural energy decomposition analysis components of water dimer for various basis sets, from239. Basis sets are as follows: (1) STO-3G, (2) 4-31G, (3)6-31G*, (4) 6-31G**, (5)6-31+G**, (6) 6-31 ++G**, (7)6-31++ G(2d,p), (8) 631 + +G(2d,2p), (9) cc-pVDZ, (10) aug-cc-pVDZ, (11) ccpVTZ, (12) aug-cc~pVTZ.

Quantum Chemical Framework

37

Another, and generally older, philosophy partitions the total energy into contributions that are associated with one or two atomic centers241,242. A strength of this approach is the ability to focus upon bond strengths via the latter term. This technique has been applied to analysis of chemical phenomena such as the Cope rearrangement243, internal rotation244, and the nature of aromaticity245,246. 1.7.3 Perturbation Schemes Still another means of partitioning the interaction energy is based on standard RayleighSchrodinger perturbation theory. It takes as its starting point a wave function which is the simple product of those of the isolated molecules and uses the interaction between the two molecules as the perturbation222,247-250. A difficulty inherent in this approach is enforcing the proper symmetry into the wave function. The MSMA expansion, due to Murrell and Shaw251, and Musher and Amos252, incorporates this symmetry only upon the wave function used as a starting point for the iterative process. The technique is commonly referred to as symmetry-adapted perturbation theory (SAPT). By treating the interaction as a perturbation, the interaction energy itself can be written as an infinite series of terms of higher and higher order. The first-order terms are similar in nature to the electrostatic and exchange energies in the other partitioning approaches. For example, the perturbational definition of exchange energy differs from the aforementioned MO description by terms of fourth order in the overlap integrals222. To a good approximation, the first-order exchange energy is proportional to the square of the overlap between the two subunits at their equilibrium separation. Accurate evaluation of this term for long distances requires a good representation of the tails of the valence orbitals. Second-order terms include first an induction energy which corresponds roughly to the sum of polarization and charge transfer in the KM scheme. Dispersion energy makes its first appearance at second order. Also present here are two manifestations of electron exchange in the form of induction-exchange and dispersion-exchange. Their sum is commonly referred to as polarization exchange energy. Whereas empirical expressions can be used to approximate dispersion or induction, the latter two exchange-related phenomena are more difficult to approximate. It should be emphasized here that various components occur repeatedly at progressively higher orders of perturbation theory. Dispersion, for example, is not limited to second order but is also present in higher order terms. It is hoped in most cases that the terms beyond second order are small enough to be safely ignored, but this is not always the case. A prime advantage of the perturbational approach is that the individual terms are evaluated explicitly, rather than as the difference between very much larger quantities as is true of supermolecule approaches. These SAPT terms are free of basis set superposition error. More important, each term corresponds to a well-defined physical phenomenon, which permits an insightful analysis. Each term can be evaluated using a different basis set, most appropriate for that particular component. For example, dispersion requires orbitals of high angular momentum whereas electrostatics can usually be derived with only a moderate basis set. It is possible to express the second-order induction energy in terms of the multipole moments of any small molecules involved and their static polarizability tensors253 but further simplification is difficult for a pair of polyatomic subunits. A similar analysis permits the dispersion to be placed within the context of dynamic polarizabilities. In the case of a pair

38

Hydrogen Bonding

of spherically symmetric atoms, the dispersion reduces to a series in even powers of 1/R, beginning with R - 6 , with the polarizabilities buried within the coefficients of the series, referred to as van der Waals constants222. While the expansion is generally divergent, damping factors may be invoked to permit obtaining useful coefficients based on experimental data or accurate ab initio computations. Some of the correspondence between the KM and perturbational values of the various components can be seen in Table 1.7. The sum of polarization, charge transfer, and mixing energies is roughly comparable to the induction energy, IND, which first appears at second order in perturbation theory. The values in Table 1.7 indicate this correspondence is fairly good, but not exact. It is useful to point out also that when combined together in this manner, the unphysical enlargements of the polarization and mixing terms pointed out earlier cancel one another to provide a reasonable net induction energy. Chalasinski and Szczesniak254 have provided a means of decomposing the correlation contribution to the interaction energy into four separate terms. Their philosophy takes the electron exchange operator as a second perturbation in the spirit of many-body perturbation theory, with molecular interaction as the first perturbation in their intermolecular M011erPlesset perturbation theory (IMPPT). At the level of second order of the correlation operator, they obtain a number of separate terms. The first is the dispersion energy, disp (20), correct through second order of correlation. ES(12) refers to the effect of correlation upon the Hartree-Fock electrostatic energy. The remaining terms represent the change in the deformation and exchange energies, relative to their SCF values. The third row of Table 1.7 repeats the SCF electrostatic energy of the H 3 NH +... NH 3 system, and is followed by the correction to this term that occurs when correlation is included to the wave function. It is apparent that these correction terms are of fairly small magnitude and of opposite sign to the Hartree-Fock level Coulomb energy. The last row lists the dispersion energy computed for this ionic system and shows it to be a negative quantity that grows quickly as the two species are brought toward one another. Cybulski et al.255 furnish an example of the sensitivity of the various perturbation components of the H-bond energy to the choice of basis set. In their study of the dimer of HF, 6-31G** refers to a standard split-valence set, with polarization functions. GD is similar in character but was designed to specifically address the dispersion energy more accurately. The S2 set was proposed by Sadlej to produce reliable dipole moments and polarizabilities of the monomers, augmented by extended polarization functions ( on F; d on H). Well-

Table 1.7 Comparison between Kitaura-Morokuma and perturbational components of the interaction energy of H 3 NH +... NH 3. a R(A) 3.25 E PL+ ECT+ EM

IND EES (12) ES

(20) disp a

IX -5.6

-5.0 -18.2 0.5 -1.7

3.0

-8.8 -8.3 -23.5

0.6 -2.9

2.75 -14.8 -14.9 -31.6 0.8 -5.0

All values in kcal/mol. Data 224, calculated using large basis set (S + f).

2.50 -27.2 -28.7 -44.8 1.0 -8.9

Quantum Chemical Framework

39

tempered sets were tested as well; referred to as WTS2 when polarized as in the S2 set. The first row of Table 1.8 illustrates good consistency of the electrostatic term at the SCF level. The exception is 6-31G** which was not formulated with good monomer properties as a prime goal. One may expect poorer results with smaller basis sets as the electrostatic term is fairly sensitive in this regard. The other SCF terms are much less sensitive so moderatesized basis sets would generally be appropriate, provided polarization functions are included. The authors expressed surprise at the insensitivity of the deformation energy and postulated that the "ghost orbitals" of the partner help to make up for deficiencies in the description of each subunit. They also point out the similarity between the perturbational ind(20) and variational EdefSCF quantities that address the same property from different perspectives. As a result of the relative constancy of exchange and induction, the full SCF interaction energy mirrors the sensitivity to basis set of the electrostatic term alone. As in the SCF case, the correlation correction to the electrostatics, ES(12), indicates a poor result with 6-31G** but better consistency with the other three basis sets. This term is repulsive, which is consonant with the general trend in H-bonded systems and is attributed to a correlation-induced reduction in monomer multipole moments. Indeed, it is common to find that correlation reduces charge separation within a variety of molecules256. The dispersion term is probably most difficult to saturate with sufficient diffuse polarization functions, so exhibits a continued rise in magnitude as the basis set is enlarged. The/functions included in S2 and WTS2 account for the most negative values there. Exchange-correlation and deformation-correlation effects are repulsive at the second order, and do not show very much sensitivity to basis set. On the other hand, they are not insignificant so should be included wherever possible. The authors were finally emphatic in pointing out that their results are much less meaningful if basis set superposition errors are left uncorrected. It might be noted finally that a perturbational approach offers the possibility of a rigorous definition of nonadditive terms within clusters257. Such unambiguous definitions are useful in understanding cooperativity, that is, the manner in which one molecule can influence the interaction between two others.

Table 1.8 Perturbation components to interaction energy of HF dimer at equilibrium geometry with various basis sets.a 6-31G**

GD

S2

WTS2

SCF level ES(10) HL

exch

ind(20)

EdefSCF ESCF

-7.51 4.39 -1.6a9 -1.58 -4.70

-6.36 4.16 -1.79 -1.72 -3.92

-6.19 4.14 -1.84 -1.81 -3.86

-6.22 4.14 -1.86 -1.83 -3.92

0.54 -1.41 0.59 -0.28

0.49 -1.43 0.69 -0.26

MP2 level (I2)

ES

disp

(20)

exchange/deformation E(2) a

All values in kcal/mol255.

0.08 -0.90 0.76 -0.06

0.55 -1.17 0.62 0.01

40

Hydrogen Bonding References

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134. Kollman, P., McKelvey, J., Johansson, A., and Rothenberg, S., Theoretical studies of hydrogenbonded dimers. Complexes involving HF, H2O, NH3, HCl, H2S, PH3, HCN, HNC, HCP, CH2NH, H2CS, H2CO, CH4, CF3H, C2H2, C4H4, C 6 H 6 , F-, and H3O+, J. Am. Chem. Soc. 97, 955-965 (1975). 135. Allen, L. C., A simple model of hydrogen bonding, J. Am. Chem. Soc. 97, 6921-6940 (1975). 136. Dill, J. D., Allen, L. C., Topp, W. C., and Pople, J. A., A systematic study of the nine hydrogenbonded dimers involving NH3, OH2, and HF, J. Am. Chem. Soc. 97, 7220-7226 (1975). 137. Kollman, P., A general analysis of noncovalent intermolecular interactions, J. Am. Chem. Soc. 99, 4875-4894 (1977). 138. Umeyama, H. and Morokuma, K., The origin of hydrogen bonding. An energy decomposition study, J. Am. Chem. Soc. 99, 1316-1332 (1977). 139. Morokuma, K., Why do molecules interact? The origin of electron donor-acceptor complexes, hydrogen bonding, and proton affinity, Ace. Chem. Res. 10, 294-300 (1977). 140. Pople, J. A., Intermolecular binding, Faraday Discuss. Chem. Soc. 73, 7-17 (1982). 141. Hobza, P. and Zahradnik, R., Van der Waals molecules: Quantum chemistry, physical properties, and reactivity, Int. J. Quantum Chem. 23, 325-338 (1983). 142. Astrand, P.-O., Karlstrom, G., Engdahl, A., and Nelander, B., Novel model for calculating the intermolecular part of the infrared spectrum for molecular complexes, J. Chem. Phys. 102, 3534-3554 (1995). 143. Del Bene, J. E., Mettee, H. D., Frisch, M. J., Luke, B. T., and Pople, J. A., Ab initio computation of the enthalpies of some gas-phase hydration reactions, J. Phys. Chem. 87, 3279-3282 (1983). 144. Levine, I. N., Physical Chemistry; 3rd ed.; McGraw-Hill: New York, 1988. 145. Espinosa, E., Lecomte, C., Ghermani, N. E., Devemy, J., Rohmer, M. M., Benard, M., and Molins, E., Hydrogen bonds: First quantitative agreement between electrostatic potential calculations from experimental X-(X + N) and theoretical ab initio SCF models, J. Am. Chem. Soc. 118,2501-2502(1996). 146. Koch, U. and Popelier, P. L. A., Characterization of C—H—O hydrogen bonds on the basis of the charge density, J. Phys. Chem. 99, 9747-9754 (1995). 147. Carroll, M. T. and Bader, R. F. W., An analysis of the hydrogen bond in BASE-HF complexes using the theory of atoms in molecules, Mol. Phys. 65, 695-722 (1988). 148. Boyd, R. J., Hydrogen bonding between nitrites and hydrogen halides and the topologicalproperties of molecular charge distributions, Chem. Phys. Lett. 129, 62-65 (1986). 149. Politzer, P. and Truhlar, D. G., eds. Chemical Applications of Atomic and Molecular Electrostatic Potentials; Plenum, New York (1981). 150. Murray, J. S., Ranganathan, S., and Politzer, P., Correlation betwen the solvent hydrogen bond acceptor parameter fl and the calculated electrostatic potential, J. Org. Chem. 56, 3734-3737 (1991). 151. Murray, J. S. and Politzer, P., Correlation betwen the solvent hydrogen-bond-donating parameter a and the calculated molecular surface electrostatic potential, J. Org. Chem. 56, 6715-6717 (1991). 152. Hagelin, H., Murray, J. S., Brinck, T., Berthelot, M., and Politzer, P., Family-independent relationships between computed molecular surface quantities and solute hydrogen bond acidity/basicity and solute-induced methanol O—H infrared frequency shifts, Can. J. Chem. 73, 483-488 (1995). 153. Ditchfield, R., Self-consistent perturbation theory of diamagnetism. L A gauge-invariant LCAO method for N.M.R. chemical shifts, Mol. Phys. 27, 789-807 (1974). 154. Ditchfield, R., Theoretical studies of magnetic shielding in H2O and (H2O)2, J. Chem. Phys. 65, 3123-3133(1976). 155. Ditchfield, R., GIAO studies of magnetic shielding in FHF- and HF, Chem. Phys. Lett. 40, 53-56(1976). 156. Ditchfield, R. and McKinncy, R. E., Molecular orbital studies of the NMR hydrogen bond shift, Chem. Phys. 13, 187-194 (1976).

Quantum Chemical Framework

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157. Rohlfing, C. M., Allen, L. C., and Ditchfield, R., Proton chemical shift tensors in neutral and ionic hydrogen bonds, Chem. Phys. Lett. 86, 380-383 (1982). 158. Chesnut, D. B. and Phung, C. G., Functional counterpoise corrections for the NMR chemical shift in a model dimeric water system, Chem. Phys. 147, 91-97 (1990). 159. Kutzelnigg, W., Theory of magnetic susceptibilities and NMR chemical shifts in terms of localized quantities., Isr. J. Chem. 19, 193-200 (1980). 160. Schindler, M. and Kutzelnigg, W., Theory of magnetic susceptibilities and NMR chemical shifts in terms of localized quantities. II. Application to some simple molecules, J. Chem. Phys. 76, 1919-1933(1982). 161. Gauss, J., Calculation of NMR chemical shifts at second-order many-body perturbation theory using gauge-including atomic orbitals, Chem. Phys. Lett. 191, 614-620 (1992). 162. Gauss, J., Effects of electron correlation in the calculation of nuclear magnetic resonance chemical shifts, J. Chem. Phys. 99, 3629-3643 (1993). 163. Buhl, M., Thiel, W., Fleischer, U., and Kutzelnigg, W., Ab initio computation of 77Se NMR chemical shifts with the IGLO-SCF, the GIAO-SCF, and the GIAO-MP2 methods, J. Phys. Chem. 99, 4000-4007 (1995). 164. Gauss, J. and Stanton, J. R, Perturbative treatment of triple excitations in coupled-cluster calculations of nuclear magnetic shielding constants, J. Chem. Phys. 104, 2574—2583 (1996). 165. van Wiillen, C. and Kutzelnigg, W., Calculation of nuclear magnetic resonance shieldings and magnetic susceptibilities using multiconfiguration Hartree-Fock wave functions and local gauge origins, J. Chem. Phys. 104, 2330-2340 (1996). 166. Eggenberger, R., Gerber, S., Huber, H., Searles, D., and Welker, M., The use of molecular dynamics simulations with ab initio SCF calculations for the determination of the oxygen-17 quadrupole coupling constant in liquidwater, Mol. Phys. 80, 1177-1182 (1993). 167. Ludwig, R., Weinhold, R, and Parrar, T. C., Temperature dependence of hydrogen bonding in neat, liquidformamide, J. Chem. Phys. 103, 3636-3642 (1995). 168. Kestner, N. R., He-He interaction in the SCF-MO approximation, J. Chem. Phys. 48, 252-257 (1968). 169. Boys, S. F and Bernardi, R, The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors, Mol. Phys. 19, 553-566 (1970). 170. Karlstrom, G. and Sadlej, A. J., Basis set superposition effects on properties of interacting systems. Dipole moments and polarizabilities, Theor. Chim. Acta 61, 1—9 (1982). 171. Dacre, P. D., A calculation of the helium pair polarizability including correlation effects, Mol. Phys. 36, 541-551 (1978). 172. Dacre, P. D., On the pair polarizability of helium, Mol. Phys. 45, 17-32 (1982). 173. Fowler, P. W. and Buckingham, A. D., The long range model of intermolecular forces. An SCF study of NeHF, Mol. Phys. 50, 1349-1361 (1983). 174. Kurdi, L., Kochanski, E., and Diercksen, G. H. R, Determination of the basis set superposition error with "DZP" basis sets in SCF calculations: CO + H2, NH3 + H2, H2 + H2, Chem. Phys. 92, 287-294 (1985). 175. Latajka, Z. and Scheiner, S., Primary and secondary basis set superposition error at the SCF andMP2 levels: H3N..Li+ and H 2 O .. Li + , J. Chem. Phys. 87, 1194-1204 (1987). 176. Johansson, A., Kollman, P., and Rothenberg, S.,An application of the functional Boys-Bernardi counterpoise method to molecular potential surfaces, Theor. Chim. Acta 29, 167-172 (1973). 177. Schwenke, D. W. and Truhlar, D. G., Systematic study of basis set superposition errors in the calculated interaction energy of two HF molecules, J. Chem. Phys. 82, 2418-2426 (1985). 178. Daudey, J. P., Claverie, P., and Malrieu, J. P., Perturbation ab initio calculations of intermolecular energies. I. Method, Int. J. Quantum Chem. 8, 1—15 (1974). 179. Collins, J. R. and Gallup, G. A., The full versus the virtual counterpoise correction for basis set superposition error in self-consistent field calculations, Chem. Phys. Lett. 123, 56-61 (1986).

48

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180. Yang, J. and Kestner, N. R., Accuracy of counterpoise corrections in second-order intermolecularpotential calculations. 2. Various molecular dimers, J. Phys. Chera. 95, 9221-9230(1991). 181. Tolosa, S., Espinosa, J., and Olivares del Valle, F. J., Computation of spectroscopic properties of van der Waals systems from post-SCF ab initio potentials including the EICP alternative counterpoise technique, J. Comput. Chem. 5, 611-619 (1991). 182. Alagona, G., Ghio, G., Cammi, R., and Tomasi, J., The effect of "full" and "limited" counterpoise corrections with different basis sets on the energy and the equilibrium distance of hydrogen bonded dimers, Int. J. Quantum Chem. 32, 207-226 (1987). 183. Loushin, S. K., Liu, S., and Dykstra, C. E., Improved counterpoise corrections for the ab initio calculation of hydrogen bonding interactions, J. Chem. Phys. 84, 2720-2725 (1986). 184. Loushin, S. K. and Dykstra, C. E., Polarization counterpoise corrections to correlated hydro gen bond interactions, J. Comput. Chem. 8, 81-83 (1987). 185. Sadlej, A. J., Exact perturbation treatment of the basis set superposition correction, J. Chem. Phys. 95, 6705-6711 (1991). 186. Ostlund, N. S. and Merrifield, D. L., Ghost orbitals and the basis set extension effects, Chem. Phys. Lett. 39, 612-614 (1976). 187. Gutowski, M., van Lenthe, J. H., Verbeek, J., van Duijneveldt, F. B., and Chalasinski, G., The basis set superposition error in correlated electronic structure calculations, Chem. Phys. Lett. 124, 370-375 (1986). 188. Gutowski, M., van Duijneveldt, F. B., Chalasinski, G., and Piela, L., Does the Boys and Bernardi function counterpoise method actually overcorrect the basis set superposition error?, Chem. Phys. Lett. 129, 325-330 (1986). 189. Gutowski, M., van Duijneveldt, F. B., Chalasinski, G., and Piela, L., Proper correction for the basis set superposition error in SCF calculations of intermolecular interactions, Mol. Phys. 61, 233-247 (1987). 190. van Duijneveldt, F. B., van Duijneveldt-van de Rigdt, J. G. C. M., and van Lenthe, J. H., State of the art in counterpoise theory, Chem. Rev. 94, 1873-1885 (1994). 191. Sordo, J. A., Sordo, T. L., Fernandez, G. M., Gomperts, R., Chin, S., and Clementi, E., A systematic study on the basis set superposition error in the calculation of interaction energies of systems of biological interest, J. Chem. Phys. 90, 6361-6370 (1989). 192. Cook, D. B., Sordo, J. A., and Sordo, T. L., Some comments on the counterpoise correction for the basis set superposition error at the correlated level, Int. J. Quantum Chem. 48, 375—384 (1993). 193. Alagona, G., Cammi, R., Ghio, C., and Tomasi, J., Noncovalent interactions of medium strength. A revised interpretation and examples of its applications, Int. J. Quantum Chem. 35, 223-239 (1989). 194. Sokalski, W. A., Roszak, S., and Pecul, K.,An efficient procedure for decomposition of the SCF interaction energy into components with reduced basis set dependence, Chem. Phys. Lett. 153, 153-159(1988). 195. Alagona, G. and Ghio, C., Basis set superposition errors for Slater vs. gaussian basis functions in H-bond interactions, J. Mol. Struct. (Theochem) 330, 77-83 (1995). 196. Sellers, H. and Almlof, J., On the accuracy in ab initio force constant calculations with respect to basis set, J. Phys. Chem. 93, 5136-5139 (1989). 197. Nowek, A. and Leszczynski, J., Ab initio study on the stability and properties of XYCO . . . HZ complexes. HI. A comparative study of basis set and electron correlation effects for H 2 CO ... HCl, J. Chem. Phys. 104, 1441-1451 (1996). 198. Mayer, I., Towards a "chemical" Hamiltonian, Int. J. Quantum Chem. 23, 341-363 (1983). 199. Mayer, I., On the non-additivity of the basis set superposition error and how to prevent its appearance, Theor. Chim. Acta 72, 207-210 (1987). 200. Mayer, I. and Vibok, A., A BSSE-free SCF algorithm for intermolecular interactions, Int. J. Quantum Chem. 40, 139-148 (1991). 201. Mayer, I., Vibok, A., Halasz, G., and Valiron, P., A BSSE-free SCF algorithm for intermolecular

Quantum Chemical Framework

202. 203. 204. 205. 206. 207. 208. 209.

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49

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50

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225. Hobza, P., Sponer, J., and Polasek, M., H-bonded and stacked DNA base pairs: cytosine dimer. An ab initio second-order M0ller-Plesset study, J. Am. Chem. Soc. 117, 792-798 (1995). 226. Claverie, P., In Intermolecular Interactions: From Diatomics to Biopolymers; Pullman, B., ed.; Wiley, New York (1978) pp 69-305. 227. Morokuma, K. and Kitaura, K., In Chemical Applications of Atomic and Molecular Electrostatic Potentials; Politzer, P. and Truhlar, D. G., eds.; Plenum, New York (1981) pp 215-242. 228. Sokalski, W. A., Roszak, S., Hariharan, P. C., and Kaufman, J. J. Improved SCF interaction energy decomposition scheme corrected for basis set superposition effect, Int. J. Auantum Chem 23, 847-854 (1983). 229. del Valle, F. J. O., Tolosa, S., and Espinosa, J., Basis set superposition effects in electronic populations calculatedon hydrogen bonded systems, J. Mol. Struct. (Theochem) 120,277-283 (1985). 230. Bonaccorsi, R., Cammi, R., and Tomasi, J., Counterpoise corrections to the components of bimolecular energy interactions: An examination of three methods of decomposition, Int. J. Quantum Chem. 29, 373-378 (1986). 231. Cammi, R., Del Valle, F. J. O., and Tomasi, J., Decomposition of the interaction energy with counterpoise corrections to the basis set superposition error for dimers in solution. Method and application to the hydrogen fluoride dimer, Chem. Phys. 122, 63-74 (1988). 232. Alagona, G., Ghio, C., Latajka, Z., and Tomasi, J., Basis set superposition errors and counterpoise corrections for some basis sets evaluated for a few X-...M dimers, J. Phys. Chem. 94, 2267-2273 (1990). 233. Olszewski, K. A., Gutowski, M., and Piela, L., Interpretation of the hydrogen-bond energy at the Hartree-Fock level for pairs of HF, H2O, and NH3 molecules, L Phys. Chem. 94, 5710-5714 (1990). 234. Alagona, G., Ghio, C., Cammi, R., and Tomasi, J., The decomposition of the SCF interaction energy in hydrogen bonded dimers corrected for basis set superposition errors: An examination of the basis set dependence, Int. J. Quantum Chem. 32, 227-248 (1987). 235. Stevens, W. J. and Fink, W. H., Frozen fragment reduced variational space analysis of hydrogen bonding interactions. Application to the water dimer, Chem. Phys. Lett. 139, 15-22 (1987). 236. Reed, A. E., Weinhold, F., Curtiss, L. A., and Pochatko, D. J., Natural bond orbital analysis of molecular interactions: Theoretical studies of binary complexes of HF, H2O, NH3, N2, O2, F2, CO and CO2 with HF, H2O, and NH3, J. Chem. Phys. 84, 5687-5705 (1986). 237. Reed, A. E., Curtiss, L. A., and Weinhold, F., Intermolecular interactions from a natural bond orbital, donor-acceptor-viewpoint, Chem. Rev. 88, 899-926 (1988). 238. Reed, A. E., and Weinhold, F., Natural bond orbital analysis of near Hartree-Fock water dimer, J. Chem. Phys. 78, 4066-4073 (1983). 239. Glendening, E. D. and Streiwieser, A., Natural energy decomposition analysis: An energy partitioning procedure for molecular interactions with application to weak hydrogen bonding, strong ionic, and moderate donor-acceptor interactions, J. Chem. Phys. 100, 2900-2909 (1994). 240. Jensen, J. H. and Gordon, M. S., Ab initio localized charge distributions: Theory and a detailed analysis of the water dimer-hydrogen bond, J. Phys. Chem. 99, 8091-8097 (1995). 241. Fischer, H. and Kollmar, H., Energy partitioning with the CNDO method, Theor. Chim. Acta 16, 163-174(1970). 242. Kollmar, H., Partitioning scheme for the ab initio SCF energy, Theor. Chim. Acta 50, 235-262 (1978). 243. Dewar, M. J. S. and Lo, D. H., Ground states of -bonded molecules. XIV. Application of energy partitioning to the MINDO/2 method and a study of the Cope rearrangement, J. Am. Chem. Soc. 93,7201-7207(1971). 244. Gordon, M. S., A molecular orbital study of internal rotation, J. Am. Chem. Soc. 91, 3122-3130 (1969). 245. Ichikawa, H. and Ebisawa, Y., Hartree-Fock MO theoretical approach to aromaticity. Interpretation of Hiickel resonance energy in terms of kinetic energy of electrons, J. Am. Chem. Soc. 107, 1161-1165 (1985).

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5I

246. Hiberty, P. C., Shaik, S. S., Lefour, J.-M., and Ohanessian, G., Is the delocalizedn-system of benzene a stable electronic system, J. Org. Chem. 50,4657-4659 (1985). 247. Jeziorski, B. and van Hemert, M., Variation-perturbation treatment of the hydrogen bond between water molecules, Mol. Phys. 31, 713-729 (1976). 248. Arrighini, P., Intermolecular forces and their evaluation by perturbation theory, vol. 25. Springer-Verlag, Berlin (1981). 249. Szalewicz, K., Jeziorski, B., and Rybak, S., Perturbation theory calculations of intermolecular interaction energies, Int. J. Quantum Chem. QBS18, 23-36 (1991). 250. Williams, H. L., Mas, E. M., Szalewicz, K., and Jeziorski, B., On the effectiveness of monomer-, dimer-, and bond-centered basis functions in calculations of intermolecular interaction energies, J. Chem. Phys. 103, 7374-7391 (1995). 251. Murrell, J. N. and Shaw, G., Intermolecular forces in the region of small orbital overlap, J. Chem. Phys. 46, 1768-1772 (1967). 252. Musher, J. I. and Amos, A. T., Theory of weak atomic and molecular interactions, Phys. Rev. 164,31-43(1967). 253. Dalgarno, A., Adv. Phys. 12, 143 (1962). 254. Chalasinski, G. and Szczesniak, M. M., On the connection between the supermolecular Mft/llerPlesset treatment of the interaction energy and the perturbation theory of intermolecular forces, Mol. Phys. 63, 205-224 (1988). 255. Cybulski, S. M., Chalasinski, G., and Moszynski, R., On decomposition of second-order M0llerPlesset supermolecular interaction energy and basis set effects, J. Chem. Phys. 92, 4357-4363 (1990). 256. Carpenter, J. E., McGrath, M. P., and Hehre, W. J., Effect of electron correlation on atomic electron populations, J. Am. Chem. Soc. 1 l l , 6154-6156 (1989). 257. Moszynski, R., Wormer, P. E. S., Jeziorski, B., and van der Avoird, A., Symmetry-adapted perturbation theory of nonadditive three-body interactions in van der Waals molecules. I. General theory, J. Chem. Phys. 103, 8058-8074 (1995).

2

Geometries and Energetics

n this chapter, we focus our attention on the equilibrium geometries of various H-bonds, and how the formation of the complex alters the internal structure of each subunit. The energetics of hydrogen bonding are also stressed. Comparison is made to experimental information where available. Whereas many geometries have been evaluated to high precision, energetic data in the gas phase, to which the calculations directly pertain, have been harder to obtain. One of the handicaps against experimental evaluation of H-bond energies in the gas phase has been the difficulty in accurate evaluation of equilibrium constants for formation of complexes involving a pair of neutral species. Advances in methodology, using Fourier transform IR spectrometry1, promise to alleviate this problem in the future. Preliminary results indicate a close equivalence between the equilibrium constants of formation in the gas phase and those obtained in inert solvent like CC14. The hydrides AHn provide a good forum by which to extract the hydrogen bonding characteristics of the A atom, both as proton donor and acceptor. A nomenclature is introduced here so as to systematize the presentation. X is used to represent the halide atoms F, Cl, and so on so the hydrogen halides are referred to as HX in the general case. H2Y corresponds to H2O, H2S, and so on while NH3 and its congeners in lower rows of the periodic 54 are represented by ZH3. This chapter is organized by system type. The complexes pairing HX with ZH3 are simplest in that it is obvious, due to their differences in acidity, which molecule will act as proton donor and which as acceptor. There is little ambiguity about the geometry of the H-bond since HX has a single proton and ZH3 only one lone pair. When HX is paired with YH2, there are two lone pairs on the latter so guessing the relative orientaion becomes less trivial. The other heterogeneous pairing, between YH2 and ZH3, is most complicated in that the acidities can be similar enough that one could imagine cither molecule acting as the donor in certain situations. There are also a fair number of possibilities in terms of numbers of protons and/or lone pairs so that the nature of the geometry is not intuitively obvious.

I

52

Geometries and Energetics

53

Following the foregoing discussion of heterogeneous pairs, the homogeneous pairs are considered wherein both molecules of the dimer are the same, or at least of the same type, for example H2Y. The pairing of two HX molecules forces one to act as proton acceptor, despite the poor basicity of the halide atoms. While the X—H . . . X arrangement will clearly tend toward 180°, it is not entirely clear from first principles how the acceptor molecule will align itself. That is, the dipole-dipole interaction would clearly favor a fully linear X—H . . . X—H even though there are no lone pairs on the X atom directly opposite the H—X bond. The situation is more complicated in dimers of H2Y, where one could imagine cyclic and bifurcated arrangements where more than one H-bond could be formed at the same time. Since NH3 is so weakly acidic, it is not obvious that the ammonia dimer would form a H-bond at all. This has indeed been a point of debate, as discussed in this chapter. The discussion extends beyond the simple hydrides and delves into some of the functional groups as well. The carbonyl oxygen is interesting in that it contains a pair of equivalent lone pairs, displaced to either side of the C=O bond. But it is not clear whether a proton donor would prefer to interact with one of these lone pairs or with the electron density directly opposite the C=O bond. It is also of interest to compare the proton accepting ability of the carbonyl and hydroxyl oxygens. Combining the C==O with a —OH on a single entity yields the carboxyl group. Its acidity makes it a potent proton donor, but it is interesting to examine the proton accepting ability of the C=O group here and on the simpler aldehyde or ketone. Of interest also are alternate bonding schemes for the nitrogen atom. The C=N double bond in imines is analogous to the carbonyl oxygen; it is interesting to examine whether the triple bond in nitriles hampers the nitrogen's ability to accept a proton in a H-bond. But of perhaps greater interest is the acidity of the C—H group in HC N; the triple bond endows the C with strong proton donating potential. The chapter concludes with a discussion of the amide group which combines a C=O and N—H on the same species. Of especial importance is the competition for H-bonding between the amide and water, due to its relevance to protein structure. Also discussed is the ability of the "larger" groups like carboxyl and amide to establish more than one H-bond in simple dimers. The emphasis in this chapter is on the fundamental properties of these complexes. Of particular interest are the trends in geometry and energetics: Are there clear patterns in terms of H-bond strength on going from one type of complex to the next? How are the properties affected by going down a column in the periodic table? What sort of correlations might exist between various features of the geometries and the energetics? The final sections of this chapter discuss two interesting points. One of the limitations of ab initio methods is the rapid increase in computer resource demands as the size of the system grows. The accelerating pace of improvements in computer hardware and code development has permitted these methods to be extended up to the range of nucleic acid base pairs. This extended range is demonstrated in this chapter, where it is shown that the computed results are in excellent agreement with experiment. The last section addresses a fundamental question dealing with isotopic substitution. Is the D-bond stronger or weaker than the H-bond? That is, if the normal protium of a H-bond is replaced by its heavier deuterium isotope, how does this affect the properties of the interaction, especially the energetics? 2.1 XH...ZH3

The simplest type of H-bond would be one in which the proton donor molecule contained only one hydrogen that could participate and the acceptor only one lone pair capable of in-

54

Hydrogen Bonding

Figure 2.1 Equilibrium geometry of XH...ZH3.

teracting with the bridging hydrogen. These respective criteria are met by hydrogen halides like HF or HC1 and by molecules of the ammonia family, ZH3, where Z is nitrogen or any other atom below it in the periodic table. In accord with the aforementioned expectations, the equilibrium geometry adopted by these complexes has the H—X axis coincident with the line joining X and Z, placing the bridging hydrogen directly along the H-bond axis as illustrated in Fig. 2.1. The entire complex belongs to the C3v point group. The interaction energies calculated for this series where X=F,Cl,Br and Z=N,P,As2 are listed in Table 2.1. These are uncorrelated values with a moderate sized basis set so should not be taken as definitive. Nonetheless, the data illustrate the important trends in going down a column of the periodic table. Greater electronegativity in the proton donor X atom makes for a more polar X—H bond which creates a stronger electrostatic pull on the lone pair of the acceptor. The more acidic nature of HX can also act to better release the proton toward the acceptor, again favoring a stronger H-bond. These expectations are confirmed by the larger H-bond energies as one reads from right to left in Table 2.1. With regard to the proton acceptor molecule, ZH3, the enlargement from N to P greatly diminishes the H-bond energy. This is not surprising as the low electronegativity of P makes it a poor candidate for proton acceptor. The combination of the high acidity of HF and the strong basicity of NH3 makes the H 3 N ... HF complex the most strongly bound of this series and indeed, of any complex composed of a pair of simple hydrides. It is a general observation that correlation adds to the H-bond energy in most complexes. Some representative data are reported in Table 2.2 for pairs of NH3 or PH3 with HF or HC13,4. Comparison of the binding energies, computed at the SCF and MP2 levels in the first and second rows, respectively, reveals the correlation-induced strengthening of the Hbond. This effect is proportionately greater for the weaker complexes containing PH3 where it can account for as much of a contribution as the entire SCF interaction itself. The distance between the nonhydrogen atoms, optimized with and without correlation, is listed in the next two rows of Table 2.2. The correlation-induced strengthening is also reflected in a small shortening of the H-bond. R(Z..X)MP2, is reduced by anywhere from 0.03 A, relative to R(Z..X)SCF, for the most strongly bound H 3 N ... HF complex to a maximum of 0.36 A for the weakest H3P...HC1.

Table 2.1 Electronic contributions to binding energies, - Eelec, of H-bonds of type H3Z...HX, calculated using DZP basis set at SCF level. Data in kcal/mol2. H3Z H3N H3.P

H 3 As

HF 11.8

3.7 3.6

HC1 7.3 3.3 1.9

HBr 5.7 1.6 1.5

Geometries and Energetics

55

Table 2.2 Calculated binding energies (— Eelec in kcal/mol) and bond lengths (A) of H-bonded complexes3,4. H 3 N ... HF

H 3 N ... HC1

11.8 15.1 2.728 2.693 0.022 0.028

11.0 3.297 3.144 0.023 0.040

- ESCF __

EMP2

R(Z..X)SCF R(Z..X)MP2

r(HX)SCF,a r(HX)MP2

H3P...HF

9.3

H3P...HCl

4.1 6.0

2.1 4.4

3.455 3.291 0.006 0.012

4.166 3.802 0.004 0.011

a

Stretch of HX bond caused by formation of complex.

One typical aspect of the formation of a H-bond is the stretching of the bridging proton away from the donor atom. This stretch can be calculated as the difference in r(XH) between the isolated HX molecule and in the complex. The values listed for r(HX) in Table 2.2 indicate some relation between the stretch and the strength of the H-bond. The stretches calculated here range from 0.004 A for H3P...HC1 up to 0.02 A for the complexes containing NH3. The effects of correlation are particularly important for accurate assessment of the degree of stretching; uncorrelated values can underestimate by several-fold. One can take the H 3 N ... HF system to illustrate the potential effects of basis set superposition error upon the calculated interaction energies. The results in Table 2.3 are taken from Latajka and Scheiner5 where basis sets of the general split-valence type were modified in an effort to minimize this error. The first two entries in the table illustrate that the superposition error, calculated by the counterpoise technique, is close to 1 kcal/mol at the SCF level, and a comparable amount is added at the correlated level. The values reported for Eelec in Table 2.3 refer to the separate SCF and MP2 contributions to the interaction energy, uncorrected for BSSE, followed by their sum. The last three columns illustrate these same properties following counterpoise correction. This correction reduces the SCF dissociation energy from 11.8 to 10.7 kcal/mol but has an even more dramatic effect on the MP2 contribution, lowering it by a factor of five, from 1.5 to 0.3 kcal/mol. The combined effect, illustrated by the last columns, is that counterpoise correction of the full SCF+MP2 interaction energy reduces it from 13.3 to 11.0 kcal/mol, a drop of 17%. The next two rows of Table 2.3 belie a common notion in the literature that superposition error drops as the ba-

Table 2.3 Basis set superposition errors and their effect on interaction energies of H3N...HF. Data5 in kcal/mol. - Eelec

-BSSE

-- ( Eelec -- BSSE)

Basis Set

SCF

MP2

SCF

MP2

SCF + MP2

SCF

MP2

SCF + MP2

6-31G**

1.07 1.21 1.48 0.69 0.35 2.43

1.16 1.21 1.59 1.23 0.79 1.87

11.79 11.85 11.08 11.31 10.42 12.27

1.48 1.71 2.28

13.27 13.56 13.37 12.94 12.08 14.20

10.72 10.64 9.61 10.62 10.07 9.83

0.32 0.51 0.69 0.40 0.87 0.07

11.04 11.15 10.30 11.02 10.94 9.90

+ 2d +-VP S +-VP s (2d) s 6-311G**

1.63 1.66 1.93

56

Hydrogen Bonding

sis set is enlarged. In fact, the opposite can occur, as the next two rows attest. The + symbol signifies the addition to the standard 6-31G** basis set of a set of diffuse sp-functions on nonhydrogen centers N and F. These functions produce an increase in the BSSE at both the SCF and MP2 levels, as do the second set of polarization functions, denoted as "2d." Standard basis sets are typically constructed by optimizing the exponents within the context of each individual atom. Incorporation of the atoms into a molecule, such as HF, would change the requirements on the basis set. With this in mind, Latajka and Scheiner reoptimized the exponents of the 6-31G** basis set within the context of the individual subunits, HF and NH35. The results of such an optimization, including the diffuse sp-set on nonhydrogen atoms, are reported in the next row of Table 2.3, labeled +VPS. While this approach reduced the SCF BSSE from 1.07 to 0.69 kcal/mol, very little change was observed in the correlated BSSE which remained at 1.2 kcal/mol. More dramatic are the improvements when the same prescription is applied to the doubly polarized basis set. The SCF BSSE for this + VPs(2d)s basis set is only 0.35 kcal/mol, 1/4 the value for the 2d set. The reduction at the correlated level is significant but not as marked, dropping from 1.6 kcal/mol for the unoptimized 2d down to 0.8 for + VPs(2d)s. Such a lowering of the correlated error is important because of the smallness of this contribution. That is, if one were to compute the interaction energy of this system with the 6-31G** basis set, augmented by a second set of dfunctions, the MP2 contribution would be 2.28 kcal/mol, but fully 70% of this amount consists of the superposition error. The BSSE contamination is less than 50% if the exponents are reoptimized for the molecules, yielding the +VPs(2d)s basis set. These results underscore the difficulty in lowering superposition errors at correlated levels, a problem with which researchers are still wrestling. The last row of Table 2.3 reveals the profound difficulties in using the 6-311G** basis set where the triple split of the valence set might normally be expected to be an improvement over 6-31G**. Instead, the SCF BSSE is more than doubled, and an increase of similar magnitude occurs in the correlated superposition error. Indeed, the correlation component is completely distorted by superposition effects: Essentially all the (1.93 kcal/mol) stabilization predicted by MP2 with this basis set is due to the artifact of superposition. Removal of this error leaves only a net stabilization of less than 0.1 kcal/mol. Although they failed to correct their interaction energy for this very substantial error, Sadlej and Miaskiewicz6 did compute a useful value of the zero-point vibrational energy of the complex. They found that the complex contains 3.1 kcal/mol more of this type of energy than the sum of the isolated complexes, using the 6-311G** basis set at the SCF level. Del Bene7 has computed the binding energy of this complex at the MP4 level, using a 6-31G type basis set, augmented by two sets of d-functions on heavy atoms and two sets of p on hydrogen. After making the required corrections (but not accounting for superposition), she obtained a binding enthalpy at 298° K of —10.8 kcal/mol. This value is likely overestimated by several kcal/mol due to its contamination by BSSE, especially at the correlated level. 2.1.1 BSSE The H 3 N ... HF complex has also furnished a model system for investigation of the spatial attributes of BSSE and the effects of secondary basis set superposition error. In other words, the magnitude of the superposition error will depend on how close together the two subunits arc and their angular orientations. This issue was considered by Latajka and Scheiner8 who allowed a ghost center to approach a HF molecule. They found that BSSE is negligible for separations of 3 A or greater. Closer approach leads to a rapid increase in this error,

Geometries and Energetics

57

climbing up to several kcal/mol for chemical bonding distances. This strong distance dependence suggests that geometry optimizations that do not correct for the BSSE are likely to be in error with respect to equilibrium separation. The approximate midpoint of the FH bond acts as a sort of central point from the standpoint that the BSSE depends on the distance of the test center from this point, and is relatively independent of direction. This near isotropy suggests that BSSE will have a negligible effect upon the angular aspects of a given H-bonding interaction. The result also suggests that "bond functions," centered not on a nucleus but rather in the space between them, may offer an efficient means of quenching superposition problems in the future. The same authors investigated secondary BSSEi by determining the effect of ghost functions upon the dipole moment of their prototype molecule, HE. When these functions are placed on the F side of the molecule, the moment increases but undergoes a decrease when the functions are situated on the hydrogen side. This trend can most simply be explained on the basis of partial transfer of electron density from HF to the ghost functions. When the negative charge associated with this electronic shift is located beyond the F atom, it acts to enhance the normal dipole of the molecule which is -F—H + . The presence of a negative charge cloud near the hydrogen will dampen the dipole, causing the decrease noted. While the two directions are opposite in sign, they are not equal in magnitude. The maximal effect of ghost functions upon the dipole occurs when they are placed about 1 A from the F atom, along the H—F axis. The effects of these functions die off as they are drawn away from the HF molecule, but persist to longer distances on the F side. One may infer that secondary basis set superposition is highly dependent on angular orientation and can easily influence the directional character of H-bonding. On the other hand, the opposite sign of secondary BSSE on the two sides of a molecule can act to lower the net effect of this phenomenon in the following way. Consider the coming together of two molecules, preparatory to formation of a H-bond. The upper portion of Fig. 2.2 illustrates the initial approach of HX and ZH3, and includes their dipole moments as the arrows. As the two monomers approach one another, the orbitals of each can act as ghost functions for the partner. This effect is indicated by the circle, and the negative charge which shifts into these orbitals by the negative sign. As a consequence of this negative charge, the dipole moment of the HX molecule is reduced, indicated by the shorter arrow, relative to that in the isolated monomer. Analogously, the dipole moment of ZH3 is enhanced since the negative charge is on the side of the molecule which is already negative. Since the dipole of one molecule is lowered and the other raised, the net result is that the dipoledipole interaction is not much affected by the secondary effect. This expectation is confirmed by Latajka and Schemer's calculations of H3N...HF,8 but their results caution against overgeneralization. The first column of data in Table 2.4 reports the change in the dipole moment of HF resulting from the ghost functions of NH3, placed just as they would occur in the complex in its equilibrium geometry with R(NF) = 2.66 A (the experimental value). The negative values are consistent with the effects just described. One should note, however, that one basis set yields a small increase in the dipole moment of HF. This discrepancy is likely due to the presence of very diffuse functions which may permit a good deal of density to be drawn from the H atom, which would tend to increase the moment of HF. The changes in the NH, moment are more erratic and are only positive in certain cases. One should conclude that a thumbnail prediction of the effects of secondary superposition are not always possible. The next column of Table 2.4 lists the total dipole moment computed for the H 3 N ... HF complex by each basis set. As mentioned, the formation of the H-bond causes a panoply of electron density shifts, both within each monomer and some from one to the other. It is in-

58

Hydrogen Bonding

prior to close approach of monomers

effect of ghost functions

Figure 2.2 Approach of XH to ZH3, indicating effects of ghost functions. Molecular dipole moments are indicated by arrows. Any electron density which accumulates in the ghost orbitals is represented by the negative sign.

formative to compare the total moment of the complex with that which would arise if these charge shifts were prohibited, that is with the sum of the moments of the two isolated monomers. These differences are reported in the penultimate column of Table 2.4 and show that the formation of the H-bond leads to an enhancement of the dipole moment, relative to two unreacting subunits, of about 1 D. The last column of the table addresses the question of how much of this moment enhancement is due to secondary BSSE. In other words, the Table 2.4 Secondary basis set superposition errors on the dipole moments of H3N--HF. Data8 all in D. Basis set 6-31G** 6-311G** 6-31+G** dif(2d) +VPS +VP s (2d) s

(HF)

(NH3)

-0.022 -0.028 -0.026 0.018 -0.031 -0.021

+0.078 +0.072 -0.079 +0.083 -0.098 -0.055

(FH-NH3)

4.781 4.719 4.773 4.453 4.758 4.661

a corrb

0.918 0.932 0.860 1.099 0.838 0.897

0.862 0.888 0.965 0.998 0.967 0.982

Geometries and Energetics

59

first two columns describe the change in subunit moment resulting not from any genuine interaction, but only from the presence of the orbitals of the partner. After subtraction of this artifact from both monomers, we are left with a corrected dipole moment enhancement, listed in the last column. Whereas some of the corrected moment changes are greater than their uncorrected counterpart, the opposite is true for a number of basis sets, underscoring the difficulty in predictions of even the sign of this effect. Latajka and Scheiner9 further elaborated on secondary superposition by considering the effects of a set of ghost orbitals along the C3 symmetry axis of NH3, using a variety of different basis sets. It was found that some of these basis sets would increase the calculated dipole moment and others would diminish this quantity. Moreover, the dependence of these changes upon the distance of the ghost functions from the N center were rather erratic. In some cases, the moment change would vary from positive to negative as the functions approached the NH3 molecule, while others simply pass through a maximum. The effects at the correlated level were far from negligible. For example, change in the MP2 moment induced by these ghost functions was found in one case to exceed the true MP2 contribution to the dipole moment. The authors noted, however, that much of this erratic behavior could be damped by including diffuse functions in the basis set of the NH3. In contrast to the dipole moment, the polarizability of the NH3 molecule always increases when ghost functions are added9. Such increases can be considered beneficial as these same basis sets underestimate the polarizability. These increases are larger for components along the C3 direction, where the ghost functions are placed and several times smaller for perpendicular components. Consistent with the dipole trends, the ghost functions have more of an absolute effect upon the SCF segment of the polarizability than the correlated contribution. But again, despite its smaller value, the MP2 ghost orbital effect on the polarizability cannot be ignored as it is competitive in magnitude with the genuine MP2 contribution. The primary BSSE is considered as an energy term while secondary effects are usually placed within the context of molecular properties such as dipole moment or polarizability. In order to have some basis of comparison on the same scale, one can consider the interaction between NH3 and an ion like Li+. Any artifact that changes the dipole moment of the neutral NH3 by an amount Au. will produce an energy increment of

where R represents the intermolecular separation, due to the ion-dipole term of the electrostatic interaction. It was found9 that this estimation of the secondary basis set superposition can be comparable to, and in many cases larger than, the primary BSSE. Whereas the primary effect is always negative, the secondary term can take either sign, belying any assumption of cancellation between the two in the general case. Of course, the energetic consequences of error in the dipole moment would be less severe when the partner is a neutral molecule, as in a typical H-bond, rather than the ionic Li+. Nonetheless, the results provide a cautionary note in calculations of this type. Clearly the issue of how to handle secondary superposition error is a thorny one. Szczesniak and Scheiner10 have demonstrated that it is possible to avoid the problem with a judicious choice of basis set. Focusing again on the strong interaction between NH3 and Li + , they demonstrated that a well-tempered basis set can not only reproduce fairly accurately the molecular properties of the subunits, but can also yield good total SCF energies. Superposition errors are also quite small. Positioning of ghost orbitals, even as close as 2 A from the NH 3 , yields a negligible change (<0.05%) in its calculated dipole moment; the

60

Hydrogen Bonding

quadrupole moment is stable to less than 1%. Most impressive were the correlation components of the polarizability of NH3 which were altered by basis set extension by less than 1 %. The authors conclude that, at least for a small system such as NH 3 ... Li + , it is possible to calculate results uncontaminated by superposition errors at either primary or secondary levels. They advise against any basis set which does not describe the core well (e.g., 6-311G) as large primary BSSE is likely to ensue. Nor should the basis be chosen with minimization of the superposition error as the major criterion since this prescription can lead to an inflexible set which does not adequately handle second-order properties. On the other hand, limited computational resources do not always allow the use of a long list of basis functions for a H-bonded system of interest, so one should be prepared for certain compromises between accuracy and feasibility. 2.1.2 Substituent Effects In the simple case where the ZH3 molecule belongs to the C3v point group like NH3, the equilibrium complex with a linear HX is of the same symmetry. However, deviations can occur if the three groups bonded to the Z atom differ, as for example, in the case of substituted amines. The loss of the C3 rotation axis in the amine slightly clouds the precise location of the lone pair, and the bridging hydrogen may deviate by several degrees from the X...Z internuclear axis. Latajka et al. 11 computed the equilibrium geometries of complexes pairing HBr and HI with mono and dimethylamine. The 0 (NXH) angles may be seen from the first rows of Table 2.5 to take the proton less than 3° from the H-bond axis in these cases.

Table 2.5 Properties of complexes of HBr and HI with substituted amines, computed with polarized split-valence basis sets, uncorrected for BSSE. All vaues of E refer to electronic contribution to binding energy 11 .

0(NBrH), degs R(N..Br)SCF, A R(N..Br)MP2, A r(HBr)SCF, A r(HBr)MP2, A - ESCF, kcal/mol - EMP2, kcal/mol - EMP3, kcal/mol - EMP4, kcal/mol

(NIH), degs R(N..I)SCF, A R(N .. I) MP2 , A r(HI)SCF, A r(HI)MP2, A - ESCF kcal/mol - EMP2, kcal/mol - EMp3, kcal/mol - KMP4, kcal/mol

MeH 2 N ... HBr

Me2HN...HBr

Me,N...HBr

2.5 3.361 2.952 0.036 0.298 7.65 13.14 10.19 10.22

0.5 3.288 2.961 0.048 0.392 7.90 17.47 14.56 14.37

0.0 3.049 2.972 0.561 0.393 17.24 20.13 17.15 17.13

MeH2N...HI

Me2HN...HI

Me 3 N ... HI

1.4 3.659 3.210 0.030 0.497 5.08 16.89 13.73 13.19

0.2 3.572 3.212 0.043 0.512 5.40 22.76 19.45 18.89

0.0 3.313 3.219 0.663 0.534 21.83 25.91 22.75 21.19

Geometries and Energetics

61

The next several columns illustrate the effects of progressive methyl substitution upon the geometric and energetic aspects of the complex formed by an amine with hydrogen halides. As more hydrogens are replaced by methyl groups, the basicity of the amine increases, making it an improved proton acceptor. This change is reflected in a number of trends. The H-bond contracts, revealed by the reduction in R(N .. X) at the SCF level, as one moves across the appropriate row of Table 2.5. A big jump in this quantity occurs between the di- and trimethyl amine due to a fundamental change that occurs. The greater acidity of the trimethylamine is sufficient to cause the bridging proton to transfer across from the HBr or HI molecule, leaving the complex as an ion pair N H + . . . - X . This transition is most obvious in the HX stretching parameter r(HX)SCF and is also responsible for the large increase in SCF binding energy when the third methyl group is added. One way of conceptualizing the latter increase is the added electrostatic interaction between the two ions once the proton has transferred across to the amine. (Backskay and Craw have recently demonstrated that the basicity of the trimethylamine is sufficient to extract the proton from HC112.) The MPn data in Table 2.5 underscore the difference in fundamental character that is associated with incorporating correlation into the treatment of these systems. As will be discussed in greater detail in chapters to follow, the transition from neutral to ion pair that occurs with progressively higher methyl substitution of the amine is a smooth one when described at the correlated level, as opposed to the sharp transition from one type of complex to the other without correlation included. It is for this reason that the MP2 intermolecular separations do not obey the same pattern as the SCF distances. Nonetheless, one can still discern that the increased basicity of the methylated amine yields a greater interaction energy, albeit between ions rather than between neutral molecules. The much larger correlated versus SCF interaction energies are due in large measure to the aforementioned conversion to an ion pair and the additional stabilization that arises from the ion-ion forces.

2.2 XH...YH2 When the proton acceptor molecule contains an atom like O or S, a second lone pair is present that makes prediction of the equilibrium geometry less obvious. If one assumes an sp3 hybridization, the two lone pairs are disposed as in Fig. 2.3A, which would yield an angle P, between the HYH bisector and the X..Y axis, in the vicinity of 125°. The alternate type of hybridization, sp2, leaves one of the lone pairs in a p-orbital, oriented 90° from the other lone pair. This arrangement would lead to a 180° angle. Geometry B is also favored by certain electrostatic arguments. Specifically, it would permit the dipole moment of YHL, collinear with the HYH bisector, to align itself with the dipole of the X—H molecule.

Figure 2.3 Dispositions of molecules and lone pairs in HX + H2Y.

62

Hydrogen Bonding

There appears to be a delicate balance between the factors favoring configurations A and B. For example, systematic analysis of a large set of diffraction data indicates the tendencies toward structures A and B are very nearly equal in the solid state13. It should be understood that while hybridization is indeed a useful concept, most molecules cannot be categorized as simply sp2 or sp3. If one insists on that language, an analysis of the wave function will usually yield nonintegral values, for example sp2.1 or sp2.77. It is thus an oversimplification to consider the electronic configuration of YH2 as simply one or the other of those pictured in Fig. 2.3. The electron cloud would more accurately be visualized as a "smear" extending from the region above the Y atom to below, with fluctuations of density as one goes around. Nor should it be thought that the geometry is controlled solely by factors of electron density of the acceptor molecule. An alternate description (see later) places more emphasis upon the electrostatic factors. Another factor has to do with the linearity of the H-bond itself. The C2v symmetry of Fig. 2.3B would place the bridging hydrogen directly along the X...Y axis, but it is plain that the forces above and below this axis in Fig. 2.3A are not equal. It should therefore not be surprising to see the proton deviate from the H-bond axis in such cases, with nonzero values of a. It is likely the proton would be above the line as drawn, based on the direction of the YH2 dipole moment in this configuration. 2.2.1 Comparative Aspects Hinchliffe published in 1984 a comprehensive theoretical comparison of hydrides of the H 2 Y-HX type14. The calculations reported in Table 2.6 were all undertaken at the SCF level and assumed Cs geometries, with a fully linear Y-HX arrangement. The patterns are consistent with the aforementioned data for H3Z..HX complexes. The strength of the Hbond diminishes with lowered electronegativity of the donor atom F > Cl > Br. H2O forms the strongest H-bonds; the distinction between H2S and H2Se is a small one. Any of these H2Y proton acceptors form a weaker complex than the corresponding H3Z molecule of the same row of the periodic table. Hinchliffe also found that the complexes involving H2O were of type B while those incorporating H2S or H2Se were of type A, with angles of about 110°. An additional set of SCF data emerged from calculations by Hannachi et al.15 who paired water with each of the hydrogen halides listed in Table 2.7, optimizing the geometry using a pseudopotential basis function of polarized split valence quality. The data echo the above trends of a weaker H-bond as the X atom of HX comes from a lower row of the periodic table. This progressive weakening is reflected in smaller stretches of the H—X bond. It is also worth noting that the small nonlinearity of the H-bond and the preferred angle of the proton acceptor water molecule are virtually independent of the nature of the HX molecule.

Table 2.6 Electronic contributions to binding energies (— Eelec kcal/mol) of H-bonds of type H2Y...HX, calculated using DZP basis set at SCF level14. H2Y H2O H2S H2Se

HF 9.0 3.5 3.4

HC1 5.4 2.4 2.0

in

HBr 4.3 1.6 1.1

Geometries and Energetics

63

Table 2.7 Energetic and geometric aspects of complexes of water with HX calculated at SCF level15.

- Ee|ec, kcal/mol R(O .. X), A Ar(XH), A a, degs , degs

H2O...HF

H2O...HC1

H 2 O ... HBr

H 2 O ... HI

8.2 2.702 0.012 3.8 135.9

4.9 3.268 0.013 2.4 140.1

4.1 3.496 0.010 2.5 139.4

2.5 3.830 0.006 2.6 139.0

Correlated data for a set of four of these complexes are listed along with SCF values in Table 2.816-18. These data were collected using basis sets of double-valence quality, and augmented with two sets of polarization functions on all atoms. The SCF energies agree fairly well with those in Table 2.6 and reinforce the same trends. Correlation acts to strengthen all interactions. This effect is proportionately larger as the number of secondrow atoms is increased. The energetic data seem to parallel experimental results pretty well. For example, the MP2 electronic binding energy of H2O...HF of 9.6 kcal/mol is only slightly smaller than an experimental determination of 10.2 kcal/mol for De, based upon absolute intensities of rotational transitions19. If one extrapolates an enthalpy of dissociation for the H2O...HF complex using the vibrational, rotational, and translational corrections derived above, a calculated value of — H298 = 6.9 kcal/mol is obtained, within the uncertainty of the experimental estimate of 6.2 ± 1 kcal/mol. A recent calculation of the H 2 S ... HF complex with a very large polarized basis set comprising 169 functions, obtained a binding energy — Eelec of 4.7 kcal/mol at the MP2 level, with counterpoise correction20. MP2-optimized intermolecular separations R(Y..X) are reduced relative to SCF distances, consistent with the strengthening role of correlation. The separations optimized at the MP2 level furnish fairly good reproductions of experimental estimates, typically too short by 0.005 A or better. It is intriguing to note the similarity between entries for H2O...HC1 and H2S...HF, both of which contain one first-row and one second-row atom.

Table 2.8 Calculated energetic (uncorrected for BSSE16J7.

- ESCF, kcal/mol - EMP2, kcal/mol R(Y..X)SCF A R(Y..X)MP2, A R(Y .. X) expt,a , A r(HX)SCF, A r(HX)MP2, A SCF degs MP2 , degs SCF , degs MP2 , degs a

See Reference 18.

Eelec) and geometric aspects of H-bonded complexes. Data

H 2 O ... HF

H2O...HC1

H 2 S ... HF

H2S...HC1

7.8 9.6 2.71 2.65 2.66 0.012 0.017 140 129 3.1 4.5

4.2 6.6 3.37 3.19 3.21 0.009 0.015 140 130 2.8 0.9

3.9 6.3 3.36 3.20 3.25 0.007 0.011 100 98

2.2 5.0 4.09 3.75 3.81 0.005 0.011 101 92 1.5 1.2

1.3 -0.3

64

Hydrogen Bonding

The next two rows reveal that correlation also enhances the stretch that occurs within the X—H bond upon formation of the H-bond. 2.2.2 Angular Features The values of listed in Table 2.8 illustrate that these complexes all adopt a "pyramidal" geometry, closer to A shown earlier than to "planar" configuration B. This is particularly true of the complexes containing H2S where the angles approach 90°. Indeed, early microwave spectra of H2S...HF21,22 suggested the proton acceptor molecule was oriented nearly perpendicular to the donor; a similar result was obtained for H2S...HC118. The nearly perpendicular arrangement of complexes containing H2S, as compared to H2O, has been confirmed recently at higher levels of theory. MP2/6-311+ +G(d,p) optimizations of H2Y..HF found angles of 140° and 112°, for Y=O and S, respectively23. One means of rationalizing the trends in the angle is via electrostatic arguments24. As mentioned earlier, the planar geometry B, with = 180°, is favored by dipole-dipole interactions between HX and YH2. This preference is illustrated in the left part of Fig. 2.4. On the other hand, the two units are certainly close enough together that the quadrupole moment of YH2 can play a role as well. The bonding pattern of YH2 leads one to expect a negative element in the direction perpendicular to the molecule. The attraction between the negative charges of this quadrupole tensor element and the positive end of the HX dipole will tend toward a 90° value for , illustrated by the right part of Fig. 2.4. The end result can be considered a compromise between these two trends toward large and small angles. The fact that the MP2 values of are smaller may be attributed in part to the correlation-induced reduction of the dipole moment of water, which would diminish the pull toward large angle. Of course, this is an oversimplification and a thorough analysis of the reasons for the directions would have to take into account forces other than electrostatic. Nonetheless, insights gained from Coulombic concepts are extremely valuable, and can be superior to predictions based on detailed analysis of the wave function23. One can also think more quantitatively about the energetic difference between pyramidal and planar geometries. The first column of Table 2.9 illustrates that the fully planar C2v structure of H 2 O ... HF is higher in energy than the optimized pyramidal geometry by only 0.1 kcal/mol at the SCF level, increasing to 0.5 kcal/mol with MP216,17 (an experimental estimate is 0.4 kcal/mol25). The latter correlated barrier was later confirmed with a larger basis set26. Similar values are found for H2O...HC1. Much higher energy differences arise for complexes where H2O is replaced by H2S. These increases are consistent with the preference toward much smaller values of p.

Figure 2.4 Interactions between multipole moments. Dipole moments are indicated by arrows, and an element of the quadrupole tensor by the double lobe.

Geometries and Energetics

65

Table 2.9 Energy required to bend each complex from its equilibrium pyramidal structure into a planar C2v arrangement. Values in kcal/mol16,17.

SCF MP2

H 2 O ... HF

H20...HC1

H2S...HF

H S...HC1

0.13 0.49

0.15 0.33

2.96 3.65

1.59 2.52

In cases where the pyramidal equilibrium geometry differs little in energy from the planar configuration, what are the consequences for experimental observation? This question may be addressed by considering the potential function for bending. In the case where the planar structure is most stable, there is little question but that experimental observations would confirm this. The situation is less clear when the planar geometry is less stable than a pyramidal structure but only marginally so. For example, Legon et al. concluded from their gas-phase rotational spectroscopic measurements that the equilibrium geometry of H2O...HX, X=Br,Cl was either planar or, if pyramidal, that the inversion barrier was very low27,28. Figure 2.5 illustrates a number of different cases that are possible. The potential energy surface is illustrated as a double-well potential with respect to the flipping of the YH2 molecule about the planar position. In the case where the barrier to this flipping is high enough that the lowest vibrational level occurs well below the top of the barrier, the square of the wave function resembles that depicted by the lowest function in the figure. There are two

Figure 2.5 Vibrational wave functions corresponding to each of several energy levels in a double-well potential, with respect to "flipping angle" p.

66

Hydrogen Bonding

maxima in the probability density, each associated with a pyramidal structure. On the other extreme is the case where the barrier is so low that the ground vibrational level is well above its top. The highest of the three probability density functions in Fig. 2.5 illustrates that an experimental measurement would indicate a planar structure despite the appearance of a shallow maximum in the potential energy. The situation is most ambiguous when the vibrational level is below the barrier top, but only slightly so. The probability density function retains two maxima but these are close to the center and poorly defined. The function may perhaps be better described as a single flat maximum extending on either side of the planar structure, = 180°. The atomic motions would correspond to large-amplitude deviations from a planar structure. It thus appears that the experimental elucidation of the precise nature of the potential energy function for this sort of wagging motion may not be a trivial task. The simple fact that the vibrational level occurs below the maximum in the potential function does not insure that one can detect a double-well potential. The preceding discussion has been simplified by assuming that the wagging of the YH2 molecule is separable from other wags and stretches; a more thorough analysis would not make this assumption. The small energy barriers computed for the H2O...HX systems correspond to the cases where the lowest vibrational level is near the top of the barrier16. The higher barriers that occur when H2O is replaced by H2S allows unambiguous determination of a pyramidal structure. The last two rows of Table 2.8 indicate that there is only a small amount of nonlinearity in the equilibrium geometries of the H2Y...HX H-bonds. The deviations from fully linear arrangements are typically less than 5°. In most cases, the bridging hydrogen lies "above" the Y...X axis, in the sense of configuration A above, as indicated by the generally positive values of a. This trend is consistent with the attempt by the proton to better align itself with the dipole moment of the H2Y, when its hydrogens are bent down. Szczesniak et al.16 have pointed out an interesting relationship between the stretch of the hydrogen away from the X atom and the energetics of the interaction. They have shown that Ar is very nearly linear, over a range of HX stretches, with respect to the contribution made by electron correlation to the H-bond. The authors assumed the latter is dominated by dispersion, and so concluded that the stretch of the H—X bond causes an increase in the molecule's polarizability. They hence infer that a molecule whose polarizability is sensitive to the X—H bond length can enhance its ability to form a H-bond by permitting a greater stretch of the bond upon complexation. 2.2.3 Alternate Complexes and Geometries The greater acidity of HX than of H2Y leads to the normal supposition that the former molecule will act as the donor in any interaction between the two molecules. Szczesniak and Scheiner17 tested this presumption in the complex between HF and H2S. A structure in which the normal bonding pattern is reversed, namely H2S donates a proton to HF, was indeed found to be a minimum on the potential energy surface. However, it was found to lie some 3 kcal/mol higher in energy than FH...SH2. Novoa26 later explored the same question for the HF,HOH pair, using a high quality basis set. They concluded that the potential energy surface probably does not contain a minimum corresponding to the "reverse" complex wherein HF acts as the proton acceptor, although the question was not answered definitively as they did note a "plateau" in that region of the surface. Substitution of the hydrogen atoms by alkyl groups appears to exert a minimal impact on the angular aspects of the calculated geometries. As an example, Amos et al.29 optimized

Geometries and Energetics

67

the geometry of the dimethylsubstituted Me2O...HCl and found difficulty in determining the equilibrium value of since the energy profile for bending away from 180° was extremely flat. Calculations by Bouteiller et al.30 found only very minor differences in the equilibrium geometries of H,O...HF and Me2O...HF. Experimentally, the geometries of H2O...HC1 and MeHO...HCl are quite similar as well31. Hannachi et al.32 calculated the relative stability of base..HX versus base...XH for complexes in which water is the base. They refer to the former geometry as "H-bonded" and the latter as "van der Waals," also known as "anti-H-bonding." In the case of the complex with HC1, they only find one minimum on the potential energy surface, corresponding to the Hbonded H2O-HC1. However, both types of complex were identified as minima for HBr and HI. The first row of Table 2.10 illustrates the closer approach of the two subunits in the antiH-bonding arrangments. The energetics indicate that the H-bonding structure is greatly preferred for HBr; this preference is much weaker for HI, such that both structures might be observed experimentally. The authors also monitored the shifts in electron density in the monomers which accompanied the formation of the two types of complexes. For either Hbonding or anti-H-bonding, the lone pairs of the oxygen atom suffer a loss of density, albeit stronger in the former case. Unlike the case of a H-bond, density is shifted from I toward H in the HI subunit of H,O..IH. This shift acts to induce a dipole moment in IH which aligns favorably with the moment of H2O. In fact, two separate geometries have been observed for the complex between HI and H2O by matrix isolation IR spectroscopy, one Hbonded and the other not33, although the details of the structure were not established. The basis set superposition errors of the H2Y..HX complexes are comparable to those observed for H3Z..HX, listed earlier in Table 2.3 for a variety of basis sets5. One interesting difference is that whereas the MP2 contribution to the binding energy of H 3 N .. HF is attractive, albeit by less than 1 kcal, the correlation contribution in H2O..HF is close to zero, with some basis sets yielding a small repulsive contribution after primary BSSE is accounted for. 2.2.4 Energy Components Backskay et al.34 have partitioned their total interaction energies into components, using a scheme similar to Kitaura-Morokuma, but with some modifications. Their results are displayed in Table 2.11 where it may be seen that the electrostatic component is the dominant attractive term in all cases. ES is particularly large for the two bases with a first-row atom, H3N and H2O. It is this pair of complexes which are most strongly bound, as indicated by the last column of Table 2.11. Of comparable magnitude to ES, but of opposite sign, is the exchange repulsion. As the only repulsive element, EX keeps the two subunits of each complex from collapsing together. The polarization and charge transfer energies are particularly

Table 2.10 Comparison of H2O..HX and H2O..XH32. E refers to electronic contribution only.

R(O .. X), A - ESCI , kcal/mol - EMP2, kcal/mol

H2O..HBr

H2O..BrH

H2O..HI

3.496 4.10 5.04

3.228 0.22 0.59

3.830

2.47 3.35

H2O..IH

3.192 1.71 1.99

68

Hydrogen Bonding

Table 2.1 I Components of interaction energy of complexes involving HC1, calculated at experimental geometries. Values in kcal/mol34.

H 3 N ... HC1 H2O...HC1 H3P...HC1 H2S...HC1 a

ES

EX

POL

CT

Totala

-17.36 -9.21 -4.33 -4.49

17.67 6.94 5.16 4.50

-2.96 -1.44 -0.69 -1.35

-3.88 .-1.62 -1.47 -1.08

-6.65 -5.35 -1.37 -2.45

Total — Eelec includes also "unassigned" contribution, so is not equal to sum of other terms.

large for H3N...HC1, and vary between —0.7 and —1.6 kcal/mol for the other complexes. The authors also examined how the various terms behave as the two subunits in H 3 N ... HC1 are pulled apart. The electrostatic attraction dominates the interaction at long distances. Slightly closer approach brings an exponential rise of the exchange repulsion. It is not until intermolecular separations of about 3.5 A or less that the polarization or charge transfer energies become significant. Much accumulated data like that above have provided evidence that electrostatics is a primary force which orients the bridging proton of a H-bond along the intermolecular axis. While an oversimplification, this force may be thought of as composed of the alignment of the dipole moments of the donor and acceptor molecules. Let us now consider a situation where this force is steadily decreased. Table 2.12 reports data which illustrate that as the halide atom of the HX molecule advances to lower rows of the periodic table, the molecular dipole moment decreases32. Hence, as one changes this molecule from HC1, to HBr, to HI, one can expect that the electrostatic drive toward a base...HX orientation will be similarly weakened. At the same time, the molecules with the larger halide also are most polarizable, particularly along their molecular axis, as indicated by the values of zz in Table 2.12. This greater longitudinal polarizability leads the interaction between the two molecules to contain progressively greater amounts of induction and dispersion energy. Both of the latter forces become rapidly more attractive as the molecules approach one another. In fact, by approaching in an "anti-H-bonding" orientation, that is, heavy atom first as in base"XH, the base can more closely approach the more polarizable halogen end of the HX molecule. And as the X atom changes from Cl to Br to I, there is more to gain by approaching this way, and less electrostatic energy to lose since the HX dipole moment becomes so small.

Table 2.12 Experimental values of dipole moment, , and dipole polarizabilities, a, of hydrogen halide HX molecules32.

.D a , aua xx, a

au

HC1

HBr

1.094 21.1 19.6

0.819 28.5 22.3

The molecular axis is defined as the Z-axis.

HI

0.447 44.4 32.9

Geometries and Energetics

69

2.3 HYH...ZH3 The importance of hydrogen bonds between amino and hydroxy groups has been amplified in recent years by the finding that such interactions can guide the formation of well-ordered supramolecular structures35,36. Because the ZH3 molecules are stronger bases than YH2, one expects the former to act as proton acceptor in complexes with the latter. This has indeed been found to be the case in the complex between water and ammonia, the most studied of systems of this type. One of the two hydrogens of the YH2 is used to bridge the two molecules in a classic H-bond that is nearly linear, incorporating the single lone pair of the ZH3 molecule as illustrated in Fig. 2.6. The geometrical aspects of the HOH...NH3 complex are reported in Table 2.13 at various levels of theory37. (The +VPS basis set is related to 6-31+G**, except that orbital exponents have been reoptimized so as to reduce BSSE.) The intermolecular distance elongates somewhat as the basis set is enlarged but diminishes to 2.94 A upon inclusion of electron correlation. This distance is just slightly shorter than estimated by microwave/farIR data which lead to a value of 2.97-2.99 A38,39. The covalent bond to the bridging proton stretches by around 0.01 A upon formation of the H-bond, less at the SCF level, more at MP2. This hydrogen lies within about 5° of the H-bond axis. The last two rows of Table 2.13 indicate that the NH3 molecule turns its lone pair up toward the connecting hydrogen since the (ONHc) angles are larger by some 15° than (ONHt). Another study40 made the interesting observation that the structure depicted is a true minimum in the MP2 potential energy surface, but is slightly less stable than that in which the NH3 is rotated 60° around the H-bond axis when the surface is uncorrelated. That is, the "staggered" geometry is the minimum in the correlated surface but an "eclipsed" structure is preferred at the SCF level. The energy differences in either case are exceedingly small, so the rotational barrier can be considered negligible. This finding is consistent with experimental estimates of a barrier of only 0.03 kcal/mol38. There is no evidence of a minimum for which the roles of proton donor and acceptor are reversed, such as H2NH...OH2. Calculated values for the binding energy of HOH...NH3 are listed for several basis sets in Table 2.1437. SCF values of — E are in the 4.6-5.6 kcal/mol range. Correlation adds to this amount, bringing the binding energy up near 6 kcal/mol. Del Bene41 performed a similar set of calculations, but all values were somewhat higher due to BSSE which was left uncorrected. Her results were nonetheless valuable in that they illustrated that MP2 values were nearly identical to full MP4 interaction energies. The best value for Eelec seems to be about —5.5 kcal/mol at this time. A lower-bound for — H of 2.9 kcal/mol comes from molecular-beam electric-resonance optothermal spectroscopy42. Del Bene7 has applied a more flexible basis set to this complex, with MP4 consideration of correlation. At the

Figure 2.6 Geometry of HYH...ZH3.

70

Hydrogen Bonding

Table 2.13 Calculated geometry of HOH...NFL complex37. SCF

R(O..N), A r(OH), A a, degs (ONHt), degs (ONHc), degs

MP2

6-31G**

+VPS

+ VPs(2d)s

+VPM

3.050 0.008 2.1 100.5 116.2

3.074 0.008 3.7 100.8 116.0

3.096 0.007 4.6 101.1 116.1

2.942 0.013 4.8 101.8 117.7

MP4/6-31 +G(2d,2p) level, a binding enthalpy at 298 K was calculated to be -4.7 kcal/mol, which included vibrational corrections, and so forth. The HOH...NH3 complex served as a recent test for symmetry-adapted perturbation theory (SAPT). Basing their work on earlier formalism43, which was further elaborated, Langlet et al.44 observed that a pure perturbation approach yielded an intermolecular separation that was somewhat too long, and underestimated the binding strength of the complex. Better correlation with experimental quantities, as well as with other accurate computations, is obtained by a "hybrid" approach, wherein the dispersion energy, computed by SAPT, is added to the (counterpoise corrected) SCF portion of the interaction energy. This conclusion was found to apply not only to HOH...NH3, but also to the homodimers of HF, H2O, and NH3. The complex between H2O and H3P is barely bound at all, with H298 only —0.8 kcal/mol41. In fact, this value might become positive were counterpoise corrections made to the binding energy. Attempts at identifying another minimum on the surface in which H2O and H3Z reverse their roles to proton acceptor and donor, respectively, failed for both Z=N and P, which suggested there is no such local minimum. A pairing of H2S with NH3 did yield a minimum, containing a linear H-bond with H2S as donor7. This geometry conforms to molecular beam electric resonance data45 which yields an intermolecular R(S .. N) of 3.639 A. This H-bond is somewhat shorter than a prior ab initio computation of 3.79 A46. The HSH...NH3 complex is bound at the MP4/631 +G(2d,2p) level, relative to the isolated monomers, by 3.6 kcal/mol. Again, this value would likely be diminished by inclusion of zero-point energy and superposition error corrections. Recent electric-resonance optothermal spectroscopic measurements47 place an upper bound of 2.8 kcal/mol on the binding energy of HSH...NH3, including zero-point vibrational corrections. This complex has a slightly smaller energy barrier to proton exchange as compared to HOH...NH3, 1.5 versus 2.0 kcal/mol, which is taken as evidence of a less directed H-bond in the former.

Table 2.14 Electronic contribution to binding energy of HOH...NH3 complex. Data in kcal/mol37.

- ESCF _ E MP2

6-31G**

+VPS

+VPs(2d)s

5.61 6.35

5.10 5.92

4.59 5.72

Geometries and Energetics

71

2.3.1 Substituents Alkylating ZH3 might be expected to make this molecule a better proton acceptor, as the larger substituents can better delocalize any charge accumulation. Calculated data are presented in Table 2.15 for the complex of methylamine with water48. Comparison with the data in Tables 2.13 and 2.14 suggests that the methyl group on the N does in fact enhance the H-bond energy by a small amount, along with a shortening of the intermolecular distance. Correlation enhances this binding, as in most other complexes of this type, but the amount cannot be easily discerned because Zheng and Merz did not remove their BSSE, which is apt to be rather appreciable with this basis set, in particular at the MP2 level. Nonetheless, the last row indicates that AG is probably positive for the binding reaction, due to the large negative AS. The sensitivity to basis set is evident from a comparison with prior calculations using the 6-31G basis set49. The deletion of the polarization functions on O and N, reduces the intermolecular separation by 0.1 A and increases the SCF interaction energy by 2.2 kcal/mol. A similar sort of analysis, but this time alkylating the hydroxyl group to form CH3OH...NH350, again confirmed very little influence of the methyl group. Another type of substitution places an aromatic group on the proton-donor oxygen atom. SCF/6-31G** optimization of the complex between phenol and ammonia51 yields a R(O..N) H-bond length of 2.891 A, somewhat shorter than the value of 3.050 A computed at the same level of theory for HOH...NH337. The enhanced proton-donating capability provided by the aromatic group is verified by the energetics of binding. Eelec is — 8.5 kcal/mol for this complex, as compared to —5.6 kcal/mol for HOH..NH3. After adding in MP2 correlation, zero-point vibrational energies, and correcting for BSSE, the binding energy for the phenol-ammonia complex is computed to be Do = 7.0 kcal/mol51. MP2-level correlation is responsible for a contraction of the H-bond by 0.11 A52. 2.4 XH...XH The structure of the complex between a pair of hydrogen halide molecules is depicted in Fig. 2.7 where three lone electron pairs are placed on the proton-accepting molecule. In the classical case of sp3 hybridization, one might expect an angle of some 109°. a measures the nonlinearity of the H-bond as in the above cases. A nonzero value of a might be expected based on the direction of the dipole moment of the acceptor molecule.

Table 2.15 Optimized H-bond length and energetics of complex between HOH and CH3NH2, calculated with 6-31G* basis set48. Energetics not corrected for BSSE. SCF

R(O..N), A Eelec, kcal/mol H, kcal/mol S, cal mor--1' deg 1 G, kcal/mol

3.015 -6.5 -5.3 -26.7 2.7

MP2

2.902 -9.1 -7.8 0.1

72

Hydrogen Bonding

Figure 2.7 Dispositions of molecules and lone pairs in HX dimer.

The principal features calculated for the geometry of the HF dimer are reported in Table 2.16 from which it may be seen that the angles are predicted reasonably well, even with rather small basis sets and without correlation53-58. The equilibrium interfluorine distance is approximately 2.76 A. The bridging hydrogen lies within at least 10° of the F..F axis, probably more like 5°. The angle made by the proton acceptor molecule is somewhat more sensitive to details of the calculation but appears to fall within the 112°-120° range. These predictions conform to the experimental measurements reported in the last row of Table 2.16. The binding energies reported in the last column of data in Table 2.16 indicate some sensitivity to the type of basis set and method of computing correlation. A better feel for these trends may be obtained from the data in Table 2.17, calculated by Del Bene59 for a range of different basis sets, all within the M011er-Plesset scheme of correlation. Although there is some degree of erratic behavior due to the failure to remove BSSE, there are some clear patterns in evidence nonetheless. Addition of a single set of diffuse .sp-functions to F yields a marked reduction in the interaction energy, probably due to the reduction in superposition error. In contrast, addition of a second set of d- or p- functions has very little effect on E. Enlarging from double-valence to triple valence in the core lowers the interaction energy, again likely due to reduced BSSE. In all cases, correlation enhances the binding energy, with MP2 being a satisfactory substitute for much more expensive full MP4. The best estimate of Eelec achieved is —4.7 kcal/mol which would likely be reduced by incorporation of a counterpoise correction. The level of theory was raised once again in a recent set of calculations wherein a different type of basis set was used, in conjunction with coupled-cluster means of considering electron correlation60. Table 2.18 lists the geometrical parameters optimized for the HF dimer, with and without counterpoise corrections. These results were obtained with a very large correlation-consistent (cc) set: [6s5p4d3f2g/5s4p3d2f]. The data indicate that MP4 and coupled-cluster singles and doubles (with triples approximation) yield very similar results. The interfluorine distance is some 2.73 A at either level. However, counterpoise correction does lengthen the equilibrium value to something closer to 2.75 A. The angular aspects of the equilibrium geometry are consistent from one type of correlation to the next. The proton acceptor is rotated some 110° from the H-bond axis and the nonlinearity within this bond is just under 7°. The best theoretical estimate of the binding energy De is 4.5 kcal/mol, with about 3.7 kcal/mol arising from the SCF level alone. This result is in excellent accord with an experimental estimate of 4.6 kcal/mol in the gas phase, based on absolute infrared line strengths61, and another estimate of 4.5 kcal/mol62. The authors conclude that the MP2 method offers a computationally efficient means to obtain the more accurate results which require much more computationally demanding approaches. The above computed results were confirmed to good accuracy by another correlated study that made use of basis sets such as triple- plus double polarization functions and a set of higher angular momentum functions 63 .

Table 2.16 Geometrical and energetic aspects of (HF)2 calculated at various levels. R(F..F) (A)

(degs)

(degs)

2.687 2.788 2.81 2.82 2.83

8.1 7.9 9 7.2 6.0

2.768 2.762 2.759 2.72

6.4 6.9 5.5 10±6

- Eelec (kcal/mol)

Reference

124.1 117 115 116.5 123.2

8.0 4.7 4.7 3.7a 3.8

[53] [54| |53] [551 [53J

120.1 — 112 117±6

4.6 5.7 5.0

[56J [571 [57] [58J

SCF 4-31G 6-31+G* 6-311G* +VPs(2d)s [11s7p2d/6slpl

Correlated CC/TZP MP2/6-31+G* MP2/6-311 + +G(2d2p) expt a

Corrected for BSSE

74

Hydrogen Bonding

Table 2.17 Calculated binding energies of HF dimer (— Eelec), in kcal/mol59. Basis set

SCF

MP2

MP3

MP4

6-31G(d,p) 6-31+G(d,p) 6-31G(2d,p) 6-31 + G(2d,2p) 6-311G(d,p) 6-311+G(2d,2p)

5.97 3.98 5.87 3.75 5.06 3.71

7.45 4.69 7.58 4.61 6.22 4.66

7.02 4.62 7.13 4.61 5.82 4.64

7.34 4.71 7.52 4.66 6.15 4.71

The data in Table 2.19 indicate that the H-bond between a pair of HC1 molecules is somewhat weaker than in (HF)254,55,64-68. Best theoretical estimates of the binding energy are less than 2 kcal/mol; the gas-phase estimate is 2.3 kcal/mol61. Part of the discrepancy is likely due to the fact that dispersion is very important to this interaction. The latter phenomenon requires particularly flexible basis sets for its saturation. The H-bond is likely less linear than in (HF)2, with some estimates for a above 10°. Notable also is the smaller value of p in (HC1)2 wherein the two HC1 molecules are nearly perpendicular to one another. Far IR spectroscopic measurements confirm the near perpendicular nature of this complex in the gas phase, with equal to about 100-110°69. An important contrast between the two systems is the strong effect of correlation in reducing the interchlorine separation. It is not immediately obvious which molecule would be the proton donor and which the acceptor in a complex pairing HF with HC1. Calculations suggest the two possibilities are nearly equal in energy55. This supposition was confirmed by later observation of both in the gas phase70. The pertinent features of the complexes are reported in Table 2.20, from which it may be observed that the stabilization energies of the two differ by less than 0.1 kcal/mol55. The binding energy is intermediate between the two homodimers (HF)2 and (HC1)2. A recent state-to-state photodissociation study of this mixed complex71 yielded a dissociation energy Do of 1.83 kcal/mol. Bearing in mind that the latter includes vibrational energies, which the electronic contributions to the binding energy listed in Table 2.20 do not, the computed values seem quite reasonable. The complex in which HF acts as proton donor has a slightly shorter R(F..C1). Indeed, the MP2 H-bond lengths of 3.29 and 3.37 A for HC1...HF and HC1...HF, respectively, are quite close to the experimental values of 3.28 and 3.37 A reported later70. It may be noted as well that the HC1 acceptor molecule is nearly perpendicular to the H-bond axis, with = 93°, as in (HC1)2.

Table 2.18 Geometrical and energetic aspects of (HF)2 computed with aug-cc-pVQZ basis set. Counterpoise-corrected values are indicated by "cc" notation60. R(F .. F)(A)

SCF MP2 MP4 CCSD CCSD(T)

no cc 2.821 2.737 2.735 2.745 2.732

cc 2.824 2.753 2.749 2.759 2.745

(degs)

(degs)

6.8 6.4 6.6 6.7 6.7

119.7 111.6 110.4 112.1 110.8

De (kcal/mol) no cc 3.71 4.63 4.68 4.53 4.72

cc 3.66 4.38 4.44 4.31 4.49

Geometries and Energetics

75

Table 2.19 Geometrical and energetic aspects of (HC1)2 calculated at various levels. R(Cl..Cl) (A)

(degs)

(degs)

- E

(kcal/mol)

Reference

SCF

4-31G [6s4pld/2s1pld] 6-31+G* 6-31G** +VPs(2d)s

3.986 3.96 4.156 4.111 4.210

7 2.6 14.6 10.3 11.3

102 83.0 — 97.3 90.3

2.1 3.6a 1.1 1.0" 0.5a

[64] [65] [54] [55] [55]

— — 91.1 91.4 90.0

1.4a 1.6a 1.7 1.7 2.0

[55] [55] [66] [66] [67] [68]

Correlated MP2/6-31G** MP2/+VPs(2d)s ACPF/[652/42] ACPF/[6531/42] MP2/[8s6p3d/6s3p]b expt

3.876 3.838 3.912 3.887 3.78 3.80

— — 6.6 6.1 8.0

a

Corrected for BSSE. Bond functions added.

b

There are of course a host of different means of including electron correlation into the computation of binding energies. It would be useful at this point to make a comparison of some of these techniques. Table 2.21 lists the interaction energies computed for the HF and HC1 homodimers, all with the same 6-31+G(d,p) basis set72. LCCM refers to a linearized coupled cluster technique73,74 and ACPF to an approximate coupled-pair functional approach75. The configuration interaction technique, truncated after all single and double excitations is designated CISD. As the latter approach is not size-consistent it is completely unsuitable for study of molecular interactions unless some further steps are taken. Davl and Dav2 indicate corrections proposed by Davidson76,77 which multiply the correlation energy by a factor which includes the coefficient of the Hartree-Fock configuration in the normalized CISD wave function. A further scaling, indicated by (s), was added to include the number of correlated electrons in the expression. The last type of correction considered is due to Pople78 and is designed for identical 2-electron systems. The first few rows of Table 2.21 show the enhancement of the SCF interaction energy arising when M011er-Plesset correlation is added. MP2, 3, and 4 are little different from one

Table 2.20 Geometrical and energetic features calculated for complex pairing HF with HC155. Eelec corrected for BSSE. HC1-HF R(F..C1)SCF, A R(F..C1)MP2, A (degs) (degs) - ESCF (kcal/mol) - EMP2 (kcal/mol)

3.465 3.294 7.4 93.2 1.77 2.39

HC1...HF 3.499 3.365 8.2 119.7 1.93 2.45

76

Hydrogen Bonding

Table 2.21 Comparative binding energies ( Eelec, in kcal/mol) computed with different correlated schemes, all with the 6-31 +G(d,p) basis set72.

SCF MP2 MP3 MP4 LCCM ACPF CISD Davl Davl(s)

Dav2 Dav2(s)

Pople

(HF),

(HC1),

-4.3 -5.0 -4.9 -5.0 -4.9 -4.9 +4.4 -2.4 -2.8 -3.3 -3.7 -4.5

-0.8 -2.1 -1.9 -1.9 -1.9 -1.9 +7.6 + 1.0 +0.5 -0.2 -0.6 -1.7

another, a common observation. However, it might be noted that the correlation-induced enhancement is likely exaggerated as no corrections were made for basis set superposition. The LCCM and ACPF methods yield results remarkably similar to MR CISD, on the other hand, predicts that both complexes would be strongly unbound, with positive values of E. This result is not a surprise, as the CISD method is not size-consistent. The various Davidson corrections seem to improve the energetics, particularly the Dav2 variant, which is nearly as attractive as the methods in the preceding rows. Even better is the Pople correction in the last row of Table 2.21. Due to the computational efficiency of the M011er-Piesset technique, it would appear to still represent a very cost-effective workhorse for study of H-bonding interactions. A later study also focused on various means of computing the correlation contribution to the interaction energy in the HF dimer79 and reached very similar conclusions. All of the correlated methods (MP2, MP4, CCSD(T) and CISD) based on the Hartree-Fock reference configuration gave essentially the same binding energy. The results deteriorate when multireference methods are used. There have been calculations that extend the set of hydrogen halides investigated to various combinations of HBr, HI, and HO80. The studies were limited to the SCF level, and made use of core pseudopotentials. The optimized geometries are reported in Table 2.22 along with the interaction energies, corrected for BSSE. The H-bond lengths exhibit the expected increases as one moves to bigger atoms as in Cl < Br < I. In the case of the proton donor, the increment from Cl to Br is 0.1 A, but a larger stretch of nearly 0.3 A occurs upon going from Br to I. The increments are larger, around 0.3 and 0.4 A, for the proton acceptor. As the proton donor changes from Cl to Br to I, the H-bond becomes progressively more linear; the smallest a angles of 3° are associated with HI. The linearity of this bond is influenced, albeit to a lesser degree, by the character of the acceptor, with a becoming larger for the heavier atoms. All of these complexes are very nearly perpendicular in the sense that is close to 90°. Experimental confirmation for such a shape for the HI dimer comes from recent high-n Rydberg time-of-flight measurements 81 . There is a definite trend for the interactions to weaken as the proton acceptor atom is enlarged but little dependence upon the

Geometries and Energetics

7777

Table 2.22 Properties of binary complexes, calculated at the SCF level, using core pseudopotentials80.

HC1...HC1 HCl...HBr HC1...HI HBr...HCl HBr...HBr HBr...HI HI...HC1 HI...HBr HI-HI a

R(A)

(degs)

(degs)

4.11 4.22 4.49 4.38 4.48 4.76 4.77 4.87 5.14

13.4 7.8 3.2 16.0 8.2 3.2 18.2 8.0 3.3

90.9 90.4 92.2 88.3 89.8 92.1 85.7 89.8 91.7

- Eeleca (kcal/mol) 1.0 1.1 0.9 0.8 0.8 0.7 0.4 0.4 0.4

Counterpoise corrected.

nature of the donor. HI is the poorest acceptor, with - Eelec equal to 0.4 kcal/mol in all cases. It is legitimate to question whether the interactions listed in Table 2.22 represent true Hbonds or might be better described in terms of simple electrostatic or dispersive interactions. Indeed, the stretches undergone by the HX bonds as a result of formation of each complex are all 0.002 A or less. And the red shifts undergone by this bond are below 20 cm- 1 . On the other hand, these calculations were limited to the SCF level and thereby ignore some of the correlation effects that become particularly important for the heavier hydrogen halides. Nor is the basis set very large, another source of error. For this reason, the interaction energy computed for (HC1)2 by this work is only about half of that obtained at higher levels (see earlier). One might conclude that the geometries of these complexes are consistent with certain patterns observed in true H-bonds but the energetics and other features are weaker than would normally be associated with such a bond. A microwave structure for the complex between HF and HI82 indicates that the angle becomes even more acute in HF...HF, as small as 70°. The I atom lies along the H—F axis, with R(I..F) = 3.66 A. This "triangular" structure would argue against the presence of a Hbond here.

2.5 HYH...YH2 The ubiquitous occurrence of water and its importance as a solvent medium have motivated a great deal of research into the fundamental nature of the interaction between water molecules by theoretical as well as experimental means. Some of the more recent work has been summarized in a review article83. Prior to experimental elucidation of the geometry of the water dimer in the gas phase or to the ability of calculations to provide an unambiguous resolution to this question, a number of different candidate structures were considered. In addition to the "standard" linear arrangment wherein the bridging hydrogen lies near the O...O axis in Fig. 2.8, cyclic and bifurcated structures were considered as illustrated in Fig. 2.9. It is now widely accepted that the linear geometry is in fact the equilibrium structure, although the energy cost in assuming other configurations remains under debate84-86. A rather

78

Hydrogen Bonding

Figure 2.8 Dispositions of molecules and lone pairs in H2Y dimer.

extensive comparison of the details of the equilibrium geometry and binding energy obtained by different basis sets and levels of theory has been contributed to the literature by Frisch et al.57. The results are summarized in Table 2.23 which illustrates nicely that the small basis sets like STO-3G and 3-21G strongly underestimate the intermolecular separation. There is a nearly consistent trend of more flexible basis sets yielding longer R(OO); in most cases correlation reduces this distance. The H-bond energies in the last two columns echo this trend in that larger basis sets yield a lower absolute value of Eelec, but that correlation enhances the binding energy. While there is substantial scatter in the calculated equilibrium (3 angles, most values are within or close to the experimental range of 113-133°. 2.5.1 Binding Energy of Water Dimer More recent calculations have further improved on the theoretical method and yielded refined values for the interaction energy of the water dimer. The Hartree-Fock limit of the electronic contribution to E has been placed at —3.73 ± 0.05 by Szalewiczetal.87, a value which was confirmed by others88. A slightly smaller estimate of —3.55 kcal/mol emerged from studies of Feller89 whose basis sets included h functions on O and g on H. In contrast to the SCF value which is relatively simple to obtain with moderate sized basis sets, the authors were more pessimistic about correlation components, in particular the dispersion energy. Later work90 placed a fully correlated binding energy of — Eelec at 4.5-4.6 kcal/mol, confirmed by others incorporating bond functions into the basis set88 or using other correlation techniques91, van Duijneveldt-van de Rijdt optimized the geometry with BSSE corrections included. Correlation seems to have only a minor influence upon angular features of the equilibrium geometry88. Kim et al.92 found a binding energy of 4.66 kcal/mol at the

Table 2.23 Equilibrium geometries and binding energies (uncorrected for BSSE) calculated for the linear water dimer at various levels of theory57. R(O-O) (A) Basis set STO-3G 3-21G 6-31G* 6-31 +G* 6-311 ++G** 6-311 + +G(2d,2p) 6-31l + +G(3df,2pd) expt

- E elec (kcal/mol)

(degs)

SCF

MP2

SCF

MP2

SCF

MP2

2.740 2.797 2.971 2.964 2.999 3.035 3.026 2.98 = 0.01

2.802 2.913 2.901 2.910 2.911 —

124.0 124.6 117.5 130.3 143.1 130.8 133.2 123±10

107.9 102.7 128.9 135.8 123.2 —

5.9 10.9 5.6 5.4 4.8 4.1 4.0

12.6 7.4 7.1 6.1 5.4 —

Geometries and Energetics

79

MP2 level, with a counterpoise correction and a (13s8p4d2f/8s4p2d) basis set. Following appropriate additional terms, they computed a binding enthalpy H of —2.86 kcal/mol. When combined with their S of -17.7 cal mol-1 K-1, a G of +3.72 kcal/mol was finally derived. Their optimized R(OO) of 2.958 A was in nice agreement with the experimental value of 2.976 A. Feller's best correlated E is —5.1 kcal/mol89, somewhat larger than other workers have found, but supported by recent calculations93. In a recent effort94, bond functions, centered on regions between atoms rather than on nuclei, have been added to the basis set. The results lead to an MP2 binding energy of —4.7 kcal/mol, following correction for basis set superposition error. The Hartree-Fock portion of this interaction, —3.6 kcal/mol, is consistent with prior work. Full optimization at the MP2 level, with counterpoise corrections included, yield R(OO) = 2.94 A. The nonlinearity parameter, a, is 6° and the proton-accepting water molecule makes an angle of 123° with the O..O axis. A followup of this work95 suggested that the bond functions were unimportant here, as any stabilization produced by their presence was largely cancelled by the large BSSE that they introduce. Another calculation made use of a very large basis set, as many as 574 functions96. Correlation was included by a method that approaches the MP2 method in an approximate fashion. The authors concluded that their best estimate of — Eelec is 5.0 ± 0.1 kcal/mol, which leads then to a binding enthalpy at 375 K of 3.2 ± 0.1 kcal/mol. Hence despite the inordinate attention paid to the water dimer and the application of state-of-the-art methods, there remains some lingering ambiguity concerning the binding energy, from both an experimental and theoretical perspective. The largest uncertainty probably lies in the dispersion part of the interaction energy which appears most resistant to saturation by enlarged basis sets97. At the present time, it is probably safe to say that the electronic contribution to E is in the range between —4.5 and — 5.0 kcal/mol. About 1.0-1.5 of this amount arises from correlation. 2.5.2 Complexes Containing H2S While experimental measurements of the water dimer in the gas phase had yielded an unambiguous linear structure, the results for (H2S)2 were less clear98. Cyclic and bifurcated geometries of the type shown in Fig. 2.9 were also proposed for this dimer. Following earlier ab initio investigation of this question, van Hensbergen et al. applied first-order exchange perturbation theory, coupled with an "effective-electron" model99. This work was unable to clearly differentiate the more stable between the cyclic and linear geometries, but found bifurcated clearly higher in energy. Later ab initio calculations54 found linear to be preferred to bifurcated, but only by a very small amount. The respective values of Eelec were —0.9 and —0.7 kcal/mol at the SCF level with a 6-31 + G* basis set. Taking the theory up to MP4SDQ raised the binding energies to — 1.4 and —1.1, still quite close to one another. Indeed, addition of zero point energies left the two in a dead heat at -0.6 kcal/mol. The intersulfur distance of the linear structure was optimized at the SCF level to 4.524 A, with the bridging hydrogen within 1.4° of the S...S axis. Later calculations carried the optimization to the MP2 level and found a strong bond contraction accompanied inclusion of correlation; as seen in Table 2.24100, R(SS) diminished by 0.44 A. The proton acceptor molecule is nearly perpendicular to the H-bond axis. The last column emphasizes the weakness of the interaction. Other calculations101 involved a detailed comparison of the three types of arrangement above using a 6-31G* basis set and found bifurcated to be more stable than linear at MP2, but only by an amount less than 0.05 kcal/mol; cyclic was less stable than linear by about 0.4 kcal/mol. A model potential which computes the total bind-

80

Hydrogen Bonding

Figure 2.9 Proposed structures of H2Y dimer.

ing energy as a sum of terras after making some simplifying assumptions identified the linear arrangement as more stable than bifurcated and cyclic by a significant margin102. One must consider the relative stability of the bifurcated and linear structures of (H2S)2 an open question at this point, particularly since the geometry optimizations of both have been limited to SCF. A more recent examination of the linear H2S dimer, using a heavily polarized [9s,6p,3d,f] basis set for S103, yielded an MP2 interaction energy Do of 1.7 kcal/mol, corrected for BSSE. The authors projected an infinite-basis set limit of 1.9 kcal/mol for this quantity. This calculation also confirmed the lack of any real binding at the SCF level. Because of their similar proton affinities, it is unclear whether H2S or H2O would be the proton acceptor in a complex combining the two. Calculations confirm the difficulty of answering this question. Amos104 carried out SCF calculations for various combinations of these two molecules including the mixed dimer, all in the linear arrangement. While he found H2S to be the preferred proton donor, the difference in energy versus the case where H2O is the donor was small enough that Amos considered the calculations not definitive. The data in Table 2.25 do provide a meaningful comparison of molecular geometries, how-

Table 2.24 Equilibrium geometries and binding energies (uncorrected for BSSE) calculated for the linear structure of (H2S)2 with 6-31G(2d) basis set100.

SCF MP2

R(S-S) (A)

(degs)

4.600 4.161

6.8 3.2

(degs) 97.1 89.6

Eelec

(kcal/mol) -0.8 -2.3

(

E+ZPVE)

(kcal/mol) 0.1 -1.1

Geometries and Energetics

81

Table 2.25 Equilibrium geometries calculated for complexes of H2S and H2O with 6-31G** basis set at SCF level104.

R(YY) (A) a (degs) (degs)

HOH-OH2

HOH-SH2

HSH--OH2

HSH-SH2

2.922 5 117

3.811 0.2 100

3.622 5 131

4.489 4 104

ever. One can see the clear progression toward longer intermolecular distances as each O is replaced by S. Note that in the mixed complex, R(SH) is shorter by perhaps 0.2 A when S acts as proton donor. All bridging protons lie within about 5° of the Y-Y axis. It is also worth noting that, regardless of the identity of the proton donor, the proton acceptor H2S molecule is nearly perpendicular to the H-bond axis, with angles around 100°. A somewhat later calculation of the mixed dimers was also unable to resolve the nature of the most stable complex. After applying the necessary corrections to energetics computed at the MP4 level, Del Bene41 found binding enthalpies of the HOH-SH2 and HSH--OFL, complexes of -1.3 and — 1.5 kcal/mol, respectively. It is unlikely that higher levels of theory will clearly differentiate between the stabilities of these two conformers, since Del Bene's work went up toMP4/6-31 + G(2d,2p). 2.5.3 Substituent Effects A minor perturbation on a molecule such as water would be the replacement of one of the hydrogens by an alkyl group. An early investigation of the interaction between methanol and water, with the former acting as the donor, yielded an interaction energy of some 6 kcal/mol105, and suggested that dispersion effects would be quite small. A later, more comprehensive study including substitution by methyl and ethyl resulted in the binding energies reported in Table 2.26, computed at the SCF level with a 6-31G** basis set106. These results show remarkable insensitivity to such substitutions. The most strongly bound of the set, where ethanol replaces water as proton acceptor, is only 0.3 kcal/mol stronger than the water dimer. The very similar energies of the water-methanol complexes, where the two

Table 2.26 SCF/6-31G** binding energies (electronic contributions, in kcal/mol) without BSSE correction 106. Complex

~AE

HOH-OH2 HOH-OHCH3 HOH-OHC2H5 CH3OH OH2 CH3OH--OHCH3 C2H5OH--OH2 C2H5OH--OHC2H5

5.54 5.42 5.85 5.52 5.42 5.44 5.66

82

Hydrogen Bonding

molecules reverse their roles as donor or acceptor, has been confirmed by IR spectral observation: whereas water will act as donor in the gas phase or in Ar matrix107,108, it is the methanol that is the donor in N2 matrix109. Calculations of a similar nature110 have demonstrated that replacement of both hydrogens of water, yielding dimethyl ether, also has only a minor effect upon the nature of the H-bond in the water dimer. With their polarized basis set, and with inclusion of corrections for BSSE, dispersion, and intramolecular correlation effects, these authors found the first methyl substitution raises the binding energy by 0.5 kcal/mol and the second by 0.6. The authors cautioned that an unpolarized basis set would fail to pick up these small effects, which they attribute to Coulomb and dispersion components of the interaction. Methyl groups have been added to the sulfur analog as well. When water is combined with S(CH3)2, the water molecule donates a proton to the S111 in a complex with C2v symmetry. The bridging hydrogen is computed to lie some 2.727 A from sulfur at the SCF/631G* level and stretches away from the oxygen by 0.003 A as a result of forming the Hbond. At the MP2/6-311 + +G** level, the binding enthalpy of this complex is -3.5 kcal/mol. This value is significantly larger in magnitude than a prior computation of HOH SH241 so the methyl groups would appear to make the sulfur a better proton acceptor. Other effects of substituents on the character of the H-bonding have been studied in one investigation of complexes involving water with silanol112. Either molecule can act as proton donor, but the greater acidity of silanol makes the latter the preferred proton donor. With a doubly polarized triple- basis set, the complex with silanol as proton donor is bound by 4.8 kcal/mol at the SCF level, 6.2 at MP2, but without correction of BSSE. Due to the greater acidity of silanol, this complex is more strongly bound than the water dimer. The comparable values of — E for the arrangement where water acts as proton donor are 3.1 and 4.6 kcal/mol, respectively, quite similar to that of the water dimer. The authors computed entropic contributions to the binding, enabling them to arrive at the thermodynamic quantities listed in Table 2.27 from which it may be seen that replacement of the proton-accepting water by silanol has little effect upon the energy or enthalpy of binding but that a significant boost is obtained if the proton-donating molecule is changed to silanol. In all cases, the Gibbs free energy of complexation is positive. Ugliengo et al.113 have optimized the geometries of various pairs of CH3OH and SiH3OH using basis sets of polarized double- quality. The geometries are all of the linear variety with all (O—H-O) angles within 10° of 180°. The results are listed in Table 2.28 and indicate that the substitution with a SiH3 group makes for a stronger proton donor, since the most strongly bound complexes involve this function for silanol. Comparison of the energetics with the 6-31G* data for the water dimer in Table 2.23 is clouded by the failure to remove BSSE from the latter. Taking instead the BSSE-uncorrected data of Ugliengo et al. with a 6-31G** basis set, the binding energies for the water and methanol dimers are vir-

Table 2.27 Calculated thermodynamics of binding of complexes involving water and silanol. Data in kcal/mol112.

EclccMP2 H° G°

HOH-OH2

HOH-OHSiH3

-4.70 -2.96 2.97

-4.56 -2.94 4.97

SiH3OH-OH2 -6.25 -4.64 3.13

Geometries and Energetics

83

Table 2.28 Energetics of complexes involving methanol and silanol. Data in kcal/mol113 are corrected for BSSE.

- EelecSCF - EelecMP2 - H (0K)

SiH3OH--OHCH3

SiH3OH-OHSiH3

CH3OH-OHCH3

CH3OH OHSiH3

5.69 6.60 5.19

4.59 5.59 4.30

4.21 5.09 3.90

3.61 4.30 3.20

tually indistinguishable, suggesting methyl substitution has little effect upon the strength of the H-bond. A similar conclusion has been drawn in comparisons of the binding of acetonitrile or formamide with either HOH or HOCH3114, albeit with a small basis set. Substitution of one hydrogen of HOH with an aromatic group leads to a phenol molecule. When paired with methanol, phenol acts as the proton donor molecule in a structure very much akin to the water dimer itself115. At the SCF/6-31G* level, the interoxygen distance is 2.89 A. The electronic contribution to the binding energy is computed to be 6.0 kcal/mol, after removal of BSSE, and 7.1 kcal/mol at the MP2 level with the same basis set. Correction of the correlated result by ZPVE yields a Do of 5.8 kcal/mol, leading to the conclusion that phenol is a more potent proton donor than is water. The structure of the phenol-water complex in the gas phase has been elucidated by microwave spectroscopy 116,117 and the phenol is indeed the proton donor in this complex. The interoxygen H-bond length was measured to be in the 2.88-2.93 A range. Prior computations with basis sets as large as 6-311 + + G(d,p) were consistent with this structure118,119; R(O O) was computed to be 2.94 A at the SCF level118, and one can expect a significant reduction upon reoptimizing the structure at a correlated level. Indeed, another computation finds a distance of 2.83 A in the MP2/6-31G** optimized structure120. These calculations suggested that this structure is indeed the global minimum on the surface of the phenol-water complex120. The BSSE-corrected interaction energy, Eelec, is calculated to be — 6.1 kcal/mol at the MP2 level118. Addition of vibrational terms yields a Do of 4.3 kcal/mol. Eclcc is computed with a more flexible aug-cc-pVTZ basis set to be —6.6 kcal/mol at the MP2 level, somewhat deeper and indicating some lingering basis set sensitivity. A secondary minimum represents a reversal in the donor-acceptor roles in that water donates a proton to the phenol120. (There was some evidence of a third minimum, wherein the water oxygen atom approaches one of the phenyl C—H bonds.) Another type of substitution, and one which is likely to have a stronger effect, is the replacement of one of the H atoms of water by a more electronegative atom like Cl. This substitution enhances the proton-donating ability of the water, so that HOC1 is the donor when combined with a water molecule. Unlike the water dimer itself, for which the anti conformer is the only stable minimum, both syn and anti arrangements represent minima on the surface of C1OH OH2121 as illustrated in Fig. 2.10. There are no minima corresponding to a reversal in which HOC1 acts as proton acceptor. However, the reader should be cautioned that HOH--OHC1 may appear to be a minimum, even at fairly high levels of theory. It required MP2/6-311 + +G(d,p) to demonstrate it not to be a true minimum. At all correlated levels of theory, the syn geometry is found to be slightly more stable than anti. At the MP4//6-311 + + G(3df,3pd)//MP2/6-311 + + G(d,p) level, the electronic contribution to the binding energy computed for syn is 8.2 kcal/mol, compared to 7.9 for anti. After a 2.3 kcal/mol ZPVE correction, these binding energies are reduced to 5.9 and

84

Hydrogen Bonding

Figure 2.10 Syn and anti conformers of H2O + HOC1.

5.6 kcal/mol, respectively. The interoxygen distances are 2.78 A in these two geometries, considerably shorter than in (H2O)2, and the H-bond is within 4° of linearity. The stronger binding here, as compared to the water dimer, is commensurate with the greater acidity of the hydrogen on HOC1.

2.6 (ZH3)2 Perhaps more than any other complex, the ammonia dimer has provided the most intriguing puzzle in piecing together its equilibrium geometry. It was presumed early on that this dimer would form a H-bond much like the other small hydrides such as HF and H2O, even if perhaps somewhat weaker. Indeed, there were indications from experimental work that the equilibrium structure was in fact linear122. For that reason, most of the early ab initio calculations focused on the linear type of structure. A representative group of data is collected in Table 2.29 which indicates the sensitivity of the calculated energetics and geometry to basis set123. The first row illustrates the inappropriateness of a minimal basis set, especially STO-3G, for this system. Most of the interaction energy is composed of superposition error, leaving only 0.8 kcal/mol of "true" binding energy. The intermolecular distance is grossly underestimated. The split-valence 431G represents an improvement in that R(NN) elongates to a more realistic value. Nevertheless, this basis is also subject to a large BSSE, about as much as its real binding energy. Addition of polarization functions lowers the BSSE to about 1/2 kcal/mol, and yields binding energies of 2.6 kcal/mol. While [541/31] does include polarization functions, the results are surprisingly poor, with a large BSSE, and small corrected interaction energy. Probably the best results emerge from the basis set in the last row of the table that contains two sets of d-functions, permitting superior treatment of polarization energy as well as electrostat-

Table 2.29 Calculated properties of linear geometry of ammonia dimer at the SCF level. Data123; all energies in kcal/mol.

STO-3G 4-31G 6-31G* 6-31G** [541/31] 6-3IG(2d,lp)

R(NN) (A)

- Eelec .

BSSE

3.08 3.31 3.44 3.44 3.44 3.54

3.8 4.1 2.9

3.05 1.97 0.53 0.55 1.04 0.68

3.1 2.4 2.4

-( E + BSSE) 0.8 2.1 2.4 2.6 1.4 1.7

(NH3) (D)

1.66 2.28 1.93 1.87 1.84 1.50

Geometries and Energetics

85

ics. It is worth mentioning that the dipole moment calculated for the ammonia monomer with this basis set, listed in the final column of Table 2.29, is quite close to the experimental measurement of 1.47 D124. Using a number of approximations123, it was estimated that the conversion from Eelec in Table 2.29 to E at 298 K requires the addition of 1.6 kcal/mol. Hence, at the SCF level, the ammonia dimer would probably not be expected to be bound at all, and if so by only a very small amount. The view of the ammonia dimer changed radically in the mid 1980s with the report by the Klemperer group that their microwave measurements argued against a linear arrangement125-128. They contended that their data supported a geometry akin to a cyclic structure. The "classic" linear and cyclic geometries are depicted in Fig. 2.11, along with the structure proposed from the microwave data. This interpretation of the microwave spectrum stimulated a flurry of activity to unearth the correct equilibrium geometry. It was suggested, for example, that photoelectron spectroscopy was not inconsistent with a cyclic structure 129 , whereas infrared photodissociation and matrix infrared measurements suggested the two molecules are not equivalent130,131 and supported the microwave equilibrium geometry132. Another set of measurements led to the notion that a tunneling motion, similar to that in the HF dimer, which interchanges the roles of proton donor and acceptor, was responsible for the two IR bands observed in the gas phase133. State selection in a hexapole electric field indicated that the dimer has a small dipole moment and that it is not a symmetric top structure134. On the computational front, Latajka and Scheiner135 considered an extensive region of the entire potential energy surface as a function of the two angles which describe the orientations of the two molecules, as well as the internitrogen distance. The only true minimum located on this surface corresponded to a cyclic structure in which the two H-bonding protons are displaced 42° from the N-N axis. A very shallow trough leads from this

Figure 2.1 I Three candidate equilibrium geometries of the ammonia dimer.

86

Hydrogen Bonding

geometry to a linear structure, predicted to lie only 0.2 kcal/mol higher in energy at the MP2/6-31G(2d,lp) level. This energy difference is unaffected by the inclusion of counterpoise corrections for BSSE. The conversion from cyclic to linear stretches R(NN) from 3.15 to 3.34 A. As a rationale for the microwave spectral data that seem to point toward a geometry that is not quite cyclic, the authors suggested motions on the surface which might tend to drive the vibrationally-averaged structure part of the way along the path from cyclic towards linear. Nearly simultaneously, Frisch et al. addressed the same problem54 but drew a different conclusion. They found the linear geometry to be the only minimum on their potential energy surface. The cyclic structure represents a transition state for conversion between the two linear arrangements, but only 0.2 kcal/mol higher in energy. While this work went up to fourth-order M011er-Plesset, and included zero-point vibrational energy, it did not attempt to remove BSSE at any level. Frisch et al.57 reoptimized the geometries of the cyclic and linear geometries at the MP2 level and confirmed their contention that linear is most stable. Other than a 0.15 A contraction of R(NN), the MP2-optimized linear structure differs very little from the SCF geometry. After making counterpoise corrections, other calculations136 confirmed the very nearly equal energy of the linear and cyclic arrangements, but their calculations were confined to the SCF level. Sagarik et al.137 incorporated correlation into the potential via a coupled-pair functional (CPF) approach which is size-consistent; their basis set was contracted to [642/31]. The dimerization energies calculated by the ab initio calculations were fit to an analytical sitesite potential function for purposes of molecular dynamics. While the linear geometry was most stable at the SCF level, CPF calculations yielded a cyclic unsymmetrical structure as the minimum, wherein the angles of the two monomers differ by 14° with respect to the N N axis. Although their surface was not adjusted for BSSE, counterpoise correction was made to the global minimum, after which their binding energy was calculated to be 2.8 kcal/mol. The results confirmed the very shallow nature of the surface along the pathway between linear and cyclic. Another set of calculations in 1986 attacked the surface of the dimer by first computing the electrical properties of the monomer at a high level138. By incorporating these properties such as multipole moments, polarizabilities, and hyperpolarizabilties into standard formulae of molecular interactions, the authors were able to extract an "electrical interaction" which includes not only electrostatic forces, but also polarization and dispersion energies. However, since the exchange which prevents collapse of the complex does not appear in their formalism, it was not possible to determine optimal intermolecular distances, and hence to identify the global minimum, so the authors focused their attention upon the angular features of the potential energy surface. Their results emphasized the flatness of the potential energy surface, in agreement with prior ab initio calculations. An attempt was made to distinguish the cyclic from the linear H-bond geometry of the ammonia dimer by modifying the basis set so as to drastically minimize the superposition error5. The results at the MP2 level favored the cyclic structure by a small amount, some 0.2 kcal/mol, with a doubly polarized basis set, containing one set of diffuse functions. It was noted that a basis of the same general quality, but unmodified to reduce BSSE, could yield the opposite conclusion as to which geometry was the more stable. Computations with a larger 6-311G(2d,2p) basis set in 198759 agreed that the cyclic was preferred to the linear structure by 0.2 kcal/mol at the MP4 level, but the two became indistinguishable when a set of diffuse functions were added to the nitrogens. Later gradient search of the ammonia dimer PES with the 6-31G* basis set7 led to a cyclic geometry; there was no minimum iden-

Geometries and Energetics

87

tified on the surface for the linear H-bond. Application of an MP4/6-31+G(2d,2p) electronic energy to this geometry, in combination with SCF/6-31G* vibrational frequencies, led to a binding enthalpy, H298, of —1.6 kcal/mol, roughly a third of the earlier experimental estimate of this quantity139. The level of theory was increased in 1991 by Hassett et al.140, with basis sets as large as 6-311 +G(3d',2p) and correlation by MP2 and QCISD. They learned that the characterization of a given geometry as a particular type of stationary point was subject to small changes in the type of calculation. For example, while the cyclic structure is indeed a stationary point, it is a minimum if the basis set does not contain diffuse functions; it is otherwise a transition state for NH3 rocking motions. In agreement with most other calculations, the microwave geometry could not be located as a minimum on any surface. The energy difference between cyclic and linear is very small indeed; at their highest level of theory, QCISD(T), the De values are 3.06 and 3.15 kcal/mol, respectively. Correction for BSSE makes these energies identical. Inclusion of zero-point vibrational energies yields a binding energy of 1.44 kcal/mol. The authors criticized the earlier Sagarik calculations137 as failing to identity true minima on the surface and imposing certain assumptions on the geometry of the dimer. Very large basis sets, including several polarization functions and bond functions, in concert with MP2-4 for correlation, have been applied by Tao and Klemperer141, along with counterpoise corrections. They find very little energetic separation between the cyclic and linear structures, with the preference dependent upon presence or absence of bond functions. Their findings echo the earlier work of Latajka and Scheiner135 in that, at their highest levels, the cyclic geometry is most stable, but a truly shallow energy path leads to the linear structure. Table 2.30 reiterates the nearly equal stability of the two geometries. Using their very flexible basis set, the authors found linear to be preferred by some 0.1 kcal/mol at the SCF level, but this difference is eliminated at any of the correlated levels, leaving an essential dead heat. The data further confirm the excellent correspondence between MP2 and MP4 dimerization energies. The "microwave" geometry proposed by Klemperer et al. was found to be rather unstable. Cybulski142 considered the ammonia dimer using even larger basis sets, as many as 200 functions, also with bond functions included. He noted a delicate balance between the specific functions used, or the centers of the bond functions, and the relative stability of the linear and cyclic geometries and warns against using bond functions without first carefully

Table 2.30 Variation of interaction energy (— Eelec) of ammonia dimer upon level of correlation. Data, in kcal/mol, were calculated with [753/41] basis set, augmented by bond functions. Both geometries fully optimized at MP2 level, including BSSE corrections141. Level SCF MP2 MP3 MP4SDQ MP4

Linear

Cyclic

1.67 2.86 2.78 2.66 2.85

1.56 2.88 2.78 2.67 2.87

88

Hydrogen Bonding

balancing the functions centered on nuclei. He concludes the linear structure is more stable by only 0.03 kcal/mol. Echoing earlier findings, the basis set superposition error was found capable of introducing distortions into the PES which might appear to favor one conformation over another, in this case linear versus nonlinear143. A CASSCF treatment leads to a linear structure as the minimum prior to BSSE correction, but nonlinear is clearly preferred following such correction. The apparent paradox between theory and experiment has been largely resolved by careful measurements of the Saykally group, reported in 1992144,145. These workers deduced from analysis of over 800 new far-IR absorption lines that the appropriate molecular symmetry group for the ammonia dimer is G144. Hence, interpretation of the spectra must allow for three different types of tunneling motion, including the unexpected umbrella inversion tunneling. In addition, surprisingly large interchange tunneling splittings were observed. These findings questioned the assumption made earlier by the Klemperer group that the umbrella tunneling was completely quenched and donor-acceptor interchange tunneling nearly so. In conjunction with six-dimensional vibration-rotation-tunneling dynamics calculations, Saykally et al. deemed it unlikely that the microwave geometry advanced by Klemperer was an equilibrium structure at all. More recent work has examined this question by formulating an empirical potential for the ammonia dimer, based on experimental and theoretical data146,147, and includes the effects of off-diagonal Coriolis interactions and octupole moments for the electrostatic interactions. Improved results for VRT states, far-IR frequencies, and properties of the protiated and deuterated dimers were obtained. The authors add further support to the contention that the ammonia dimer is highly nonrigid, and the essential nature of including vibrational averaging effects148. Their potential contains an energy barrier to interchange of only 7 c m - 1 . Olthof et al. go on to demonstrate that the data which led the Klemperer group to incorrectly conclude the dimer was rigid would be better explained as the competing effects of: (1) a stronger localization of (ND3)2 near one of its minima, and (2) its smaller ortho-para difference. The authors conclude that ab initio calculations are probably not yet up to the task of the 7 cm-1 accuracy necessary for truly accurate representation of such a weakly bound dimer. Based on the potential which best fits the experimental data, the authors147 hypothesize that there are two equivalent minima in the PES that correspond to nearly linear Hbonds. The top of the barrier for their conversion is a cyclic type of geometry. VRT-averaging of the ground state leads to a geometry that is nearly cyclic. In their view, this is not a H-bond, even though some of the features of a H-bond are present. This work is counterpointed by a study of the analogous dimethylamine dimer in the gas phase149 in which the authors deduce a geometry that appears to contain a distorted linear H-bond. That is, both the H atom of the proton donor molecule and the lone pair of the acceptor are bent to one side of the N N internuclear axis. But this distortion is substantial, and the designation as linear type is not entirely clear; indeed, the authors refer to their structure as "cyclic" although only one of the hydrogens can be conceivably involved in a Hbond. Other gas phase work has indicated the possibility for ammonia acting as proton donor in a H-bond as well. Held and Pratt150 examined the complex of 2-pyridone with ammonia by rotationally resolved spectra and found what they consider two H-bonds holding the two molecules together. The shorter, and probably stronger of the two, has the N—H group of the 2-pyridone acting as proton donor to the N of ammonia, with an H N distance of 1.99 A. A longer bond connects one of the protons of ammonia to the carbonyl oxygen of 2-pyridonc, with R(H--O) = 2.91 A. The designation of the latter as a H-bond is questionable due

Geometries and Energetics

89

to its length, but the possibility cannot be dismissed out of hand, especially as the authors found evidence of its existence in the fluorescence excitation spectrum; the torsional barrier to rotation of the ammonia is quite a bit larger than steric barriers normally encountered in complexes of this type with no H-bond. It is the unanimous conclusion of all high-level theoretical work that the potential energy surface of the ammonia dimer is quite flat, particularly the region connecting the cyclic and linear structures. While there remains some disagreement as to which of the latter two is more stable, the energy difference between them appears to be vanishingly small. It would hence be more realistic to speak not so much of the single equilibrium geometry of the ammonia dimer, as of a vibrationally averaged structure, with high-amplitude vibrational motions. The binding energy is rather low, probably less than 2 kcal/mol after vibrational energies are included. It is questionable whether this dimer should be classified as bound by a hydrogen bond in the usual sense. The presence of a H-bond in the PH3 dimer is even less likely. Frisch et al.54 found the equilibrium structure to be of cyclic type with no other minima on the surface. The two P atoms are 4.346 A from one another and each of the bridging hydrogens is located 88° from the P P axis. The electronic contribution to the binding energy is less than 1 kcal/mol. After reductions of this quantity by zero-point vibrational energies and removal of BSSE, it is questionable whether this dimer would be bound at all. A calculation of the mixed complex of NH3 with PH3 locates two minima on the surface7. Both contain what appears to be a linear H-bond; NH3 is the proton donor in one minimum and PH3 in the other. At the MP4/6-31 +G(2d,2p) level, both complexes are bound by 1.0 kcal/mol. However, since no correction has yet been made for superposition error, nor is there any computation of zero-point vibrational energy, it is likely that these minima will effectively disappear when these corrections are made.

2.7 Carbonyl Group

The doubly bonded oxygen atom of the carbonyl group presents an interesting contrast to the hydroxyl of water and related molecules. Lewell et al.151 have optimized the geometry of the complex between formaldehyde and water using a variety of fairly small basis sets and find that the latter molecule acts as the proton donor and the carbonyl as the acceptor. The bridging proton does not lie along the C=O axis but is off to one side, along a "lone pair direction" as indicated in Fig. 2.12. The trends in Table 2.31 illustrate that the carbonyl group obeys trends much like the simpler hydrides 151 . As the basis set becomes more flexible, the interaction energy is lowered and the intermolecular distance lengthened. Much of this trend is due to the reduction

Figure 2.12 Pairing of formaldehyde with water, containing nearly linear H-bond.

90

Hydrogen Bonding

Table 2.31 Optimized geometries (A and degs) and energetics (kcal/mol) of H-bond between water and formaldehyde151. Results at SCF level, not corrected for BSSE. See Fig. 2.12 for atom numbering scheme. Basis set

r(O2H1)

STO-3G 3-21G 6-31G 6-31G**

1.88 1.97 2.04 2.12

r(CO2)

r(O1H1)

0.002 0.006 0.005 0.005

-0.001 0.002 0.004 0.004

(O1H1O2)

- E

177.9 141.0 139.4 146.3

3.38 9.14 6.69 5.25

of the artificially attractive BSSE, which was not corrected by the authors. STO-3G yields anomalous results here, as in other systems. The covalent bond between O1 and H1 undergoes a small stretch, as a result of the formation of the H-bond. A new feature here is the stretch of the carbonyl bond between C and O2. This stretch can be rationalized within the context of this bond losing double-bond character as the bridging hydrogen moves toward it. The authors pointed out also that, whereas the STO-3G and 6-31G basis sets yield a fully planar geometry for the complex, 3-21G and 6-31G** predict that the peripheral hydrogen of water will swing out of the plane by some 60°. The penultimate column indicates that there is a fair degree of nonlinearity predicted within the H-bond for all basis sets except STO-3G. Calculations on this system were extended to include correlation in 199090, although internal geometries of each subunit were held rigid. With a polarized double- basis set, the interoxygen distance is calculated to be 3.00 A at both the SCF and CEPA-1 levels, only slightly shorter at 2.96 A for MP2. The (COO) angle is 100 ± 2° at any level; there is a 20° nonlinearity in the H-bond. The binding energies are surprisingly insensitive to the inclusion of correlation. At all three levels, — Eelec is calculated to be in the range 4.2-4.4 kcal/mol, with counterpoise corrections included. This interaction energy is quite similar to the binding computed by the same authors for the water dimer. Kumpf and Damewood152 examined the entire surface of the water-formaldehyde complex so as to consider all possible candidate geometries for the lowest energy. They found structure II in Fig. 2.13 to be slightly more stable than structure I earlier. Structure II differs from I in that the H-bond is far from linear, that is, the (O1H1O2) angle is not close to 180°. This nonlinearity is compensated to some extent by the proximity of the water oxygen toward one of the CH2 hydrogens. Another way of thinking of the reason for its stability is that the dipole moments of the two subunits are nearly antiparallel. The details of the optimized geometries of I and II are compared in Table 2.32, where it may be seen that the interoxygen distance is shorter in II, but the distance from the bridg-

Figure 2.13 Alternate pairing of formaldehyde with water, with strongly bent H-bond.

Geometries and Energetics

91

Table 2.32 Optimized geometries (A and degs) of geometries 1 and II for the complex between water and formaldehyde; data calculated at the SCF level with 6-31G** basis set152.

R(O..O) r(O2..H1) (O1H1O2) r(CO2) r(O1H1)

I

II

3.042 2.096 176.7 0.004 0.003

2.945 2.107 146.7 0.005 0.004

ing hydrogen, H1, to the acceptor oxygen is longer152. The H-bond is nonlinear by some 35° in II. Table 2.33 illustrates the very similar energies of the two structures, and how the relative stability can in fact shift from one level of theory to the next. For example, structure II is more stable with the 6-31G** basis set, but the situation becomes murkier for 6311+G**. It is difficult to draw any certain conclusions, especially since the authors did not attempt to remove BSSE, but structure II does appear to be at least slightly more stable. The same workers considered a number of other types of geometries. However, it was unclear as to how many represent true minima since a number of geometries were optimized under symmetry constraints and the numbers of imaginary frequencies were not reported. Dimitrova and Peyerimhoff153 focused their work on geometry I and helped provide a more accurate assessment of its interaction energy. Their highest level of theory stopped at MP2 but incorporated a 6-311 + +G(2d,2p) basis set. The SCF part of the interaction energy is —4.79 kcal/mol, reduced to —4.04 when corrected for BSSE. The superpositioncorrected contribution from MP2 correlation is —0.97 kcal/mol, adding up to a value of Eelec = — 5.0 kcal/mol, quite similar to the best estimates for the water dimer. When zeropoint vibrational corrections are added, this quantity lowers in magnitude to —3.3 kcal/mol. Ramelot et al.154 probed the nature of the minimum with the highest level of correlation to date, fully optimizing the geometries of stationary points. They verify the nature of structure II as a true minimum. At their highest level of theory, CCSD with a doubly-polarized

Table 2.33 Interaction energies (- Eelec, in kcal/mol) of geometries I and II for the complex between water and formaldehyde; data152 not corrected for BSSE.

SCF/6-31G** MP2/6-31G** SCF/6-311 + G** MP2/6-311 + G** MP3/6-311 + G** MP4SDQ/6-311 + G** SCF/6-31G** + ZPVE MP2/6-31G** + ZPVE

I

II

4.6 5.6 4.0 4.7 4.6 4.5 2.9 3.9

5.2 6.7 3.9 5.0 4.9 4.8 3.0 4.6

92

Hydrogen Bonding

triple- basis set, the bridging hydrogen is 2.007 A from the carbonyl oxygen and the (O 1 H 1 O 2 ) angle is 150°. The two O atoms are separated by 2.881 A. The authors disputed the earlier claim by Kumpf and Damewood of a second weaker H-bond between the water oxygen and a CH2 hydrogen, noting that their electron density plots indicated little of the perturbation characteristic of a H-bond. While Ramelot et al. find structure II most stable, they confirm the earlier conclusions by Kumpf and Damewood that a structure like I, with a more linear H-bond, is only marginally less stable. Indeed Ramelot et al. find only a 0.05 kcal/mol difference in energy. The authors recommend that diffuse functions should be added to this system whenever possible. The complex pairing H2CO with HF was examined recently at a high level of theory155. Optimization of the geometry at the MP2/6-311 + + G(2df,2pd) level yielded an intermolecular R(O-F) distance of 2.627 A, close to an experimental value of 2.66 A156. The Hbond is within 13° of linearity, with 0 (FH-O) = 167°. This H-bond is clearly shorter than that in H2CO-H2O. It is stronger as well, with a binding energy — Eelec of 7.57 kcal/mol (including counterpoise correction). Following the standard corrections, most notably the zero-point vibrations, the full — E is computed to be 4.99 kcal/mol. Raising the level of theory to MP4 lessens the latter quantity by 0.1 kcal/mol. The authors noted that inclusion of diffuse (+) functions provided an important component to their interaction energy, adding about 1 kcal/mol. Inclusion of these functions serves another important purpose. They reduce the BSSE by a factor of three. Consequently, computations with such diffuse functions are recommended to avoid certain spurious effects which might not be fully corrected by the counterpoise technique. Correlation was incorporated into the geometry optimization of the complex of formaldehyde with HC1157 in 1988 using basis sets as large as doubly polarized triple- . Table 2.34 lists the relevant properties of this complex at the SCF and two different correlated levels, MP2 and a coupled-pair functional (CPF) approach. The best value obtained for the electronic contribution to the binding energy from this study would appear to be around 5.0 kcal/mol, which is reduced to about 3 kcal/mol after zero-point vibrations are considered. Counterpoise corrections would likely further reduce this interaction, were they to be introduced. A later work explicitly accounted for basis set superposition and found these effects to indeed be important 158 . The binding energy, including ZPVE corrections, was computed to be 2.65 kcal/mol at the MP4/6-31 l + +G(2df,2pd) level; a CCSD treatment of correlation yields a similar value.

Table 2.34 Optimized geometries (A and degs) and energetics (kcal/mol) of H-bond between HC1 and formaldehyde157. Energetics not corrected for BSSE. SCF

R(Cl--O)

r(C02) r(ClH) (C1H,02) (CO 2 H,) - Eelec

-( E + ZPVE)

MP2

CPF

DZP

TZ2P

DZP

TZ2P

DZP

TZ2P

3.356 0.003 0.008 172.7 132. 1 4.3 2.9

3.375 0.003 0.008 169.1 124.2 3.6 2.2

3.166 0.004 0.018 168.0 113.7 6.2 4.3

3.124 0.005 0.021 164.4 107.3 5.7 3.8

3.246 0.004 0.011 167.8 115.7 5.3 3.6

3.208 0.004 0.014 163.2 109.0 4.8 3.1

Geometries and Energetics

93

A recent set of IR spectra in the gas phase159 suggest that the interaction energy is not very sensitive to substitution on the carbon atom. The measured interaction energies of HC1 with acetone, 2-butanone, methyl formate, and methyl acetate are all within 0.5 kcal/mol of one another. The energetics in the last two rows of Table 2.34 illustrate that correlation acts to significantly enhance the binding, even more for MP2 than for CPF157. This general trend is evident in the optimized geometrical parameters as well. For example, the SCF equilibrium R(Cl-O) distances are about 3.36 A, but are contracted by correlation, more so by MP2 than by CPF. The CPF/TZ2P distance is in fact very close to the experimental measurement of 3.21 A160. Correction of the potential for BSSE increases the R(Cl--O) H-bond length by some 0.05 A158. The C=O bond stretches by some 0.004 A upon complexation, accompanied by a longer stretch of the H—Cl bond. These bond stretches are highly sensitive to correlation and the method of its inclusion. The CPF/TZ2P r(HCl) stretch is 0.014 A; it is even longer, 0.020 A, at the MP2/6-311 + +G(2df,2pd) level158. The bridging proton is predicted to lie within 15° of the H-bond axis and the HC1 molecule approaches along the general direction of a carbonyl lone pair, with (COH) angles of 120 ± 13°. The authors emphasize that an angle of just this magnitude would be expected based upon the electrostatics of the two subunits. In the analogous complex between formaldehyde and HF, the rotational spectrum indicates a similar structure156, with a (COH) angle of 115°, although high-level computations yield a slightly smaller angle in the 102-110° range155. Perhaps more important, the energy of the H2CO-HC1 complex is fairly insensitive to this angle. The authors also draw a parallel between the optimized value of this angle and the dipole moment computed for H2CO157. It stands to reason that the dipole-dipole interaction between H2CO and HC1 will drive the complex toward a linear arrangement with 0 (COH) equal to 180° so deviations from this large angle will hinge upon a smaller moment. Similar arguments pertain to the much smaller angle of 91° computed for the sulfur-analog, H2C=S-HF, at the MP2/6-311 + +G(d,p) level23. This nearly perpendicular arrangement is in fact consistent with a survey of H-bonds in crystals where the 0 (C=S H) angle distribution peaks at about 110°, as compared to a maximum probability of almost 130° for 6 (C=O H) angles23. The authors note that their results of a smaller angle for the sulfurcontaining systems are best explained by a set of Coulombic interactions in the complex. While there does appear to be some degree of preference for proton acceptors to approach the carbonyl oxygen atom along a lone pair direction, there is not. much energy cost to an approach of the proton donor along the C==O axis. In fact, the entire region in between the two lone pairs can be considered "fair game" for formation of a H-bond. When the H-bond formed by H2CO and the proton donor is weakened by the replacement of HF by HCN, the (COH) angle enlarges nearly 30° to 138°,161 and the energy barrier to increasing this angle to 180° is observed to be low. This sort of result is confirmed by surveys of crystal structures13'162-166. 2.7.1 Substituent Effects A recent work has systematically examined the effects of replacing one of the H atoms of H2CO by F or Cl, then pairing this proton acceptor with each of HF or HC1167. Geometries were fully optimized at the MP2 level, using a flexible 6-311G(2d,2p) basis set. The essential features of these complexes are listed in Table 2.35 from which it may be seen that the H-bonds are somewhat shorter for the complexes containing HF, as compared to HC1.

94

Hydrogen Bonding

Table 2.35 Geometries (A) and energetics (kcal/mol) of complexes pairing HXCO with HX (X = F or Cl). Data computed at MP2/6-311 G(2d,2p) level l67 . R(O-H) HFCO-HF HC1CO-HF HFCO-HC1 HC1CO-HC1

1.867 1.871 2.087 2.088

r(HX)

r(C=O)

0.007 0.007 0.010 0.009

0.006 0.010 0.005 0.009

Eelec -4.57 -4.12 -3.45 -3.22

Eelec +

ZPVE

-2.52 -1.84 -1.85 -1.60

More specifically, the equilibrium distance between the carbonyl oxygen and the bridging proton is smaller by 0.2 A, as evident in the first column of data. The next column indicates that the HF bond elongates by less than does the HC1 bond in forming these complexes. Incorporation of a Cl atom makes the C=O bond more flexible: formation of the H-bond causes this bond to stretch by twice as much in HC1CO, as compared to HFCO. The energetics in the last two columns reveal a slightly stronger interaction for HF as compared to HC1. There is also a smaller tendency for HFCO to form stronger H-bonds than HC1CO. It is important to note a lingering basis set sensitivity. Comparable computations with a slightly smaller 6-311G(d,p) basis set, containing fewer polarization functions167, indicate different trends, favoring HClCO HCl as the most strongly bound, and HFCO HCl the weakest. Comparison of the results in the last column of Table 2.35 with unsubstituted H2CO HC1, at a similar level of theory158, indicate that unsubstituted H2CO is a better proton acceptor; the counterpoise-corrected Eelec, with ZPVE included, amounts to —3.12 kcal/mol for H2CO HC1. The H-bond in H2CO HC1 is also shorter. R(O H) is equal to 1.85 A in H2CO HC1158, as compared to 2.09 A in HXCO HC1. The reduction in protonaccepting ability, that appears to be associated with replacement of a hydrogen by a halogen, can be understood simply on the basis of the greater electronegativity of the halogens which drain electron density away from the carbonyl oxygen atom.

2.8 Carboxylic Acid The carboxylic acid group is of particular interest as it contains both the hydroxyl —OH and carbonyl C=O functionalities on the same molecule. Its acidic nature is exhibited, for example, in crystal structures where its H-bonds tend to be shorter and straighter than those of other proton donors168. The geometries of complexes pairing HCOOH with HF and HC1 have been optimized at the SCF level169 and the results presented in Table 2.36, based on the geometrical parameters described in Fig. 2.14. Note that when in position I, the HX molecule acts as proton donor to the carbonyl oxygen, and as donor to the hydroxyl in site II. The H-bond lengths are of course longer for HC1 than for HF, due principally to the larger size of the Cl atom. The 4-31G basis set yields a shorter bond than 6-31G**. Of greatest import, site I yields a uniformly shorter H-bond than site II, indicating a stronger H-bond for the former. This expectation is confirmed by the energetics in the lower part of Table 2.36, where the HX molecule at site I is bound by an energy greater than that of site II by a factor of approximately two. We conclude that the doubly bonded oxygen acts as a much

Table 2.36 Optimized geometries (A and degs) and energetics (kcal/mol) of H-bond configurations I and II (see Fig. 2.14) between HF or HC1 and formic acid169. E refers to electronic contribution, not corrected for BSSE. HF

HC1

I

R(O X) r(XH) r(C=O 1 ) r(C-O2) 6(X-OC) (HX-O) _

ESCF

_AE MP2

II

I

II

4-31G

6-31G**

4-31G

6-31G**

4-31G

6-31G**

4-31G

6-31G**

2.586 0.019 0.014 -0.020 97.5 24.7 14.52

2.636 0.015 0.013 -0.016 98.3 24.6 11.75 14.72

2.618 0.009 -0.005 0.018 112.8 22.8

2.740 0.004 -0.003 0.012 110.3 27.5 5.03 6.70

3.180 0.011 0.008 -0.011 105.9 24.7 6.98

3.308 0.009 0.006 -0.009 114.3 17.2 5.08 6.64

3.262 0.008 -0.004 0.011 126.0 2.0 4.06

3.435 0.004 -0.003 0.008 129.1 1.7 2.53 3.86

8.60

96

Hydrogen Bonding

Figure 2.14 Possible configurations of HCOOH + HX.

more effective proton acceptor than does the hydroxyl oxygen, and will be the preferred site in most cases. The preference of site I over II is confirmed by observation of the HCOOH---HF complex in solid Ar170. The interaction energies in the last row of Table 2.36 would probably be reduced by 1 or 2 kcal/mol were BSSE corrected. One may hence consider the electronic contribution of the binding energy of HF to formic acid to be perhaps 12-13 kcal/mol, stronger than the case where formaldehyde is the acceptor molecule. Likewise, the interaction of formic acid with HC1 would be some 5 kcal/mol, only slightly stronger than the comparable data for formaldehyde in Table 2.34. Other indicators of the stronger interaction with the carbonyl oxygen emerge from the perturbations of the internal geometries. Considerably larger stretches of the HF or HC1 bond are seen for site I, as much as 0.015 A for HF with the larger basis. As in the case of formaldehyde, H-bonding to the carbonyl oxygen stretches the C=O bond. Interestingly, in the case of formic acid, a concomitant contraction of the C—O bond to the hydroxyl oxygen is observed as well. This contraction is larger in magnitude than the C=O stretch. Analogously, formation of a H-bond to the hydroxyl oxygen stretches the C—O bond and shrinks C=O. There is an interesting difference in angle of approach amongst the various types of Hbonds. First, the 9 (X OC) angles are smaller for site I than for II. (These angles are insensitive to basis set choice.) The specific halogen atom makes a difference in that the angles are appreciably larger for HC1 than for HF. Another intriguing distinction is the greater deviation from linearity for HF, as indicated by the larger values of the (HX O) angle. Zheng and Merz48 have paired HCOOH with water which serves as the proton donor. The geometry optimized with a 6-31G* basis set is like geometry I for HCOOH plus HX, and is illustrated in Fig. 2.15 as the syn geometry. Table 2.37 indicates that correlation strengthens the H-bond and shortens the interoxygen distance by some 0.07 A. Comparison with Table 2.36 indicates the interaction of HCOOH with water is slightly weaker than with HF, but certainly stronger than with HC1. The last row indicates that HOH and HCOOH are bound with respect to one another, that is, G<0, only after correlation has been added. Calculations at the same level 171 emphasized the "cyclic" nature of the syn geometry, in the sense that O acts as a proton acceptor from HCOOH while the water proton is donated to the carbonyl oxygen. The inability of such a geometry to form when the carboxylic acid is in its anti configuration was stressed. The data represented an improvement over the earlier data48 in that Nagy et al. included a counterpoise correction. This correction is quite large since it reduces the SCF interaction energy of the syn structure from — 10.7 to —8.8

Geometries and Energetics

97

Figure 2.15 Syn and anti arrangements of HCOOH + HOH.

kcal/mol, and the MP2 value lowers from —14.3 to —10.1 kcal/mol. These results are essentially unchanged when the basis set is enlarged to 6-311 + + G**. The interaction energy in the syn structure is greater than that in the anti by some 1.3 kcal/mol at either the SCF or MP2 level. Instead of water or HF, one may pair HCOOH with a nitrogen base. The data on the right side of Table 2.37 indicate that the greater basicity of methylamine versus water leads to a stronger H-bond at either SCF or MP2 level of theory. Note, however, that the internuclear distances are comparable. The Cl atom in CH3C1 can unsurprisingly act as a proton acceptor when paired with formic acid. In conjunction with this particular interaction, one of the H atoms of methyl chloride can "curl around" and approach an oxygen atom of HCOOH172. Several geometries of the dimer which contain this second potential H-bond are illustrated in Fig. 2.16. The notation lists the proton donor group of the formic acid (which can be either the OH or CH), followed by the proton acceptor oxygen (—O or —O). The computed binding energies of these various configurations are listed in Table 2.38 at various levels of theory. All methods agree that the most stable is the OH/=O geometry.

Table 2.37 Optimized interoxygen distance and thermodynamics of H-bond between formic acid and water or CH3NH2, calculated with 6-31O* basis set48. Energetics not corrected for BSSE. Water

R(O-Ba) (A) Eelec (kcal/mol) H (kcal/mol) S ( c a l m o l - 1 (leg - 1 ) C (kcal/mol) a

B = O or N.

CH3NH2

SCF

MP2

SCF

MP2

2.796 -10.8 -8.3 -30.1 0.7

2.726 -14.5 -12.5

2.834 -11.5 -9.6 -29.9 -0.7

2.718 -16.3 -14.4

-3.5

-5.5

98

Hydrogen Bonding

Figure 2.16 Geometries of stationary points on surface of complex pairing HCOOH with CH3C1.

This result is not surprising as the OH group is certainly more acidic than is CH of formic acid and the carbonyl oxygen should serve as a better proton acceptor than the hydroxyl oxygen. The electronic contribution to the binding energy is some 6.0 kcal/mol, reduced to 5.0 when zero-point vibrational energies are added. There is some ambiguity as to the next most stable configuration, but it appears to be OH/—O, again due to the acidity of the carboxyl hydrogen. This structure is less stable than OH/=O in part because of the strain incurred by the OH---C1 grouping so as to enable the H atom of CH3C1 to approach the hy-

Geometries and Energetics

99

Table 2.38 Interaction energies, Eelec, (in kcal/mol) computed for various configurations of complex between HCOOH and CH3Cl172. Notations from Fig. 2.16. Basis set 6-311+G** 6-31G**

6-31 + G*

SCF MP2 MP2 + ZPVE MP2 MP2 + ZPVE

OH/=O

OH/-O

CH/=O

-3.89 -6.09 -5.14 -5.98 -5.03

-2.01 -3.32 -2.75 -3.82 -3.25

-2.35 -3.60 -2.92 -3.45 -2.77

CH/-O -1.39 -2.43 -1.92 -2.63 -2.12

droxyl oxygen. Comparable in stability is CH/=O where the CH of formic acid substitutes for OH as the proton donor group. The same is true of the least stable geometry, labeled as CH/—O. The authors did not correct their interaction energies for BSSE, but noted that the corrections are all 0.40 kcal/mol or less at the SCF level, while no attempt was made to assess the MP2 error. One can take the results above to argue that the CH group can act as a proton donor. In the OH/=O and OH—O configurations, the CH group of CH3C1 does approach an acceptor atom so as to resemble a H-bond. Granted, it is true that this interaction can be considered as simply auxiliary to the "main" or conventional H-bond between the OH of formic acid and the Cl atom. However, both of the H-bonds present in each of the other two configurations, CH/=O and CH/—O, involve a CH group as proton acceptor, with no "conventional" H-bonds at all. (The possible existence of CH H-bonds will be discussed in greater detail in Chapter 6.) 2.8.1 Carboxylic Acid Dimers When two carboxylic acids are paired together, it is possible to form two H-bonds, within an eight-membered ring, as illustrated for formic acid in Fig. 2.17. Much of the early work on this complex, applying SCF-type calculations to small unpolarized basis sets, illustrated the sensitivity of the intermolecular separation to the basis set173-175. The geometrical parameters optimized for this dimer are reported in Table 2.39 for a variety of higher levels of ab initio theory176. With respect to the internal geometry of each monomer, there is not much change ongoing beyond the polarized double- basis set. Electron correlation elongates both the carbonyl C=O bond and the OH bond. When the dimer is formed, both of these bonds become somewhat longer. The C=O bond stretches by 0.14-0.18 A, with the stretches on the longer end of this spectrum obtained with correlation included. Similarly for the OH bond, stretches of as much as 0.025 A are observed at the MP2 level. Consistent with these greater intramolecular perturbations at the correlated level, the H-bonds are

Figure 2.17 Geometry of formic acid dimer.

Table 2.39 Optimized parameters (A and degs) of the geometry of the formic acid dimer 176 . Basis set DZP

TZ2P expt a

Relative to monomer.

SCF MP2 SCF MP2

r(C=O)

r(C=O)a

r(OH)

r(OH)a

R(O O)

6(OH O)

1.191 1.221 1.190 1.219 1.217

0.014 0.018 0.015 0.018

0.967 0.998 0.961 0.998 1.03

0.015 0.025 0.014 0.021

2.767 2.670 2.781 2.672 2.70

172.3 177.8 174.6 180.0

Geometries and Energetics

101

also somewhat shorter, as measured by R(O O), and more linear with values of (OH O) closer to 180°. The energetics of the complex are reported in Table 2.40176-182. The results indicate that the electronic contribution to the binding energy is of the order of 12-13 kcal/mol with the DZP and TZ2P basis sets. The effect of correlation is inconsistent from one basis to the next. Note also that the results in Table 2.40 have been corrected for basis set superposition error. Uncorrected data are inflated by as much as 3-4 kcal/mol, nearly 8 kcal/mol at the MP2 level. After addition of zero-point vibrational energies and other factors, the binding enthalpies at 300 K are computed to lie in the range of 10.4-11.7 kcal/mol. These results have not yet reached convergence. When a larger basis set is applied to the problem, the calculated binding enthalpy reaches 13.1 kcal/mol. The latter result is in reasonable coincidence with experimental estimates that are in the 14-15 kcal/mol range. From their explorations with various geometries, the authors conclude that a MP2 calculation of the energy, using a geometry optimized at the SCF level, will typically lead to errors of less than 1 kcal/mol for this dimer. They recommend a set of DZP quality for this identification of an appropriate geometry. Two sets of polarization functions should provide accuracy of 0.3 kcal/mol in the SCF binding energies (provided counterpoise corrections are made). Correlation is more resistant to convergence; diffuse and semidiffuse d and f functions can be recommended for quantitative accuracy. From computations with basis sets of the polarized 6-31G type, at SCF and MP2 levels, the change from formic acid dimer to acetic acid dimer has only a very minor influence upon the energetics of binding183. This conclusion is indeed verified by resonant laser photoacoustic spectroscopic measurements179,181,182 which find further that trifluoroacetic and propionic acid dimers have a very similar binding enthalpy to formic dimer. One may extrapolate that the formic acid dimer H-bond energy is probably applicable for most carboxylic acids with alkyl chains replacing the CH group.

2.9 Nitrile Another bonding arrangement of the nitrogen atom involves the nitrile group, as in HC=N. As the single lone pair of the nitrogen atom is collinear with the rest of the molecule, as well as with the molecular dipole moment, there is little question as to the angular preferences for H-bonding. It is interesting to compare and contrast the H-bonds formed by this

Table 2.40 Energetics of binding (kcal/mol) in the formic acid dimer. Data corrected for BSSE176. Basis set DZP

TZ2P VQZ2PP expt a,b

SCF MP2 SCF MP2 MP2

- Eelec,

- H(300° K)

12.9 12.0 12.1 13.3 14.7

11.2 10.4 10.5 11.7 13.1 14.1-15.2

"See references 177-181. b Result for acetic acid dimer is 15.0 kcal/mol l82

102

Hydrogen Bonding

group with amines wherein the nitrogen has three separate bonding partners. Moreover, the hydrogen in HCN is acidic enough that the molecule may act as an effective proton donor. Several H-bonded complexes involving HCN were optimized at the SCF level with a doubly polarized triple- basis set by Somasundram et al.184. When paired with NH3, HCN acts as proton donor, with R(C-N) = 3.292 A. HF is the proton donor in FH-NCH, with R(F-N) = 2.878 A. The latter bond length is only slightly longer than the value of 2.848 A obtained in an earlier calculation185 with a polarized triple- basis set, with electron correlation treated by an approximate coupled cluster approach186. The computed binding energy of FH---NCH agrees quite nicely with a high resolution FTIR measurement of 6.9 kcal/mol for Dc187. While FH---NCH is the more stable, it is also possible to trap the reverse complex, NCH---FH, in Ar matrix188. A dimer of HCN is fully linear and the C and N atoms are separated by 3.307 A. Binding energies for NCH---NH 3 and FH---NCH were reported to be 7 kcal/mol, in comparison to the weaker binding of 5 kcal/mol for the HCN dimer. A later calculation189 optimized the geometry of NCH---NH 3 at the correlated MP2 level and with a much enlarged [6-31 +G(3df,2p)] basis set and confirmed its C3v structure. The R(C-N) distance is 3.139 A in the complex, which is bound by 6.06 kcal/mol relative to the isolated monomers, following counterpoise correction. This binding energy drops to 4.37 kcal/mol when zero-point vibrational effects are included. Curiously enough, a second minimum was identified on the PES pairing HCN with NH3. A geometry in which one of the H atoms acts as a bridge to the N atom of HCN is a local minimum on the MP2 correlated surface. This complex is much more weakly bound than the conventional structure where NH3 is the proton acceptor. The binding energy of the structure in Fig. 2.18 is only 1.1 kcal/mol at the SCF/6-31G* level, with counterpoise corrections. It is likely not to be bound at all following incorporation of vibrational effects. The latter geometry has not been detected in experimental studies of this complex190,191 which have confirmed the presence of a C3v geometry for NCH-NH3. Replacement of NH3 by PH3, maintains the linear character of the H-bond in NCH--PH3, although R(C P) is lengthened to 3.913 A in a microwave spectrum192,193. The presence of a true H-bond is also questionable in NCH---SH 2 where a very long R(C S) of 3.809 A is observed194. Using measured intermolecular stretching force constants as a criterion, the interaction in this complex is the weakest of any complex of the type AH---YH2 (A=NC, Cl, F;Y=O,S) 194 . SCF/6-31G** calculations by Boyd195 showed that substitution of a methyl group for the hydrogen of HCN increases the binding energy with HC1 from 4.3 to 5.3 kcal/mol, suggesting the larger molecule is a better proton acceptor. Del Bene et al.196 also paired HC1 with acetonitrile. The geometry optimized at the MP2/6-31 + G(2df,2pd) level contains a fully linear H-bond, with a R(Cl--N) distance of 3.316 A. The latter distance compares with 3.301 A measured by microwave spectra197. Applying MP4 to this geometry yields a binding energy, De, of 5.9 kcal/mol, lowered to 5.2 after correction for BSSE. This result is in excellent agreement with a FTIR photometric measurement of 5.2 kcal/mol198. As expected, the H-bond is stronger in FH---NCCH3, with a binding energy measured to be 6.9

Figure 2.18 Alternate geometry for HCN + NH3.

Geometries and Energetics

103

kcal/mol199. In fact, computations suggest certain other substitutions might further strengthen this interaction, for example, replacing NCCH3 by NCLi200. The dimer of HCN was studied in more detail, with much improved basis sets, ranging up near Hartree-Fock quality. Specifically, the largest set employed by Kofranek et al.201 was doubly polarized on C and N, and had a single set of p-functions on H. The complex is fully linear, as illustrated in Fig. 2.19. Beginning with the SCF data, the salient geometric and energetic aspects of this dimer are listed in the upper part of Table 2.41201,202. Enlargement of the basis set leads to a progressive lengthening of the intermolecular distance and weakening of the binding energy. The H-bond computed with the biggest basis set is too long by some 0.1 A, not surprising since the attractive effects of correlation have not been considered. One can see also the usual stretching of the bridging hydrogen away from the C atom, and small changes in the internal C N bonds. These trends are duplicated when correlation is included, and the intermolecular distance comes closer to the experimental value, as do the energetic quantities.

2.10 Imine

Intermediate between the N atom of amines, which is involved in single bonds, and the triply bonded N in nitriles, lie the imines with their double bonds. Recent calculations203 have combined methyleneimine, and its methylated derivatives as proton acceptor, with water and with methanol as donor. The general arrangement of water with methyleneimine is illustrated in Fig. 2.20 where it is emphasized that the hydrogen of the water not participating in the H-bond prefers to lie outside of the molecular plane of H2C=NH. The same is true of the methyl group when water is replaced by methanol. The sensitivity of the energetic aspects of the H-bonds to the level of theory is reported in Table 2.42. The inclusion of polarization functions reduces the interaction energy somewhat, probably due largely to reduction of BSSE. With these d-functions included, there is little difference between water or methanol as proton donor. Correlation enhances the binding strength, as noted in other systems. Comparison with other complexes is difficult as the authors did not remove the BSSE from their data. On the other hand, the authors have compiled a list of thermodynamic quantities for a number of related systems, all at the SCF/631G level, so there is some basis for comparison here. Comparison of the first two rows of Table 2.43 indicates that the doubly bonded N in the imine is a better proton acceptor than is the doubly bonded carbonyl oxygen in formaldehyde. This trend is consistent with the single bonded analogues wherein amines are more basic than hydroxyls. The succeeding rows address methyl substitution at various sites. It appears that methyl substitution at either the proton-donating water or proton-accepting nitrogen slightly diminishes the H-bond energy. The two effects reinforce one another when both are alkylated. In contrast, the H-bond is strengthened when one of the hydrogens of the CH2 group of the imine is replaced by methyl. Unlike the carbonyl bond, which stretches by around 0.005 A upon forming a H-bond, the double bond between C and N in the imine stretches by less than 0.002 A.

Figure 2.19 Linear HCN dimer.

Table 2.41 Optimized geometries (A) and energetics (kcal/mol) of the HCN dimer. Data201,202 contain energetics not corrected for BSSE. See Fig. 2.19 for atomic labeling scheme. Basis set [53/3] [64/4] [641/41] [752/41] [752/421

SCF

[641/41] [752/41] [752/421 expt

CPF

R(C N)

r(CdHd)

r(CdNd)

r(CaAa)

3.293 3.345 3.392 3.405 3.401

0.006 0.007 0.005 0.005 0.005

0.001 0.001 0.002 0.001

3.364 3.342 3.331 3.29

0.005 0.005 0.005

- Eelec

-AH

0.001

-0.002 -0.001 -0.001 -0.001 -0.002

5.60 4.71 4.39 4.19 4.21

4.72 — 3.59 3.39 3.42

0.000 0.000 0.000

-0.001 -0.001 0.000

4.37 4.35 4.54 4.4

3.63 3.61 3.80 2.7-3.8

Geometries and Energetics

105

Figure 2.20 H-bond arrangements in methyleneimine-water complex.

2.1 I Amide

The prevalence of amide groups as structural elements in proteins has motivated a good deal of interest in its H-bonding ability204. An early calculation, using unpolarized minimal and split-valence basis sets205 noted the near equivalence of the strengths of the amide-amide and amide-water H-bonds. Other computations illustrated the sensitivity of the formamide dimer binding energy to the particular basis set. A range of 10 to 23 kcal/mol for this quantity was obtained for basis sets varying from minimal (STO-6G) to 6-31G*206,207. These values were undoubtedly inflated by the failure to correct them for superposition error. There are various ways in which two amide molecules can come together to form a Hbonded complex. Ostergard et al.208 assumed the general configuration shown in Fig. 2.21 and carried out geometry optimizations at various levels. The data in Table 2.44 illustrate the stretching of the intermolecular separation that is typically seen as small basis sets are enlarged. The 5 angle which characterizes the placement of the proton donor relative to the carbonyl oxygen is rather sensitive to the choice of basis set or to inclusion of correlation. The orientation of the donor molecule, denoted by |3, is clearly in the range between 116° and 122°. None of the interaction energies are at a high enough level to be considered definitive, particularly since no BSSE corrections have been made. SCF computations with the 6-31G** basis set209 indicate that the interaction energy is rather insensitive to the 8 angle and there is little dependence on the NH---O angle, provided it remains within about 30° of linearity. Again taking formamide as a model amide, other calculations examined its modes of binding to water210. Four separate minima were located on the potential energy surface, generated via full geometry optimizations with DZ and DZP basis sets. The most stable structure found is illustrated as I in Fig. 2.22. It contains two H-bonds in a cyclic arrangement, permitting H2O and HCONH2 to act as both proton donor and acceptor simultaneously. The carbonyl oxygen is proton acceptor in Structure II, whereas the NH proton is

Table 2.42 Binding energies (— Eelec in kcal/mol) of complexes of memyleneimine with water and methanol. BSSE was not removed in data203. SCF

H 2 C=NH-HOH H 2 C=NH--HOCH 3

MP2

6-31G

6-31G*

6-31G

6-31G*

8.2 7.8

6.1 6.2

9.2

8.2

106

Hydrogen Bonding

Table 2.43 Thermodynamic quantities calculated203 for various H-bonded complexes. All entries in kcal/mol, except AS in cal mol-1 deg - 1 . Eelec

H 2 C=O HOH H 2 C=NH-HOH H2C=NH-HOCH3 H2C=NCH3--HOH H2C=NCH3 HOCH3 CH3CH=NCH3--HOH CH3CH=NCH3--HOCH3 CH3CH=NH-HOH CH3CH=NH-HOCH3

-6.7 -8.2 -7.8 -7.9 -7.6 -8.7 -8.3 -9.0 -8.7

AH°

AS

AG°

-4.9 -6.2 -6.0 -6.0 -5.8 -6.7 -6.5 -6.6 -6.9

-24.2 -25.9 -27.2 -25.8 -27.1 -27.3 -28.6 -27.2 -28.7

2.3 1.5 2.0 1.7 2.3 1.4 2.0 1.0 1.7

donor in III and IV. The results of the calculations, and the existence of geometry I, were later confirmed by microwave spectral data211. The experimental structure contains slightly shorter H-bonds than calculated, but is otherwise quite similar. The complex pairing formamide with HF is again similar to I in that the primary H-bond has HF donating a proton to the carbonyl oxygen, with a secondary interaction between F and the NH group212. The electronic contributions to the binding energy, — Eelec, for the various geometries are listed in Table 2.45, computed at both the SCF and correlated (SCEP) levels210. The results are apt to be inflated since BSSE was not removed. It is nonetheless clear that the presence of two H-bonds in Structure I makes it the most stable by 2-3 kcal/mol. The carbonyl appears to be a slightly better proton acceptor than the NH2 group is a donor, since geometry II is more stable than III and IV. The similarity of the latter two indicates there is little difference between the syn and anti hydrogen atoms. There is some question concerning the ability of peptide groups to form H-bonds in aqueous versus apolar solvents213,214. A recent effort215 compares the strengths of H-bonds between peptide groups (modeled by N-methylacetamide, NMA) and water. The former differs from formamide principally in eliminating some of the possibilities of H-bonding to one of the NH2 hydrogens, so a cyclic structure like I in Fig. 2.22 is excluded. Cognizant of the similarities in strength between these sorts of bonds, the authors employed a polarized double- basis set for full geometry optimizations and extraction of vibrational energies; energies were then computed with a much larger basis set at the correlated MP2 level, and BSSE corrections evaluated. Several configurations were considered in which a water molecule is paired with NMA. The first pair of structures illustrated in Fig. 2.23 has the two methyl groups of NMA trans

Figure 2.21 Formamide dimer, illustrating the definition of two intermolecular angles.

Geometries and Energetics

107

Table 2.44 Optimized geometries (A and degs) of formamide dimer and interaction energy (kcal/mol)208; parameters defined in Fig. 2.21. R(O-N)

5

- Eelec

SCF STO-3G 4-31G 6-31G** 6-31 + +G

2.778 2.949 3.074 3.089

122.4 163.7 143.1 158.2

4-31G 6-31G**

2.988 3.016

126.3 125.

121.8 116.1 118.7 118.9

5.7 8.4 6.3

117.7 121.7

8.8 —

MP2

to one another, while they are cis in the second two geometries. In WTd, the trans NMA molecule acts as proton donor to the water molecule, while it is the acceptor in WTa. NMA is the acceptor in WCa, but the proximity of the carbonyl oxygen and NH group of NMA in its cis geometry permits a pair of H-bonds to be formed with a single water molecule. NMA is hence simultaneously proton donor and acceptor in WCad. The water-NMA H-bond is compared with the bond between a pair of amide units in the last two structures in Fig. 2.23. A single H-bond is formed when two trans geometries are paired in TdTa whereas a pair of cis molecules can form two H-bonds such that each molecule is simultaneously donor and acceptor, as in CdaCda.

Figure 2.22 Structures of four minima identified on H2O + HCONH 2 PES.

Table 2.45 Binding energetics (— Eelec in kcal/mol) of complexes of formamide with water. Data computed with DZP basis set and uncorrected for BSSE210.

I II III IV

SCF

SCEP

-7.9 -5.8 -5.2 -5.2

-9.5 -6.7 -6.0 -6.2

Figure 2.23 Configurations pairing NMA with water or another molecule of NMA.

Geometries and Energetics

109

Table 2.46 lists the energetics of binding for the various complexes. Comparison of the first two rows indicates that the amide interacts more strongly with a water molecule when the amide is the proton acceptor, that is, water is not a good proton acceptor. The similarity of the second and third rows suggests that proton donation to the carbonyl oxygen is unaffected by the cis or trans character of the amide. (Other calculations had indicated a similarly small sensitivity of the H-bonding interaction to the rotations of the two methyl groups216.) However, the cis arrangement provides another option to the complex in which there are two H-bonds present. The stronger binding of the WCad configuration confirms that despite any angular distortions, these two H-bonds are cumulatively stronger than a single H-bond. When a pair of amides interact, the single H-bond in TdTa is similar in strength to that in WTa or WCa. In other words, the N—H of the amide is comparable to HOH as a proton donor. Again, the appearance of a second H-bond, as in CdaCda, strengthens the interaction, this time by quite a bit. The binding energy in the latter configuration is roughly twice that of TdTa where there is only one such bond. This near additivity is probably due in part to the low degree of distortion of each H-bond in the NMA dimer, compared to the bent H-bonds when one of the molecules is the smaller water. The aforementioned trends are noted at either SCF or MP2 levels, and apply not only to the electronic segment of the binding energy, but also to AH where vibrational contributions have been considered. The authors conclude that a single amide-amide H-bond is similar in strength to that between an amide and a water molecule. If one is interested in a binary complex, there is a preference for the syn geometry of NMA, as it then becomes possible to form more than one H-bond. Of course, this preference would be removed as additional molecules are added to the system, as discussed by Guo and Karplus217. This reasoning is amplified by inversion transfer 13C NMR spectroscopic data of model systems204 which indicate that the amide-water interaction is stronger than that between a pair of amides. (The work points out the anomalous character of formamide which may not make it a good model for the peptide units in proteins.) Since the data suggest that amides form stronger H-bonds with water than with other amides, the prevalence of peptide-peptide H-bonding in folded proteins is attributed to the cooperativity that occurs as these peptide units form multiple H-bonds. (See Chapter 5 for a detailed discussion of the cooperativity phenomenon.) As a last parenthetical note, there is a recent set of calculations218 that lead to the intriguing suggestion that interaction

Table 2.46 Binding energetics of complexes of amide with water or another amide (see Fig. 2.23 for definitions). All data computed with aug-cc-pVDZ basis set and corrected for BSSE, in kcal/mol215. - H298

- Eelcc

WTd WTa

wc. wcad T

dTa

cdacda

SCF

MP2

SCF

MP2

3.5 5.7 5.6 6.8 5.4 10.5

4.8 7.0 7.0 8.9 7.2 14.0

2.0 4.0 3.9 4.9 3.8 8.8

3.3 5.3 5.2 7.0 5.6 12.3

110

Hydrogen Bonding

of an amide with multiple water molecules will have a large effect on its internal geometry in the excited * electronic state, and further, that this effect will be threefold larger and in an opposite sense to geometry changes that occur in the ground state. As reported above for the pair of N-methylacetamide molecules, two amides can form a cyclic complex containing two H-bonds as the favored geometry. In the case of NMA, the stability of this cyclic structure is very nearly equal to twice the binding energy of each individual H-bond. This question was examined for formamide as well176 and the results reported in Fig. 2.24 and Table 2.47. Fig. 2.24a illustrates the two equivalent H-bonds within the dimer in its optimized geometry. This type of geometry is consistent with the crystalline

Figure 2.24 Geometries of the formamide dimer, computed at the SCF/DZP level 176 . Bond lengths refer to distances between nonhydrogen atoms.

Geometries and Energetics

I !I

Table 2.47 Binding energies (— Eelec in kcal/mol) of various geometries of formamide dimer (see Fig. 2.24). Data calculated with DZP basis set, corrected for BSSE176.

SCF MP2

a

b

e

d

10.6 11.4

5.6 5.6

7.1 7.1

3.8 4.0

structure of formamide219-221, and the computed H-bond length is only slightly longer than the solid-phase value of 2.93 A. Another study has examined the effect of level of theory upon the H-bond length in this structure222. Correlation typically contracts the H-bond, but the amount of this reduction is sensitive to basis set quality. For example, the SCF and MP2 H-bonds are nearly identical in length for a double- basis set, whereas simply adding d-functions results in a correlationinduced contraction of 0.08 A. The magnitude of this contraction generally grows as the basis set is further improved, reaching as high as 0.18 A for TZ(2df,p). Once polarization functions have been added, the H-bond length is reasonably stable with respect to changes in basis set222. At the SCF level of optimization, R(N--O) remains in the range between 2.99 and 3.02 A as the basis set is enhanced from DZ(d) to TZ(2df,2pd); the MP2 range is 2.84-2.92 A. The NH--O bond stretches by 0.06 A when the two molecules are disposed as in a so that only a single H-bond can form between them176. The binding energy of the cyclic dimer is not quite twice that of the single H-bond in b, indicating little apparent cooperativity in the energetics in this case. This result is consistent with the data described above for the dimethylated amides215. The MP2 binding energy of the cyclic formamide dimer (a) of 11.4 kcal/mol, compares with a comparable calculation with the 6-31G** basis set223, which led to a value of 12.4 kcal/mol. Following correction of this result by ZPVE, the enthalpy of binding at 0° K comes to 9.6 kcal/mol. Another computation evaluated the binding enthalpy at 298 K of the formamide dimer to be 12.1 kcal/mol224 at a fairly high level of theory: MP2/6-311 + +G(2d,p), using a geometry optimized at the MP2 level. This value, however, likely overestimates the true value as the authors failed to correct for BSSE. Based on another set of computations222, enlargement of the basis set beyond the polarized doubletype is unlikely to have much influence on the computed binding energy. Unlike NMA, formamide contains a CH group which has the potential to act as a proton donor in a H-bond. The ability of the CH group to substitute for N—H in this manner is explored via structure (c)176. The R(C O) distance is rather long, and the (NH--O) distance is 0.01 A longer than in structure (a). The weaker binding in (c) as compared to (a) is confirmed by the energetic data in Table 2.47 where the (CH--O) interaction adds only about 1.5 kcal/mol to the binding energy of a single H-bond (b). The longest H-bonds of all occur when both H-bonds are of the (CH--O) variety in (d). Note that, even summed together, these two H-bonds are cumulatively weaker than a single (NH--O) interaction, as in (b). It is worth mentioning that the relative populations of linear and cyclic structures can be shifted by temperature. A recent work which couples experimental measurements with ab initio calculations of quadupole coupling constants225 determined that rings of six formamide molecules dominate in the liquid state at low temperatures, but are replaced by linear tetramers as T approaches 400° K.

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2.11.1 Interaction with Carboxylic Acid and Ester The NH2 group of formamide contains two hydrogens which could be donated in a H-bond, and the oxygen atom could serve as a proton acceptor. When paired with formic acid, which also contains a separate proton donor and acceptor group, it is not surprising to see two Hbonds formed, within an eight-membered ring, as illustrated in Fig. 2.25a176. Since oxygen is more electronegative than nitrogen, and formic acid more acidic than formamide, the fact that the R(OH--O) H-bond is shorter than R(NH--O) by some 0.25 A is also not unexpected. By considering the open (noncylic) structures depicted in Fig. 2.25b and 2.25c, it was possible to assess the relative strengths of these two H-bonds. Comparison with Fig. 2.25a indicates that the cyclization of the complex, that is, the formation of two simultaneous Hbonds, results in a contraction of each. R(O-O) is shorter by 0.05 A in the cyclic complex and R(N-O) shorter by 0.10 A.

Figure 2.25 Geometries optimized for complex of HCOOH + HCONH2, at SCF/DZP level 176 . Bond lengths refer to distances between nonhydrogen atoms.

Geometries and Energetics

II3

The energetics, too, illustrate the cooperative nature of the two H-bonds. As reported in Table 2.48, the formamide and formic acid molecules are bound together in the cyclic complex by some 12.6 kcal/mol at the MP2/DZP level. The (OH-O) H-bond of structure (b) contributes 7.0 kcal/mol while 3.8 kcal/mol more arises from the (NH--O) interaction. Together, these two separate H-bonds add up to less than the full interaction in the cyclic structure. (Structures (b) and (c) do not represent true minima on the potential energy surface.) It is interesting to note that correlation plays little apparent role in the computed binding energies of this particular complex. In addition to the eight-membered cyclic structure of Fig. 2.25a, one might envision a 7-membered ring such as pictured in Fig. 2.25d in which the NH2 proton-donating group is replaced by the CH group. This structure is computed to be 3.0 kcal/mol less stable than the global minimum (a). On the other hand, the (CH--O) interaction does appear to add something to the (OH--O) H-bond. Complex (d) is bound 2.6 kcal/mol more strongly than (b); moreover, formation of the cycle shortens R(OH--O) by 0.04 A. Structure (e) substitutes NH2 for =O as the proton-accepting group in the other H-bond, also representing a 7-membered ring. This complex was found to be unbound, relative to the pair of isolated monomers, and does not represent a minimum on the PES. The studies of formic acid and formamide suggest that the CH group can act as a proton donor. While there is some question as to whether it constitutes a true H-bond, there are clearly certain stabilizing interactions present. Formamide was paired with a small ester, methyl acetate, at the SCF and MP2 levels in order to compare the strength of this interaction with that between a pair of amides226. Three minima were identified on the surface, all containing a H-bond wherein the NH2 group of formamide acts as donor. The optimal acceptor is the carbonyl oxygen atom, as compared to the alkoxy oxygen of the ester. As reported in Table 2.49, the former H-bond is shorter than the latter by 0.14 A and is more linear. The energetics indicate an approximate 2 kcal/mol preference for the carbonyl oxygen atom as acceptor. Note also that correlation appears to contribute perhaps 3 kcal/mol to the total interaction energy of the complex.

2.12 Nucleic Acid Base Pairs The nature of the interaction between nucleic acid bases has been a continual source of fascination ever since the nature of the genetic code was unraveled. However, the numbers of atoms and electrons in these systems erected a high barrier to the application of accurate quantum mechanical methods for many years. For example, the guanine-cytosine (GC) pair contains 29 atoms and 136 electrons.

Table 2.48 Binding energies ( — Eelec in kcal/mol) of complexes of formamide with formic acid. Data calculated with DZP basis set, corrected for BSSE176.

SCF MP2

A

B

C

D

E

12.4 12.6

6.9 7.0

4.0 3.8

9.8 9.6

unbound unbound

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Table 2.49 Hydrogen bonding parameters calculated with 6-31G* basis set for complex pairing formamide with carbonyl and alkoxy oxygen atoms of methylacetate226. Geometry optimized at SCF level. Counterpoise corrections for BSSE added to all energies, reported in kcal/mol.

R(N-O)(A) (NH-O) (degs) - Eelec (SCF) - H298 (SCF) - Eelec(MP2) - H298 (MP2)

Carbonyl

Alkoxy

3.073 177.7 5.9 3.9 8.1 6.1

3.215 173.1 2.9 1.4 5.6 4.0

From semiempirical227-230 to ab initio231-236 the levels of treatment of this interaction over the years parallel the development of computational quantum chemistry. Some of the recent computations were able to optimize the molecular geometries using gradient procedures237-240 while others have fit the results of ab initio calculations on smaller systems to empirical functions so as to then study the full nucleic acid base pairs241-242. There has also been work that relates the effect of adding or removing an electron to the base pair upon the nature of the interaction243,244. We will focus here on the "state of the art." That is, it is now possible in practice as well as in theory to perform reliable ab initio calculations of a nucleic acid base pair. By this, it is meant that polarized basis sets can be used, and the effects of electron correlation can be included explicitly. A case in point is a recent series of computations that compared the stability of 30 different arrangements of DNA base pairs245. A 6-31G** basis set was used to obtain gradient-optimized geometries, including normal harmonic vibrational modes and energies. This vibrational analysis also confirmed the presence of a minimum or stationary point of higher order. The interaction energies were then computed using the MP2 method to include correlation, at the SCF-optimized geometries. Counterpoise corrections were added at each level. Addition of the ZPVE led to an estimate of the binding enthalpy of each pair. To provide some estimate of the scale of these computations at this point in time, each SCF geometry optimization, including vibrational analysis, required some 85-430 hours of CPU time on a Cray Y-MP supercomputer. Computation of the MP2 energy at a single geometry is much quicker, needing only 1-5 hours. As a push toward the outer limit of current technology, optimization of the geometry of the cytosine dimer at the MP2 level required on the order of several weeks of CPU time. The authors estimated that were they to attempt to compute electron correlation effects by the more time-consuming CCSD(T) approach on the cytosine dimer, enforcing C2h symmetry and using a 6-31G* basis set lacking polarization functions on hydrogen, such a calculation would require days of CPU time, 25 GB of scratch disk space, and one full GB of memory. Table 2.50 lists the binding energies of 13 of the 30 pairs studied in this work245,246. For any given pair of bases, G and C for example, only the most strongly bound is included in the table. The first two columns report the electronic contributions to the binding at the SCF

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Table 2.50 Calculated energetics (kcal/mol) of binding of selected nucleic acid base pairs. Data245 computed with 6-31G* basis set. - Eelec

GCWC GG CC GAa GTa AC TAH TARH TAWC TARWC AA TCa

TT

-AH

SCF

MP2

calc

exptb

23.4 22.5 15.9 11.5 12.6 11.0 10.2 10.2 9.6 9.5 8.1 8.3 8.4

23.8 22.2 17.5 14.1 13.9 13.5 12.7 12.6 11.8 11.7 11.0 10.7 10.0

21.9 21.5 15.5 13.3 13.3 11.7 11.4 11.3 10.5 10.4 9.3 9.5 9.1

21 16

13

9

a

Not a true minimum; one negative eigenvalue in Hessian. Train reference 246.

and MP2 levels, respectively, followed by the full enthalpy (using the MP2 binding energy) in the last columns. There are various ways that the bases can be paired together, maintaining good H-bonding contacts. In addition to the Watson-Crick arrangement, abbreviated as WC by the authors and illustrated in Fig. 2.26 for G—C and A—T, there is also reverse Watson-Crick (RWC), Hoogsteen (H), and reverse Hoogsteen (RH). These conformations are depicted for the adenine-thymine pair in Fig. 2.27. The first row of Table 2.50 indicates that the Watson-Crick pairing of guanine and cytosine is predicted to be the most strongly bound of any pair, with a AH of — 21.9 kcal/mol. There are three H-bonds in this complex: One NH---O H-bond length is 3.02 A and the other is 2.92 A; R(NH-N) = 3.02 A. Another factor contributing to the strength of the interaction is the very near linearity of the placement of the bridging proton: all three H-bond angles are within 4° of 180°. Although it only contains two H-bonds, the GG pair is only slightly less strongly bound than GC. This is probably due in part to the large dipole moment (7.1 D) computed for G and the ability of the two dipoles to align in an antiparallel fashion in the GG complex. Also, the two H-bonds are short and linear; both are of NH---O type with a bond length of 2.87 A and angle of 178°. It might be noted that both the GCWC and GG pairs have only minor correlation contributions to their total binding energies. Like GG, the CC pair also contains only two H-bonds. But the binding enthalpy is significantly smaller, down to 15.5 kcal/mol. This weaker binding may be due to the lesser protondonating ability of the NH group in the two NH--N H-bonds, each of which are 3.05 A in length. The GA and GT combinations are somewhat less strongly bound. It was noted by the authors that these structures are not true minima. The single negative eigenvalue probably refers to an out-of-plane motion of a side group (the authors enforced full planarity), and so is probably not a serious concern to this discussion. These two dimers have in common the presence of a pair of H-bonds; nitrogen serves as the proton donor in all of these.

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Figure 2.26 Watson-Crick pairing of guanine-cytosine and adenine-thymine. Only hydrogens involved in H-bonds are shown explicitly.

Of particular interest is the comparison of the four types of TA dimers that were examined. The calculations245 indicate a clear preference for Hoogsteen pairing over WatsonCrick; in both cases, it does not matter whether these are standard or reverse form. One cannot attribute the discrepancy to the type of H-bonds present. In all four instances, there is one NH—N and one NH---O bond. The adenine homodimer and thymine homodimer are both slightly less strongly bound. In summary, the purine-pyrimidine pairs follow the trend in binding energy:

If one compares purine/purine or pyrimidine/pyrimidine pairs:

Restricting the latter to homodimers only:

The authors stress that the ability of G and C to form the strongest interactions with other bases may be traced to their large dipolc moments. They summarize that neither the total number of H-bonds, nor their length or linearity, can be taken as sole arbiters of the strength of the overall interaction, but that all factors must be considered together.

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Figure 2.27 Reverse Watson-Crick (RWC), Hoogsteen (H), and reverse Hoogsteen (RH) pairings of adenine with thymine.

As a final test of the influence of correlation, the CC geometry was reoptimized at the MP2 level, using a 6-31G** basis set245. The resulting changes, with respect to the HartreeFock structure, accounted for a further stabilization of only 1.1 kcal/mol, or a 6% increase, despite the contraction of one of the H-bonds from 3.05 to 2.92 A, a full 0.13 A. It was noted that an estimate of the dispersion energy by the empirical London formulation does not reproduce the correlation contribution to the interaction as computed by MP2. Comparison of the computed binding enthalpies in Table 2.50 with the experimental quantitites in the last column 246 confirm the ability of ab initio methods to treat such extended systems with a high degree of accuracy. The experimental enthalpies were obtained

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by mass spectrometry and provide no information about the actual geometries. The correspondence with the calculated structure is made under the presumption that the structure observed experimentally is in fact the most stable computed geometry. Not only do the calculations correctly predict the relative order of the GC, CC, TA, and TT pairs, but the binding enthalpies are within about 1 kcal/mol. A similar set of calculations was limited to the Watson-Crick structures of GC and AT, and included also a Hoogsteen conformation of AT236. Although the computations were carried out at the MP2 level, counterpoise corrections were limited to SCF only. Their best computed enthalpies of interaction are all somewhat greater than those in Table 2.50. The two studies agree, however, in the relative ordering. The Watson-Crick GC pair is most strongly bound, and the Hoogsteen AT pair is bound by about 1 kcal/mol more than the Watson-Crick geometry. While one might expect that H-bonded conformations of nucleic acid base pairs would be most stable, there is also the interesting possibility of stacked structures in which the planes of the two bases are parallel to one another. Correlated calculations at the MP2/631G* level suggest these stacked geometries are less stable than the H-bonded structures247. Taking the cytosine dimer as an example, the interaction energy of the stacked structure is roughly half that of the H-bonded conformer. While a large fraction of the binding energy in the more stable structure originates in Coulombic attraction, it is dispersion which is chiefly responsible for the binding in the stacked structure. Since dispersion is relatively insensitive to the mutual orientations of the two molecules within their respective parallel planes, the anisotropy of the electrostatic term guides the two molecules into their optimal positions.

2.13 H-Bonds versus D-Bonds

The question sometimes arises as to whether a hydrogen bond is stronger or weaker than a deuterium bond. This question might better be posed in pairing a HF molecule with DF: will FH--FD be more or less stable than FD---FH? Since H and D are chemically indistinguishable, the electronic part of the energies of these two dimers will be identical. The differences are associated with the masses, and thence to the vibrational energies. The points can best be illustrated by analyses of the vibrational frequencies of FH---FD and FD---FH. The data calculated at both the SCF and MP2 levels248 are reported in Table 2.51, along with the total zero-point vibrational energy (in c m - 1 ) in the last row. The first two rows list the intramolecular HF and DF stretching frequencies. It is obvious that the HF stretch goes down if this molecule is the proton donor. On the other hand, a red shift of comparable magnitude occurs in the DF stretch when DF acts as donor. Consequently, these two effects pretty much cancel one another, as may be seen in the third row of Table 2.51 where the total intramolecular zero-point vibrational energies of FH---FD and FD---FH are very nearly identical, within 10 cm - 1 . The source of the difference in energy between these two isotopic isomers may be traced to the intermolecular frequencies in the lower part of Table 2.51. The intermolecular stretching frequencies of FH---FD and FD---FH are close to one another, as are the first of the two in-plane bends. The biggest discrepancies occur in the next two intermolecular bending modes. Both of these represent primarily the wagging motion of the bridging hydrogen. In the case of the D-bond, that is, FD---FH, it is a heavy D nucleus which is moving, so the frequency is reduced relative to the case of FH---FD where it is protium that acts as the bridge. The reductions in each of these bending frequencies are of the order of 100 c m - 1 . As a re-

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119

Table 2.51 Vibrational frequencies and zero-point vibrational energies (cm - 1 ) of the two isotopic dimers248. SCF HF stretch DF stretch intramolecular ZPVE iritermolecular stretch in-plane bend out-of-plane bend in-plane bend inter-molecular ZPVE total ZPVE

FH-FD 4371 3202 3787 126 169 439 495 615 4401

MP2 FD-FH 4418 3169 3794 138 183 319 407 524 4316

FH-FD 3993 2952 3473 149 194 500 568 706 4178

FD-FH 4073 2895 3484 163 210 362 466 601 4084

sult, the total vibrational energy of the D-bonded complex is lower by about 100 cm-1 compared to the H-bond. The total difference in zero-point vibrational energies of the FH---FD and FD---FH complexes, displayed in the last row of Table 2.51, is 85 cm-1 at the SCF level, and 94 c m - 1 at MP2. It is interesting that this difference is relatively insensitive to correlation. The size of the basis set is not crucial either, as earlier calculations with a smaller basis set249 had obtained a value of 109 c m - 1 . While the potential energy surface of the dimer can perhaps be calculated reasonably well, the biggest source of error in this analysis is the assumption of harmonic frequencies. Analysis of high resolution near-IR data for the Cl analogue250 is consistent with a "stronger" D-bond here as well: C1D---C1H is more stable than C1H---C1D by 16±4 c m - 1 . Despite the errors, the calculations permit a simple interpretation of the greater stability of D-bonds versus H-bonds. The highest intermolecular frequencies are the bends that involve the rocking of the proton donor molecule. The higher mass of D lowers these frequencies when DF is the proton donor, and thereby reduces the total zero-point vibrational energy. There are increases in the other two intermolecular frequencies associated with placing the lighter H nucleus on the acceptor molecule, but these changes are very much smaller than those in the aforementioned bends. Changing the subunit from HX to H2Y adds a number of new vibrational modes to the dimer. Nonetheless, the D-bonded form of the water dimer is more stable than the H-bonded complex251. The energy difference of 60 cm-1 is attributed chiefly to the out-of-plane motion that shears the bond which is of higher frequency for a proton than a deuteron, consistent with the observations for the HF dimer. The preference for D- versus H-bonding persists in solid matrices as well. When HDO is paired with NH3, HOH, formaldehyde, or formamide, it is the D atom of the former molecule which acts as the bridge252-254. The same holds true for complexes betwen HDO and olefins, where there is some question as to whether there is a true H-bond present255. In the latter case, the preference for D acting as a bridge rather than H is quite small, inasmuch as both isotopomers are observed. The difference in energy is estimated as less than 0.1 kcal/mol. An even smaller preference is noted between NCH---NCD and NCD---NCH, both of which have been observed in the gas phase, leading to the suggestion that their energy difference is only several c m - 1 at most256.

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Whereas the D-bond is stronger in some sense than the H-bond, the question of its length is more complicated. In contrast to their greater length in the solid state, gas-phase results indicate D-bonds are shorter than their protium analogs257. This discrepancy diminishes as the H-bond becomes stronger. The authors explained the gas-phase shortening on the basis of the high-frequency bending and stretching modes. The preference for D-bonding versus placement of a protium in the bridging position appears to reverse in ionic H-bonds. Calculations at the SCF and MP2 levels of various isotopomers of the H 2 O-H + --OH 2 system258 indicate that any D atoms would tend to migrate away from the bridging position, and toward one of the four peripheral positions. The equilibrium constant for this preference lies in the neighborhood of 2-3, based on a strict harmonic treatment of the ab initio force field. Using a temperature of 298° K, an equilibrium constant of 2 translates into a free energy preference for D to act as bridging atom of 0.4 kcal/mol. These results confirm an earlier experimental study based on fractionation factors in the gas phase259. Similarly, in the case of an analagous anion, like (CH 3 O-H--OCH 3 ) - , ab initio calculations with a 4-31G basis set260 indicate that there is a preference for a protium over D in the bridging position. It is found from ion cyclotron resonance spectroscopic measurements that the reaction

lies to the left, with an equilibrium constant of 0.3260. A similar preference for H versus Dbonding has been observed in the gas phase for (CI-H--C1)- as well261. 2.13.1 Water Molecules Recent ab initio calculations have attempted to probe the fundamental source of the reversal of H/D preference in ionic as compared to neutral systems, using water as a test base262. A harmonic analysis of the potential energy surface of the water dimer, computed with a 631G** basis set, indicates that the preference for D in the bridging site can be explained in a manner similar to that described earlier for HF--HF. The frequency of the bending motion of the bridging atom is sensitive to its mass: this effect leads to a lower vibrational energy of some 0.2 kcal/mol when the heavier D undergoes this motion. The computations indicated that electron correlation has little effect upon this conclusion, even its quantitative aspects. While the treatment was purely harmonic in nature, other calculations263 have indicated that anharmonicity effects yield very little distinction between one isotopomer and the next. The calculations262 considered higher levels of deuteration than single substitution and found that this central conclusion remains unchanged. Several manifestations of this preference for a D-bond are: 1. When DOD is paired with HOH, it is the former molecule that acts as proton donor. 2. HOD--OH2 is lower in vibrational energy than is DOH OH. 3. DOD--OD2 is more strongly bound than is HOH--OH2. The energetic preference for placement of a deuterium at a bridging position in the water dimer carries over nearly unchanged in the trimer262. This complex forms a triangular structure, almost equilateral. Each H-bond is distorted from linearity by the constraints of the structure, with 6 (OOH) angles close to 20°. Nonetheless, it is again the vibrational modes which distort this bridging atom from the H-bond axis that can be traced as the source

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of the lower vibrational energy for isotopomers where D acts as bridge as opposed to H. The close similarity between dimer and trimer in this respect leads one to suppose that the principle would remain valid for larger aggregates, including the liquid state as well. In fact, recent calculations verified the insensitivity of fractionation factors for the bridging hydrogens in oligomers of formamide264. Ions of both positive and negative charge were analyzed as well, using (H2OH+--OH2) and (HOH--OH)- as prototypes262. As in the case of the neutral dimer, deuterosubstitution of the bridging atom lowers the total intermolecular zero-point vibrational energy (ZPVE) of (H2OH+--OH2) by 0.2 kcal/mol more than if the replacement occurs at a nonbridging (terminal) site. But the situation is quite different with respect to the intramolecular modes. Whereas the intramolecular ZPVE of the neutral dimer is quite insensitive to the site of substitution, in the cation there is a 0.7 kcal/mol greater lowering of this quantity when D occurs at a terminal, as opposed to a bridging, site. The latter effect outweighs the intermolecular preference, so that in the case of (H 2 OH + --OH 2 ), a deuterium is favored to occupy a terminal position by some 0.5 kcal/mol. This same trend extends to multiple deuterosubstitution. The calculations indicated a remarkably small influence of correlation upon the trends discussed above. This lack of sensitivity is particularly notable since the equilibrium geometry of (H2OH+-OH2) is fundamentally different at the SCF and MP2 levels. The proton transfer potential contains a pair of minima in the former case, whereas there is only a single, centrosymmetric minimum present after correlation is included. The computations were extended to the larger proton-bound dimer of methanol, and the results were found to be similar, indicating that water is typical of general systems bound together through hydroxyl groups. The results of deuterosubstitution in the anion are of smaller magnitude, but do appear to confirm the trend that H-bonds are slightly stronger than D-bonds. Inclusion of aspects of the H-bond energy other than the electronic contribution, as well as entropic effects, permit inferences to be drawn about the magnitude of AG for formation of a H-bond at room temperature. It was found262 that the computed AS of the D-bridged water dimer is more negative than for the H-bridge by perhaps 0.2-0.3 cal mol-1 deg - 1 . As a result, the energetic preference for the D-bond in the neutral complex will slowly diminish as the temperature rises, eliminating some (but not all, by any means) of the 0.2 kcal/mol preference. Entropy has a similar "push" toward the H-bond in the ions as well, in this case reinforcing the energetic preference for the H-bond. While not directly relevant to H-bonds per se, a recent work compared the H/D fractionation factors for a number of small molecules, as computed at various levels of theory265. The results indicated that the relative free energy of protiated versus deuterated species is rather insensitive to choice of basis set. Best results are achieved if polarization functions are added to all atoms, and correlation is recommended, but even SCF computations with a basis set as small as 3-21G* can provide quite reasonable results.

2.14 Summary

The strength of the hydrogen bond rises with the acidity of the proton donor and the basicity of the acceptor. Of the simple hydrides, then, complexes of the H3Z--HX type are the strongest, with an electronic contribution to the binding energy of about 11 kcal/mol. Alkylation of the amine can improve its H-bonding ability, leading to H-bond energies on the order of 20 kcal/mol. Indeed, when the acidity and basicity of the donor and acceptor, re-

122

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spectively, reach large enough proportions, the character of the complex can change to an ion pair when the bridging proton is transferred across to the base. The strongest H-bonds are formed between first-row atoms F, O, and N. The decrements in going from the second to lower rows are smaller ones. Associated with stronger H-bonds are typically shorter distances between the nonhydrogen atoms. One of the shorter contacts is R(N---F) of about 2.7 A in H3N---HF; a similar bond length is characteristic of H2O—HF as well, although the latter is not quite as strongly bound. The formation of the H-bond stretches the A—H covalent bond, again by a larger amount for stronger interactions. This stretch amounts to some 0.03 A in H3N HF. Electron correlation is a rather important phenomenon in H-bonding. Its neglect can lead to underestimation of the strength of the bond and its associated phenomena. Correlation becomes progressively and proportionately more important for atoms of the second and third rows of the periodic table. It is also important to correct for basis set superposition error, particularly at the correlated level, as this phenomenon can lead to spurious inflation of the H-bond energy by several kcal/mol. Zero-point vibrational energies will typically reduce the dissociation energy as the complex contains more vibrational modes than the pair of isolated monomers. This reduction is about 3 kcal/mol for a strongly bound complex like H3N HF. When a H2Y molecule acts as the proton acceptor, there is a delicate balance between one orientation, in which the proton donor approaches along the direction of a Y lone electron pair, and another in which the donor is attracted toward the negative end of the H2Y molecular dipole moment. Consequently, the energy profile for transition from a pyramidal to planar complex is typically a rather flat one. Atoms of lower rows of the periodic table, that is, S and Se, show a marked proclivity for pyramidalization while water tends more toward planarity. In the case where the planar structure is only slightly higher in energy than the pyramidal one, vibrational motions may hide the energy barrier to inversion around the O atom, and experimental observation may suggest a planar equilibrium geometry. In the case of a heavy atom such as I, an "anti-H-bonded" geometry such as H2O IH can become competitive in energy to the H-bonded, albeit weakly so, H2O-HI geometry. While the latter is stabilized by the normal electrostatic interaction between the two subunits, dispersion is maximized in the former. When paired with a stronger H3Z base, H2Y acts as proton donor. While this interaction has all the characteristics of a H-bond, it is considerably weaker than H3Z HX or even H2Y--HX, suggesting H2Y is not a very effective proton donor. The electronic portion of the binding energy of H3N--HOH is about 6 kcal/mol, only about half that in H3N---HF, and the H-bond is longer by some 0.2 A. When two HX molecules are placed together, they do not line up in collinear HX---HX fashion. Rather, the proton acceptor rotates so as to present one of its three lone electron pairs toward the bridging hydrogen. This reorientation makes for an angle in the 110°-120° range. Because of the poor proton-accepting quality of HF, the dimerization energy of (HF)2 is only about 4.5 kcal/mol (the electronic contribution), comparable to, but slightly weaker than, H3N--HOH. The thorough testing of various levels of theory suggest that MP2 is an excellent means to evaluate the contribution of electron correlation, with results quite similar to those achieved with MP4 or coupled cluster approaches. The interfluorine distance is about 2.75 A. The HC1 dimer is more weakly bound, with a — Eelec of less than 2 kcal/mol. The acceptor is nearly perpendicular to the donor in (HC1)2. In the mixed complex pairing HF with HC1, either molecule can act as donor or acceptor; there is little difference in energy between HP--HC1 and HC1---HF. The binding energy of either is slightly

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greater than in the HC1 dimer. HX dimers involving Br and I are even more weakly bound. In fact, it is questionable whether any of the HX dimers, other than (HF)2, really contain a true H-bond. Despite a great deal of speculation over the years as to the possibility of "bifurcated" or "cyclic" H-bond geometries in the water dimer, it appears that the classic linear structure is the equilibrium structure, and perhaps the only true minimum on the potential energy surface. Calculations with large basis sets and thorough account of correlation have enabled a particularly accurate estimate of the energetics of this H-bond. The SCF limit of the electronic part of the binding energy is 3.5-3.7 kcal/mol. Correlation adds another kcal/mol or so, leading to a theoretical value of 4.5-5.0 kcal/mol. Thus, the water dimer is bound at about the same strength as (HF)2, although the H-bond in (H2O)2 is about 0.2 A longer. After correction for vibrational terms, AH comes to —2.9 kcal/mol for the water dimer. Since the entropy of binding is negative, the Gibbs free energy becomes positive at 298° K, about + 3.7 kcal/mol. The equilibrium geometry of (H2S)2 remains unclear at this time, with linear and bifurcated likely candidates. In either case, this dimer is weakly bound, with a binding energy of 1 kcal/mol or so after zero-point vibrational corrections. When mixed together, it is not clear whether H2O or H2S would be the proton donor and which the acceptor. Alkyl substitution of the hydrogens of H2O has little influence upon the H-bond energetics. Replacement by an electronegative atom like Cl, on the other hand, significantly enhances the binding energy (by several kcal/mol) as HOC1 is a considerably stronger proton donor than is HOH. The potential energy surface of the ammonia dimer is so flat that the concept of a single equilibrium geometry loses some of its meaning. The linear type of H-bond easily interconverts with a cyclic geometry with very little change in energy. As a result, the presence of a true H-bond in this dimer is questionable. The system offered an opportunity to examine how small modifications in theoretical method can shift the balance between one geometry and another, and provided a caution against drawing conclusions of relative stability based upon small differences in energy. The electronic part of the interaction energy appears to be some 4 kcal/mol, slightly weaker than the water dimer. Enlarging the types of systems studied beyond the simple hydrides adds interesting possibilities. Proton donors appear to prefer to approach the carbonyl oxygen atom along one of its lone pairs (6(C=O-H)~120°) rather than directly along the C=O axis, although this preference is not a pronounced one. In addition to the stretch of the A—H bond within the donor molecule, noted for the hydrides, the C=O bond of the acceptor elongates as well, and by a comparable amount. The carbonyl oxygen of formaldehyde is comparable in strength as a proton acceptor to the O of water. By sacrificing a linear O—H O arrangement, it is possible for a proton donor molecule like H2O to align itself with H2CO such that a second stabilizing interaction is formed: the oxygen of the water can approach the C—H of H2CO. The energetic cost of bending the H-bond is nearly equally compensated by the second interaction. The two O atoms are a little closer together in the H 2 O—H 2 CO complex than in the water dimer. The carboxylic group is considerably more acidic than water. Nonetheless, the two O atoms of COOH act as proton acceptors when paired with a hydrogen halide, due in part to the poor proton accepting ability of the halogen atom. Not unexpectedly the C=O oxygen is a considerably better proton acceptor than the —OH oxygen. Indeed this C=O within the context of the carboxyl group is a superior acceptor to the C=O in formaldehyde; forming a H-bond with HF in excess of 10 kcal/mol. On the other hand, this interaction energy is not solely the result of a single H-bond; some arises as a result of a secondary H-bond as

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the F accepts a proton from the —OH group of carboxyl. The primary H-bond is rather short as well, around 2.6 A. Other measures of the stronger H-bond are enhanced lengthening of the HF and C=O bonds. When HF is replaced by the better proton acceptor, HOH, the "secondary" H-bond, wherein HOH accepts a proton from the —OH of COOH, is strengthened at the expense of a somewhat weaker interaction with the C=O group where HOH acts as donor. As a result, the complex takes on a bit more of a "cyclic" character with two nearly equivalent H-bonds. The H-bonds with HCOOH donating a proton to either an O or N acceptor appear to be some 2.7 A in length, with an interaction energy in excess of 10 kcal/ mol in either case. A pair of carboxylic acids will form a dimer, containing two strong O—H O=C H-bonds. The total interaction energy approaches 15 kcal/mol, with each Hbond about 2.7 A in length. Although the N atom of NH3 or amines is not a good proton donor, it becomes much more electronegative when involved in a triple bond as in a nitrile. The N of HCN will accept a proton from the strong donor HF. The ensuing H-bond is of normal length, less than 2.9 A, with a strength of some 7 kcal/mol. Replacing the hydrogen of HCN by an alkyl group enhances the ability of the N to accept a proton. But more important than the protonaccepting ability of the nitrile N is the acidity of the C H group. There is evidence of the formation of the reverse complex, that is, NCH---FH, as a secondary minimum on the PES. NCH clearly donates a proton when combined with NH3 Although this H-bond is fairly long, perhaps 3.14 A, it amounts to some 6 kcal/mol. And a standard H-bond is present as well in the dimer of HCN, albeit slightly weaker and longer than in NCH—NH 3 . (HCN)2 shows evidence of stretching of the C—H bond upon complexation, with a H-bond energy of about 4 kcal/mol. When involved in a double bond, the N atom of an imine is not as good a proton acceptor as in amines, but better than the doubly bonded O of the carbonyl group. The amide group contains a carbonyl C=O and amine group, directly connected to each other so that they perturb one another's properties. One can draw a resonance structure in which a double bond connects the C and N atoms, leaving a negative charge on O and positive charge on N. This interaction enhances the proton accepting ability of the former and makes the N—H group a strong donor. When an amide like formamide interacts with HOH or some such molecule, a cyclic arrangement occurs which allows the N—H to donate to the O while the carbonyl C=O accepts a water proton. The overall interaction energy is in the range of 9 kcal/mol. This cyclic structure is 2 kcal/mol more stable than one in which only a single H-bond is formed to the carbonyl oxygen; a single H-bond to the N-H group of the amide is weaker still by the same amount. Even stronger is the interaction between a pair of amides. If there is a free N—H cis to the C=O group, the two molecules can form a pair of NH O=C H-bonds, neither of which is badly bent. The total interaction energy here amounts to some 14 kcal/mol, twice the energy of a single H-bond between a pair of amides. The N—H group in an amide appears to be comparable in strength as a proton donor to the O—H of water. An analogous pair of H-bonds can be formed, within the context of a cyclic geometry, when one of the amide molecules is replaced by a carboxylic acid. Ab initio calculations are now capable of being applied directly to systems as large as pairs of nucleic acid bases. One can obtain reliable data, with binding enthalpies within about 1 kcal/mol of experiment, using polarized basis sets and MP2 correlation, applying counterpoise correction, and fully optimizing the geometries of the entire pair using gradient procedures. The computations are able to provide information about the native binding abilities of these bases with one another, in isolation from the other components and geometrical constraints of the DNAbiopolymer. The results illustrate the strong binding of the guanine-cytosine pair, followed closely by the guanine homodimer. It also appears on the

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basis of the calculations that the Hoogsteen pairing of adenine and thymine is probably somewhat more stable than the Watson-Crick combination, although this difference amounts to only about 10% of the full binding energy. In any case, the binding enthalpy of the AT pair is only about half as large as the GC pair. Comparison with a series of other pairs makes it clear that this difference cannot be attributed solely to the presence of three Hbonds in GC and only two in AT. While the interaction energy of the stacked bases is certainly weaker than the coplanar H-bonded geometries, the stacking energy can represent a significant contribution to the overall stability of the DNA molecule. Of course isotopic substitution has no effect on the potential energy surface of H-bonded complexes or any system. However, atoms of different mass will leave the entire system with differing normal modes of vibration and their associated frequencies. In simple systems like FH--FH, the zero-point vibrational energy is lowered more if the bridging hydrogen is replaced by deuterium, as compared to the terminal atom. That is, FD--FH has less vibrational energy than does FH---FD. This difference can be traced to one of the intermolecular modes, the wagging motion of the bridging atom. Since this normal mode consists chiefly of the motion of the hydrogen, its effective mass is lowered significantly by the replacement of H by D. The energy difference amounts to some 100 cm - 1 , or about 0.3 kcal/mol. Similar results have been obtained for the water dimer, albeit a slightly smaller energetic preference for the D-bond versus the H-bond. The trend reverses for strong ionic complexes such as (H2OH OH2)+ or (HO H OH)- where it is the H that is preferred over D to adopt the bridging position. References 1. Marco, J., Orza, J. M, Notario, R., and Abboud, J.-L. M., Hydrogen bonding of neutral species in the gas phase: The missing link, J. Am. Chem. Soc. 116, 8841-8842 (1994). 2. Hinchliffe, A., Ab initio study of the hydrogen-bonded complexes H3N HBr, H3P—HBr, H 3 As-HF, H 3 As-HCl, and H 3 As-HBr, J. Mol. Struct. (Theochem) 121, 201-205 (1985). 3. Latajka, Z. and Schemer, S.,Ab initio study of FH—PH3 and CIH—PH3 including the effects of electron correlation, J. Chem. Phys. 81, 2713-2716 (1984). 4. Latajka, Z. and Schemer, S., Ab initio comparison of H bonds and Li bonds. Complexes of LiF, LiCl, HF, and HCl with NH3, J. Chem. Phys. 81, 4014-4017 (1984). 5. Latajka, Z. and Scheiner, S., Basis sets for molecular interactions. 2. Application to H3N—HF, H3N-HOH, H20-HF, (NH3)2, and H 3 CH-OH 2 , J. Comput. Chem. 5, 674-682 (1987). 6. Sadlej, J. and Miaskiewicz, K., Ab initio calculations of the vibrational spectra of the ammonia and fluoramide complexes with HF, J. Mol. Struct. (Theochem) 236, 427-441 (1991). 7. Del Bene, J. E., An ab initio molecular orbital study of the structures and energies of neutral and charged bimolecular complexes of NH3 with the hydrides AHn (A = N, O, F, P, S, and Cl), J. Comput. Chem. 10, 603-615 (1989). 8. Latajka, Z. and Scheiner, S., Three-dimensional spatial characteristics of primary and secondary basis set superposition error, Chem. Phys. Lett. 140, 338-343 (1987). 9. Latajka, Z. and Scheiner, S., Primary and secondary basis set superposition error at the SCF and MP2 levels: H 3 N—Li + a n d H 2 O - L i + , J. Chem. Phys. 87, 1194-1204 (1987). 10. Szczesniak, M. M. and Scheiner, S., Accurate evaluation of SCF and MP2 components of interaction energies. Complexes of HF, OH2, and NH3 with Li+, Coll. Czech. Chem. Commun. 53, 2214-2229 (1988). 11. Latajka, Z., Scheiner, S., and Ratajc/.ak, H., The proton position in amine-HX (X = Br,I) complexes, Chem. Phys. 166, 85-96 (1992). 12. Bacskay, G. B. and Craw, J. S., Quantum chemical study of the trimelhylamine-hydrogen chloride complex, Chem. Phys. Lett. 221, 167-174 (1994).

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237. Aida, M., Characteristics of the Watson-Crick type hydrogen-bonded DNA base pairs: An ab initio molecular orbital study, J. Comput. Chem. 9, 362-368 (1988). 238. Dive, G., Dehareng, D., and Ghuysen, J. M., Energy analysis on small to medium sized H-bonded complexes, Theor. Chim. Acta 85, 409-421 (1993). 239. Jiang, S.-P., Raghunathan, G., Ting, K.-L., Xuan, J. C., and Jernigan, R. L., Geometries, charges, dipole moments and interaction energies of normal, tautomeric and novel bases, J. Biomol. Struct. Dyn. 12, 367-382 (1994). 240. Florian, J. and Leszczynski, J., Theoretical investigation of the molecular structure of the k DNA base pair, J. Biomol. Struct. Dyn. 12, 1055-1062 (1995). 241. Pranata, J., Wierschke, S. G., and Jorgensen, W. L., OPLS potential functions for nucleotide bases. Relative association constants of hydro gen-bonded base pairs in chloroform, J. Am. Chem. Soc. 113, 2810-2819(1991). 242. Trollope, K. L, Gould, I. R., and Hillier, L H., Modelling of electrostatic interactions between nucleotide bases using distributed multipoles, Chem. Phys. Lett. 209, 113-116 (1993). 243. Colson, A.-O., Besler, B., Close, D. M., and Sevilla, M. D., Ab initio molecular orbital calculations of DNA bases and their radical ions in various protonation states: Evidence for proton transfer in GC base pair radical anions, J. Phys. Chem. 96, 661-668 (1992). 244. Aida, M., Kaneko, M., and Dupuis, M., An ab initio MO study on the thymine dimer and its radical cation, Int. J. Quantum Chem. 57, 949-957 (1996). 245. Sponer, J., Leszczynski, J., and Hobza, P., Structures and energies of hydrogen-bonded DNA base pairs. A nonempirical study with inclusion of electron correlation, J. Phys. Chem. 100, 1965-1974(1996). 246. Yanson, I. K., Teplitsky, A. B., and Sukhodub, L. F., Experimental studies of molecular interaction between nitrogen bases of nucleic acids, Biopolymers 18, 1149-1170(1979). 247. Hobza, P., Sponer, J., and Polasek, M., H-bonded and stacked DNA base pairs: cytosine dimer. An ab initio second-order Moller-Plesset study, J. Am. Chem. Soc. 117, 792-798 (1995). 248. McDowell, S. A. C. and Buckingham, A. D., Isotope effects on the stability of the carbon monoxide-acetylene van der Waals molecule and the hydrogen fluoride dimer, Chem. Phys. Lett. 182, 551-555 (1991). 249. Curtiss, L. A. and Pople, J. A., Ab initio calculation of the force field of the hydrogen fluoride dimer, J. Mol. Spectrosc. 61, 1-10 (1976). 250. Schuder, M. D. and Nesbitt, D. J., High resolution near infrared spectroscopy of'HCl—DCl and DCl—HCl: Relative binding energies, isomer conversion rates, and mode specific vibrational predissociation, J. Chem. Phys. 100, 7250-7267 (1994). 251. Engdahl, A. and Nelander, B., On the relative stabilities of H-and D-bonded water dimers, J. Chem. Phys. 86, 1819-1823 (1987). 252. Nelander, B. and Nord, L., Complex between water and ammonia, J. Phys. Chem. 86, 4375-4379 (1982). 253. Engdahl, A. and Nelander, B., The intramolecular vibrations of the ammonia water complex. A matrix isolation study, J. Chem. Phys. 91, 6604-6612 (1989). 254. Nelander, B., Infrared spectrum of the water formaldehyde complex in solid argon and solid nitrogen, J. Chem. Phys. 72, 77-84 (1980). 255. Engdahl, A. and Nelander, B., Water-olefin complexes. A matrix isolation study, J. Phys. Chem. 90,4982-1987 (1986). 256. Fillery-Travis, A. J., Legon, A. C., Willoughby, L. C., and Buckingham, A. D., Rotational spectroscopy of 15 N-hydrogen cyanide dimer: Detection, relative stability and D-nuclear quadrupole coupling of deuterated species, Chem. Phys. Lett. 102, 126-131 (1983). 257. Legon, A. C. and Millen, D. J., Systematic effect ofD substitution on hydrogen-bond lengths in gas-phase dimers B . . . HX and a model for its interpretation, Chem. Phys. Lett. 147, 484-489 (1988). 258. Edison, A. E., Markley, J. L., and Weinhold, F., Ab initio calculations of pmtium/deuteriumfractionation factors in O 2 H 5 + clusters, J. Phys. Chem. 99, 8013-8016 (1995).

Geometries and Energetics

137

259. Graul, S. T., Brickhouse, M. D., and Squires, R. R., Deuterium isotope fractionation within protonated water clusters in the gas phase, J. Am. Chem. Soc. 112, 631-639 (1990). 260. Weil, D. A. and Dixon, D. A., Gas-phase isotope fractionation factor for proton-bound dimers of methoxide anions, J. Am. Chem. Soc. 107, 6859-6865 (1985). 261. Larson, J. W. and McMahon, T. B., Isotope effects in proton-transfer reactions. An ion cyclotron resonance determination of the equilibrium deuterium isotope effect in the bichloride ion, J. Phys. Chem. 91, 554-557 (1987). 262. Scheiner, S. and Cuma, M., On the relative stability of hydrogen and deuterium bonds, J. Am. Chem. Soc. 118, 1511-1521 (1996). 263. Astrand, P.-O., Karlstrom, G., Engdahl, A., and Nelander, B., Novel model for calculating the intermotecular part of the infrared spectrum for molecular complexes, J. Chem. Phys. 102, 3534-3554 (1995). 264. Edison, A. E., Weinhold, E, and Markley, J. L., Theoretical studies of protium/deuterium fractionation factors and cooperative hydrogen bonding in clusters, J. Am. Chem. Soc. 117, 9619-9624(1995). 265. Harris, N. J., A systematic theoretical study of harmonic vibrational frequencies and deuterium isotope fractionation factors for small molecules, J. Phys. Chem. 99, 14689-14699 (1995).

3

Vibrational Spectra

n contrast to the deep minimum in the potential energy surface that corresponds to the equilibrium geometry of a standard molecule, the equilibrium structure of a typical Hbonded complex resides in a much shallower minimum. The PES of the complex is overall much flatter, yet commonly contains other minima, albeit not quite as stable, which are energetically accessible from the global minimum. This chapter focuses on the nature of the potential energy surface in the vicinity of the equilibrium structure, deferring until later a discussion of more extended regions of the surface. A principal question addressed here is the steepness of the surface as one moves away from the global minimum in various directions, and the source of this slope. This point is directly examined by theoretical calculations which can determine the energy of any given geometrical distortion of the structure. The shape of the minimum in the surface is experimentally probed by vibrational spectroscopy. It is here that the computations can make direct connection with experimental information. Formation of the H-bond from a pair of isolated molecules converts three translational and three rotational degrees of freedom of the formerly free pair of molecules into six new vibrations within the complex. The frequencies of these modes are indicative of the functional dependence of the energy upon the corresponding geometrical distortions. But rather than consisting of a simple motion, for example, H-bond stretch, the normal modes are composed of a mixture of symmetry-related atomic motions, complicating their analysis in terms of the simpler motions. In addition to these new intermolecular modes, the intramolecular vibrations within each of the subunits are perturbed by the formation of the Hbond. The nature of each perturbation opens a window into the effects of the H-bond upon the molecules involved. The intensities of the various vibrations carry valuable information about the electron density within the complex and the perturbations induced by the formation of the H-bond.

I

138

Vibrational Spectra

I 39

3.1 Method of Calculation Most ab initio analyses of vibrational spectra invoke a "double-harmonic" assumption wherein the potential energy surface in the vicinity of the minimum is fit to a function that involves only quadratic dependence of the energy with respect to the nuclear motions. The intensities of the normal vibrational modes are extracted from the derivatives of the dipole moment, taken as linear with respect to nuclear coordinates. Within this approximation, the intensities of the fundamentals are proportional to the square of the dipole moment derivatives with respect to normal coordinates 1-3. The harmonic approximation ignores third and higher-order terms in the dependence of the energy upon nuclear coordinates. In the case of most molecules, the approximation of the shape of the well as a parabola is a reasonable one, at least for small displacements from equilibrium. It is a straightforward matter for computer programs that evaluate the energy derivatives to go on and elucidate the normal vibrational modes and their associated frequencies, using a standard GF matrix formulation4. Analysis of the modes allows one to determine the real nature of a given vibration, for example, stretching, bend, or some combination. The frequency provides a measure of the "stiffness" of the potential with respect to the particular motion involved. Experience has shown that SCF-level calculations typically overestimate the magnitude of the vibrational frequencies by some 5-10%, even with very large flexible basis sets. This exaggeration is due in part to the inability of the SCF wave function to dissociate properly. A compilation of a great deal of data5 has suggested that SCF/6-31G* frequencies should be multiplied by a scaling factor of 0.893 for best reproduction of experimental values, while the factor for MP2/6-31G* is 0.943. Such a prescription is estimated to reproduce experimental frequencies to 50 cm-1 as a root mean square error. In the case of H-bonds, the intermolecular part of the potential is subject to greater degrees of anharmonicity, so harmonic frequencies should be treated with caution, even if the energetics are computed with a high level of theory. The symbol v is normally used to express the frequency of any given vibrational mode, in units of c m - 1 . Since most calculations are restricted to the harmonic approximation, the use of this symbol in the computational literature likewise refers generally to harmonic frequencies. In those cases where anharmonicity is added to the computations, the notation can become confusing in that v usually refers to the anharmonic value, with reserved for the harmonic approximation to this frequency. The reader should therefore exercise some caution in scanning the original literature. In this text, we will adopt the convention that v will represent the harmonic frequency; in those cases where anharmonicity is included, the distinctions and notation will be clearly delineated. The integrated infrared band intensity of the ith mode is measured experimentally as

where C and L refer to the concentration and the optical path length, respectively, and Io and I the intensities of the incident and transmitted light. This same quantity may be expressed within the confines of the double harmonic approximation as

140

Hydrogen Bonding

where Qi is the normal coordinate and the dipole moment vector of the molecular system. The sum extends over any g degeneracies of the mode. NA is Avogadro's constant, c the speed of light, and o the vacuum permittivity. The ratio (vi/ i) allows for incorporation of anharmonic effects if desired, but is generally taken as unity. If the summed expressions are reported in the usual units, the intensity can be expressed in km mol-1 with the leading coefficient as below.

Somasundram et al.6 have derived an analogous expression for Raman intensities, based upon derivatives of the isotropic, a, and anisotropic, , polarizabilities, with respect to the normal coordinate Q:

As in the preceding equations, the symbols A and S will be used to refer to infrared and Raman intensities, respectively, throughout this text. Nuclear quadrupole coupling constants (NQCCs) constitute another window into the fundamental attributes of the hydrogen bonding phenomenon. Bacskay et al.7 decompose the NQCC, , at a nucleus A of a H-bonded complex in terms of this same property in the unperturbed molecule, o, and its shift due to complexation, .

represents the instantaneous angle between the molecule HA and the inertial axis. The brackets indicate librational motion averaging. For small angles, can be approximated as

The NQCC can be related to the electric field gradient through the nuclear quadrupole moment at atom A, eQ(A), and the vibrationally averaged component of the electric field gradient vector

where the inertial axis is defined as the z-axis. One then can write an expression for the latter vibrationally averaged quantity as

in terms of the electric field gradient at A in free HA and its change due to formation of the complex. There are two vibrational modes that are of particular interest in H-bonding studies. The first is the stretching vibration within the proton donor molecule, involving chiefly the bridging hydrogen. This mode is commonly referred to as v . Over the years, it has been learned that the shift which this frequency undergoes when the H-bond is formed, relative to the uncomplexed A—H molecule, correlates very well with the strength of the H-bond. This correlation is typically referred to as the Badger-Bauer rule 8-11 .

Vibrational Spectra

141

The other mode of particular interest is the stretching between the two subunits. This stretch, denoted v , can be contaminated by minor amounts of internal distortions but is of much lower frequency. Other intermolecular vibrations involve combinations of stretches, wags, and bends; they are referred to by various nomenclature systems.

3.2 Accuracy Considerations Before directly considering the perturbations of each subunit that occur upon formation of a H-bond, it would be worthwhile to first establish the sorts of accuracy that one may expect in calculating vibrational spectra of individual molecules. A number of popular basis sets, ranging from minimal to extended, were examined for this purpose12. Frequencies computed for the HF stretch and for the three internal vibrations of H2O are listed in Table 3.1, along with experimental estimates in the final row. It is immediatly clear that SCF-level calculations overestimate the frequencies whether they are stretches or bends. This is not surprising as it is known that correlation is required to obtain better agreement with experiment. The largest sets seem to be converging to a limit of accuracy. However, this limit does not require very large basis sets, as comparable results can be achieved even with double- polarized sets. Polarization functions seern to be required for good accuracy, even if only a single set of such functions. With respect to the smaller sets, the minimal basis set provides particularly large exaggerations of the frequencies. 3-21G seems to do surprisingly well for a set of this size, at least for this pair of molecules. The MP2 and CISD correlated methods reduce the frequencies a good deal, lowering the overestimation to only a few percent. MP2 seems superior to CISD.

Table 3.1 Computed frequencies (cm - 1 ) of vibrational bands in HF and H2O. Data are at SCF level unless otherwise indicated12. H2O Basis set

HF

v1

STO-3G 3-21G DZ TZ 6-31G(d) MP2

4475 4061 4228 4224 4358 3941

4140 3812 4028 3990 4070 3748

v2

2170 1799 1711 1723 1827 1682

v3

4391 3946 4204 4156 4189 3894

CISD

4000

3800

1712

3920

DZP 6-31+G(d)

4490 4314

4152 4071

1750 1797

4267 4190

TZP 6-31 + +G(d,p)

4454 4492

4114 4143

1748 1726

4226 4245

TZ2P

4457

4116

1751

6-31l + +G(3d,3p) expt

4486 3962

4129 3657

1757 1595

4221 4229 3756

142

Hydrogen Bonding

Table 3.2 Computed intensities (D 2. amu -1 • A - 2 ) of vibrational bands in HF and H2O. Data are at SCF level unless otherwise indicated12. H2O Basis set

HF

STO-3G 3-2 1G DZ TZ 6-31G(d) MP2 CISD DZP 6-31+G(d) TZP 6-31 + +G(d,p) TZ2P 6-311 + +G(3d,3p) expt

0.90 0.78 2.28 1.95 3.35 3.19 2.84 4.11 4.28 3.68 4.54 3.90 3.74

v 1

1.05 0.001 0.08 0.02 0.43 0.28 0.21 0.52 0.55 0.42 0.60 0.35 0.35 0.06

V 2

0.17 1.89 3.21 3.11 2.54 2.52 2.44 2.40 2.82 2.32 2.03 2.28 2.22 1.2-1.6

V 3

0.71 0.22 1.54 1.26 1.38 1.68 1.27 1.81 2.06 1.57 2.09 2.14 2.10 1.0-1.4

Table 3.2 contains analogous information concerning the intensities of the various vibrational modes. Comparison with the data in the last row illustrates the difficulty in computing experimental intensities to a high degree of accuracy. As in the case of the frequencies, it appears that the DZP basis set seems to perform about as well as any of the much more extended sets in a number of instances. Deletion of the polarization functions leads to erratic fluctuations in the intensities, particularly the V1 symmetric stretching frequency of water. Minimal and 3-21G are especially bad and should be avoided. Correlation does not influence the intensities in an obvious predictable manner. A different study 13 provides some further information concerning the effects of electron correlation upon calculated intensities. In addition to SCF and MPn type methods, these authors considered also the coupled-cluster model which is an infinite-order generalization of MP. Coupled-cluster, limited to single and double excitations (CCSD), was considered, as well as CCSD + T which includes the effects of connected triple excitations. The doubleharmonic approximation was made, consistent with most calculations of H-bonded systems. The [5s3pld/3slp] basis set is of the polarized double- variety, within reach of most workers. Results computed for the HF stretch and for the three internal vibrations of H2O are listed in Table 3.3, along with experimental estimates of the intensities in the final row. Comparison with the first row indicates that SCF intensities seem to be consistently too high, sometimes by a factor of less than two, but by an order of magnitude in the case of the symmetric stretch of water. Inclusion of correlation immediately lowers all intensities. MP2 appears satisfactory for all modes, with the exception of v1 for H2O. Higher levels of correlation beyond MP2 introduce smaller reductions, in some cases even lower intensities than experimentally observed. Comparison of various modes in additional molecules led to the conclusion that the calculations of stretches of terminal A—H bonds are particularly demanding. With regard to basis set requirements, there seems to be little benefit in enlarging the set beyond the DZP

Vibrational Spectra

143

Table 3.3 Computed intensities (km/mol) of vibrational bands in HF and H2O. All data calculated with [5s3pld/3slp] basis set13. H2O Method

SCF MP2 MP4 CCSD CCSD+T expt

HF

V1

v2

v3

168.8 116.7 94.2 99.4 93.3 100-102

19.2 8.9 4.8 5.8 4.6 2.2

84.6 64.9 61.7 64.1 62.0 54-64

57.7 45.7 30.4 31.9 29.2 40-48

level. On the other hand, it is essential that every atom, hydrogen included, contain at least one set of polarization functions. The intensities are quite insensitive to the values chosen for the exponents of these polarization functions. Similar sorts of conclusions apply to the frequencies. A systematic study14 found that a DZP basis set yields vibrational frequencies within about 9% of experimental (harmonic) values. The discrepancy diminishes to 4% when correlation is included via CISD and to 2% with a coupled cluster treatment. Another set of calculations15 confirmed the cost-effectiveness of the MP2 treatment of vibrational frequencies, indicating better agreement with experiment than MP3 on some occasions. Certain types of modes can be more sensitive to the level of theoretical treatment than others. For example, out-of-plane bending motions for -bonded systems can require triple- plus two sets of polarization functions, as well as a set of f-functions in the basis set16. In general, the results just reported, as well as other work,17-19, indicate that infrared intensities are much more sensitive to the level of theory than are the vibrational frequencies. In certain cases, correlation-induced changes can be small, but this is not a general rule. Without polarization functions, the vibrational spectra are apt to contain errors so large as to render the data potentially useless.

3.3 (HX)2

As a first example, let us consider the simplest possible H-bond between a pair of diatomics, as in (HF)2. Prior to formation of the complex, each molecule has a single vibrational mode, with a particular frequency and intensity. Table 3.4 collects data calculated over the years at various levels, for the harmonic frequencies in the monomer and the dimer20-27. The upper part of the table illustrates the expected overestimate of the vibrational frequency by SCF calculations. The correlated frequencies for the monomer in the first column of data better match the experimental quantity in the last row of the table, although there still remains some scatter from one method to the next. It does appear that a correlated treatment such as MP2, in conjunction with a large flexible basis set, can successfully reproduce the harmonic aspects of the experimental spectrum. Of greater interest than the frequency of the monomer is the change this property undergoes as a result of formation of a H-bond. The next two columns of Table 3.4 indicate that the frequencies of both the proton donor and acceptor groups shift toward the red. One

144

Hydrogen Bonding

Table 3.4 Vibrational frequencies (cm - 1 ) of HF in the monomer and changes undergone in the dimer. v, dimer v, monomer

Method

acceptor

donor

Reference

SCF DZ 6-31G** 6-31+G* DZP +VPs(2d)s

4103 4495 4314 4440 4488

CI/DZ CI/DZP SCEP/TZP MP2/6-31+G* MP2/6-311 + +G(2d,2p) CCSD(T)/TZ2P(f,d) expt

-33 -41 -35 -43 -40

-82 -89 -72 -91 -94

[20] [211 [22| [201 [23]

3732 4150 3994 3941 4170 4157

-10 -47 -18 -36 -43 -38

-40 -105 -74 -89 -116 -107

[20] [20] [24] [25] [25] [26]

4139a

-31

-94

[27]

correlated

a

Harmonic frequency

may take this frequency drop as an indicator that each internal H—F covalent bond is weakened somewhat by the complexation. There is a good deal of scatter amongst the calculated data. In fact, in some respects, the SCF frequency shifts seem superior to the correlated data, at least in the aggregate. For example, the correlated donor shifts vary in magnitude between 40 and 116 c m - 1 , whereas the SCF values are in the narrower range of 72-94 c m - 1 . Because of their sensitivity to small redistributions of electron density, the computation of the intensities of vibrational modes has proven to be more demanding than the frequencies. Table 3.5 reports calculations of the intensities at the SCF and correlated levels. The intensity of the internal vibration of the proton-acceptor molecule is changed only little by the perturbation, but that of the donor undergoes a large increase by a factor of three or four. The latter intensification is characteristic of H-bonds and will be seen repeatedly. When two diatomics such as HF are combined, there are new intermolecular vibrational modes generated. These modes can sometimes be easily recognized as a particular sort of

Table 3.5 Intensities of vibrational modes (km mol - 1 ) of HF in the monomer and dimer. Dimer Method

Monomer

6-31G** +VPs(2d)s

132 168

TZ2P(f,d)

103

acceptor

SCF 182 167 CCSD(T) 120

donor

Reference

376 459

[21] [23]

427

[26]

Vibrational Spectra

145

stretch or bend, but in many cases are instead a complex mixture of various types of distortion. In the case of (HF)2, there are three vibrations occurring within the plane of the dimer and one out-of-plane distortion. In an early work, Curtiss and Pople28 carried out a detailed analysis of the composition of these normal modes. The results were computed with only a 4-31G basis set and at the SCF level but are illustrative of the motions nonetheless. The intermolecular mode of frequency 226 c m - 1 in Table 3.6 is composed largely of the stretch between the two F atoms, and can be designated as v . But the contamination by bending of the two HF molecules is not negligible. There is also a good deal of this H-bond stretching motion in the lowest-frequency mode, which represents primarily the bending by the proton acceptor molecule. The mode at 588 cm-1 can be identified with bending of the proton donor molecule, but not without appreciable bending of the acceptor as well. The fourth mode is of different symmetry, and is a pure torsional motion, that is, an out-of-plane (oop) bend. The reader should thus be alerted that designations of certain vibrational motions can be misleading since they are generally a combination of various pure motions. How do these notions about intermolecular modes vary with larger basis sets and with correlation effects? The frequencies listed in Table 3.7 indicate that the fundamental natures of these modes change little. The intermolecular modes have frequencies in the range between about 120 and 600 cm - 1 , and as such are clearly separated from the much higherfrequency intramolecular vibrations. The highest-energy distortion mode is an in-plane bend. Analysis of this mode shows it to be composed largely of a wagging of the proton donor molecule, distorting the linearity of the H-bond. The torsional or out-of-plane motion is perhaps 100 cm-1 smaller, followed by the other two in-plane distortions. That mode which seems to most closely resemble an intermolecular stretch appears at around 200cm - 1 . The intensities reported in Table 3.8 appear to be quite sensitive to particular features of the basis set or to inclusion of electron correlation. For example, the intensity of the lowest-frequency mode changes by a factor of 25 upon going from 6-31G** to the larger +VP s (2d) s . Some of this difference is due to the change in the nature of the motion itself since the two basis sets are associated with different potential energy surfaces. It might be stressed finally that the particular nomenclature, that is, assignment, of certain bands can be nettlesome. For example, both the V and in-plane bending modes of (HF)2 contain elements of F—F stretch and bends. Table 3.6 Composition of intermolecular vibrational modes in (HF)228.

v, cm-1

171 226 588 519

-2.6( R) + 0.3(r d -3.2 ( R) - 0.3 (rd -0.1 ( R) - 1.2(r d 0.06(R )

d)

- 1.0 (ra a) + 0.6 (r a) d) - 0.4 (ra a) d)

146

Hydrogen Bonding

Table 3.7 Intermolecular vibrational frequencies ( c m - 1 ) of HF dimer. ip Bend

DZ 6-31G** 6-31 + G* DZP +VP s (2d) s

519 601 551 529 522

CI/DZ CI/DZP SCEP/TZP MP2/6-31+G* MP2/6-311 + +G(2d,2p) CCSD(T)/TZ2P(f,d)

523 607 420 607 582 567

v

ip Bend

Reference

189 230 221 193 206

165 127 154 143 142

[20] [21] [22] [20] [23]

196 218 167 243 231 157

166 156

[20] [20] [24] [25] [25] [26]

oop Bend SCF 475 455

463 442 433 correlated 459 486 — 497 516 458

127 174

163 210

Bunker et al.29,30 have attempted to calculate anharmonicities and tunneling splittings for the HF dimer. Starting with a large number of single point energies on the potential energy surface, the authors fit an empirical analytic expression with 39 adjustable parameters. Their calculated anharmonic frequencies for V1 and V2 are 3926 and 3874 cm - 1 , respectively, which compare rather well with the experimental values of 3931 and 386831. Some of the vibrational frequencies of the analogous HC1 dimer have been computed by Karpfen et al.32 using a correlated scheme and rather large basis sets. The data in Table 3.9 indicate trends similar to (HF)2, albeit of lesser magnitude. That is, the stretches of both the acceptor and donor are red-shifted, the latter by a greater amount. The intensity of the donor stretching mode is considerably greater than in the case of the acceptor molecule. The corresponding data for the intermolecular modes are reported in Table 3.10. Comparison with data in Tables 3.7 and 3.8 for the HF dimer reveal uniformly smaller frequencies for all modes. The H-bond stretching motion is the lowest in frequency for (HC1)2, unlike (HF)2 where one of the bends is of lower frequency. v is quite small, less than 70 c m - 1 . In addition, this band has a very weak intensity of only 0.1 km/mol, several orders of magnitude weaker than in (HF)2. Indeed, all of the intermolecular bands in (HC1)2 are quite weak, with integrated intensities of less than 50 km/mol. As a point of interest, Karpfen et al.32 have also computed the vibrational frequencies of the cyclic geometry, which represents not a minimum on the PES, but rather a transition state, and hence the imaginary frequency listed in Table 3.11 for the bending motion that would transform this cyclic geometry into one of the minima. It might be noted that the asymmetric and symmetric internal HF stretching modes in the cyclic dimer are of very simTable 3.8 Intensities of intermolecular modes (km m o l - 1 ) of (HF)2.

SCF/6-31G** SCF/+VPs(2d)s CCSD(T)/TZ2P(f,d)

ip Bend

oop Bend

167 219 160

279 230 188

v 111

160 25

ip Bend

Reference

154 6 141

[21] [23] [26]

Vibrational Spectra

147

Table 3.9 Vibrational frequencies (cm - 1 ) and intensities (km/mol) of HC1 in the monomer and dimer32. v, dimer Method ACPF/[652/42] ACPF/[6531/42]

A, dimer

v, monomer

acceptor

donor

acceptor

donor

3005 2993

-11 -6

-32 -30

39 44

165 188

ilar frequency. It is also of interest that the various intermolecular modes are of lower frequency than is the case for the minimum energy configurations in Table 3.10. In the absence of spectral information in the gas phase, it is common to compare calculated features of the vibrational spectrum to data measured in rare gas matrix, the premise being that the latter medium perturbs the H-bonded system as little as possible. The influence of the medium was considered33 via a self-consistent reaction-field formalism wherein inductive interactions between the polar system and the polarizable medium are incorporated into a model Hamiltonian34. The calculations made use of the 6-31G** basis set at the SCF level. The immersion of (HF)2 into a simulated Ar or Xe matrix led to a more linear configuration, lowering a in Fig. 3.1 from 13.6° to 8°. The proton acceptor molecule also rotated toward a less perpendicular arrangement, with increasing from 103° to 110°. There were only very small changes observed in the internal bond lengths, although a small stretch of the interfluorine distance of the order of 0.01 A occurred. The net result is a 10% increase in the dipole moment of the (HF)2 complex. Ar matrix has little effect upon the gas-phase properties of (HC1)2, whereas immersion in Xe yields results much like those for (HF)2. The changes calculated to occur in the stretching frequencies of the proton donor and acceptor HX molecules are reported in Table 3.12, along with experimental values in parentheses. It seems clear that this theoretical approach is a disappointment as it strongly underestimates the effects of placing the complex within the matrix. As the authors point out, this effort represents merely a first step toward an improved treatment of matrix effects. While their formalism includes purely inductive effects, it neglects repulsive and dispersion forces which are likely important as well. Another direction toward improvement might be permitting some geometrical relaxation of the matrix and complex, which would allow coupling of vibrational and rotational motions.

Table 3.10 Intermolecular vibrational frequencies (cm - 1 ) and intensities (km/mol) of (HC1)232. ACPF/[652/42] Mode bend torsion bend stretch v

ACPF/[6531/42]

V

A

v

A

290 195 98 63

15 25 43 0.1

297 204 129 66

13 19 31 0.1

148

Hydrogen Bonding

Table 3.11 Intermolecular vibrational frequencies (cm - 1 ) and intensities (km/mol) of the cyclic geometry of (HC1)2, which represents a transition state on the surface32. ACPF/[652/42] Mode

Asymmetric stretch Symmetric stretch Bend Torsion Stretch v Bend

v

ACPF/[6531/42] A

Intramolecular 2989 76 2986 0 Intermolecular 250 0 157 49 60 0 75i

V

A

2982 2979

87 0

252

0 46 0

160 60

Hi

3.4 H3Z-HX The linearity of the H-bond in a complex like H 3 N ... HF simplifies the analysis of the vibrational frequencies in some ways. The vibrational data for this complex are reported in Table 3.13 from data computed at the SCF level with a 6-31G** basis set21. One may first note that the red shift of the proton donor HF molecule is more than 400 c m - 1 , about five times greater than in (HF)2. This larger shift is concordant with the stronger H-bond interaction in H3N...HF. The intensification of this band is also more profound in the latter system. The ratio of intensity in the complex versus that in the HF monomer is 7.1 in H3N...HF, as compared to 2.8 in (HF)2, with the same theoretical framework. With regard to the proton acceptor molecule, the vibrational frequencies of NH3 are largely unaffected by the complexation. The exception is the symmetric bending motion which is blue-shifted by nearly 100 c m - 1 . This change is in contrast to the HF proton donor, whose stretching frequency is red-shifted. In (HF)2, the intensity of the vibration of the proton acceptor molecule was not changed much in comparison to the monomer. When NH3 acts as proton acceptor, the result is quite different. Both stretching modes undergo strong intensification, by factors of 16 and 35. The bending modes keep their intensities largely intact upon complexation. One might expect the principal result of the replacement of NH3 by PH3 to be an overall weakening of the trends due to the latter's weaker basicity. This is indeed found to be the case for a number of the spectroscopic properties. Comparison of the first rows of Tables 3.14 and 3.13 shows a marked drop in the red shift of v , as well as its intensification which drops from 7 to 4. Curious, though, are the changes within the base molecule. Whereas the stretching frequencies of NH3 are essentially unaffected by complexation, both symmetric and asymmetric stretching modes of PH3 shift to higher frequency by some 26-30 c m - 1 . Instead of a strong blue shift, as in H3N...HF, the symmetric bend of PH3 shifts slightly to the red, while its asymmetric bend moves to higher frequency by 94 cm - 1 . Also obeying different trends are the intensities of these modes. In contrast to the very strong intensification of the stretching vibrations in H 3 N ... HF, the internal stretches of PH3 are made less intense by the complexation.

Figure 3.1 Definition of and

angles in (HX)2 2

Table 3.12 Changes in stretching vibrational frequencies (cm ') induced by placement of complex in Ar or Xe matrix. Calculated with 6-31G** basis set33. Experimental values in parentheses. Acceptor

(HF)2 (HC1)2

Donor

Ar

Xe

Ar

Xe

16(35) 1(24)

15 4(40)

11 (42) 2 (39)

9 2(40)

Table 3.13 Frequencies (in c m - 1 ) and intensities (km m o l - 1 ) of HF and NH3, and the changes resulting from formation of the H 3 N ... HF complex. Data, calculated at SCF level with 6-31G** basis set21. Mode

Vmon

v

Amon Adim/Amon

HF

4495

2432

3706 3844

132

7.1

NH3

V bend (a 1 )

1141

0 -2 93

Vbcnd(e)

1812

-7

v str (a 1 ) vstr(e)

0.2 0.8 217 21

35 16 1.0 1.2

Table 3.14 Frequencies (in c m - 1 ) and intensities (km mol - 1 ) of HF and PH3, and the changes resulting from formation of the H 3 P ... HF complex. Data, calculated at SCF level with 6-31G** basis set21. Mode

V mon

v

A A

mon

Adim/Amon

A

HF

4495

-134

132

4.0

2596 2600 1112 1250

26 30 -13 94

106 128 36 21

0.4 0.7 1.4 1.0

PH3 Vstr(a1)

vstr(c) Vbend(a1)

vbend(e)

150

Hydrogen Bonding

3.4.1 Analysis of Intensities Because the intensity of a given vibrational mode is connected with the changing molecular dipole moment associated with that particular motion of the atoms, analysis of these intensities offers valuable insights into charge redistributions within the system. One can partition the dipole changes into contributions from various atoms using an "atomic polar tensor" (APT) formalism35-36 which is defined for an atom a as

where u^ is the ith (x, y, or z) component of the dipole moment vector of the entire system, and qj represents the jth coordinate of atom a. It is generally most useful to define a local coordinate system for each atom such that the z-axis corresponds to a bond. Also useful are characteristics of the tensor which are invariant to choice of coordinate system37. An effective charge x is related to all nine elements of the APT, defined as the square root of 1/3 the sum of squares of the elements. The diagonal elements are stressed in the mean dipole derivative, p, equal to 1/3 the trace of the matrix. Anisotropy is defined in terms of these two quantities, as is the deformability of the electron cloud. This type of analysis provides useful clues as to the behavior of the intensities in Tables 3.13 and 3. 14 as follows. Of particular relevance is the Pzz element for each hydrogen atom which describes how much a stretch of the H atom along its bond axis will affect the molecular dipole moment in that same direction. The sharp increase in intensity of the HF stretch upon H-bonding can thus be associated with a bigger increase in that accompanies motion of the proton in the dimer, as compared to the monomer. More intriguing than the HF stretch is the qualitatively different behavior of the NH3 and PH3 internal vibrational modes. Indeed, the intensification of the NH3 stretching modes are greater in a relative sense than the HF stretch when the H 3 N ... HF complex is formed. Again, the behavior observed for the intensities is mirrored by the Pzz element of the APT. Further insight is provided by separating Pzz into two components. One can evaluate the change in molecular dipole moment that would result from the simple motion of the hydrogen nucleus, along with its static partial charge. Subtraction of this quantity from PZZ is a measure of how much the dipole moment is affected by redistribution of electron density, that is, how much the partial charge of the hydrogen is changing as it is moved. This type of analysis revealed that the hydrogens of NH3 in the monomer suffer a loss of positive charge as they move away from the N center, that is, they pick up additional electron density. Hence, the stretch away from the N engenders two competing effects. Motion of a H atom with a partial positive charge acts to increase the dipole moment, but this is counteracted by the shift of excess electron density in the same direction. This cancellation is responsible for the small intensities of the N — H stretches in the monomer noted in Table 3. 13. The situation changes when the ammonia is bound to HF. Formation of the H-bond transfers some electron density from NH3 to HF, causing an increase in the positive charge of the three hydrogens of NH3. The larger positive charge causes a greater change in the molecular dipole moment when the proton moves. Another effect of the presence of the HF is to "lock up" the electron density of NH3, that is, it makes it more difficult for the density to follow the ammonia hydrogens as they move away from N. This less flexible density lowers the counteracting influence noted in the free ammonia, adding to the value of z/ z. What accounts for the very different behavior in PH3? First, the P— H stretches are already rather intense, even in the isolated monomer. This difference can be traced to the

Vibrational Spectra

151

charges of the H atoms, which are slightly negative compared to their positive charge in NH3. The different charges of hydrogen in the two molecules can be associated with the lesser electronegativity of the P atom versus N. The negatively charged hydrogens lead to like changes for the Pzz APT elements since motion of the hydrogen away from the P atom will diminish the molecular dipole. As in NH3, the charge flux again acts to augment the electron density around the shifting hydrogen. Instead of making the hydrogen less positive, as in NH3, this atom now becomes more negatively charged as it moves away from P. So rather than counterbalancing one another, which made the intensities of the NH3 stretches so weak, the charge flux and partial charge of the H atoms in PH3 reinforce each other, leading to the strong P—H stretches in the monomer. Binding to the HF molecule again draws electron density away from the proton acceptor molecule. But as opposed to NH3 where the hydrogens were made more positive, this removal decreases the negative charges of the hydrogens of PH3. The stretching motion of the latter hydrogens hence yields a smaller decrement of the molecular dipole than in the isolated monomer, and the result is a weaker intensity. Extending the same partitioning to the HF stretches helps us understand the intensification of vs. The charge on the H atom is positive in the monomer, and becomes slightly more positive in the complex. But more important is the charge flux term. Upon formation of the H-bond, the electron density can less effectively follow the proton as it moves away from the F atom. The stretch therefore increases the hydrogen's positive charge and causes a large increment in the molecular dipole. This finding is consistent with other calculations wherein motion of the bridging hydrogen induces a flow of electron density in the opposite direclion38,39. The data calculated for the intermolecular modes of H 3 Z . . . F are reported in Table 3.15. The 876 cm-1 frequency of the bending of the proton donor molecule in H 3 N ... HF is clearly higher than any of those calculated for (HF)2, another indication of the stronger H-bond. This motion corresponds primarily to imparting nonlinearity into the H-bond by pulling the proton off of the axis. The wagging motion of the acceptor requires less energy, partially explaining its lower frequency. The H-bond stretching frequency, v , is similar in magnitude to the acceptor bend. v is a relatively pure mode in the H3Z...HF complexes, unlike other cases where the intermolecular stretch is combined with substantial contributions from other stretches and bends. The patterns of intensities can be understood on the basis of how much a given vibrational motion changes the molecular dipole moment. The donor bend, for example, acquires its high intensity because it turns the HF molecule with its large molecular moment in such a way as to give the C3v complex a nonzero moment perpendicular to the H-bond axis. In contrast, the stretch of the two subunits away from each other in the v mode does little to

Table 3.15 Vibrational spectra calculated for intermolecular modes21 at SCF/6-31G** level. Frequency ( c m - 1 ) Mode H-bond stretch (v ) donor bend acceptor bend

Intensity (km mol - 1 )

H 3 N ... HF

H 3 P ... HF

H 3 N ... HF

H3P...HF

240 876 237

113 467 112

3 208 10

I 134 6

I 52

Hydrogen Bonding

change either the magnitude or direction of the complex's moment. Analysis carried out by Kurnig et al.21 indicated that of the total intensity of this mode in H3N...HF, 15% arises from small mixing in of an internal HF stretch and rocking and internal bending of NH3. The remaining 85% arises from the interactions between the two molecules, for example, the changing amount of charge transfer as the two subunits are pulled apart. Indeed, low intensities are expected in any H-bonded complex for v which is constituted primarily of the intermolecular stretch. What lends intensity to this band in a complex like (HF)2 is the contribution of bending motions that change the dipole moment vector direction, and internal HF stretches that alter the subunit's moment. For example, Swanton et al. had determined in their analysis of the water dimer40 that 70% of the v intensity can be attributed to internal motions within the two water subunits.

3.4.2 Anharmonicity Bouteiller et al. have devised a means of treating the anharmonicities of H-bonded complexes41 by expanding the potential energy function as a fourth-order polynomial, fit to the ab initio calculations. These investigators use two degrees of freedom, corresponding to the X—H distance, r, and the intermolecular separation between X and Z, R.

The variational method is next used to solve the Schrodinger equation of vibrational motion, taking eigenfunctions of the harmonic oscillator as expansion functions. A group of selected expansion coefficients are listed in Table 3.16 to illustrate the degree of coupling and anharmonicity. With regard first to the second-order coefficients in the first three rows, the much larger values of a20 reflect the greater sensitivity of the energy to small changes in r(H—X) than to the intermolecular distance. But it is significant that a11 is comparable in magnitude to a02, reflecting the coupling between r and R. The large value of a30 is another source of anharmonicity as is the fourth-order term represented by a40. The frequencies that arise from this treatment are listed for the three complexes in Table 3.17. Note first that the anharmonic frequencies of vs (XH in Table 3.17) are quite a bit smaller than the harmonic values in the upper part of the table, differing by about 400 cm - 1 .

Table 3.16 Expansion coefficients calculated for H-bonded complexes at SCF level (CI values in parentheses)41. All values in units of mdyn/A p+q+1 . apq a20 a11 a02

a30 a21

a12 a03 a40

C1H..NH 3 2.382(1.938) 0.197(0.213) 0.106(0.132) -5.572 (-4.503) 0.379(0.510) -0.459 (0.004) - 0.08 (-0.240) 4.879 (3.835)

C1H ..NH2CH3

BrH .. NR 3

2.239 0.190 0.109 -5.309 0.805 -0.193 -0.160 4.917

1.816 0.144 0.078 -3.765 0.566 -0.165 -0.106 3.098

Vibrational Spectra

153

Table 3.17 Vibrational transitions calculated41 at SCF level, except those in parentheses derived at CI level. All values in units of c m - 1 . CIH NH2CH3

CIH NH3

BrH NH3

Harmonic approximation

XH X..N

2888 (2606)

XH X..N

2501 (2249) 162 (201)

172(192)

2800 146 Anharmonic data 2365 144

2516 135 2084 144

Nonetheless, there is a strong parallel between these vs frequencies and the a20 coefficients in Table 3.16, with either the harmonic or anharmonic treatments. There is little such parallel in the v stretches (X .. N) and the a02 coefficients, providing a warning against using simple force constants to estimate the frequency of this intermolecular mode. It is interesting, however, that the harmonic and anharmonic values of the X..N stretch are quite similar. With regard to various levels of theory, it is significant that inclusion of correlation (values in parentheses for C1H..NH3) lowers the vs frequency but has the opposite effect of increasing v . It is also worth noting that the SCF and CI frequencies can differ by as much as 300 c m - 1 , harmonic or anharmonic. Indeed, anharmonicity and electron correlation effects are additive here: the anharmonic CI vs frequency is smaller than the SCF harmonic value by a full 639 c m - 1 . A later work by Bouteiller et al.42 considered the set of complexes of NH3 with FH, FD, and FLi. Comparison of the results of the first two illustrates isotope effects while substitution with lithium introduces a different sort of bonding. The Vibrational transition energies calculated by the above anharmonic approach, extracted from energy surfaces at the SCF and MP2 levels, are listed in Table 3.18. The correlated F—H and F—D stretching frequencies are considerably smaller than the SCF values, whereas a small increase results when correlation is included for FLi. Similar discrepant behavior in terms of correlation effects has been noted in geometries and energetics. There does not seem to be much of an isotope effect on the F..N stretch, as results for H and D are nearly identical. Correlation exerts a still significant effect here, raising the frequencies by perhaps 10%, but again the effect is opposite for FLi. The values in parentheses refer to the frequencies computed when all coefficients apq are ignored when p + q > 2, again at the correlated level. One can see that neglect of anharmonicity yields F—L frequencies that are too high, for all F—L including FLi. The effects are much smaller on the F..N stretches.

Table 3.18 Vibrational transitions calculated42 in units of c m - 1 . Harmonic values in parentheses. ..

..

F-L F..N

..

FD NH3

FH NH3

SCF

MP2

SCF

3773 240

3331(3514) 263(265)

2752 236

MP2

2427(2527) 255(262)

FLi NH3

SCF

MP2

932 292

947(1010) 281(272)

154

Hydrogen Bonding

The Bouteiller approach to anharmonicity also permits extraction of energies of excitation to vibrational levels beyond the first excitation. The various progressions are reported in Table 3.19, arising from the correlated potentials. Proceeding down each column, the spacing between successive overtones decreases as the quantum number rises, resulting from the mechanical anharmonicity. 3.4.3 Other Properties As mentioned earlier, there are additional properties of H-bonded systems accessible to calculations of this sort. For example, Bacskay et al.7 have computed the vibrationally averaged component of the electric field gradient (EFG) tensor along the symmetry axis of C1H...NH3 and C1H...PH3. Table 3.20 reports the change in this component, at the Cl nucleus, caused by formation of the H-bond, as Vzz(Cl), as well as the change in the molecular dipole moment, also along the symmetry axis. Also listed in the last column is the root mean square angle between HC1 and the symmetry axis, after vibrational averaging. The calculations used a basis set of moderate size, with polarization functions for all atoms. The first column of Table 3.20 indicates threefold more change in the EFG at the Cl nucleus for the stronger complex with NH3 than with PH3. As a percentage of the EFG in the uncomplexed HC1 monomer, the next column indicates a 17% change for C1H...NH3, in reasonable agreement with an experimental measurement of 23%. The 5.5% change calculated for C1H...PH3 is also lower than the experimental estimate of 7.4%. The dipole moment of the complex changes much more for the stronger H-bond, too. The last column illustrates the "floppier" character of the complex for a weaker H-bond. The root mean square angle made by the HC1 molecule with the symmetry axis, resulting from vibrational averaging is some 13° for C1H...NH3 but 19° for C1H...PH3. The authors noted that the basis set superposition error was negligible for their EFG and other electronic properties. Bacskay et al.7 also concerned themselves with the possible effects of electron correlation on the aforementioned electronic properties. They consequently compared their SCF data with results obtained using an approximate coupled pair functional (ACPF) approach to correlation. As the authors had noted that geometries optimized at the SCF level typically

Table 3.19 Overtones and combination bands calculated with anharmonicity at the correlated level. Data in cm-1. 42 vn v'n' 00 00

01

00

03

00 00

04 05

00

10

02

00 00

11

00 00

13 14 15

00

12

FH NH3

FD..NH3

FLi..NH3

263 509 739

255 494 717 931

281 557 829

1096

1141 2427 2735

1272

958 1172 3331 3644 3921 4180 4425 4658

2995 3239 3470 3691

Note. v and n refer, respectively, to the F-L and F-N stretching modes.

947

1551 1822

Vibrational Spectra

155

Table 3.20 Changes of electric field gradient and dipole moment caused by H-bonding (atomic units) and rms HC1 librational angle (degs) calculated with [642/531/31] basis set for [C1,P/N/H]7. - Vzz(Cl)

- Vz//Vzz°

0.618 0.196

0,17 0.06

..

C1H NH3 C1H..PH3

-

2 1/2

<

z

0.412 0.250

>

12.7 18.8

had the two subunits too far apart, they evaluated their properties at the experimental geometries in Table 3.21. Comparison of Tables 3.20 and 3.21 indicates a significant increase in the magnitude of Vzz results from bringing the two molecules closer together, as well as forcing each subunit into its experimental monomer geometry. For example, this property has increased from 0.62 to 0.92 atomic units when the experimental geometry of C1H...NH3 is adopted. Indeed, the sensitivity of this quantity to the intermolecular separation is embodied by the Vzz/ R term in Table 3.21. The 0.83 value for C1H...NH3 indicates that a stretch of only 0.1 A would alter the EFG at the Cl nucleus by as much as 0.15 atomic units. Comparison of the first two columns of data in Table 3.21 reveals that correlation reduces to some extent the sensitivity of the EFG to the H-bonding interaction by 7-17%. The values in parentheses are not much different than their preceding values, indicating very little basis set superposition influence upon this property. The dipole moment change resulting from the H-bond formation has surprisingly little sensitivity to inclusion of electron correlation. 3.4.4 Relationship between H-Bond Strength and Spectra An example of the Badger-Bauer relationship between the strength of the H-bond and the red shift of the X—H stretching frequency is provided by recent correlated computations of complexes pairing HC1 with a series of 4-substituted pyridines43. As illustrated by the solid line in Fig. 3.2, the change in this frequency is very nearly linearly related to the calculated strength of the H-bond. The dashed line refers to the intensity of this mode in the dimer, as compared to the HC1 monomer. This property, too, is strongly correlated with the strength of the H-bond. Note the magnitude of this enhancement, making the intensity in the complex some two orders of magnitude higher than in the monomer.

Table 3.21 Changes of electric field gradient and dipole moment caused by H-bonding (atomic units) calculated at experimental geometries with a [642/531/31] basis set for [C1,P/N/H]7. - Vzz (Cl)

..

C1H NH3 C1H- PH a

SCF

ACPF(+CC) a

Vzz/ R SCF

0.924 0.340

0.862(0.848) 0.285 (0.282)

0.83 0.30

With full counterpoise correction.

-

z

SCF

ACPF

0.486 0.304

0.499 0.279

156

Hydrogen Bonding

Figure 3.2 Illustration of the Badger-Bauer relationships for HCl...base where the base is a sesries of 4-substituted pyridines. Data43 calculated at MP2/6-31 +G(d,p) level. The dashed line represents the magnification of the intensity in the dimer, relative to the moment. E refers to the electronic contribution to the binding energy.

3.5 H2Y...HX

The complex combining water with HF furnishes a good example where the proton acceptor has two lone pairs. Somasundram et al.6 and Amos et al.44 have examined this complex with two different basis sets, one singly and the other doubly polarized. The results for both basis sets are reported in Table 3.22 and illustrate the expected strong red shift of the proton donor stretching frequency, albeit by a lesser amount than in the more strongly bound H3N...HF. The shift calculated at the SCF level of around 260 cm-1 underestimates the experimentally observed value of 353 c m - 1 , but the correlated shifts are much closer to experiment. The frequency changes in the proton acceptor are fairly consistent from one basis to the next: Both stretches (V1 and V3) are red shifted by some 7-10 c m - 1 whereas the bending motion is only slightly affected. This behavior contrasts with NH3 in its complex with HF where the stretching frequencies are altered less than the bends. The trends are more or less intact after correlation, but the red shift of the highest frequency mode is larger. Somasundram et al.6 computed the intensities not only for IR but also for Raman bands and the results are listed in Table 3.23. The intensification of the IR proton donor stretching band is by a factor of nearly 5 for H 2 O ... HF, a little smaller than the magnification of 7 in H3N...HF. Whereas the two stretching motions in H3N were strongly intensified by complexation with HF, these increases are much smaller in H2O. In both cases, the intensities

Vibrational Spectra

157

Table 3.22 Frequencies (in cm-1 )of HF and OH2, and the changes resulting from formation of the H2O ...HF complex6,44.

MP2

SCF TZ2P

DZP

Mode

mon

v

TZ2P

DZP

Vmon

Av

4471

-264

Vmon

V Vmon

vmon

V

HF

4511 4166 1752 4289

V1 V2 V3

-259

H2O -9 0 -10

4128 1760 4228

-7 5 -10

-340

4221

3913 1671 4059

2 -3 -18

4154 3858 1657 3980

-363 -10 -3 -20

of the bending motions are insensitive to the H-bond. The Raman bands undergo some minor modifications upon complexation but none larger than 1.6. The intermolecular modes are listed in Table 3.24 along with their IR and Raman intensities, all calculated with the doubly-polarized TZ2P basis set. The SCF H-bond stretching frequency of 220 cm-1 is surprisingly similar to the experimental measurement of 198 cm - 1 , considering the harmonic and other approximations involved in its calculation. This quantity is comparable to that in H3N HF, but the IR band is considerably more intense in HLjO—HF. This greater intensity is linked to the coupling into the mode of bending motions which permit a greater change of the dipole moment. Indeed, the weakest band in this part of the IR spectrum, corresponding to a bending motion, probably owes its low intensity to a certain degree of H-bond stretching motion. Of particularly low intensity are the Raman bands, all less than 2 A4/amu. This finding is not surprising as there are only small changes within the covalent bonds of the monomers that accompany these intermolecular motions. The authors expressed their belief that their IR and Raman intensities of the intermolecular vibrations are correct within an order of magnitude. It should be noted that correlation acts to increase the frequencies of all the intermolecular modes by amounts varying from 20 to 100cm -1 .

Table 3.23 IR and Raman intensities of HF and OH2, and the changes resulting from formation of the H2O...HF complex. Data calculated with TZ2P basis set6. IR (km m o l - 1 ) Mode

Amon_

Raman (A4/amu)

A A dim /A mon

Smon_

Sdim/Smon

HF

147

4.6

26

1.6

70 4 29

1.0 0.7 1.0

H2O v1 v2 v3

14 92 70

5.5 1.0 1.6

158

Hydrogen Bonding

Table 3.24 Vibrational spectra calculated for intermolecular modes of H2O...HF6,44 with TZ2P basis set. Mode H-bond stretch (v ) bend bend shear shear

v S C F (cm - 1 )

vMPZ(cm-1)

A IR (km mol - 1 )

220 182 234 644 786

270 232 252 742 862

87 155 3 226 194

SRamen

(A4/amu) 0.4 0.8 2 1 0.2

Latajka and Scheiner45 carried out a vibrational analysis of H2O...HC1 using several different basis sets. The results with their best basis, at the SCF level, are presented in Table 3.25 where the red shift of the vs band of HC1 equals 105 c m - 1 . Its intensity is magnified by a factor of 6.4 upon forming the complex. The frequencies of the proton-accepting water are little affected and intensity changes are only moderate, none increasing by more than a factor of two. These changes are all smaller here than in the more tightly bound H2O...HF. The vibrational data for the intermolecular modes are reported in Table 3.26. The H-bond stretching frequency, v , is only 118, comparable to the same quantity in the H 3 P ... HF complex. The other frequencies are also in the same range as H3P...HF. The v is of notably low intensity, as in the H 3 Z ... HF complexes, suggesting little mixing with the bending modes that would add intensity via changing the dipole moment. The other modes are of higher intensity, in the 30-80 km mol - 1 range. Hannachi et al.46 have carried out their calculations of the spectrum of the full series XH ... OH 2 , X=F,Cl,Br,I. The energies were computed with a core pseudopotential approach, specifically the PS-31G** basis set for the halogens, and standard 6-31G** for water. The first clear trend in Table 3.27 is a progressive decrease in both the vs and v frequencies as the halogen atom changes from F to Cl to Br to I. These trends are true for either the harmonic or anharmonic frequencies. While the harmonic and anharmonic values of vs are clearly different, the magnitude of this difference diminishes as one proceeds from F to I. The red shift of this band is enhanced by anharmonicity in all cases. Again, inclusion of anharmonicity effects do little to change the X..O stretching frequencies.

Table 3.25 Frequencies (in cm - 1 ) and intensities (km mol - 1 ) of HC1 and OH2, and the changes resulting from formation of the H2O...HC1 complex. Data calculated with + VPs(2d)s basis set45. Mode

V mon

V

A mon

Adim/Amon

HC1 3141 v1 v2 v3

4139 1759 4244

-105 H2O -4 1 -3

56

6.4

19 103 91

1.9 0.9 1.3

Vibrational Spectra

159

Table 3.26 Vibrational spectra calculated for intermolecular modes of H2O...HC145. Mode H-bond stretch (V ) donor bend donor bend acceptor bend acceptor bend

Frequency ( c m - 1 )

Intensity (km mol - 1 )

118 459 351 143 94

77 38 33 28

3

3.5.1 Alkyl Substituents The effects of methyl substitution upon the Vibrational spectra may be determined from comparison of the aforementioned results for H2O...HC1 in Table 3.26 with the data computed by Amos et al.44 for (CH3)2O...HC1 in Table 3.28. Bearing in mind the results were obtained with slightly different basis sets, it is nevertheless apparent that the H-bond stretching frequency is changed very little by the substitution, nor is the intensity of this band altered by much. There seems to be an increase in the frequencies for bending the proton donor, whereas the frequencies for bending the acceptor molecule are very small. This drop is due in some measure to the large increase in the effective mass for this motion when the two hydrogens of H2O are replaced by methyl groups. The red shift of the HC1 stretch, listed in the last row of Table 3.28, is considerably larger than in H2O...HC1. Nonetheless, this shift of 170 c m - 1 is only about half of the experimental quantity of 316 c m - 1 47. The same trends are observed in solid matrix. When the proton acceptor in the H2O...HC1 complex is changed to dimethyl (or diethyl) ether, the red shift of the HC1 stretch increases by several hundred cm-1 48. Similarly increased red shifts when the base is alkylated are noted for HF and HBr as proton donors. The sulfur analogs, namely, H2S, Me2S, and Et2S, obey similar patterns when paired with HF, HC1, and HBr48. 3.5.2 Other Properties As described above for complexes of HC1 with NH3 and PH3, Bacskay et al.7 have computed the vibrationally averaged component of the electric field gradient (EFG) tensor along

Table 3.27 Calculated vibrational transitions46. All values in c m - 1 . FH..OH2

C1H..OH2 Harmonic approximation 3035 150 155 Anharmonic data

BrH..OH2

IH..OH2

2656 129 120

2390 59 98

XH - vs ..

xo

4243 321 235

XH - vs

4019 370

2847 257

2501 209

2307 76

X..O

226

146

121

97

160

Hydrogen Bonding

Table 3.28 Vibrational spectra calculated for intermolecular modes of (CH3)2O...HC144 with DZP basis set at SCF level. mode H-bond stretch (v ) donor bend donor bend acceptor bend acceptor bend HO shift ( vs)

frequency, cm-1

intensity, km mol-1

107 507 399 10 35 -170

3 37 51 4 0

the inertial axis of C1H...OH2 and C1H...SH2. Table 3.29 reports the change induced in this tensor by formation of the H-bond, Vzz(Cl), and the accompanying change in the molecular dipole moment. < 2>1/2 refers to the vibrationally averaged, root mean square angle between HC1 and the inertial axis. As noted earlier, the stronger H-bond produces more of a change in the EFG and the dipole moment at the Cl nucleus. The trends in Table 3.29 are consistent with those noted for the complexes of HC1 with XH3. The 12% change in the EFG for C1H...OH2 matches very closely the experimental estimate and the 8.2% change in C1H...SH2 matches the 8.6% experimental result. The averaged librational angles for the two complexes are quite similar to values calculated for the C1H...NH3 and C1H...PH3 analogues. The effects of electron correlation on these electronic properties may be noted from Table 3.30. As in the case of the C1H...ZH3 complexes, comparison of Tables 3.29 and 3.30 indicates an increased magnitude of Vzz results from bringing the two molecules closer together, coupled with forcing each subunit into its experimental monomer geometry. Vzz/ R in Table 3.30 shows a sensitivity of the EFG to intermolecular separation, although not quite so much as for C1H...NH3. Comparison of the first two columns of data in Table 3.30 reveals that correlation reduces the sensitivity of the EFG to the H-bonding interaction by about 8%. As for the other complexes studied by the authors, there is little basis set superposition affecting this property. The dipole moment change resulting from the H-bond formation is basically independent of electron correlation. 3.6 H2Y...HYH Just as for other properties of H-bonded systems, the water dimer has been the subject of perhaps the greatest scrutiny to its vibrational spectrum. Curtiss and Pople's seminal work49

Table 3.29 Changes of electric field gradient and dipole moment caused by H-bonding (atomic units) and rms HC1 librational angle (degs) calculated with [642/531/31] basis set for [C1,S/O/H]7. - Vzz(C1) ClH..OH2 ClH .. SH 2

0.431 0.290

-Vzz/Vzz° 0.12 0.08

-z

z

0.281 0.235

< 2 > 1/2 13.8 17.8

Vibrational Spectra

161

Table 3.30 Changes of electric field gradient and dipole moment caused by H-bonding (atomic units) calculated at experimental geometries with a [642/531/31] basis set for [C1,S/O/H]7. - V zz (Cl)

ClH..OH2 ClH..SH2 a

-

vzz / R a

SCF

ACPF (+CC)

0.546 0.394

0.499 (0.488) 0.361 (0.359)

z

SCF

SCF

ACPF

0.50

0.319 0.293

0.324 0.295

0.30

With full counterpoise correction.

consisted of a FG matrix analysis to obtain the normal modes, using SCF force constants. It was learned that simple description of the intermolecular modes is complicated by a high degree of mixing between the various internal coordinates. Nonetheless, the authors were able to identify a mode which is composed largely of the hydrogen-bond stretching motion. Less obvious, but still recognizable, were librational motions associated with nonlinearity in the H-bond. One is primarily an in-plane wagging of the proton donor and the other an out-of-plane bend. A rotation of the proton acceptor molecule about its internal symmetry axis, that is, out of the H-bond plane, is of very low frequency, only 80 cm-1 or so. Some very interesting calculations50 have addressed the question of how the geometry of the H-bond directly affects the vibrational features of the complex, using the water dimer in Fig. 3.3 as a model H-bonded system. The stretching force constant, k, of the bond between Od and Hd was evaluated as a function of the intermolecular geometrical parameters R, , and . k is smallest at the equilibrium geometry, reflecting the weakening effect of the H-bond. k rises much more slowly with increasing (3, as the proton acceptor molecule swings away from its optimal angular orientation, than when the donor is rotated via an increase in a. This stretching force constant rises toward its monomer value as the H-bond is stretched. Indeed, the authors remark upon the similarities between the behavior of this particular stretching force constant and the interaction energy, E, itself. The authors go on to conclude that the red shift of the vs band in this H-bonded complex can be directly attributed to the lengthening of the O d - H d bond. By partitioning the interaction energy into various components, they show how the stretch of this bond makes it both more polar and polarizable, which in turn, increases the induction and charge transfer components of the interaction energy. Although the authors did not include correlation in their treatment, the same could be said for dispersion energy which is directly related to polarizabilities of the individual monomers. It is for this reason that a nearly linear relationship is observed between vs and r. Zilles and Person36 have reached a similar conclusion that the polarity and polarizability of the O—H bond increases upon formation of the H-

Figure 3.3 Geometry of water dimer, defining three geometrical parameters.

162

Hydrogen Bonding

bond, based upon their atomic polar tensor analysis of the wave function. Indeed, the latter authors attribute the bulk of vibrational intensity changes seen in all normal modes upon dimerization to the electron density shifts in this bond. 3.6.1 Polarizability Swanton et al.51 investigated the effect of H-bond formation upon the electronic structure of the water molecule, in particular its polarizability. These properties are related to experimentally accessible quantities via Raman bands. Using the harmonic approximation, the differential cross section perpendicular to the incident light can be described as

where g is the degeneracy of the mode and C a physical constant. The quantity in brackets is referred to as the scattering activity. mean' is the derivative of the average polarizability and ( ')2 the square of the polarizability-derivative anisotropy, where

and derivatives, indicated by prime, are taken with respect to the normal coordinate. One can also define a degree of polarization, p, as

when the incident light is directed along the x-axis, polarized in the z-direction, and scattered in the y-direction. Coupled perturbed Hartree-Fock calculations at the SCF level were used to assess the polarizability tensor elements, each of which is defined as

where represents the applied field. The authors used a [5s4pld/4slp] basis set in their calculations. In order to focus on the effects of the molecular interaction, they introduced the concept of a "noninteracting dimer" wherein the dimer wave function is a simple product (non-antisymmetrized) of the unperturbed monomer functions. The effects of the interaction are thus in evidence by comparison of the two columns in Table 3.31 from which it may be seen that the average polarizability is little affected, increasing from 16.48 to only 16.60. The anisotropy of the polarizability, however, as measured by 2, undergoes a dramatic increase. Whereas the polarizability tensor is nearly spherical in the monomer, with all ii values between 16.3 and 16.6, xx is increased up to 18 when the two molecules interact with one another. This increase is thus focused along the H-bond direction. The changes in the polarizability quantities that are associated with each of the normal intramolecular vibrational modes are presented in Table 3.32. Any changes that occur on going from the monomer to the noninteracting dimer are due to the redefinition of the normal coordinate motion within the context of the dimer, rather than any changes in electronic structure. For example, the symmetric stretching motion in the monomer couples together the two O—H bonds. But this coupling is weakened within the dimer where the first donor stretch correlates with the O—H bond of the bridging proton. This changing motion pro-

Vibrational Spectra

163

Table 3.31 Polarizability aspects of the water dimer and a dimer with identical geometry in which the two molecules are prevented from interacting with one another. Proton-donating water molecule lies in xy plane; x-axis is approximately parallel to O..O line. Data51 in units of ao2e2/Eh.

mean 2 xx yy zz

xy

Noninteracting

Interacting

16.48 1.48 16.63 16.26 16.54 -0.68

16.60 8.20 18.09 15.67 16.04 -0.02

duces some strong effects. For example, the mean polarizability derivatives in the symmetric stretches of the two molecules split from 4.75 in the monomer to 5.84 and 2.60 in the donor and acceptor molecules, respectively. The actual interaction causes a small increase in the former and a decrease in the latter, resulting in a further splitting. The antisymmetric stretching motions in the monomer do not have any effect upon the mean polarizability because of the strict coupling between the two O—H bonds. But again, placed within the context of the dimer, the two O—H stretches are uncoupled. The asymmetric stretching mode of the donor correlates with the stretch of the donor O—H bond (the H not involved in the H-bond) and causes a marked change in polarizability, as indicated by the 2.08 entry in Table 3.32. The polarizability is fairly insensitive to bending motions, either within the monomer or the dimer. The various changes described above translate into the analogous intensification and weakening of the scattering activities in the rightmost section of Table 3.32. Significant changes occur in the donor stretching modes and the acceptor symmetric stretch. The bulk of these changes can be attributed to the changes in the normal modes that accompany dimerization, with smaller effects resulting from the actual interaction between the two monomers. The greatest intensification, by a factor of about two, is noted in the O—H stretch of the donor. This increase is dwarfed by the much larger changes noted in the infrared spectrum when H-bonding occurs. The authors also studied the polarization patterns associated with the intermolecular vibrational modes. Average polarizability derivatives were calculated to be quite small, yielding small scattering activities, all below 10 (xlO -34 C 4 N -2 kg -1 ). 3.6.2 Comparison between (H2O)2 and (H2S)2 Vibrational frequencies and intensities were compared between the monomer and water dimer by Amos in 198652 using a polarized basis set of the 6-31G** type. Also calculated and reported for purposes of comparison is the analogous dimer of H2S. The Vibrational frequencies and intensities of the monomers are listed in Tables 3.33 and 3.34, respectively, along with the changes that occur upon dimerization53. One might make a preliminary note that the frequencies are overestimates compared to experiment (shown in parentheses), as are the intensities.

Table 3.32 Calculated data relevant to polarizabilities in water dimer for intramolecular vibrational modes, and Raman scattering activities. 51a

( ')2

mean'

a' a' a' a" a' a'

donor stretch (sym) acceptor stretch donor stretch (antisym) acceptor stretch donor bend acceptor bend

a

mean'

Units:

in aoe2Eh- 1

-1/2

S

P

mono

nonint

ilnt

mono

nonint

int

mono

nonint

int

mono

nonint

int

4.75 4.75 0.0 0.0 -0.24 -0.24

5.84 2.60 -2.08 0.0 -0.18 -0.27

6.28 2.33 2.04 0.0 -0.09 -0.22

26.9 26.9 51.6 51.6 1.78 1.78

18.0 38.8 48.5 51.6 2.05 1.50

49.8 34.2 54.9 50.6 1.88 2.76

0.07 0.07 0.75 0.75 0.55 0.55

0.03 0.25 0.37 0.75 0.64 0.48

0.08 0.27 0.40 0.75 0.71 0.63

704 704 211 211 8.8 8.8

970 337 312 211 9.2 8.1

1240 282 334 207 7.9 12.5

; ( ') 2 in a o 2 e 4 E h - 2

-1

; p and S in l 0 - 3 4 C 4 N - 2 kg-

Vibrational Spectra

165

Table 3.33 Calculated frequencies and changes induced by H-bonding in intramolecular modes, calculated with 6-31G** basis set, in c m - 1 . 52 Experimental values in parentheses. (H2S)2

(H2O)2

sym stretch

bend asym stretch a

V mon

vd

va

4149(3657) 1772(1597) 4259 (3756)

-45 27 -25

-5 -1 -8

v

vd

va

—5 4 -3

1 -2 1

vmon

2874 (2614)a 1321 (1183) 2887 (2619)

Taken from reference 53.

Focusing first on the water dimer, both of the stretching frequencies are lowered when the complex is formed, but these changes are much more pronounced in the donor molecule. The intensities are also enhanced, particularly for the first mode listed, corresponding to v , which is amplified by an order of magnitude in the donor. The bending frequency undergoes a significant blue shift in the donor and a slight weakening of its intensity, but the acceptor bending mode is hardly affected. Very similar patterns emerge in the H2S dimer, although the effects are smaller in magnitude. The red shift of vs is only 5 cm-1 and it is intensified threefold. The changes in the acceptor frequencies are insignificant. Qualitative differences between the two dimers are revealed in the intensities. Rather than the enhancement observed for both stretches in (H2O)2, the asymmetric stretches of donor and acceptor of (H2S)2 are both reduced in intensity, as is the symmetric stretch of the acceptor. The data in Table 3.35 refer to the intermolecular vibrational modes calculated for the two systems by Amos52. All the frequencies are uniformly higher for the water dimer, attributable to the stronger H-bond as compared to (H2S)2. For both complexes, the highest frequency corresponds to the out-of-plane bending motion. The a' stretch, most closely corresponding to v , is considerably lower, particularly for (H2S)2. But other than this frequency, the others seem to fall in approximately the same order for the two complexes. The most intense intermolecular band would appear to be the a' bend. Second most intense for (H2O)2 is the a" shear or the a" bend for (H2S)2. It is not clear exactly how distinct these two modes are since they are of the same symmetry and can consequently mix extensively. Another fairly intense mode is the a' shear. The marked difference between the two congeners with respect to the a' stretch is particularly interesting. It is unclear why this difference should arise, but may be due to some particular mixing of the a' modes.

Table: 3.34 Calculated intensities, in km/mol, and changes, Aa, induced by H-bonding in intramolecular modes52. Experimental values in parentheses. (H2S)2

(H20)2 Amon

Ad

Aa

Amon mon

1.71 1.06 1.52

6.7 5.0

sym stretch

17(2.2)

10.41

bend asym stretch

97 (54) 58 (45)

0.91 1.79

a

A is calculated as the ratio between the dimer vs. the monomer: A d i m /A m o n .

1.8

Ad

Aa

3.17 0.94 0.54

0.39 1.09 0.56

166

Hydrogen Bonding

Table 3.35 Calculated frequencies and intensities of the intermolecular modes in the dimers of H2O and H2S52. Frequency (cm - 1 )

Intensity (km/mol)

(H2O)2

(H2S)2

(H2O)2

605 375 175 145 142 121

217 130 43 67 62 40

185 78 127 64 237 142

a" shear a' shear a' stretch a" torsion a' bend a" bend

(H2S)2 20 21 0.7 9.5 48 34

3.6.3 Effects of Electron Correlation and Matrices Recent work has addressed the issue of how much correlation influences the internal frequencies of water when involved in its dimer54. First of all, comparison of the data in Tables 3.33 and 3.36 indicate that the SCF frequencies and shifts of the 6-31G** and 6-311+G(2d,2p) basis sets are very similar, suggesting the extra functions in the latter larger set have only minor effects on these quantities. Focusing now on Table 3.36 shows that correlation lowers the two stretching frequencies. While the SCF and correlated frequency shifts of the acceptor molecule are virtually identical, including correlation adds a significant increment to the red shift of the donor. The magnitude of the shift of the symmetric stretch is doubled by correlation, while the asymmetric stretch rises from —21 to —28 cm - 1 . It is further notable that extending the correlation treatment beyond MP2 appears to have no significant impact on the results. The authors were particularly concerned with the discrepancy between gas-phase and matrix data, summarized in the last two rows of Table 3.36. After noting that the calculated frequency shifts match the matrix results much better than they do the gas phase, they went on to argue for a reinterpretation of the latter measurements. The abilities of different types of correlation treatments to properly handle the vibrational frequencies were the subject of a careful inquiry55. Frequency shifts of the intramolecular modes of the water dimer, calculated at various levels with the harmonic approximation, are exhibited in Table 3.37. All computed frequencies are higher than the experimental Table 3.36 Calculated frequencies and changes (in c m - 1 ) induced by H-bonding in intramolecular modes of (H2O)2, calculated with 6-311+ G(2d,2p) basis set, in c m - 1 . 54 sym stretch V

SCF MP2 MP3 MP4 expt (gas phase) expt (Kr matrix)

asym stretch

mon

Vd

Va

V mon

Vd

Va

4146 3865 3878 3838 3657 3628

-45 -91 -87 -89 -125 -59

-7

4247 3986 3986 3947 3756 3724

-21 -28 -26 -27 -26 -23

-11 -13 -12 -12 -34 -6

0

-8 -9 -57 -3

Table 3.37 Calculated frequencies and changes (in c m - 1 ) induced by H-bonding in intramolecular modes of (H2O)2, calculated with TZ2P basis set, in cm - 1 . 55 sym stretch vmon SCF MP2 CISD CCSD(T) expt (Ar matrix)

3894 3861 3943 3845 3638

asym stretch

vd

va

-51 -95 -17 -72 -64

-5 -12 +46 -8 -4

vmon

4238 3983 4042 3951 3733

bend

vd

va

vmon

vd

va

-22 -31 +31 -26 -24

-10 -17 +44 -13 -7

1764 1663 1698 1679 1596

2 2 14 1 3

23 29 41 28 21

168

Hydrogen Bonding

values in the last row of the table, and correlated frequencies are smaller than SCF. The MP2 and CCSD(T) results are similar to one another and closer to experiment than CISD. The latter method yields especially poor shifts in the frequencies, when compared to the monomer, even to the point of predicting the incorrect sign of the shift. Overall, the SCF shifts are in surprisingly good coincidence with experiment for all three intramolecular modes. Correlated shifts are significantly larger in magnitude, with MP2 shifts larger than CCSD(T). The CISD method should be avoided for calculations of this sort. The authors conclude with an important point: "the post-HF frequency shifts are not necessarily better than the HF ones unless the calculational level is high and the basis set used is large." Calculations performed by Woodbridge et al.53 allow a clear picture of the effects of electron correlation upon the harmonic frequencies of the weakly bound H2S dimer. First of all, as may be noted from Table 3.38, correlation has a substantial lowering effect upon the frequencies of the monomer, typically by about 100 c m - 1 . Whereas the SCF calculations predict very minor changes in the frequencies upon H-bond formation, these changes are much more pronounced at the MP2 level. A 35 cm-1 red shift is calculated for the symmetric stretch in the donor, corresponding to the vs mode. These stronger changes are consistent with the strengthening effect of correlation upon the H-bond. One may conclude that correlation is very important when considering the shape of the potential energy surface in complexes like (H2S)2 which contain second-row atoms. The reader may have noted that experimental spectra of H-bonded species are commonly measured in either the gas phase or in inert gas matrices. Of course, there may be some differences as the molecules of the matrix can interact in various ways with the H-bonded complex. A recent set of measurements56 provides some estimates as to the perturbations caused by the matrix. Table 3.39 reports in the first row the frequencies of the OH stretches of the free and bridging hydrogens of the proton donor molecule of the water dimer in the gas phase. The next row indicates that a Ne matrix has only a very small effect, perhaps 10 c m - 1 . The Ar and Kr matrices produce larger perturbations, reducing the frequencies by about 30 cm - 1 . A smaller cluster of Ar atoms, averaging perhaps 50 such atoms yields a result very much like a full Ar matrix. With the single exception of the very small increase for the free OH stretch in the Ne matrix, all matrices and the Ar cluster lower the frequencies of both of the modes studied. 3.6.4 Substituent Effects With regard to changes induced by replacement of hydrogens by other groups, it has been mentioned above that when water is paired with methanol, it is not entirely clear which of Table 3.38 Calculated frequencies and changes (in c m - 1 ) induced by H-bonding in intramolecular modes of (H2S)2, calculated with 6-3lG(2d) basis set, in c m - 1 53. SCF

sym stretch bend asym stretch

V mon

vd

2842 1332 2857

-1 +4 +2

MP2 va

+1 -1 - 1

v

mon

2694 1233 2720

vd

vd

-35 12 -10

-5 -3 -5

Vibrational Spectra

169

Table 3.39 Frequencies (cm - 1 ) measured for the proton donor molecule of the water dimer in various media56.

gas phase Ne matrix Ar matrix Kr matrix ArN cluster

free OH

bridging OH

3730 3734 3709 3701 3714

3601 3590 3574 3569 3576

the two will act as proton donor and which as acceptor. Table 3.40 provides vibrational spectral data57 to illustrate the very different frequencies of the two competing complexes. In the situation where water acts as proton acceptor, all three of the water intramolecular frequencies are red-shifted by small amounts. This contrasts with the values in the last column of the table where the shifts would be larger, and that of the bend would be to the blue, were the water to serve as proton donor. The patterns in the methanol molecule are even more discrepant. When water is the proton acceptor, the OH stretching frequency in the methanol molecule is strongly diminished, and a small increase is noted for the C—O stretch. The latter would be red-shifted if water were donor, and there would be little change in the OH stretch. Whereas experimental assessments of the frequency of the OH stretch in the donor molecule of the water dimer cover a range between 3500 and 3600 cm-1 in the gas phase58-60, the assignment is clearer in the methanol dimer, at 3574 cm - 1 . 6 1 A recent work has optimized the geometries of the dimers of water, methanol, and silanol at the MP2 level62. The vibrational frequencies include correlation by this approach, and are then corrected for BSSE and anharmonicity. The basis sets applied were DZP, as well as a triple- set, and is polarized under the rubric VTZ(2df,2p). The data in the first row of Table 3.41 refer to the red shifts of the OH stretching normal mode for each of the three dimers. (The HOD dimer was used instead of (HOH)2 so as to

Table 3.40 Calculated frequencies and changes (in c m - 1 ) induced by H-bonding in complex between water and methanol. The role of water as either proton acceptor or donor is indicated by PA and PD, respectively. Data, computed with 6-31G** basis set57. PD

FA

SCF

MP2

obsd

SCF

-2 -3 -7

-8 -5 -23

-16 -4 -14

-50 + 28 -28

12 -72

33 -111

14 -128

-16 -0

H2O V1 V2

CH3OH vCO vOH

170

Hydrogen Bonding

Table 3.41 Calculated frequency shifts (in c m - 1 ) of the OH stretch in the dimers of water, methanol, and silanol. Data computed with VTZ(2df,2p) basis set62. HDO..HOD

normal decoupled

CPa anharmonic

CPa

(CH3OH)2

(SiH3OH)2

SCF

MP2

SCF

MP2

SCF

MP2

-74 -76 -73 -89 -87

-133 -136 -123 -164 -150

-71 -72 -71 -86 -85

-156 -142 -191 -177

-111 -112 -111 -136 -135

-208 -196 -259 -245

_

a

With counterpoise corrections.

decouple the OH and OD stretches that exist as symmetric and asymmetric stretches in HOH.) The next rows consider the OH stretch in isolation from the other motions within each monomer, using a one-dimensional treatment. Similarity of the first and second rows illustrates this to be a valid approximation. The third row is similar to the second, except that counterpoise corrections have been used to modify the potential. Such corrections appear to have a small, albeit significant, lowering influence upon the magnitude of the SCF red shift. The last two rows add anharmonicity effects into the OH stretching frequency. The anharmonicity magnifies the red shift quite appreciably, particularly at the correlated MP2 level. The authors conclude with their contention that a correlated treatment of the OH stretch, in isolation from other motions, and corrected by a one-dimensional anharmonic approach, can produce frequency shifts within 10 c m - 1 of experiment. Comparison of the three systems indicates the OH stretching frequency suffers somewhat of a larger red shift in the methanol dimer than in the water dimer; however this difference might not be observed at the SCF level. The shifts in the silanol dimer are quite a bit larger in magnitude. Recent calculations of the phenol-methanol pair provide results that compare remarkably well with experiment63. The experimental frequencies listed in Table 3.42 were obtained by spectral hole burning and dispersed fluorescence spectroscopy which permitted assignment of the various bands. The agreement with the SCF/6-31G* frequencies is all the more impressive due to the absence of electron correlation and anisotropy effects.

Table 3.42 Comparison of calculated and experimental intermolecular frequencies (in c m - 1 ) in the phenol-methanol complex. Data calculated at the SCF/6-31G* level63.

H-bond stretch, rocking, p2 torsion, wagging, 2 rocking, 1 wagging, 1

Calc

Expt

158 17 30 55 70 90

162 22 35

55 65 91

Vibrational Spectra

171

When one of the H atoms of a water molecule of (H2O)2 is replaced by Cl, the interaction is strengthened64. In this complex, HOC1 acts as proton donor, due to its enhanced acidity. The red shift of the OH stretch is calculated to be 185-195 cm - 1 , in nice agreement with an experimental measurement of 229 cm-1 ,65 and considerably larger than in (H2O)2. 3.6.5 NMR spectra There have been some calculations carried out to consider the NMR spectra of the water dimer and related systems. Using a GIAO approach, and a small basis set66, it was demonstrated that the most shielded direction for the bridging proton generally coincides with the H-bond axis, in agreement with experiment. The calculated isotropic shifts for the water dimer and its ionic counterparts fall in the same region as experimental data. The anisotropy typically increases as the molecules are brought together. Other calculations67 have shown that the deshielding of the bridging hydrogen can be attributed to three factors. The proton loses overall density as the H-bond is formed. The proton acceptor molecule deshields the perpendicular components and shields the parallel ones. The proton donor atom has a more varied effect but also deshields the perpendicular components. The linear correlation noted by these authors between isotropic and perpendicular shifts was later confirmed by multipulse NMR data68. A means of incorporating a counterpoise correction for BSSE was later developed69 and applied to the water dimer, using a larger 6-311G** basis set. These BSSE corrections appear to be negligible for the bridging hydrogen, but effects upon the other hydrogens are significant.

3.7 Expected Accuracies

For anyone considering carrying out ab initio calculations of vibrational spectra, or simply interested in a thumbnail critique of a given paper in the literature, it would be particularly useful to have at hand some information as to what level of accuracy might be expected with a given basis set. We therefore take a brief interlude to explore this question for the dimers of HF and H2O, since both of these cases have witnessed the heaviest barrage of all levels of theory over the years. A comparison of a broad range of basis sets, varying from minimal to quite extended, was carried out several years ago for the HF and water dimers70. The limitation of this study is that it did not go beyond the SCF level, nor did it include anharmonicity, so comparisons with experiment are tenuous. Nevertheless, the data do illustrate the trends and provide useful information as to the types of errors likely to be incurred for a H-bonded system with any given basis set type. 3.7.1 HF Dimer Data computed for the intramolecular vibrational modes of HF and its dimer are reported in Table 3.43. Taking the values in the last row, computed with a very extended basis set, as a benchmark, it is immediately apparent that frequencies computed with small unpolarized basis sets are several hundred c m - 1 too small. 3-21G is probably the worst offender in this regard. Once polarization functions have been added, even a single set, the frequencies are more in line with those of the better basis sets. The same patterns are observed in the intensities which are significantly underestimated with the small unpolarized basis sets.

172

Hydrogen Bonding Table 3.43 Calculated frequencies (in cm - 1 ) and intensities (km m o l - 1 ) of HF and (HF)2. A and D refer to proton acceptor and donor, respectively70. v

A (HF)2

(HF)2

Basis set

HF

A

D

MINI-1 3-21G 4-31G

4183 4064 4124 4103 4440 4494 4314 4485 4488 4452

4171 4041 4102 4070 4397 4453 4279 4449 4448 4418

4131

DZ

DZP 6-31G(d,p) 6-31 + G(d) +VPS +VPs(2d) Extended

3958 4047 4021 4349 4406 4242 4401 4394 4366

A

D

23 79

279 371

77

110

395

133

182

374

171 168

170 167

450 459

HF

8.6 33

The limits of accuracy were probed in a study which focused on the gradient and force constants in diatomics like HF71. As more basis functions are added to one of 6-31G** type, changes of the order of 1 % occur in the force constant, at both correlated and SCF levels. The superposition contribution to the gradient is fairly large, and can account for variation of as much as 0.004 A in the bond length. If one concedes that SCF vibrational spectra, with no correction for anharmonicity, are unlikely to reproduce experiment, the next question would concern whether such calculations are capable of reproducing changes that occur in each molecule upon formation of the H-bond. Table 3.44 lists the shifts in the HF stretching frequency that result upon dimerization, along with the intensification, expressed as a ratio between that of the dimer versus that of the monomer. The results are in many ways a confirmation of the absolute val-

Table 3.44 Frequency shifts and intensification ratios (dimer/monomer) resulting from dimerization of HF70. v(cm-1) Basis set MINI-1 3-21G 4-3 1G DZ DZP 6-31G(d,p) 6-31 + G(d) + VPS s

+VP (2d) Extended

Ad/Ain

A

D

A

D

-12 -23 -22 -33 -43 -41 -35 -36 -40 -34

-52 -106 -77 -82 -91 -88 -72 -84 -94 -86

2.7 2.4 1.4

32 11 5.1

1.4

2.8

1.0 1.0

2.6 2.7

Vibrational Spectra

I 73

ues in Table 3.43. That is to say, the results with the unpolarized basis sets are undependable. The minimal basis set greatly underestimates the frequency shifts of both molecules. 3-21G and 4-31G are both double-valence type sets; nonetheless, they yield very different shifts in the donor molecule, and the acceptor shifts are both too small. The DZ set yields good results, but it is difficult to say if this is fortuitous. The polarized sets, on the other hand, all yield frequency shifts in decent agreement with the extended set. Turning to the magnifications in the intensities resulting from dimerization on the right side of Table 3.44, MINI-1 and 3-21G both fare especially poorly. Unlike the larger sets that yield little change in the intensity of the acceptor, and an enhancement of 2-3 in the donor, these two sets are off by orders of magnitude. 4-31G is somewhat better, as is 6-31G(d,p). It would appear that the intensity enhancements are somewhat more demanding in terms of basis set quality than are the frequency shifts. 3.7.2 Water Dimer The water dimer adds a number of new dimensions to the problem since each water molecule contains three vibrational frequencies instead of one. The two stretching modes are labeled v1 and v3; v2 refers to the symmetric bending motion. The frequencies computed for the water monomer are reported in the first three columns of Table 3.45, followed by the corresponding frequencies in the dimer. As in the case of (HF)2, the unpolarized basis sets strongly underestimate the stretching frequencies in the monomer. On the other hand, the bending frequency is computed reasonably well with all of the sets, albeit the small unpolarized sets do yield a bit of an overestimate. Rather similar patterns are evident in the dimer as well. The unpolarized sets underestimate v1 and v3 and yield a small overestimate, by less than 100 c m - 1 , of the frequency for v2. The shifts in each intramolecular vibrational frequency that occur upon dimerization of water are described in Table 3.46. Whereas even very small basis sets seem capable of predicting qualitatively correct shifts in the HF dirner, (H2O)2 apparently represents a more stringent test. For example, the three larger basis sets predict that both stretching frequencies of the proton-acceptor molecule will be lowered, and are in good agreement as to the amounts. The smaller sets in the first three rows of Table 3.46, on the other hand, yield erratic results. All three make the erroneous prediction of a blue shift in v1; there is no consistency at all for v3. Correct prediction of the behavior of the bending frequency is apparently more demanding. Only the two polarized basis sets which also contain diffuse + functions are in agreement. Even 6-31G(d,p) yields an incorrect sign. It might be stressed at this point that reproduction of the large basis set results are rendered particularly difficult in the case of the acceptor, due to the small magnitudes of the shifts involved, all less than 10 c m - 1 . As in the case of (HF)2, the shifts are larger in the donor molecule. All basis sets, even the smallest, agree that both stretches suffer a red shift and that the frequency of the bend is increased. There is discrepancy concerning the magnitudes of these shifts. The three polarized sets concur that the red shifts are some 40-50 cm-1 for v1 and 20-30 cm-1 for v3. The shifts predicted by the three smaller basis sets are erratic and generally undependable. Turning now to the intensities of the various modes, the data in Table 3.47 indicate that a polarized basis set like 6-31 G(d,p) offers a reasonable alternative to a much more extended set such as 6-31 + +G(2d,2p), even if not quantitatively very accurate. The split-valence 431G is considerably poorer, and the data with 3-21G are much worse, even though formally of split-valence type as well. The intensity magnification ratios induced upon dimerization

Table 3.45 Calculated frequencies (in km m o l - 1 ) of H2O and (H2O)27

(H2O)2 H2O Basis set MINI-1 3-2 1G 4-3 1G 6-31G(d,p) 6-31+G(d) 6-3l + +G(2d,2p) HF-limit

D

A

v1

v2

V3

3897 3811 3960 4149 4071 4128 4130

1816 1799 1767 1770 1797 1746 1747

4127 3945 4098 4267 4190 4235 4231

v1

V2

3900 3817 3979 4144 4068

1806 1785 1771 1767 1806

4123

1752

V3

v1

V2

V3

4115 3945 4121 4258 4181 4226

3843 3712 3907 4102 4028 4076

1852 1845 1813 1798 1826 1767

4071 3891 4085 4240 4165 4213

Vibrational Spectra

175

Table 3.46 Calculated frequency shifts (in c m - 1 ) occurring upon dimerization of water70. D

A

Basis set

V1

V2

V3

V1

MINI-1 3-21G 4-3 1G 6-31G(d,p) 6-31 +G(d) 6-31 + +G(2d,2p)

3 6 19 -5 -3 -5

-10 -14 4 -3 9 6

-12 0 23 -9 -9 -9

-54 -99 -53 -47 -43 -52

V2

36 46 46 28 29 21

V3

-56 -54 -13 -27 -25 -22

are listed in Table 3.48. If one is interested in these properties, the 6-31G(d,p) basis would appear satisfactory in most cases. 4-31G does fairly well, the primary exception being its five-fold exaggeration of the enhancement of the first stretching mode in the donor molecule. 3-21G should be avoided. In summary, calculation of vibrational frequencies can be meaningful, even if restricted to the SCF level and with no account of anharmonicity. The frequencies are less demanding of basis set quality than are the intensities. In some cases, one can compute reasonable estimates of dimerization-induced frequency shifts with basis sets of 4-31G type, although polarization functions are strongly recommended for uniform quality of results. Intensity calculations without polarization functions can be expected to yield only the crudest of estimates. Reasonable results can be achieved with only one set of such functions on each atom.

3.8 HYH...NH3 Somewhat more complicated than the complexes discussed earlier is the combination of water with ammonia. The IR spectral characteristics of this complex were calculated using a variety of basis sets and the results are presented in Table 3.4972. Also presented in this table are comparable data computed at a correlated (MP2) level73, for purposes of comparison. The red shift of the proton donor water molecule vs band is 103 c m - 1 , quite a bit lower

Table 3.47 Calculated intensities (in km mol - 1 ) of H2O and (H2O)270. (H2O)2 H2O

Basis set 3-21G

4-31G 6-31G(d,p) 6-31++G(2d,2p)

v1

A

v2

v3

v1

v2

v3

v1

98 172 112 110

45 131 89 103

80

9

5

4 16

125 105

54

8 26

14

91

0.05

58 81

D

22

v2

v3

282

98

283

138

52 110

184 191

99 65

104 127

176

Hydrogen Bonding Table 3.48 Magnifications of intensity of vibrational bands occurring upon dimerization of water70. A

Basis set 3-21G 4-3 1C 6-31G(d,p) 6-31 + +G(2d,2p)

D

V1

V2

V3

V1

V2

V3

100 1.9 1.6 1.6

1.2 1.4 1.1 1.2

5 2.4 1.5 1.3

5640 71 11 14

1.2 1.1 0.9 0.7

6 2.0 1.8 1.6

than in H3N...HF, and even smaller than in H 3 P ... HF or H2O...HF. This shift is similar to that in H2O...HC1 where HC1 acts as the donor. Correlation increases the red shift to 167 c m - 1 , close to the 197-209 c m - 1 observed in Ne, Ar, and Kr matrices74-75. The intensification of this band is surprisingly large at 14, and grows to 88 when correlation is added. The patterns of frequency shifts and intensity changes for water in H 3 N ... HOH are very different than when water acts as the proton acceptor as in H2O...HF. This set of differences may act as a simple marker when there is some question as to the nature of a given complex in a spectrum. As a proton donor, the bending frequency of the water molecule is blue shifted and the other stretch reduced by a small amount. Note that correlation increases the magnitude of these shifts. When complexed with HF, the stretching bends in NH3 were unshifted but intensified by upwards of thirty-fold. The situation is different when NH3 is bound to H2O, as the intensifications are much smaller, only an approximate doubling at the SCF level. However, correlation drastically enhances these bands, intensifying them by an order of magnitude. (Note that the lower symmetry of the H3N...HOH complex breaks the degeneracy of the pairs of vibrational levels in NH3.) The effects of complexation on the bending modes of NH3 are similar when either HF or H2O acts as proton donor. The lower-frequency bend is Table 3.49 Frequencies ( c m - 1 ) and intensities (km mol - 1 ) of H2O and NH3, and the changes resulting from formation of the H 3 N ... HOH complex. SCF data72 calculated with + VPS basis set; MP2 values73 computed with 6-31G**. V

V

A

mon

Mode

SCF

SCF

v1

-103

V2

4129 1727

V3

4242

MP2

Adim/Amon

n mo

SCF

SCF

MP2

25 94 85

14 1.1 1.2

88 0.4 1.5

0.4

2.5

6

1.9

17 17 46 0.9 1.3 3.7

H2O

Vstr Vstr

37 -16

-167

60

-45 NH3

3709 3847

Vbend

1096

Vbend

1794

1 4 -15 83 0 -2

3

2 0

2.0

36

242

0.9

-4 -13

29

0.5 0.8

Vibrational Spectra

177

blue shifted and the other nearly unaffected; small changes in intensity are predicted, with or without correlation. The H-bond stretching frequency is calculated in Table 3.50 to be 115 c m - 1 , comparable to that of H 3 P ... HF or H2O...HC1. The intensity of this mode is low, only 3.3 km mol - 1 , again similar to H2O...HC1 or indeed any of the H3Z...HF complexes, suggesting that this mode represents a fairly pure H-bond stretching motion in all of these. The numerical value does not reproduce very well the experimental measurement of 202 cm-1 in matrices74. The other intermolecular modes are generally a little higher frequency and intensity than those in H 2 O ... HCl. The two donor bending frequencies of 669 and 443 c m - 1 match the experimental values of 662 and 430 surprisingly well. The same is true of one of the acceptor bends but the other is significantly in error. Replacement of one of the hydrogens of H2O by an aromatic ring was indicated in the preceding chapter to strengthen the H-bond with NH3. The effects upon the calculated vibrational spectrum are consistent with this observation76 in that the H-bond stretching frequency of C6H5OH...NH3 is calculated to be 165 c m - 1 , somewhat larger than the 115 c m - 1 computed for H3N...HOH72. As indicated in Table 3.51, the computed frequencies match the available experimental data for this complex exceedingly well, despite the lack of correlation or anharmonicity effects.

3.9 (NH3)2 In 1987, Sadlej and Lapinski77 carried out a force-field analysis of the ammonia dimer with the 4-31G basis set. Because of the difficulty in definitively locating the true minimum on the potential energy surface, and the dubious ability of this small basis to correctly model the interaction, the results are provided here for instructional purposes and should be taken in that spirit only. Note first from the results in Tables 3.52 and 3.53 that the linear arrangement is a transition state on the SCF/4-31G surface, indicated by the single imaginary frequency. The symmetry of the linear structure allows one to assign each intramolecular mode as primarily occurring within the donor or acceptor while this is not possible in the cyclic geometry with a pair of equivalent molecules. It is stressed that assignment of certain bands as "bends" is tenuous and does not differentiate between scissoring, rocking, or wagging

Table 3.SO Vibrational spectra calculated for intermolecular modes of the H3N...HOH complex with +VP S basis set. Data at SCF level72. v(cm-1)

A(km mol - 1 ) a

Mode

calc

expt

H-bond stretch (v ) donor bend donor bend acceptor bend acceptor bend torsion

115 669 443 181 219 50

202 662 430 180 411 20

a

See reference 74.

calc 3.3 215 124 59 37 0.5

178

Hydrogen Bonding

Table 3.51 Vibrational frequencies (cm - 1 ) for intermolecular modes of the H3N...HOC6Hg complex. Calculated data at SCF/6-31G** level with +VPS basis set76. V

Mode

calc

expt

H-bond stretch (v ) wagging rocking wagging rocking torsion

165 305 242 64 31 37

162

62

motions. Red shifts of 50-60 cm-1 occur in the stretching modes of the proton donor molecule in the linear structure, while the cyclic arrangement has smaller shifts. The work indicates red shifts in all stretching modes and shifts to higher frequency for internal bends, whether cyclic or linear. The H-bond stretching motion in either configuration is in the neighborhood of 100-150 cm - 1 , but again this mode is not pure by any means. It is interesting finally to note that the total zero-point vibrational energies of the two geometries are quite similar, 47.4 kcal/mol for the linear arrangement and 47.6 for the cyclic.

Table 3.52 Frequencies (cm - 1 ) and intensities (D2 A- 2 a m u - 1 ) calculated for the ammonia dimer in its linear configuration77. Mode

a", stretch a' , stretch a', stretch a', stretch a", bend a', bend a', bend a', stretch a", stretch a', bend a", bend a', bend

vmon proton donor 3929 3898 3738 3710 1866 1830

822 proton acceptor 3926 3922 1836 1829

817

v

A

-28 -59 -22 -50 45 11 200

0.09 1.78 0.01 1.52 0.83 0.77 10.07

-31 -35 15 8 195

0.36 0.38 1.15 1.17 15.71

intermolecular Mode a', v

a' a', bend a", bend

a' a", bend

V

112 134 381 314 112 imaginary

A 0.76 1.16 1.71 1.20 0.76

Vibrational Spectra

I 79

Table 3.53 Frequencies (cm - 1 ) and intensities (D2 A-2 amu - 1 ) calculated for the ammonia dimer in its cyclic configuration77. Mode bu stretch au, stretch bg, stretch ag, stretch bu, stretch ag, stretch a u ,bend bg, bend ag, bend b u ,bend a g ,bend bu, bend/torsion Mode

ag, bend au, bend bg,bend au, bend bu, bend

vm o n intramolecular 3917 3916 3916 3916 3728 3724 1856 1836 1829 1821 869 771 intermolecular v 137

477 255 159 96 93

v

-40 -41 -41 -41 -38 -36 35 15 8 0 247 149

A

1.81 0.42 0 0 0.27 0 1.90 0 0 1.77 0 27.0 A 0

0 2.06 0 0.61 5.55

Yeo and Ford78 worked out the atomic polar tensors of the cyclic and linear ammonia dimers to help interpret the spectral intensities. They found that dimerization produced only very minor changes, with the exception of the proton donor N and H atoms, particularly the latter atom, for the linear structure. The cyclic geometry sees the largest changes in the two bridging hydrogens. It is the charge flux that appears to be most important in the intensity changes seen upon dimerization. The vibrational spectra of the linear and cyclic geometries of the ammonia dimer were computed at the correlated level, using a polarized basis set in 199279. The striking contrasts between the spectra of these two geometries are clear from the data in Table 3.54. For example, a number of the modes in the cyclic geometry would have zero intensity in the harmonic approximation, whereas some of the others would be considerably more intense than in the linear configuration. Vibrational frequency shifts, too, exhibit discrepant behavior in the two geometries. Comparison of the data in Table 3.54 with those in Tables 3.52 and 3.53 points out some of the dramatic changes that occur when correlation effects are included, along with a polarized basis set. As an example, the SCF frequencies of all stretching modes in the linear geometry are red-shifted by 20-60 c m - 1 , whereas MP2 calculations indicate very little shift at all for a number of these stretches. The red shifts of these same modes in the linear structure are also reduced considerably upon adding correlation. 3.10

Carbonyl Oxygen

The carbonyl oxygen provides a contrast to the hydroxyl atom in a number of ways. It is interesting to examine how the C = O bond reacts when this group accepts a proton from a donor.

180

Hydrogen Bonding Table 3.54 Frequency shifts and intensity ratios of the vibrational modes of NH3 that accompany dimerization into the linear and cyclic ammonia complex. Data computed at the MP2/6-31G* level79. v(cm-1) Parent mode a1 stretch e stretch

a1 bend cbend

linear

cyclic

linear

cyclic

1 -35 0 -45 0 -11 58 13 2 2

-11 -7 -12

0.3 335 9 153 11 1.2 0.9 0.9 0.6 1.0 0.6 1.5

75 — 3 — 87 — 2.2 — 1.9 — 1.1

41 -7

3.10.1 Relationship between

A dimer /A monomer

-2

-3 -12 39

15 8 23 11 -18

E and v

Latajka and Scheiner80 considered the nature of the relationship between the stretching frequency of this covalent C=O bond and the strength of the H-bond, or indeed other sorts of interaction. For this purpose, a water molecule was brought up toward the carbonyl oxygen of H2CO and allowed to approach to within a set of specific distances. For interaction energies in the 2.5-4.5 kcal/mol range, there appeared to be a linear relationship between E and v, where the latter quantity is the red shift of the C=O vibration. The weakening of the C=O bond, implied by the lower frequency, is consistent with a picture of the H-bond wherein the group takes on a certain amount of C — O — H character. This notion is confirmed by the lengthening of the C=O bond associated with its acceptance of a proton, and discussed in the preceding chapter. What was most interesting about this particular study was the close linear correspondence between v and the H-bond energy. The small values clouded numerical precision of the relationship, so stronger interactions involving ions were considered as well. The ions that were allowed to interact with H2CO were H 3 O + , Na + , and Mg +2 . These interactions covered a wide range of E between — 8 and —75 kcal/mol. A very nearly linear relation was noted between E and v, which suggested that each 1 kcal/mol strengthening of the interaction would shift the C=O stretch to the red by approximately 2 cm - 1 . These linear relationships have foundation in experimental work, as reported by Thijs and Zeegers-Huyskens81, who also confirmed that the slope of 2 cm-1 kcal-1 mol is consistent with observation in solution. (In fact, the red shift induced by formation of a H-bond appears to occur even when the C=O group is part of a monolayer assembly82.) There is an angular dependence to the relationship between the stretching frequency and the interaction energy. When the proton donor approaches along the C=O direction, the amount of red shift is less than when it is directed along a lone pair, that is, 8 (C=O...X) ~ 120°, given the same interaction energy. More recent calculations have considered the same problem from the perspective of the proton donor83. When water was paired with a set of N and O-type proton acceptor mole-

Vibrational Spectra

181

cules, a linear relationship was noted between the shift of the O—H stretching frequency and the calculated strength of the H-bond. The slopes were somewhat different for the two types of acceptors: a 23 cm-1 shift corresponds to 1 kcal/mol change in H-bond strength for O acceptors, while the same energy enhancement in N-bases is accompanied by a 27 c m - 1 shift. (The foregoing analysis was aided by use of HOD, rather than HOH, so as to more clearly distinguish the stretches of the two hydrogen atoms of water.) In either case, the shifts associated with the proton donor are much larger than those in the acceptor molecule. 3.10.2 Formaldehyde + Water High-level calculations84 have predicted the frequency shifts for the various internal modes when water binds with formaldehyde. The equilibrium geometry is of type II (see section 2.7) in which water donates a proton to the carbonyl oxygen, and the water oxygen atom also approaches within about 2.7 A of a CH2 H atom, as a secondary and weaker interaction. As in earlier cases, correlation has a marked effect upon the vibrational frequencies in Table 3.55, in most cases reducing these quantities by a substantial amount. On the other hand, the SCF and correlated frequency shifts arising from complexation are not very dissimilar. The trend of red shifts of one of the O—H stretches of water and of its bending mode, matches the same trend in the proton donor of the water dimer. Indeed, the quantitative aspects are rather similar to those in Table 3.33 particularly given the different basis sets. The C=O stretching vibration is barely affected at all, surprising in light of the change in this bond's length upon forming the H-bond. Most altered is firstly the CH2 rocking motion, probably due to its proximity to the water molecule, which hinders this motion. The two C—H stretching vibrations undergo a blue and red shift, respectively. The intermolecular frequencies are reported in Table 3.56 at various levels of theory, along with the calculated SCF intensities in the last column. With only one exception, the correlated frequencies are somewhat higher than the SCF values. Because of the bent nature of the H-bond in H2CO--HOH, the H-bond stretching frequency is not particularly pure. Table 3.55 Frequencies (cm- 1 ) of H2O and H2CO, and the changes resulting from formation of the complex 84. CCSD/TZ2P

SCF/TZ2P(f,d) Mode

V

mon

v1, stretch V 2 , stretch v3, bend

4073 1780

V1, C—H stretch v2, C—H stretch v3, C=O stretch V4, HCH scissor CH2 rock V 5, CH2 wag V

3173 3125 1980 1645 1343 1341

a

In solid Ar.

4261

v HOH -43 18 -17 H2CO 20 -15 0 6 32 8

expta v

v

4023 3802 1722

-61 15 -20

-58 21 -24

3055 3004 1790 1550 1259 1214

25 -14 -3 7 42 7

16 -6 1 5 16

mon V

2

182

Hydrogen Bonding

Table 3.56 Vibrational frequencies and intensities calculated for intermolecular modes of the H2O-H2CO complex84. v(cm-') Mode

A (km m o l - ' )

SCF/TZ2P(f,d)

CISD/TZ2P

CCSD/TZ2P

151 329 65 457 148 39

179 359 95 501 171 29

180 354 101 493 173 20

H-bond stretch (v ) bend bend oopbend H2CO rotation torsion

expt

a

SCF/TZ2P(f,d) 31 103 50 131 0 155

250,261 435,439 68

a

ln solid Ar.

Its frequency of about 180 cm-1 is quite similar to that of the water dimer. The intensity is only 30 km/mol, considerably weaker than the same band in (H2O)2. The two in-plane bends listed in Table 3.56 are of quite different frequency. The first, of higher frequency, corresponds to a distortion in which the XO-H angle becomes straighter and the OHO angle more acute, while the lower refers to an overall straightening of both aspects of the H-bond. The other intermolecular bends are all of a" symmetry. The out-of-plane distortion is the highest frequency mode of all and the torsion the lowest. The latter frequency is quite a bit smaller than that of the torsional mode of water dimer. Agreement with available experimental observation in solid Ar matrix is moderate. 3.10.3 Formaldehyde + HX The combination of H2CO with HC1 was considered, along with two different approaches to electron correlation, by Rice et al.85. The harmonic frequencies they predicted for the intramolecular modes at various levels are reported in Table 3.57, along with the changes induced by formation of the H-bonded complex. The numbering scheme was taken directly

Table 3.57 Frequencies (cm - 1 ) of HC1 and H2CO, and the changes resulting from formation of the complex85. SCF/DZP

Mode

Vmon

SCF/TZ2P

V

Vmon

MP2/DZP

V

Vmon

CPF/DZP

V

Vmon

V

HC1 V v3, stretch 3144 -103 3130 -109 3061 -251 3022 -151 H2CO v1,CH2 a stretch V2, CH2 s stretch v4, CO stretch

3226 3149 2009

vv CH2 scissor

1659

v6, CH2rock v10, CH2wag

1370 1338

30 22 -10

-1 0 6

3168 3096 1992

28 19 -11

3128 3040 1774

1655

0

1572

1373 1339

2 7

1284 1217

46 32 -11

-4 1 4

3089 3012 1796

1566 1281 1197

43 30 -11

-3 2 7

Vibrational Spectra

183

from their paper, and refers simply to the numerical order of frequencies, grouped by symmetry. The red shift of the HC1 stretch is calculated to be between 100 and 110 cm-1 at the SCF level, very similar to that predicted when H2CO is replaced by H2O. This shift is substantially increased when correlation is included, especially by MP2, wherein the shift is more than doubled. In particular, the calculated shift of 251 cm-1 is in superb agreement with a measurement of 242 cm-1 in Ar matrix86. An examination of the sensitivity of the red shift of the HC1 stretch in this complex to basis set87 indicates it remains in the 200-300 crn-1 range for basis sets varying from 6-31G(d,p) to 6-311 + +G(2df,2pd), all at the MP2 level. The stronger H2CO--HF complex shows a red shift of 422 cm-1 at the MP2/631 l + +G(2df,2dp) level88. Both CH2 stretching modes undergo frequency increases, in contrast to H2CO--H2O where one of these modes is red-shifted. The CO stretch is shifted to the red by 10 c m - 1 , a result which is insensitive to basis set or correlation. Again, this result is in excellent agreement with experiment where a shift of 12 cm-1 has been measured86. The calculated red shift in H2CO--HF is 13 cm - 1 . 8 8 The pattern in these H2CO--HX complexes is also distinct from the H2CO--H2O complex where little shift of this stretching frequency is calculated. The rocking motion of the CH2 group is not shifted by complexation in H2CO-HC1, unlike the complex with water where a significant frequency increase is calculated and in fact observed experimentally. A systematic comparison of the shifts in the HX stretching frequency for a series of H2CO-HX complexes in Ar matrix86 shows a clear correlation with the strength of the Hbond as the shift varies in the order HF > HC1 > HBr > HI. There is no real pattern observed for the C=O stretching frequency which is in the range between 10 and 17 cm-1 for all four complexes. The intermolecular frequencies for H2CO-HC1 in Table 3.58 again show certain similarities with H2CO--H2O. The H-bond stretching frequency in the 120-127 cm-1 range for the former is quite close to the 118 calculated for the latter, all at the SCF level. Note the small but significant increases that result when correlation is added. Indeed, correlation increases the frequencies of all of the intermolecular modes of H2CO--HC1, and by substantial amounts. The greater strength of the H2CO--HC1 interaction is reflected in its intermolecular frequencies. At the MP2/6-311 + +G(2df,2dp) level 88, the H-bond stretching frequency, v , is computed to be 248 c m - 1 ; the two higher-frequency shearing motions occur in the 754-791 cm-1 range. The intensities calculated for the intramolecular vibrational modes of the H2CO--HC1 complex are reported for two different basis sets in Table 3.59. The intensification of the vs mode, v3, is estimated in the range between seven- and eightfold, similar to that predicted for H2O..HC1. Within the H2CO molecule, the two CH2 stretches both lose some intensity; increases are observed in the CO stretch and the CH2 scissor. Analogous data for the interTable 3.58 Frequencies (cm - 1 ) of the intermolecular vibrational modes of H2CO..HC185. Mode v8, in-plane stretch (v ) v7, in-plane shear v9, in-plane bend v11, out-of-plane shear v 12 , out-of-plane shear

SCF/DZP

SCF/TZ2P

MP2/DZP

CPF/DZP

127 394

120 391 36 384 122

170 554 54 521 186

155 478 45 460 158

38 399 112

184

Hydrogen Bonding

Table 3.59 Infrared intensities (km m o l - 1 ) of the intramolecular vibrational modes of H2CO..HC1, calculated at the SCF level, and the enhancement ratio as compared to the monomer85. DZP Mode

A dim

v3, stretch

412

v1, CH2 a stretch v2, CH2 s stretch V4, CO stretch V5, CH2 scissor V6, CH2 rock v10, CH 2 wag

85 57 211 21 17 1.7

A

TZ2P dim/Amon

HC1 8.4 H2CO 0.7 0.8 1.3 1.5 0.9 0.9

A

dim

A

dim /A mon

388

7.4

70 41 186 26 19 2.3

0.7 0.7 1.2 1.5 0.9 1.0

molecular modes are listed in Table 3.60 where it may be seen that the H-bond stretch is computed to have an intensity of some 16-22 km/mol. The v mode in H2O..HF is computed to be even stronger, with an MP2/6-311 + + G(2df,2dp) intensity of 31 km/mol88. The result for H2CO..HC1 is smaller than in H2CO..HOH, but nearly an order of magnitude larger than in H2O..HC1. The other intermolecular bands of H2CO..HC1 have comparable intensity to va with the exception of the out-of-plane shear, v 12 , which has a very low intensity.

3 . 1 1 Imine Migchels et al.89 have evaluated the effect of interaction with a proton-donating water or methanol molecule upon the internal vibrational frequencies of a number of imines. These perturbations are reported in Table 3.61 at the SCF/6-31G level and indicate first that the red shift of the hydroxyl group of methanol is quite a bit larger than that of water. This discrepancy may be due to the symmetric nature of the water molecule which thoroughly mixes the two O—H stretches into a symmetric and asymmetric pair. The frequency chosen by the authors as a reference point for water is 4145 c m - 1 , quite distinct from the 4032 cm-1 of the purer O—H stretch in CH3OH. Nonetheless, the large shifts in the methanol corn-

Table 3.60 Infrared intensities (km m o l - 1 ) of the intermolecular vibrational modes of H2CO..HC1, calculated at the SCF level85. Mode v8, in-plane stretch (v ) v7, in-plane shear v9, in-plane bend v n . out-of-planc shear v12, out-of-plane shear

DZP

TZ2P

16 53 12 53 0.1

22 43 12 4) 0.0

Vibrational Spectra

185

Table 3.61 Frequencies (in c m - 1 ) of imine molecules, and the changes resulting from formation of complexes with HOH or CH3OH as proton donor. Data calculated at SCF/6-31G level89. Mode

vmon

Av(HOH)

Av(CH3OH)

CH 2 =NH V

OH

V

CN

V

NH

1866 3686

-45 -5 21

-173 -5 20

-49 -4 25

-202 -3 25

cis-CH 3CH=NH V

OH

V

CN

V

NH

1884 3659

trans-CH3CH=NH V

OH V

CN

-y

NH

1896 3699

-51 -6 14

-210 -5 14

-46 -2

-177 -1

-52 -5

-213 -3

CH2=NCH3 V

OH

V

CN

V

OH

V

CN

1901 CH3CH=NCH3

1932

plexes are consistent with experimental data reported by the same authors who found this quantity to be —315 cm-1 in a complex very much like CH3CH=NCH3---CH3OH. There are very small red shifts of the C=N stretch on complexation, and there is very little difference whether the proton donor is water or methanol. There are larger blue shifts of the NH band, again insensitive to the nature of the donor. The authors discussed how the formation of the H-bond alters the nature of the C=N stretching mode. In a monomer such as CH 2 =NH, this mode includes a contribution from a scissoring motion of the CH2 group. When water forms a H-bond, this mode becomes mixed with a certain degree of N—H stretching as well.

3.12 Nitrile The nitrile group as a proton acceptor provides an interesting counterpoint to the nitrogen atom of amines which is involved in all single bonds, or the double bonds in imines. In each case, the proton donor lies directly along the direction of the N lone pair. As one example, frequencies and intensities for the complex of HCN with HF were computed in 1973 by Curtiss and Pople90 and later with more flexible basis sets by Somasundram et al.6 The intramolecular frequencies obtained are presented in Table 3.62, along with experimental information. The calculated frequencies are in error by several hundred cm -1 when compared to experimental harmonic frequencies. The enlargement of the basis set markedly increases the HF stretching frequency, but less of an effect is noted in HCN. Probably of greater importance than the frequencies themselves are the shifts arising from complexation. The red shift of the HF stretch is only 127 cm-1 with 4-31G and as large as 189 cm-1 with the big-

186

Hydrogen Bonding

Table 3.62 Frequencies (in cm-1) of HF and HCN, and the changes resulting from formation of the HCN..HF complex. Data calculated at SCF level6,90. 4-31G Mode

V

HF

V

HC

V

CN V

bend a

V

DZP

mon

V

4117

-127

3695

2384 911

-13 12 26

Vmon

4511

3638 2406 861

expta

TZ2P v

Vmon

v

Vmon

HF -189

4471

-171

4138

-251

3600 2408 869

-7

3442 2129 727

24

HCN 0

31 17

19 12

v

Harmonic frequencies.

ger sets. This result is still quite a bit smaller than the best experimental estimate. It also represents less of a shift than when HF is paired with the much more basic NH3. The CN stretching frequency of HCN is shifted toward the blue, as is the bending mode. In contrast, the HC stretch suffers a lowered frequency. The quantitative aspects of these changes are rather sensitive to basis set. The intermolecular frequencies in Table 3.63 indicate that the larger sets predict progressively smaller frequencies for the H-bond stretch, but the trends are more erratic for the two bending modes. The frequencies seem to be calculated rather well, even with the smallest 4-31G basis set, surprising in light of the low order of theory and the neglect of anharmonic effects. 3.12.1 Correlation and Anharmonicity Botschwina91 later used a doubly polarized basis set to study this complex, along with a CEPA-1 treatment of electron correlation. The ab initio energetics were fit to an analytic four-dimensional function in order to elucidate anharmonic effects. The results at various levels of theory are presented in Table 3.64 along with experimentally measured quantities. Comparison of the SCF and CEPA-1 data suggests that while correlation yields major changes in the frequencies themselves, the shifts that occur upon complexation are surprisingly insensitive to correlation. The same is true of introduction of anharmonicity with one major exception. Whereas the frequency shifts of the stretches of the HCN proton acceptor molecule are little affected by introduction of anharmonicity, the red shift of HF is increased by 46% from 168 to 245 c m - 1 . This latter result is in near perfect agreement with

Table 3.63 Vibrational frequencies (cm - 1 ) calculated for intermolecular modes of the HCN .. HF complex at SCF level6,90. Mode H-bond stretch (v ) Donor bend (shear) Acceptor bend

4-31G

DZP

TZ2P

expt

193 561 86

176 645 108

159 581 84

155±1O 55±3 7()±24

Vibrational Spectra

! 87

Table 3.64 Harmonic and anharmonic frequencies (in c m - 1 ) of HF and HCN, and the changes resulting from formation of the HCN..HF complex. Data91 calculated with doubly polarized basis set. SCF Mode

vmon

CEPA-1

expt

v

vmon

v

-164

4160 3982

-168 -245

3439 3325 2155 2128

-3 1 28 27

vmon

v

HF V

HF

4458 4295

harm anharm

HCN V

harm

V

anharm harm

HC

CN

anharm

3604 3505 2407 2385

-242

-2 0

21 19

3716

-245

3310

1

2121

24

the experimental quantity in the last column of Table 3.64. Indeed, the shifts of all the stretches are calculated to lie very close to experiment for this complex. Note however that the same cannot be said of the stretching frequencies themselves which remain in substantial error when compared to experiment. Amos et al.44 considered the same complex using comparable basis sets, and evaluated the anharmonic constants using standard second-order perturbation formulas, based upon third and fourth derivatives of the SCF energy. This treatment evaluates each vibrational frequency, Vi, in terms of a purely harmonic potential i, and anharmonic constants xij relating the various modes i and j (assuming all modes are nondegenerate).

The results furnish an interesting comparison with Botschwina's work, both because of a different means of including correlation (MP2 vs. CEPA-1) and due to the alternate treatment of anharmonicity. The Amos group also tested the feasibility of adding anharmonic corrections to the intermolecular frequencies, as indicated in Table 3.65. It is first noteworthy that the MP2 correlation treatment significantly enhances the red shift of the HF stretch, as opposed to CEPA-1 which had only marginal effects upon any of the vibrational frequencies. The anharmonic correction lowers this red shift but only by a little, bringing the final shift into reasonable agreement with experiment. The MP2 shifts of the internal frequencies of the HCN molecule are considerably different than the SCF values, with anharmonicity having an enlarging effect in one case, the CN stretch, but reducing the shift in the other two modes. In the case of the intermolecular frequencies, all the SCF values are too high, in comparison to experiment. These frequencies are further raised at the MP2 level. But when anharrnonicity is included, they are lowered. As a result, the V and shearing frequencies are quite close to experiment although the bending frequency of the proton acceptor remains too high. The authors considered the question as to whether a second-order perturbation treatment is appropriate for the case of H-bonds. They concluded that terms higher than quartic should typically be considered if possible.

188

Hydrogen Bonding

Table 3.65 Harmonic and anharmonic frequencies (in c m - 1 ) of HF and HCN, and the changes resulting from formation of the HCN .. HF complex, along with intermolecular modes. Data44 calculated with doubly polarized basis set. SCF

Mode

V

mon

MP2 V

V

mon

Anharmonic V

expt

v

Intramolecular

HF V

4322

HF

-189

3955

-238

-226

HCN V

HC

V

CN

3638 2437 878

bend acceptor bend H-bond stretch (v ) donor bend (shear)

0 3511 31 2050 17 727 Intermolecular frequencies 108 122 176 195 645 697

5 45 8

2 53 0 101 169 539

72 ±4

163 550

Both electrical and mechanical anharmonicity can be considered in the calculations. This was done92 using higher energy and dipole moment derivatives. Despite the use of relatively small basis sets, the anharmonicity constants were in surprisingly good coincidence with experiment. The infrared and Raman intensities computed for the HCN . . HF complex are listed in Tables 3.66 and 3.67 for the intramolecular and intermolecular modes, respectively. Like the red shift, the intensification of the HF stretch is smaller for this dimer than for H 3 N .. HF, indicative of the weaker binding. The Raman band is strengthened by a factor of 2.5. The three vibrations of the HCN monomer all have reasonable IR intensity, varying between 10 and 80 km m o l - 1 . While the CN stretch is intensified by about 2.6, the other two modes are relatively unaffected by complexation with HF. Raman intensities are all increased slightly. The intensity of the intermolecular H-bond stretching band is fairly small, only 3 km m o l - 1 , in HCN .. HF, which matches quite closely to the same quantity in H 3 N .. HF. This similar-

Table 3.66 IR and Raman intensities of HF and HCN, and the changes resulting from formation of the HCN . . HF complex. Data calculated with TZ2P basis set6. IR (km mol-1 ) Mode

A 147

V

ITC

V

V

CN

bend

76 9.7 66

Raman (A4/amu)

A..

/A

HF 4.8 HCN 1.1 2.6 0.9

S

S,. /S

26

2.5

14.1 48 1.8

1.1 1.1 1.1

Vibrational Spectra

189

Table 3.67 Vibrational spectral intensities calculated for intermolecular modes of HCN .. HF 6 with TZ2P basis set. AIR (km mol - 1 )

Mode H-bond stretch (V ) donor bend (shear) acceptor bend

SRaman (A4/amu) 0 4 6

3 346 30

ity indicates that the intensity is less sensitive to the strength of the H-bond as a reflection of the fact that the mode in both cases is a simple stretching motion that involves little reorientation of the subunits, and hence only small changes in dipole moment of the complex. In common with H 3 N .. HF, the bending motion of the proton donor is rather intense, followed by a weaker acceptor bend. It is notable that the latter two intensities are significantly stronger in HCN .. HF than in H 3 N .. HF. The Raman intensities are weak for all three intermolecular modes, V in particular. Bouteiller and Behrouz93 re-examined the question of anharmonicity in the HCN .. HF complex, with a particular eye toward the effects of basis set superposition. As in approaches of this sort, a two-dimensional grid of points was constructed and the energies fit to a polynomial of r(HF) and R(N .. F), up to fourth order. They found first that the SCF BSSE for this complex is 0.58 kcal/mol, as compared to an uncorrected interaction energy of 6.69 kcal/mol. The authors next reoptimized the r and R parameters of the optimized geometry and found a change of less than 0.01 A in the former while R elongates by 0.02 A, from 2.872 to 2.892 A. At the correlated MP2 level, the interaction energy is reduced 25% by BSSE and the optimized R(N .. F) stretched by 0.06 A. In the next step, the two-dimensional PES V(r,R) was refit to the calculated grid of points, but with counterpoise correction of the BSSE for each energy. The results are presented in Table 3.68 which is divided into frequencies derived using the harmonic approximation and anharmonic data taking account of the fourth-order polynomial description of the PES. It appears that the SCF harmonic frequencies are barely altered at all by BSSE. In fact, the only harmonic frequency to be affected is the vs stretch, v(FH), which is increased by 12 cm-1 at the MP2 level when the BSSE is included. When the treatment is expanded to include anharmonicity, there is again virtually no effect on either frequency from BSSE. However, MP2 calculations do show significant changes: account of superposition error raises the vs frequency by 32 cm-1 and lowers v(F..N) by a Table 3.68 Vibrational transitions (in c m - 1 ) calculated for HCN ... HF 93 with and without correction for BSSE. Harmonic

SCF SCF+BSSE MP2 MP2+BSSF

Anharmonic

v(FH)

..

v(F N)

v(FH)

v(F .. N)

4202 4202 3785 3797

167 164 201 201

4010 4009 3604 3636

168 151 201 171

190

Hydrogen Bonding

similar amount. The latter changes would be consistent with a weakening of the interaction between the two molecules, which is in fact a direct result of correcting the BSSE. Shortly thereafter, Bouteiller94 continued on this same line of reasoning and considered combination bands and intensities. The data in Table 3.69 represent transitions of some combination of the v(FH) and v(F .. N) bands, as indicated. At both the SCF or MP2 levels, inclusion of BSSE corrections reduce the transition energies for excitation of the v(F..N) mode from the ground state: ]00> -> |0n>, as may be seen from the first four rows of Table 3.69. The values listed in parentheses refer to the spacing between the successive transitions. The counterpoise corrections reduce this spacing. In all cases, this spacing diminishes with n, indicative of the mechanical anharmonicity of the H-bond. It is worth noting, however, that there is less such reduction when superposition errors are corrected. The next sets of transitions all include an excitation of the v(FH) mode by one quantum, that is, |0n> -> ln'>. The main transition, |00> -> |10>, is lowered by BSSE correction by a very small amount at the SCF level, but increases by 31 cm-1 for MP2. For any given progression, |0n> —> ln'> with fixed n, there is a reduced spacing between lines as n' rises, due again to mechanical anharmonicity. The SCF spacings are relatively unaffected by BSSE whereas correction of this error yields consistently reduced spacing at the MP2 level. Whether or not BSSE is corrected, the MP2 spacings are consistently larger than their SCF correlates. The authors also investigated the intensities of their various combination excitations. The results led to the conclusion that the greatest intensities arise from the simple |00> —> |10> transition. Also rather intense are the |0n> -> |ln> transitions wherein the v(F..N) mode remains at the same level. Del Bene et al.95 computed the vibrational spectra of the methylsubstituted complex of CH3CN..HC1. Their intermolecular frequencies for the C3v equilibrium structure were in

Table 3.69 Vibrational transitions (in c m - 1 ) calculated for HCN...HF94 with and without correction for BSSE.a Values in parentheses indicate increase relative to transition in preceding row. SCF

SCF + BSSE

MP2

MP2 + BSSE

100> 100> 100> 100>

101> 102> 103> 104>

164 315(151) 455 (140) 584 (129)

150 287(137) 415(128) 539(124)

200 388(188) 562(174) 725(163)

171 326 (155) 470(144) 611 (141)

100> 100>

110> lll>

4028 4217(189)

4019 4209 (190)

3611 3838 (227)

3642 3839 (197)

101 101> 101

110> lll> 112>

3864 4053 (189) 4216(163)

3869 4059 (190) 4211(152)

3410 3638 (228) 3841 (203)

3471 3669 (198) 3843(174)

102> 102> 102> I02>

110> 111> 112> 113>

3713 3901 (188) 4064(163) 4213 (149)

3732 3922(190) 4074(152) 4214(140)

3223 3451 (228) 3654 (203) 3844(190)

3316 3513(197) 3688(175) 3848 (160)

a

ij> refers to v(FH)and v(F . . N) excitations, respectively.

Vibrational Spectra

191

good agreement with experimental values96. For example the calculated V is 117 c m - 1 , as compared to 97±3 from the FTIR spectrum. This value is somewhat smaller than estimates of the unmethylated complex with HF in HCN..HF. The mode corresponding to bending of the proton donor has calculated and experimental frequencies of 417 and 350± 100 cm - 1 , respectively. The out-of-phase bending of the monomers is of much lower frequency, 35 cm-1 in the calculations as compared to 40±20 cm-1 in the experiment. 3.12.2 HCN as Proton Donor Unlike most C—H bonds which are poor donors, the triple bond to the carbon atom makes HCN a rather effective proton donor molecule. The frequency shifts encountered in the two subunits when HCN is combined with NH3 as proton donor are listed in Table 3.70. There is a substantial red shift of the HC stretch, corresponding to the vs band. This shift is reproduced surprisingly well with the TZ2P basis set, even though the calculations are at the SCF level. The CN stretching frequency is also lowered while the bend undergoes a substantial shift to the blue. These changes are quite a bit different than when HCN acts as proton donor, as with HF in Table 3.62. In such a case the HC stretching frequency is changed only very little and the CN stretch shifts a small amount to the blue. The bending mode is increased in either case but by a much smaller amount when HCN acts as proton acceptor. It is interesting as well to compare the behavior of NH3 when it interacts with HCN as compared to a strong proton donor like HF. In the former case, both stretching frequencies shift to the red a small amount, whereas little change occurs with HF. In either case, the a1 bending frequency increases by nearly 100 cm - 1 , while the other bending mode is essentially unaffected. The intensifications of the IR and Raman bands in Table 3.71 exhibit the expected increase of the IR band of the HC stretch, comparable in magnitude to the change of the vs band in HCN...HF. On the other hand, this same band is weakened in the Raman spectrum, opposite to the HF stretching band in HCN...HF. It is particularly intriguing to note the even stronger intensifications of both the IR and Raman bands for the CN stretching mode, again very much more exaggerated than when HCN acts as proton acceptor. Even the bending mode shows significant enhancement in NCH ... NH 3 . The effects of complexation on the IR

Table 3.70 Frequency shifts (in crn -1) of NH3 and HCN arising from formation of the NCH ... NH 3 complex. Data calculated at SCF level6. Mode

TZ2P

expta

-189

-162

-162

-29 153

-26 131

-11

DZP

HCN V

HC

V

CN

V

bend

V

a

str( 1) vstr(e) v b e n d (a 1 )

vbend(e) a

Harmonic frequencies.

NH3 -9 -18 93 0

-7 -12 68 1

192

Hydrogen Bonding

Table 3.71 IR and Raman intensities of NH3 and HCN, and the changes resulting from formation of the NCH ... NH 3 complex. Data calculated with TZ2P basis set6. Mode

V V

V

Raman

dim/smon s

HCN 4.7 7.3 1.2 NH3

HC

CN

bcnd

0.7 8.3 1.8

100a

v str (a 1 ) v str (e) v bcnd (a 1 )

0.9 1.0 0.2 0.9

3.4 1 1.1

vbend(e) a

IR A dim /A mon

Intensity in monomer too small for ratio to be meaningful.

intensities of the NH3 subunit are comparable to those in FH ... NH 3 , although there are some quantitative differences. The intermolecular frequencies in Table 3.72 are similar in pattern to other H-bonded complexes6'97. The H-bond stretching frequency, in excellent agreement with experiment, is smaller than in HCN ... HF or FH...NH3, consistent with a weaker bond. The highest intermolecular frequency is again the pivoting of the proton donor molecule. A comparison of SCF and MP2 data indicates that correlation does not have a profound effect upon these intermolecular frequencies. Its principal effect is an increase in the H-bond stretch frequency. The intensities in Table 3.73 are in keeping with comparable systems. The H-bond stretch is of low IR intensity, with the donor bend much stronger. 3.12.3 HCNDimer It is of interest also to examine the dimer of HCN where one molecule acts as donor and the other as acceptor. The results in Table 3.74 fit and confirm the patterns when HCN is involved in complexes with other molecules. When acceptor, the HC frequency is changed

Table 3.72 Vibrational frequencies (cm - 1 ) calculated for intermolecular modes of NCH ... NH 3 . SCFa

MP2b

Mode

DZP

TZ2P

6-3 1G*

6-31+G(3df,2p)

expt

H-bond stretch (v ) donor bend (shear) acceptor bend

160 323 120

146 302 113

190 308 123

158 283 132

141 ± 3

a b

Data from Reference 6. Data from Reference 97.

Vibrational Spectra

193

Table 3.73 Infrared and Raman intensities calculated for intermolecular modes of NCH ... NH 3 6 with TZ2P basis set.

Mode

AIR(km mol-1)

SRaman (A4/amu)

2 158 4

H-bond stretch (v ) Donor bend (shear) Acceptor bend

0 0 10

very little, while the other two modes are blue shifted by some 10-30 cm - 1 . (Note however, that the experimental result appears to have a small increase for the CH frequency.) The IR intensity of the CN mode is increased, but the two other modes are changed only little. When HCN acts as donor, the HC stretch undergoes a strong red shift. A smaller red shift occurs in the CN stretch and a surprisingly large blue shift is noted in the bending mode, comparable in magnitude to the vs shift. Intensity increases are observed in both stretching modes, and a smaller enhancement in the bend. The calculated intermolecular H-bond stretching frequency is surprisingly close to the experimental value in Table 3.75. This frequency of some 120 cm-1 is smaller than in the HCN..HF complex or that in NCH...NH3, confirming that HCN is neither as strong a proton donor as HF nor as good an acceptor as NH3. The IR intensity of this band is again quite weak, as in all the other complexes where symmetry restrains this mode from including much reorientation of the two subunits. The donor bend remains the mode of highest frequency and the acceptor bend the lowest. The intensity of the former vibration is by far the strongest of the intermolecular modes. Somewhat later calculations by Kofranek et al.98 were able to incorporate correlation into the spectral data of the HCN dimer. The SCF data in Table 3.7699,100 confirm the patterns obtained earlier by Somasundram et al.6 with a different basis set. The correlated results in the next two columns of the table indicate no major changes in patterns. Curiously enough, the inclusion of correlation reduces all of the red shifts while enlarging all the blue

Table 3.74 Frequency shifts and IR intensity enhancements of the two monomers in HCNH ... CN complex. Data calculated at SCF level using DZP basis set6. Frequency shifts ( c m - 1 ) Mode

V

HC

V

CN

V,bend,

V

HC

V

CN

V

bend

a

Harmonic frequencies.

calc

-69 -11 76 -2 16 13

expta HCN (donor) -65 —9

HCN (acceptor) 11 8

Intensity factor (Adim/Amon) calc

expt

4.1 4.8 1.3

1 2

0.8 4.0 1.1

1.0 3

194

Hydrogen Bonding

Table 3.75 Vibrational frequencies and IR intensities calculated for intermolecular modes of the NCH ... NCH complex at SCF level6. v (cm - 1 )

A(kmmol-1)

Mode

calc

expt

calc

H-bond stretch (v ) donor bend (shear) acceptor bend

122 159 60

119

2 122 10

40a

a

See references 104 and 105.

shifts to higher frequency. The comparison with the experimental shifts in the last column of Table 3.76 is particularly encouraging. From Table 3.77, it appears that correlation exerts only a minor effect upon the CH stretching intensities. The absolute values of the monomers are diminished somewhat, but the dimerization-induced magnification of both the donor and acceptor are increased relative to the SCF results. The internal HCN bends are virtually unaffected. These trends are consistent with the SCF results of Somasundram et al.6 achieved with a different basis set, and recorded in Table 3.74. But quite dramatic effects are observed in the CN stretches. In the first place, correlation reduces the intensity of the HCN monomer band by an order of magnitude. Regarding the intensification caused by dimerization, SCF calculations from Tables 3.74 and 3.77 predict a factor of perhaps 3-5. But this magnification enlarges when correlation is included: the dimer/monomer ratio is calculated to be 10 for the proton acceptor and 64 for the donor. Recent experimental measurements support the correlated data in that enhancements of approximately 30 are seen. Analogous information about the intermolecular frequencies and intensities are exhibited in Table 3.78. The SCF data are consistent with those in Table 3.75 for a different basis set. It appears that correlation has a surprisingly small effect on any of the frequencies. The intensity of the lower-frequency bending mode is also changed very little by correla-

Table 3.76 Harmonic frequencies (in c m - 1 ) of HCN, and the changes resulting from formation of the (HCN)2 complex. Data98 calculated with polarized basis set [641/41]. SCF Mode

V

CH

V

CN

V

bend

V

CH

V

CN

V

a

bend

V mon

-64 -11

868

60

3618 2418

-7 9 13

Harmonic freqilencies 99 . See Reference 100.

b

v

3618 2418

868

expta

CPF V mon

HCN (donor) 3493 2170 695 HCN (acceptor) 3493 2170 695

v

V mon

v

-56 -3 76

3441a 2129a 721b

-65 -6 77

-1 13 15

344 la 2129a 721b

11 8 13

Vibrational Spectra

195

Table 3.77 Calculated intensities (km mol - 1 ) of HCN, and the changes resulting from formation of the (HCN)2 complex. Data98 calculated with polarized [641/41] basis set. CPF

SCF

Mode

mon A

dim/Amon

Amon

expta

Adtm/Amon

dim/Amon

4.7 64.3 1.03

30±10

1.03 10.2 0.84

30±10

HCN (donor) V

CH

V

CN

V

bend

V

CH

V

3.7

75.2 10.3 83.0

0.93 3.1 0.88

62.3 0.12 86.7

3.9 1.0

HCN (acceptor)

CN

V

75.2 10.3 83.0

bend

62.3 0.12 86.7

a

See reference 99.

tion. On the other hand, correlation approximately doubles the intensity of the H-bond stretch, while diminishing that of the other bend by about one third.

3.13

Amide

Following earlier calculations with smaller unpolarized basis sets101, the vibrational spectrum of the complex pairing together two formamide molecules was later computed by stergard et al.102. The data obtained with their best basis set, 6-31 + +G**, are reported for the intramolecular vibrations at the SCF level in Table 3.79. We use the authors' original nomenclature for the individual modes wherein refers to a torsion, to a wag, v to a stretch, 8 to scissoring, r to rocking, and to an out-of-plane bend. The v(NH2) stretch probably most closely corresponds to the vs band of the proton donor. Table 3.79 indicates a red shift of about 30 c m - 1 , as contrasted to only 3 or 4 cm-1 in the acceptor formamide molecule. This shift is rather small for a H-bonded complex, smaller, for example, than the shift of the C—H proton in (HCN)2. Also of particular interest is the frequency of the C=O stretch in the acceptor molecule, which diminishes by 27 c m - 1 . This red shift contrasts with the same C=O stretch in the complex of H2CO with H2O, where little change is seen. In addition to these bands, there are a number of other in-plane motions whose frequencies change by 30 cm-1 or less. The largest shift occurs for the out-of-plane wag of the NH2

Table 3.78 Vibrational frequencies and IR intensities calculated for intermolecular modes of the NCH ... NCH complex98. v(cm-1)

A(km mol - 1 )

Mode

SCF

CPF

SCF

CPF

H-bond stretch (v ) bend bend

106 139 48

116 125 38

1.3 128 13

2.8 84 16

196

Hydrogen Bonding

Table 3.79 Frequencies (in c m - 1 ) of formamide, and the changes resulting from formation of the dimer.a Data calculated at SCF level with 6-31 + +G** basis set102. Mode

vvmon

donor

V

acccptor

in-plane(a')

(NCO) r(NH2) v(CN)

(CH) (NH2)

v(CO) v(CH) v(NH2) v(NH2) (NH2) (NH2)

(CH)

618 1154 1368 1547 1773 1966 3193

14 30 21 —2

19 -11 24 -30 -29

3835

3982 out-of plane(a") 252

664 1175

253 78 6

4 11 18 1 3 -27 -4 -4

88 15 5

a

refers to a torsion, to to a wag, v to a stretch, to scissoring, r to rocking, and 7 to out-of-plane bend.

group. The donor frequency doubles upon formation of the H-bond, and an increase of 88 cm-1 occurs in the acceptor. This change can be accounted for based upon the low energy cost of wagging in the isolated monomer. But this same motion would disrupt the H-bond in the dimer, so the frequency rises accordingly. Intermolecular frequencies, calculated with various basis sets, are listed in Table 3.80. The authors point out that the normal modes are far from pure and that their nomenclature is very approximate. For example, the H-bond stretch contains a strong element of donor libration. Their designation of "strain" refers to simultaneous in-phase rotations of the two molecules, while they use the term "bend" to indicate out-of-phase rotations. The imaginary frequencies of the torsional modes are evidence that the true equilibrium geometry of the formamide dimer is nonplanar. The v mode appears at about 120 c m - 1 , a little smaller than that for H2CO...HOH where a water molecule donates a proton to the carbonyl oxy-

Table 3.80 Frequencies (in c m - 1 ) of intermolecular vibrational modes of the formamide dimer. Data calculated at SCF level 102. Mode

4-31G

6-31G**

6-31 + +G**

in-plane(a') stretch, v a bend strain

144 21 43

122 23 59

114 21 48

torsion bend strain

out-of-plane(a") 3; 30 117

28; 28 72

15; 21 81

Vibrational Spectra

! 97

gen of formaldehyde, and fairly close to the V frequency in H2COH...C1. There is a moderate level of sensitivity of the frequencies to the basis set. Outside of overestimating v and the out-of-plane strain mode, 4-31G does surprisingly well. The force constants in the cyclic formamide dimer were explicitly evaluated with the 6-31G** basis set103 at the SCF and MP2 levels. The intramolecular diagonal force constants are listed in Table 3.81 for the CN, CO, and NH stretches, the latter being the H atom that forms the H-bond bridge in the dimer. If one takes the magnitude of this force constant as a measure of the bond strength, the CO bond is considerably stronger than CN (formally a single bond), which is in turn stronger than NH. The entries for k in Table 3.81 represent the changes that occur in each force constant as the dimer is formed out of its constituent monomers. The NH and CO bonds are both weakened while the CN bond becomes stronger. These changes are consistent with the shifts of the corresponding vibrational frequencies. The changes in the force constants are in the neighborhood of 7-10% at the SCF level. Correlation reduces all of the force constants, the usual expectation for bonds of any sort. Of greater interest are the effects of correlation upon the changes in the force constants caused by dimerization. While the percentage change in the CO force constant is little affected by correlation, the increase in the CN constant rises to over 10%. Even more dramatic is the N—H reduction which amounts to 18% at the MP2 level. This decrement is nearly double that observed at the SCF level. This effect is consistent with the large correlation-induced enhancement in the red shift of vs associated with H-bond formation. It might be expected that the theoretical method that predicts the strongest interaction between the two monomers should also yield the largest force constant for pulling the two molecules apart. This supposition can be tested in Table 3.82 which indicates it is not fully reliable. The minimal basis set predicts a very large force constant of 0.27 mdyn/A and a binding energy of 16.0 kcal/mol. The next larger set has a stronger interation energy but a much smaller force constant. As the method continues to improve, however, the pattern does obey the expected connection between reduced binding energy and smaller force constant.

3.14

Summary

SCF frequencies typically overestimate the various vibrational frequencies in H-bonded complexes as well as in their constituent subunits. Once correlation is added, with MP2 usually a satisfactory approach, the computations can match fairly well the experimental spectra, particularly if the basis set is large and flexible enough. Calculation of accurate vibra-

Table 3.81 Computed intramolecular force constants (mdyn/A) of the cyclic formamide dimer103. SCF Stretching mode

CN CO NH a

MP2

k

ka

k

ka

8.98 14.68 7.71

0.76 -1.10 -0.75

8.45 12.26 6.67

0.89 -0.82 -1.18

Difference between force constant in dimer versus isolated monomer.

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Hydrogen Bonding

Table 3.82 Comparison of force constant for intermolecular stretch of the cyclic formamide dimer with the computed binding energy, without BSSE correction. Data103 are at SCF level unless otherwise indicated.

k(v ), mdyne/A — Eelec, kcal/mol

MINI-1

MID1-1

4-31G

6-31G**

MP2/6-31G**

0.27 16.0

0.17 20.3

0.15 17.2

0.12 13.4

0.15 17.4

tional intensities is somewhat more demanding, and the results less stable with respect to changes in basis set. In conjunction with the stretch of the A—H bond that occurs in the donor molecule upon formation of a H-bond, the stretching frequency of this bond is lowered by a good deal, as much as several hundred c m - 1 . This red shift, coupled with a strong intensification of the band, is characteristic of H-bond formation. The H-bonded complex contains a number of intermolecular vibrational modes that do not exist within the separated monomers. Because the H-bond is so much weaker than covalent bonds, the frequencies of these modes are generally less than 1000 c m - 1 , sometimes below 100 c m - 1 . The precise nature of these modes differs from one system to the next but one can normally recognize several common ones, such as a H-bond stretch, v , and bending motions of the donor. The frequencies are usually correlated with the strength of the interaction between the two subunits. A normal mode analysis of (HF)2 reveals the mixing together of the various intermolecular stretches and bends, and underscores a certain amount of arbitrariness in their identification. The internal stretch in the proton donor molecule is calculated to shift to the red by 100 c m - 1 , in good agreement with experimental measurement. The acceptor stretching frequency is also diminished but by only about one third as much. The intensity of the donor stretch is calculated to increase by a factor of 3 upon dimerization, while that of the acceptor is unchanged. The intermolecular stretch is in the range of 200 c m - 1 ; proton donor wags are 500-600 cm~'. Vibrational spectral data for the analogous (HC1)2 are similar with the proviso that the complex is more weakly bound. The red shifts of the HC1 stretches are consequently smaller in magnitude; V is only around 60 c m - 1 , and the intermolecular donor wags amount to less than 300 c m - 1 . A complex like H 3 N ... HF is much more strongly bound so one might expect larger effects from formation of the H-bond. Indeed, the red shift in the proton donor HF molecule amounts to more than 400 c m - 1 , about five times greater than in (HF)2, at the same level of theory. The intensification of this band is greater than sevenfold, again several times larger than in (HF)2. The proton acceptor NH3 has more internal modes than does HF. Most of its frequencies are little affected by the complexation, with the exception of a 100 cm-1 increase in its symmetric bend. The intensities of the stretching modes of the NH3 molecule are very strongly increased. Replacement of NH3 by PH3 leads to a weakening of most of the patterns in H3N...HF, but there are certain anomalous changes as well. Analysis of the intensities in terms of atomic polar tensors yields insights into the electronic redistributions that accompany the formation of each H-bond. For example, the intensification of the v band is attributed to the fact that the bridging H is positive in the monomer and becomes more so in the complex. Coupled to this is the lesser ability of the charge cloud in the complex to follow the motion of this proton; indeed, the density seems to move in the

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199

opposite direction. Consequently, motion of the bridging proton yields a greater dipole enhancement than would occur in an isolated HF molecule. The intermolecular modes also provide clear evidence of the greater strength of H 3 N ... HF as compared to (HF)2. The frequency for wagging of the donor molecule is nearly 900 cm-1 and the H-bond stretch v is 240 cm - 1 . The C3v symmetry of this complex imparts to the latter mode a nearly pure stretching character. As the two molecules are pulled apart, there is little off-axis motion occurring, so the dipole moment of the complex changes very little. Consequently, the intensity of the v mode is quite weak. This sort of reasoning can be generalized to rationalize the observed intensities on the basis of the balance between stretches and bends in the mode in question. Anharmonicity corrections can be quite large in complexes of this type, up to several hundred c m - 1 . In FH...NH3 and C1H...NH3, for example, the anharmonic vs frequency is lower than the harmonic value by about 400 c m - 1 . It is interesting also that whereas correlation lowers the vs frequency, it has the opposite effect of increasing v . Nor can one expect cancellation to occur between anharmonic and correlation effects; in some cases, the two can be additive. Mechanical anharmonicity leads to a progressively smaller spacing between successive overtones of the v mode as the quantum number increases. Intermediate between the extremes of HP...HF and H3N...HF is the pairing of HF with a water molecule. The red shift of the HF stretch in H2O...HF is somewhat smaller than in H3N...HF, a little less than 400 c m - 1 ; the intensification of this band is about 5 in the former complex, as compared to 7 in the latter. Not many changes are observed in the frequencies of the proton acceptor water. The intermolecular frequencies are similar to H3N...HF; the H-bond stretch is around 200-300 c m - 1 , and the wags of the donor molecule approach 1000 c m - 1 . Changing one atom from first to second-row type seems to have similar effects, regardless of the nature of the atom chosen. The overall effect is a weakening of the H-bond, observed here in the spectral data, confirming the energetic information in the previous chapter. For example, the stretching frequency of the H2O...HC1 complex is comparable to that of H3P...HF. This weakening continues to progress as F is changed to Cl, then to Br and I. The red shift of the HX band drops by severalfold, as does the intermolecular H-bond stretch frequency. Alkyl substitution on the water changes the spectrum by only a little. There is a curious increase in the red shift of the HC1 stretch in (CH3)2O...HC1, however, as compared to H2O...HC1. Stronger H-bonds also have enhanced effects upon the electric field gradient; changes of the order of 10% are calculated for complexes like H2O...HC1. Despite significant mixing with other types of intermolecular motion in the water dimer, one can recognize a H-bond stretch as one normal mode, as well as bends of the donor and acceptor molecule. The force constant for the OH stretch in the proton donor molecule obeys trends quite similar to the H-bond energy itself. That is, k is smallest for the equilibrium geometry and rises only slowly as the proton acceptor molecule is disoriented, but more quickly for rotations of the donor. By one analysis, the drop in frequency in the proton donor stretching frequency is a direct consequence of the bond's stretch in the H-bond, with r and vs being nearly linearly related. This longer bond is more polar and polarizable, enabling it to form a stronger H-bond. Indeed explicit computations of polarizability support the notion that the formation of the H-bond induces a noticeable change in the polarizability in the H-bond direction. When HOH is the proton donor molecule, the vs band is not as pure a single X H bond stretch as for HX, since the normal modes of the HOH molecule contain a symmetric and

200

Hydrogen Bonding

antisymmetric stretch, each of which involve both H atoms. It is the symmetric stretch which corresponds most closely to a vs band in the dimer in that its frequency shifts more to the red and its intensity is increased by a greater amount. The latter intensification is computed to be an order of magnitude, but it must be understood that some fraction of this increase is due to the change in the character of the vibrational motion itself, as opposed to perturbations of the electronic structure. The bending mode in the donor molecule is shifted toward the blue and suffers a small diminution in intensity. The v frequency of the water dimer is computed to be some 175 c m - 1 , similar to that in (HF)2, just as their H-bonds are close in energy. The mode corresponding roughly to a proton donor wag is about 600 c m - 1 , also similar to the HF dimer. In concurrence with the weak bonding between HOH and NH3, the red shift of the vs band is rather small, only about 170 c m - 1 , at the correlated level, but the magnification of its intensity is surprisingly large, nearly 90-fold. The V frequency reflects the weakness of this H-bond, barely over 100 c m - 1 , although a value of double this amount is obtained from experimental measurement in a matrix. The interaction between a pair of NH3 molecules is even weaker. The vibrational spectrum of the linear arrangement of the H-bond (not a true minimum on the surface) does not lend itself readily to clear identification of intermolecular bands as bends, stretches, and so on. The frequency corresponding most closely to a Hbond stretching motion is barely over 100 c m - 1 . The largest red shift in the proton donor is less than 50 cm-1 in one of its bending modes. A comprehensive comparison of various basis sets for the homodimers of HF and H2O offers hope that calculation of vibrational frequencies can be meaningful, even when restricted to the SCF level and with no account of anharmonicity. The frequencies are less demanding of basis set quality than are the intensities. Minimal basis sets are to be avoided in most cases, as are small split-valence sets such as 3-21G. In some cases, one can compute reasonable estimates of dimerization-induced frequency shifts with basis sets of 4-31G type; however, results with unpolarized basis sets can be deceptive. Polarization functions are strongly recommended for uniform quality of results, particularly if one is interested primarily in spectral changes induced by H-bond formation. Intensity calculations without polarization functions can be expected to yield only the crudest of estimates. Reasonable results can be achieved with only one set of such functions on each atom. In some cases, it may be useful to include diffuse "+" functions as well. When engaged in a H-bond as proton acceptor, the carbonyl C=O bond is weakened somewhat, based upon a lowered stretching frequency. This red shift appears to be linearly correlated with the H-bond energy. The relationship between frequency shift and energy is linear also for the proton donor molecule, namely the O—H stretch of water. When water is paired with H2CO, the vibrational spectrum of the donor (water) is very much like that in the water dimer, as is the H-bond stretching frequency, v . The intensities are different, probably due in large part to the different nature of the modes themselves in the two complexes. Along this line, changes induced in the spectrum of other proton-donor molecules like HC1 are similar for H2CO as compared to H2O. The C=N double bond in the imine group is also shifted to the red when this group accepts a proton. In contrast to some of the aforementioned cases, the stretching frequency of the C N triple bond in the nitrile group shifts to the blue when this group accepts a proton in a Hbond. This shift amounts to some 20-30 cm-1 when HCN is paired with HF, and the band is intensified by a factor between two and three. Also shifted toward the blue is the bending frequency of the HCN molecule. The red shift in the HF stretch within the donor molecule is considerably smaller than when HF donates a proton to the more basic amines, as

Vibrational Spectra

201

is the magnification of the intensity of this mode. The H-bond stretching frequency, v , is around 150 c m - 1 . Its purity as a simple stretching motion, which has little effect upon the overall dipole moment of the complex, leads to a weak intensity, comparable to that in H3N...HF. The effects of both correlation and anharmonicity are not yet entirely clear; different means of evaluating these factors lead to different conclusions. Nonetheless, the methods concur that the C N stretch is blue-shifted on H-bond formation. The triple bond in the nitrile group makes the C of HCN electronegative enough to act as an effective proton donor. When complexed with a strong acceptor like NH3, the HC stretching frequency is reduced by some 160 c m - 1 .Further evidence of a genuine H-bond is the fivefold enhancement of its intensity. As opposed to the blue shift of the C N stretch observed when the nitrile is a proton acceptor, this frequency is diminished when HCN donates a proton, and the intensity magnified by a factor of seven. The intermolecular stretching frequency is about 140 c m - 1 , with a very low intensity, again due to the lack of any bending character in this mode. As the molecules are enlarged, it becomes progressively more difficult to identify any single normal mode as the A—H stretch, vs. So it is that in the dimer of formamide this assignment is made to a stretching mode of the NH2 group of the donor molecule. The red shift in this band incurred by H-bond formation is quite small, only about 30 c m - 1 .The C=O stretch in the acceptor also drops by a similarly small amount. Similar impurity affects the intermolecular modes; the vibration corresponding most closely to v is about 120 c m - 1 . Analysis of the force constants in the cyclic dimer is consistent with normal Lewis structures: the C=O bond is stronger than C—N, and N—H is weaker still. Formation of the H-bond strengthens C—N but weakens the other two bonds, again congruent with the conventional picture of these bonds. References 1. Swanton, D. J., Bacskay, G. B., and Hush, N.S., The infrared absorption intensities of the water molecule: A quantum chemical study, J. Chem. Phys. 84, 5715-5727 (1986). 2. Botschwina, P., Rosmus, P., and Reinsch, E. A., Spectroscopic properties of the hydroxonium ion calculated from SCEP CEPA wavefunctions, Chem. Phys. Lett. 102, 299-306 (1983). 3. Hess, B. A. J., Schaad, L. J., Carsky, P., and Zahradnik, R., Ab initio calculations ofvibrational spectra and their use in the identification of unusual molecules, Chem. Rev. 86, 709-730 (1986). 4. Wilson, E. B. Jr., Decius, J. C., and Cross, P. C., Molecular Vibrations; Dover, New York, (1955). 5. Pople, J. A., Scott, A. P., Wong, M. W., and Radom, L., Scaling factors for obtaining fundamental vibrational frequencies and zero-point energies from HF/6-31G* and MP2/6-31G* harmonic frequencies, Isr. J. Chem. 33, 345-350 (1993). 6. Somasundram, K., Amos, R. D., and Handy, N. C., Ab initio calculation for properties of hydrogen bonded complexes H 3 N . . . HCN, HCN-HCN, HCN . . . HF, H 2 O . . . HF, Theor. Chim. Acta 69,491-503(1986). 7. Bacskay, G. B., Kerdraon, D. I., and Hush, N. S., Quantum chemical study of the HCl molecule and its binary complexes with CO, C2H2, C2H4, PH3 ,H 2 S, HCN, H2O, and NH3: Hydrogen bonding and its effect on the 35Cl nuclear quadrupole coupling constant, Chem. Phys. 144, 53-69 (1990). 8. Badger, R. M., and Bauer, S. H., Spectroscopic studies of the hydrogen bond. II. The shifts of the O—H vibrational frequency in the formation of the hydrogen bond, J. Chem. Phys. 5, 839-851 (1939). 9. Rao, C. N. R., Dwivedi, P. C., Ratajczak, H., and Orville-Thomas, W. J, Relation between O—H stretching frequency and hydrogen bond energy: Re-examination of the Badger-Bauer rule, .1. Chem. Soc., Faraday Trans. 71, 955-966 (1975).

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Hydrogen Bonding

10. Millen, D. J., and Mines, G. W., Hydrogen bonding in the gas phase, J. Chem. Soc., Faraday Trans. 273, 369-377(1977). 11. Andrews, L., Fourier transform infrared spectra of HF complexes in solid argon, J. Phys. Chem. 88, 2940-2949 (1984). 12. Yamaguchi, Y., Frisch, M., Gaw, J., Schaefer, H. F., and Binkley, J. S., Analytic evaluation and basis set dependence of intensities of infrared spectra, J. Chem. Phys. 84, 2262-2278 (1986). 13. Stanton, J. F., Lipscomb, W. N., Magers, D. H., and Bartlett, R. J., Correlated studies of infrared intensities, J. Chem. Phys. 90, 3241-3249 (1989). 14. Besler, B. H., Scuseria, G. E., Scheiner, A. C., and Schaefer, H. F., A systematic theoretical study of harmonic vibrational frequencies: The single and double excitation coupled cluster (CCSD) method, J. Chem. Phys. 89, 360-366 (1988). 15. Alberts, I. L., and Handy, N. C., M ller-Plesset third order calculations with large basis sets, J. Chem. Phys. 89, 2107-2115 (1988). 16. Simandiras, E. D., Rice, J. E., Lee, T. J., Amos, R. D., and Handy, N. C., On the necessity off basis functions for bending frequencies, J. Chem. Phys. 88, 3187-3195 (1988). 17. Miller, M. D., Jensen, F., Chapman, O. L., and Houk, K. N., Influence of basis sets and electron correlation on theoretically predicted infrared intensities, J. Phys. Chem. 93,4478-4502 (1989). 18. Handy, N. C., Gaw, J. F., and Simandiras, E. D., Accurate ab initio prediction of molecular geometries and spectroscopic constants, using SCF and MP2 energy derivatives, J. Chem. Soc., Faraday Trans. 2 83, 1577-1593 (1987). 19. Michalska, D., Hess, B. A., Jr., and Schaad, L. J., The effect of correlation energy (MP2) on computed vibrational frequencies, Int. J. Quantum Chem. 29, 1127-1137 (1986). 20. Gaw, J.F., Yamaguchi, Y., Vincent, M. A., and Schaefer, H. F., Vibrational frequency shifts in hydrogen-bonded systems: The hydrogen fluoride dimer and trimer, J. Am. Chem. Soc. 106, 3133-3138(1984). 21. Kurnig, I. J., Szczesniak, M. M., and Scheiner, S., Vibrational frequencies and intensities of Hbondedsystems. 1:1 and 1:2 complexes of NH3 and PH3 with HF, J. Chem. Phys. 87,2214-2224 (1987). 22. Frisch, M. J., Pople, J. A., and Del Bene, J. E., Molecular orbital study of the dimers (AHn)2 formed from NH3, OH2, FH, PH3 ,SH2, and CIH, J. Phys. Chem. 89, 3664-3669 (1985). 23. Latajka, Z., and Scheiner, S., Structure, energetics and vibrational spectra of H-bonded systems. Dimers and trimers of HF and HCl, Chem. Phys. 122,413-430(1988). 24. Michael, D. W., Dykstra, C. E., and Lisy, J. M., Changes in the electronic structure and vibrational potential of hydrogen fluoride upon dimerization: A well-correlated (HF)2 potential energy surface, J. Chem. Phys. 81, 5998-6006 (1984). 25. Frisch, M. 5., Del Bene, J. E., Binkley, J. S., and Schaefer, H. F., Extensive theoretical studies of the hydrogen-bonded complexes (H2O)2, (H2O)2H+, (HF)2, (HF)2H+, F 2 H - , and (NH 3 ) 2 J. Chem. Phys. 84, 2279-2289 (1986). 26. Collins, C. L., Morihashi, K., Yamaguchi, Y., and Schaefer, H. F., Vibrational frequencies of the HF dimer from the coupled cluster method including all single and double excitations plus perturbative connected triple excitations, J. Chem. Phys. 103, 6051-6056 (1995). 27. Dinur, U., Bond contraction and spectral blue-shift in hydrogen-bonded dimers. An atom-based molecular mechanics analysis, Chem. Phys. Lett. 192, 399-406 (1992). 28. Curtiss, L. A., and Pople, J. A., Ab initio calculation of the force field of the hydrogen fluoride dimer, J. Mol. Spectrosc. 61, 1-10 (1976). 29. Bunker, P. R., Jensen, P., Karpfen, A., Kofranek, M., and Lischka, H., An ab initio calculation of the stretching energies for the HF dimer, J. Chem. Phys. 92, 7432-7440 (1990). 30. Jensen, P., Bunker, P. R., Karpfen, A., Kofranek, M., and Lischka, H., An ab initio calculation of the intermolecular stretching spectra for the HF dimer and its D-substituted species, J. Chem. Phys. 93, 6266-6280 (1990). 31. Pine, A. S., Lafferty, W. J., and Howard, B. J., Vibrational predissociation, tunneling, and rotational saturation in the HF and DF dimers, J. Chem. Phys. 81, 2939-2950 (1984).

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53. Woodbridge, E. L., Tso, T.-L., McGrath, M. P., Hehre, W. J., and Lee, E. K. C, Infrared spectra of matrix-isolated monomeric and dimeric hydrogen sulfide in solid O2, J. Chem. Phys. 85, 6991-6994(1986). 54. Ventura, O. N., Irving, K., and Latajka, Z., On the dlmerization shift of the OH-stretching fundamentals of the water dimer, Chem. Phys. Lett. 217, 436 (1994). 55. Kim, J., Lee, J. Y., Lee, S., Mhin, B. J., and Kim, K. S., Harmonic vibrational frequencies of the water monomer and dimer: Comparison of various levels ofab initio theory, J. Chem. Phys. 102, 310-317(1995). 56. Huisken, F., Kaloudis, M., and Kulcke, A., Infrared spectroscopy of small size-selected water clusters, J. Chem. Phys. 104, 17-25 (1996). 57. Bakkas, N., Bouteiller, Y., Loutellier, A., Perehard, J. P., and Racine, S., The water-methanol complexes. L A matrix isolation study and an ab initio calculation on the 1—1 species, J. Chem. Phys. 99, 3335-3342 (1993). 58. Wuelfert, S., Herren, D., and Leutwyler, S., Supersonic jet CARS spectra of small water clusters, J. Chem. Phys. 86, 3751-3753 (1987). 59. Page, R. H., Frey, J. G., Chen, Y.-R., and Lee, Y. T., Infrared predissociation spectra of water dimer in a supersonic molecular beam, Chem. Phys. Lett. 106, 373-376 (1984). 60. Nelander, B., The intramolecular fundamentals of the water dimer, J. Chem. Phys. 88, 5254-5256 (1988). 61. Huisken, F., Kulcke, A., Laush, C., and Lisy, .1. M., Dissociation of small methanol clusters after excitation of the O-~H stretch vibration at 2.7 u, J. Chem. Phys. 95, 3924-3929 (1991). 62. Bleiber, A., and Sauer, J., The vibrational frequency of the donor OH group in the H-bonded dimers of water, methanol and silanol. Ab initio calculations including anhannonicities, Chem. Phys. Lett. 238, 243-252 (1995). 63. Schmitt, M., M ller, H., Henrichs, U., Gerhards, M., Perl, W., Deusen, C., and Kleinermanns, K., Structure and vibrations of phenol'CH ^OH (CD3OD) in the electronic ground and excited state, revealed by spectral hole burning and dispersed fluorescence spectroscopy, J. Chem. Phys. 103,584-594(1995). 64. Dibble, T. S., and Francisco, J. S., Ab initio study of the structure, binding energy, and vibrations of the HOC1-H20 complex, J. Phys. Chem. 99, 1919-1922 (1995). 65. Johnson, K., Engdahl, A., Ouis, P., and Nelander, B., A matrix isolation study of the water complexes ofCl2, CIOCI, OCIO, and HOC! and their photochemistry, J. Phys. Chem. 96,5778-5783 (1992). 66. Rohlfing, C. M., Allen, L. C., and Ditchfield, R., Proton chemical shift tensors in neutral and ionic hydrogen bonds, Chem. Phys. Lett. 86, 380-383 (1982). 67. Rohlfing, C. M., Allen, L. C., and Ditchfield, R., Proton chemical shift tensors in hydrogenbonded dimers ofRCOOH and ROM, J. Chem. Phys. 79, 4958-4966 (1983). 68. Kaliaperumal, R., Sears, R. E. J., Ni, Q. W., and Furst, J. E., Proton chemical shifts in some hydrogen bonded solids and a correlation with bond lengths, J. Chem. Phys. 91,7387-7391 (1989). 69. Chesnut, D. B., and Phung, C. G., Functional counterpoise corrections for the NMR chemical shift in a model dimeric water system, Chem. Phys. 147, 91-97 (1990). 70. Latajka, Z., Ratajczak, H., and Person, W. B., On the reliability ofSCF ab initio calculations of vibrational frequencies and intensities of hydrogen-bonded systems, J. Mol. Struct. (Theochem) 194,89-105(1989). 71. Sellers, H., and Almlof, J., On the accuracy in ab initio force constant calculations with respect to basis set, J. Phys. Chem. 93, 5136-5139 (1989). 72. Latajka, Z., and Scheiner, S., Structure, energetics, and vibrational spectrum of'H 3 N . . HOH, J. Phys. Chem. 94, 217-221 (1990). 73. Yeo, G. A., and Ford, T. A.. Ab initio molecular orbital calculations of the infrared spectra of hydrogen bonded complexes of water, ammonia, and hydroxylamine. Part 6. The infrared spectrum of the water-ammonia complex, Can. J. Chem. 69, 632-637 (1991). 74. Engdahl, A., and Nelander, B., The intramolecular vibrations of the ammonia water complex. A matrix isolation study, .1. Chem. Phys. 91, 6604-6612 (1989).

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75. Nelander, B., and Nord, L., Complex bet\veen water and ammonia, J. Phys. Chem. 86, 4375-4379 (1982). 76. Schieflce, A., Deusen, C., Jacoby, C., Gerhards, M., Schmitt, M., Kleinermanns, K., and Hering, P., Structure and vibrations of the phenol-ammonia cluster, J. Chem. Phys. 102, 9197-9204 (1995). 77. Sadlej, J., and Lapinski, L., Ab initio calculations of the vibrational force field and IR intensities of the ammonia dimers, J. Mol. Struct. (Theochem) 150, 223-233 (1987). 78. Yeo, G. A., and Ford, T. A., Ab initio molecular orbital calculations of the infrared spectra of hydrogen bonded complexes of water, ammonia, and hydroxylamine, J. Mol. Struct. (Theochem) 168,247-264(1988). 79. Yeo, G. A., and Ford, T. A., The combined use ofab initio molecular orbital theory and matrix isolation infrared spectroscopy in the study of molecular interactions, Struct. Chem. 3, 75-93 (1992). 80. Latajka, Z., and Scheiner, S., Correlation between interaction energy and shift of the carbonyl stretching frequency, Chem. Phys. Lett. 174, 179-184 (1990). 81. Thijs, R., and Zeegers-Huyskens, T., Infrared and Raman studies of hydrogen bonded complexes involving acetone, acetophenone and benzophenone I. Thermodynamic constants and frequency shifts of the vQHandvc=o stretching vibrations, Spectrochim. Acta A 40, 307-313 (1984). 82. Nuzzo, R. G., Dubois, L. H., and Allara, D. L., Fundamental studies of microscopic -wetting on organic surfaces. L Formation and structural characterization of a self-consistent series of polyfunctional organic monolayers, J. Am. Chem. Soc. 112, 558-569 (1990). 83. Gould, I. R., and Hillier, I. H., The relation bet\veen hydrogen-bond strengths and vibrational frequency shifts: A theoretical study of complexes of oxygen and nitrogen proton acceptors and water, J. Mol. Struct. (Theochem) 314, 1-8 (1994). 84. Ramelot, T. A., Hu, C.-H., Fowler, J. E., DeLeeuw, B. J., and Schaefer, H. R, Carbonyl-water hydrogen bonding: The H2CO~H2O prototype, J. Chem. Phys. 100, 4347-4354 (1994). 85. Rice, J. E., Lee, T. J., and Handy, N. C., The analytic gradient for the coupled pair functional method: Formula and application for HCl, H2CO and the dirtier H2CO'"HCl, J. Chem. Phys. 88,7011-7023(1988). 86. Bach, S. B. H., and Ault, B. S., Infrared matrix isolation study of the hydrogen-bonded complexes between formaldehyde and the hydrogen halides and cyanide, J. Phys. Chem. 88, 3600-3604 (1984). 87. Nowek, A., and Leszczynski, J., Ab initio study on the stability and properties ofXYCO—HZ complexes. III. A comparative study of basis set and electron correlation effects for H2CO- • • HCl, J. Chem. Phys. 104, 1441-1451 (1996). 88. Nowek, A., and Leszczynski, J., Ab initio investigation on stability and properties ofXYCO'"HZ complexes. H: Post Hartree-Fock studies on H2CQ-HF, Struct. Chem. 6, 255-259 (1995). 89. Migchels, P., Zeegers-Huyskens, T., and Peeters, D., Fourier transform infrared and theoretical studies of alkylimines complexes with hydroxylic proton donors, J. Phys. Chem. 95, 7599-7604 (1991). 90. Curtiss, L. A., and Pople, J. A., Molecular orbital calculation of some vibrational properties of the complex between HCNandHF, J. Mol. Spectrosc. 48, 413-426 (1973). 91. Botschwina, P. In Structure and Dynamics of Weakly Bound Molecular Complexes; Weber, A., ed. D. Reidel (1987) pp 181-190. 92. De Almeida, W. B., Craw, J. S., and Hinchliffe, A., Ab initio vibrational spectra of'HCN-HF arid HF"HCN hydrogen-bonded dimers: Mechanical and electrical anharmonicities, J. Mol. Struct. (Theochem) 200, 19-31 (1989). 93. Bouteiller, Y, and Behrouz, H., Basis set superposition error effects on electronic and VFX, VF..N stretching modes of hydrogen-bonded systems FX . . . NCX (X=H,D), J. Chem. Phys. 96, 6033-6038 (1992). 94. Bouteiller, Y., Basis set superposition error effects on vFx, VFX ,.N stretching modes of hydrogenbonded systems FX-NCH (X=H,D), Chem. Phys. Lett. 198,491-497(1992). 95. Del Bene, J. E.. Mettee, H. D., and Shavitt, I., Structure, binding energy, and vibrational frequencies ofCH 3 CN . . . HCl, L Phys. Chem. 95, 5387-5388 (1991).

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96. Ballard, L., and Henderson, G., Hydrogen bond energy of CH3CN—HCl by FTIR photometry, J. Phys. Chem. 95, 660-663 (1991). 97. Chattopadhyay, S., and Plummer, P. L. M., Ab initio studies of the mixed heterodimers of ammonia and hydrogen cyanide, Chem. Phys. 782, 39—51 (1994). 98. Kofranek, M., Lischka, H., and Karpfen, A., Ab initio studies on structure, vibrational spectra and infrared intensities ofHCN, (HCN)2, and (HCN)r Mol. Phys. 61, 1519-1539 (1987). 99. Hopkins, G. A., Maroncelli, M., Nibler, J. W., and Dyke, T. R., Coherent Raman spectroscopy of HCNcomplexes, Chem. Phys. Lett. 114, 97-102 (1985). 100. Pacansky, H., The infrared spectrum of a molecular aggregate. The HCN dimer isolated in an argon matrix, J. Phys. Chem. 81, 2240-2243 (1977). 101. Wojcik, M. J., Hirakawa, A. Y., Tsuboi, M., Kato, S., and Morokuma, K., Ab initio MO calculation of force constants and dipole derivatives for the formamide dimer. An estimation of hydrogen-bond force constants, Chem. Phys. Lett. 100, 523-528 (1983). 102. stergard, N., Christiansen, P. L., and Nielsen, O. F., Ab initio investigation of vibrations in free and hydrogen bonded formamide, J. Mol. Struct. (Theochem) 235, 423-446 (1991). 103. Florian, J., and Johnson, B. G., Structure, energetics, and force fields of the cyclic formamide dimer: MP2, Hartree-Fock, and density functional study, J. Phys. Chem. 99, 5899-5908 (1995). 104. Jucks, K. W., and Miller, R. E., The intermolecular bending vibrations of the hydrogen cyanide dimer, Chem. Phys. Lett. 147, 137-141 (1988). 105. Jucks, K. W., and Miller, R. E., Infrared spectroscopy of the hydrogen cyanide dimer, J. Chem. Phys. 88, 6059-6067 (1988).

4

Extended Regions of Potential Energy Surface

t is a common perception that when a molecule with an available proton is placed in the vicinity of another molecule that has a lone electron pair, the two will approach one another and quickly adopt their most stable, equilibrium H-bonded geometry. While this may be true in certain instances, the potential energy surface of a typical H-bonded complex is surprisingly complex. It is not unusual to find more than one minimum; the secondary minima can be only slightly higher in energy than the global minimum. The paths connecting the various minima can be rather complex as well, as the surface is littered with stationary points of second, third, and higher order, all bunched within a few kcal/mol of one another. Whereas the previous chapters have focused their attention on the equilibrium geometries of H-bonded complexes, and their immediate surroundings, this chapter examines broader reaches of the surface. Of particular interest are the paths along the surface that convert one minimum to another. The ammonia dimer furnishes an example of an extremely weakly bound complex, wherein the presence of a true H-bond is questionable. Its surface is extremely flat, furnishing a particularly stringent test of quantum chemistry to identify the true global minimum. Complexes pairing water with a hydrogen halide molecule offer the opportunity to compare H-bonding with other sorts of interaction. For example, the H2O...HX geometry contains a H-bond, but H 2 O ... XH does not. Even in the absence of a hydrogen bonding sort of interaction, the latter can be a true minimum on the surface of certain of these complexes. The simplicity of the HX dimer permits an especially thorough search of its PES. One can consider configurations that provide interesting contrasts to the traditional H-bond. The geometry of the equilibrium structure contains a nearly linear H-bond, with the remaining hydrogen nearly perpendicular to this axis. The pathway over the surface for interconversion of the roles of proton donor and acceptor represents an especially interesting problem, with spectroscopic manifestations. The dimer of water is intriguing in that it is not obvious from first principles that the linear H-bonded structure must be the only minimum on the surface, or indeed the global minimum. Other candidates include cyclic, bifurcated, trifur-

I

207

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Hydrogen Bonding

cated, and stacked geometries. By dissecting the total interaction energy of the water dimer into its constituents, it is possible to glean some insights into the underlying reasons for the shape of its PES. The larger groups add to the complexity of the potential energy surface and to the number of potential minima and stationary points. The pairing of H2O with H2CO is a case in point. The strong H-bond resulting from the pairing of an amine with HX brings up the possibility that a proton could transfer from the acid to the base so as to yield an ion pair, as an additional minimum on the PES.

4.1 Ammonia Dimer The earlier discussion of the most stable geometry for the ammonia dimer focused on the linear and cyclic arrangements. In one of the first considerations of extensive regions of the potential energy surface, Latajka and Scheiner1 varied the two angles which describe the orientations of the two molecules, as well as the internitrogen distance. The cyclic structure was found to be the only true minimum on the surface, but a very shallow trough leads from this geometry to a linear H-bond. The conversion from cyclic to linear stretches R(NN) from 3.15 to 3.34 A. The energy rises relatively rapidly if one climbs the walls on either side of this valley. There is hence a definite direction to the high-amplitude vibrational oscillations of the ammonia dimer. On the other hand, the complex can be easily pulled apart at any point along the valley; a stretching force constant of less than 0.12 mdyn/A was computed. Torsional motions of one of the NH3 molecules about its C3 axis are essentially free rotations in the vicinity of the linear arrangement but are stiffer for the cyclic geometry, where such a torsion would break any H-bonding interactions present. The results also provided a warning against limited searches, where certain parameters are held fixed. For example, the authors found that if the search is limited to only slices of the PES, each of which has a fixed internitrogen distance, the linear geometry can appear to be a "minimum." The two symmetrically related linear structures are then connected by a cyclic "transition state." Tao and Klemperer2 also evaluated the path connecting the two equivalent linear Cs geometries of the ammonia dimer, and passing through the cyclic minimum at its midpoint. Confirming the earlier calculations', they found this path to be remarkably flat, with the energy varying by only several cm - 1 . A number of studies investigated large domains of the surface by avoiding direct ab initio computations of the energy of each point. Liu et al.3 made use of an empirical potential function based upon the electrical properties of the ammonia monomer. Unfortunately their potential did not include the exchange which is necessary to prevent the collapse of the dimer. Hence, their surface was constructed as a slice, with constant R(NN), through the complete potential. Two degrees of freedom were considered, the angles that describe the deviation of the C3 symmetry axes of the two NH3 molecules from the N..N axis. Under these constraints, the authors identified 18 symmetry-related, equivalent minima on the surface, due to the 120° periodicity of the molecular rotations. Conversion from one minimum to the next tracks over an energy barrier of some 0.7 kcal/mol, emphasizing the flatness of the surface. Sagarik et al.4 examined the surface of the ammonia dimer via an empirical potential, fit to the results of correlated ab initio calculations. The analytical site-site potential contained separate terms modeling exchange repulsion, electrostatic interactions, and dispersion. The authors found a very shallow potential for pivoting one of the two NH3 molecules from the cyclic dimer geometry.

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209

While Hassett et al.5 did not consider the entire surface, they did characterize various candidate structures as to whether or not they were true minima. Their calculations were valuable since they involved full optimization of all degrees of freedom. They found that the linear type of dimer, wherein the two nonbridging hydrogens of the proton-donating molecule eclipse the pertinent hydrogens of the acceptor, appears to be a true minimum at most levels of theory, varying from SCF/6-31 +G(d) to MP2/6-311 +G(2d',p). Rotation of one molecule to form the staggered linear complex results in a single imaginary frequency at most levels tested, as does the cyclic structure. A "trifurcated" complex wherein all three hydrogens of one molecule are oriented in the general direction of the N of the other, is of high energy and is a second-order stationary point. The same is true of a variant of the cyclic complex where each molecule contributes two hydrogens, rather than one, to the region between the nitrogens. The results of this paper also furnished a good estimate of the zeropoint vibrational contributions to the binding energies of each structure. 4.2 H2O...HX Hannachi et al.6 sampled the energy surface of a trio of complexes of the type H2O...HX, where X=C1, Br, and I. The internal geometries of the two subunits were held fixed and the surface generated in terms of the intermolecular distance R and the angle 6 which measures orientation of the HX molecule. Calculations were limited to the SCF level, using an effective core, pseudo-potential approach, with the PS-31G** basis set. The generated maps covered a range of about 1.5 A in R and 240° segments of 9. For H2O...HC1, the surface contains a single minimum. Attempts to change the angle away from 0° can destabilize the system by roughly 8 kcal/mol for a perpendicular arrangement, before the energy starts to go down again. A second minimum appears in the surface at = 180° when X=Br. H2O...HBr is more stable than H2O...BrH by 4 kcal/mol. Indeed, this secondary minimum might disappear were zero-point vibrations added to the surface. The system must surmount an energy barrier of about 6 kcal/mol to reach this second minimum; the pathway involves a small amount of O...Br stretching along the way. The two minima are much better defined in H2O...HI and are more competitive in stability, differing by only some 0.6 kcal/mol. The energy barrier for conversion appears to be 4 kcal/mol. 4.3

(HX)2

In an early effort to consider tunneling of hydrogens in the HF dimer, Curtiss and Pople7 carried out an abbreviated scan of the SCF/4-31G surface. They varied the angle made by one of the HF molecules with the P...F axis in 20° increments, optimizing the other parameters at each point. The conversion from one linear type of H-bond to its equivalent, in which the proton donor and acceptor switch roles, was found to pass through a transition state, characterized by a C2h cyclic geometry. Both HF molecules make angles of 55° with the F...F axis in this structure, which is 1.1 kcal/mol higher in energy than the minima. Michael et al. 8 considered a more extensive region of the surface in an effort to analyze the vibrational motions of the two HF molecules. The energies of over 200 configurations were computed with a polarized triple basis set; correlation was added to 73 of these points. They noted that correlation was important to obtain the proper shape of the potential. The correlation energy changes smoothly with variation in the H—F bond length, but the dependence is slightly different for the dimer than for the free monomer.

210

Hydrogen Bonding

Similar intentions motivated Redmon and Binkley9 who expanded the surface coverage to over 1300 points, all at the MP4/6-311G** level. The results were then fit to an analytic function. The authors computed the Restricted Hartree-Fock multipole moments of the HF molecule as a function of internuclear distance, important information in fitting the electrostatic part of the intermolecular potential. Fig. 4.1 illustrates that the dipole moment increases linearly with bond length, rising from 1.6 D when r(HF) = 0.7 A to 2.68 D when stretched to 1.27 A. The longitudinal and perpendicular components of the quadrupole moment also vary with r(HF), and nearly linearly. The stretching of the bond leads to a reduction in the magnitude of 6zz, but 0xx becomes larger. The computed energies show that there is a sharp maximum in the potential for a headto-head configuration: F—H . . . H—F, where the two hydrogens are pointed at one another, about 40 kcal/mol higher in energy than the minimum. The tail-to-tail H—F . . . F—H geometry is also a maximum, but much less sharp, and only about 7 kcal/mol higher than the minimum. Torsional rotation of one molecule around the F...F axis from the planar geometry is mildly destabilizing; the maximum in the rotational profile is the 180° rotation wherein the two hydrogens are located on the same side of the F...F axis. 4.3.1 Anisotropies of Energy Components Szczesniak and Chalasinski10 focused their attention on the correlation segment of the intermolecular interaction, and in particular on the angular dependence of the dispersion energy. The latter component is typically considered to be quite isotropic so the authors de-

Figure 4.1 Variation of Restricted Hartree-Fock dipole ( . in D) and quadrupole moments (0 in D.A) of HF with bond length, calculated by Redmon and Binkley9. The molecular axis is taken as the z-axis.

Extended Regions of Potential Energy Surface

21 I

cided to test this assumption on the HF dimer. They learned that there is considerable angular dependence of the dispersion. This component is smallest in the head-to-head configuration where the two hydrogens are pointed at one another, but climbs by a factor of 14 when the two fluorines approach one another. Even more anisotropic than the dispersion energy is es(12), which reflects the change in the electrostatic energy as a result of including correlation. This term switches sign: it is repulsive for the head-to-tail linear configuration and becomes attractive as the proton acceptor molecule rotates around toward the F—H . . . H—F geometry. Figure 4.2 illustrates the anisotropy of the energetics of this system as the proton acceptor molecule is rotated, keeping the distance between centers of mass fixed at 2.8 A. The solid curve indicates a shallow minimum in the SCF potential for (3 in the vicinity of 120°. As (3 diminishes, the two hydrogens are brought closer to one another and the SCF energy quickly becomes repulsive, due to the electrostatic repulsion between the two molecular dipoles as well as steric repulsions between the two hydrogen atoms. Adding in the dispersion energy stabilizes the system, as illustrated by the long-dashed curve in Fig. 4.2. On the full scale of Fig. 4.2, the SCF and SCF+disp curves are nearly parallel, indicating that the anisotropy of the SCF potential far exceeds that of the dispersion. The short-dashed curve represents the anisotropy of the full potential computed at the MP2 level. This curve is nearly coincident with the SCF potential in the region of large [5, around the minimum, suggesting that the attractive dispersion is approximately canceled by other facets of the second order correlation here, chiefly es(12). The SCF+MP2 curve falls below SCF for smaller

Figure 4.2 Anisotropy of SCF and correlated components of interaction energy in (HF)2'°. refers to the angle between righthand molecule and F..F axis. Centers of mass of two molecules are held fixed at 2.8 A.

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Hydrogen Bonding

angles, indicative that the dispersion outweighs the latter electrostatic correction for the relevant configurations. The anisotropy of the correlated SCF4+MP2 curve is apparently best mimicked by the sum of three terms, adding the dispersion and es(12) to the SCF energy, as recommended by the authors. 4.3.2 Interconversion Pathways Bunker et al.n computed 1061 points on the correlated surface of the HF dimer using a polarized basis set, including all six degrees of freedom of the complex. They then fit their results to an analytic function containing 42 adjustable parameters. Improving upon the earlier work7, the authors were able next to determine the lowest energy paths for conversion from the optimal geometry to other structures. The motion on the left of Fig. 4.3 decreases a while is increased, taking the dimer through a linear transition state (C ) on a pathway that moves each of the two hydrogens to the other side of the F..F axis from which they started. An alternate route rotates the two hydrogens in opposite directions so as to reverse the roles of the two molecules as donor and acceptor, respectively. This route passes through a cyclic C2h structure as the transition state. Fig. 4.4a illustrates the energetics of motion along either pathway. Starting from the minimum energy configuration at ~120° in the center of the figure, the energy climbs as one moves toward larger and the linear geometry (to the left), or toward smaller and the cyclic structure to the right. The two transition states are nearly equal in energy at 345 and 332 cm-1 above the minimum, respectively (1 kcal/mol). Fig. 4.4b indicates the manner in which the other angle a changes along the two low-energy paths. Again, starting from ~ 120° in the center, a quickly diminishes from 8° to 0° as the linear geometry is approached or climbs quickly toward 56° in the cyclic structure. The behavior of the interfluorine dis-

Figure 4.3 Interconversion pathways in (HF)2. The path on the right switches the roles of proton donor and acceptor molecules.

Extended Regions of Potential Energy Surface

21 3

Figure 4.4 Variation of (a) energy, (b) angle, and (c) interfluorine distance (1 au = 0.529 A) along the lowest energy path of (HF)2, all as a function of angle (3. The equilibrium structure occurs at approximately = 120° in the center of each figure11.

tance is illustrated in Fig. 4.4c from which it may be seen that R is in excess of 2.86 A for the linear geometry but less than 2.72 A for the cyclic. The transition is rather sharp: R rises steeply as increases beyond 110°, reaching its full value when this angle is only 140°. These authors also explored the potential for rotating one molecule or the other around the F..F axis. The torsional potential shows a single minimum at the equilibrium geometry and a barrier of about 0.4 kcal/mol. 4.3.3 HClDimer Karpfen et al.12 turned their attention from (HF)2 to the analogous dimer of HC1 in 1991, using an averaged coupled pair functional (ACPF) approach13 to include correlation; the two basis sets were of [6,5,2/4,2] and [6,5,3,1/4,2] quality. In addition to the minimum illustrated in Fig. 4.5a, several other stationary points were identified on the PES. The cyclic geometry in Fig. 4.5b is not a minimum but rather a first-order stationary point, representing the transition state for interconversion of the proton donors and acceptors, just as for (HF)2. The authors find this point lies only 0.2 kcal/mol higher in energy

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Hydrogen Bonding

Figure 4.5 Energies (kcal/mol) computed for various stationary points on the PES of the HC1 dimer12. The values displayed for E refer to the electronic contribution Eclec.

than the minimum. Less stable, but also a first-order stationary point is the nearly linear structure in Fig. 4.5c, also of C2h symmetry. This minimum is not present in the analogous PES for (HF)2. The two linear structures in Figs. 4.5d and 4.5e also represent transition states on the surface, barely bound with respect to a pair of HC1 molecules, with binding energies of around 0.25 kcal/mol. The authors generated contour surfaces as functions of a and (3, at a series of fixed values of R(C1..C1). They noted firstly that this surface is considerably flatter than that of the (HF)2 congener. Assuming no change in this distance, the minimum-energy path for transtunneling from the minimum energy structure 4.5a through either transition state 4.5b or 4.5c is a linear one, that is, the changes in the two angles are both proportional to the progress along the path. The problem was reexamined in 1995 using an MP treatment of correlation; the [8s6p3d/6s3p] basis set was augmented by bond functions 14. The results largely confirm those of Karpfen et al.12. The MP2 barrier to interchange occurs at a geometry akin to that in Fig. 4.5b, with a = 45°. This configuration lies 0.17 kcal/mol higher in energy than the minimum, as in the earlier work. The collinear arrangement in Fig. 4.5d was found to be bound by some 0.4 kcal/mol, and 4.5e is similarly weakly bound, with Eelec = —0.3 kcal/mol. The authors provide evidence that the MP2 anisotropy of the energetics of the complex matches quite well the MP4 results. Another surface has been generated by a fitting of microwave and far and near-infrared spectroscopic quantities 15 . This "experimental" surface confirms most of the substantive

Extended Regions of Potential Energy Surface

21 5

predictions of the higher quality calculations. The global minimum has the two centers of mass separated by 3.746 A, slightly closer together than derived from some of the correlated ab initio calculations, and with a H-bond energy stronger by almost 0.5 kcal/mol. The barrier to donor-acceptor interchange is 0.14 kcal/mol on this surface, also in excellent agreement with ab initio predictions. The pathway along this experimental surface involves a contraction in the intermolecular separation at the conversion barrier by 0.1 A, consistent with an earlier calculation16. 4.4 Water Dimer As arguably the most important of all H-bonded complexes, the water dimer has stimulated a great deal of theoretical study. The "evolution" of its potential energy surface over the years, as theoretical methods have become progressively more sophisticated and reliable, offers a fascinating case study. Of particular interest are the attempts to determine which geometry is the global minimum on the surface, and whether other candidate geometries represent local minima, transition states for conversion of one structure to another, or if they are stationary points of some higher order. 4.4.1 Characterization of Possible Minima and Stationary Points 4.4.1.1 Linear, Bifurcated, and Cyclic Despite early assumptions that the linear arrangement is the most stable conformation of the H-bond in the water dimer, a number of early works considered the possibility of other geometries as well. Most notable among the candidates were the cyclic and bifurcated structures. Beginning in the early 1970s, Diercksen17 was probably the first to discuss the issue from the perspective of a moderately large basis set, containing polarization functions. This approach yielded a dipole moment for the water monomer within about 20% of the experimental value. His searches of the potential energy surface suffered from several limitations. In the first place, the work predated gradient algorithms so the surface was obtained as a series of single points, without the possibility of unambiguous verification of any structure as a true minimum or even as a stationary point of any order. Other sources of error were the failure to remove BSSE or to include electron correlation. With these caveats established, Diercksen found the linear structure to be the probable minimum, with R(OO) ~3.00 A. The two molecules could rotate about the H-bond axis with very low energy barriers. The calculations indicated the bifurcated structure was not a minimum, but would immediately convert to the linear structure with no barrier. Matsuoka et al.18 added electron correlation to hthe comparison of linear, bifurcated, and cyclic structures. Like their predecessors, these authors did not attempt to correct the superposition error, which is now understood to typically be considerably larger for correlated contributions than for the SCF interaction. This study confirmed the earlier finding that the linear geometry was most stable, bound by 5.6 kcal/mol. The binding energies of the (distorted) cyclic and bifurcated structures were, respectively, 4.9 and 4.2 kcal/mol, significantly weaker than the linear dimer. On the other hand, the latter two structures could not be verified to be true minima on the surface. To be more specific, while certainly generating minima in certain slices through the full potential energy surface as a function of R(OO), there was insufficient probing of various angular parameters to be able to properly classify these structures.

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Hydrogen Bonding

Some of the prior problems were addressed in 1985 by Baum and Finney19 who attempted to correct the BSSE of their correlated calculations. The CI procedure they employed was not size-consistent so they added a correction for this error as well. Their basis set was of [5 s4p 1 d/3 s 1 p] quality, and the CI of the type that includes all single and double excitations (except those involving the 1s orbitals of O). While unable to employ gradient procedures, the scan of the surface permitted the workers to guess as to whether or not they had located a minimum on the surface, since angles were varied as well as intermolecular distance. They found that the trans type of linear H-bond is the most stable conformer. A cis linear arrangement is not a minimum, but can readily convert to trans. The bifurcated structure is also likely to rearrange to trans linear, but with a very small barrier. The authors believed that even uncorrected SCF computations with their basis set should be adequate to characterize the basic features of the PES. At approximately the same time, Frisch et al.20 introduced correlation through M011erPlesset theory. Although no account was taken of BSSE, their methods were able to introduce gradient optimization techniques so they could readily distinguish minima from other stationary points. The results confirmed that the linear structure is a true minimum. The bifurcated arrangement is a saddle point and a planar cyclic geometry is a stationary point of second order. At the MP4SDQ/6-31+G**//SCF/6-31+G* level, the linear minimum is more stable than the bifurcated and cyclic structures by 1.6 and 1.5 kcal/mol, respectively. However, these differences are reduced to 0.9 and 0.7 when zero-point vibrational energies are considered. Singh and Kollman21 dissected the interaction energies of the three candidate geometries using the Morokuma decomposition scheme, within the framework of a doubly polarized basis set. The ESX term in Table 4.1 is a summation of electrostatic, exchange repulsion, and "mixing." The dispersion energy, DISP, was evaluated by Singh and Kollman using a three-term expansion in 6-8-10 powers of 1/R22, and added to the SCF interaction energy to obtain the total interaction energies reported in the last column of Table 4.1. At this level of theory, the combined electrostatic and exchange term favors the cyclic over the linear geometry by a small amount, as does the dispersion energy. However, these differences are more than compensated by larger preferences of the charge transfer and polarization energies for the linear arrangement. The bifurcated geometry is less stable than the other two in all categories considered. 4.4.1.2 Stacked Hobza et al.23 considered a different sort of geometry altogether. Their "stacked" structure places the planes of the two water molecules parallel to one another. The O atoms are directly above one another, and similarly for the two hydrogens in the parallel arrangement in Fig. 4.6.a. Table 4.1 Morokuma energy components (kcal/mol) of three arrangements of the water dimer21.

linear cyclic bifurcated a

ESXa

CT

POL

DISP

Total

-2.68 -2.91 -1.96

-1.28 -0.87 -0.37

-0.90 -0.47 -0.27

-0.32 -0.42 -0.28

-5.19 -4.67 -2.87

Summation of" electrostatic, exchange repulsion and "mixing."

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Figure 4.6 Parallel and antiparallel types of stacking of pairs of water molecules.

This complex was studied by the investigators as a function of the distance between the two planes, identical to R(OO), with a variety of basis sets, ranging from minimal STO-3G to a doubly polarized, double- set. The interaction was found to be repulsive at the SCF level with all sets considered, which the authors attributed to the parallel alignment of the molecular dipole moments. There was not much sensitivity of this potential to the particulars of the basis set, except that the STO-3G results were not as repulsive as the others. In contrast to this behavior, the correlated part of the potential was rather sensitive to basis set choice. The authors found that the attractiveness of the correlation contribution was directly related to the size of the set. But in no case was the negative correlation term large enough to compensate for the strong repulsive nature of the SCF interaction, so the parallel dimer above is clearly not a minimum on the surface. The authors also investigated the antiparallel geometry illustrated in Fig. 4.6b. Since the dipoles are now more favorably aligned, this interaction can be expected to be attractive. Indeed, such an attractive interaction, albeit a weak one, was observed at the SCF level. 4.4.1.3 Trifurcated Spurred by semiempirical AM1 calculations that suggested the lowest energy structure of the water dimer might not be of the linear variety at all, Dannenberg24 applied well-correlated ab initio methods to test the validity of the AM1 findings. He made use of the standard 6-311G** basis set and went up to MP4SDQ but did not attempt to correct for BSSE or zero-point vibrational energies. It is somewhat difficult to fully interpret the results as some of the structures were only partially optimized and cannot be characterized as minima, or indeed as stationary points of any order. Dannenberg was chiefly concerned with the relative stability of the linear type of structure, as compared to what he terms a "trifurcated" H-bond which is basically a variant of the cyclic configuration except that one of the water molecules contributes two hydrogens to the region between the oxygen atoms, as illustrated in Fig. 4.7. Beginning from the AM1 structure as a starting point, Dannenberg carried out a geometry optimization at the MP2/6-31G* level to arrive at a structure like that in Fig. 4.7, with O .. HO angle equal to 129° (although it was not entirely clear from the paper that this was indeed a truly optimized structure). The binding energy of this dimer was calculated to be 5.5 kcal/mol at the MP4SDQ/6-311G** level of theory. A similar type of optimization led

218 Hydrogen Bonding

Figure 4.7 Trifurcated type of arrangement of the water dimer.

to the standard linear H-bond, bound by 5.8 kcal/mol, only 0.3 more than the trifurcated structure. The author pointed out that the experimental determination of the equilibrium geometry and binding energy had been carried out at 350 400° K. At a temperature this high, it is probable that entropy would play an important role. He argued that if the trifurcated structure were only 10 eu lower in entropy than the linear structure, this would be sufficient to render the latter one dominant at high T. The lower entropy of the trifurcated structure could easily be rationalized on the basis of "freezing" the positions of three of the hydrogen atoms in H-bonds of some type, as compared to the linear configuration which only has a single bridging proton. 4.4.1.4 Definitive Characterization Smith et al.25 added some particularly useful information to the debate with their high level study of various regions of the potential energy surface. This work employed MP4/6311 + G(2df,2p) single-point calculations at geometries optimized at the MP2/6311 + G(d,p) level. Counterpoise corrections were computed, but not added directly to the energetics, as there was no way to include such corrections directly into geometry minimizations or characterization of stationary points on the potential energy surface. These workers identified a number of stationary points and characterized the transition states separating them. Their global minimum corresponds to the linear geometry, and is bound with respect to two isolated water molecules by 3.48 kcal/mol at 373° K, after correction for BSSE and vibrational and thermal energies, which the authors compared with the thermal conductivity measurement of 3.59 ± 0.5 kcal/mol26. The process which interchanges the two hydrogens on the proton acceptor molecule consists of a rotation of that molecule, as indicated in Fig. 4.8a. The transition state for this interchange has no symmetry elements and lies 0.6 kcal/mol higher in energy than the global minimum on the potential energy surface. While the latter structure is nonplanar as indicated, its planar correlate is only 0.1 kcal/mol higher in energy. The interchange of the donor or acceptor roles of the two molecules requires slightly more energy, passing over a barrier of 0.9 kcal/mol, as illustrated in Fig. 4.8b. The transition state geometry is cyclic in nature, with O..OH angles of 112°. The two hydrogens not participating in the H-bond lie alternately above and below the plane; placing them both in this plane raises the energy by 0.4 kcal/mol. It takes more energy to interchange the two protons in the donor molecule since one of them participates directly in the H-bond. A bifurcated geometry, in which two protons participate simultaneously in a pair of bent H-bonds, serves as the transition state to this switch, with a barrier of 1.9 kcal/mol (Fig. 4.8c). Smith et al. also reinvestigated the question of the trifurcated structure and found it to be a second-order saddle point, nearly 2 kcal/mol higher in energy than the minimum. This work reinforced the notion that the potential energy surface is rather flat, not sur-

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Figure 4.8 Paths for interconversion of one linear dimer to a symmetrically related one. Energies of transition states for each path are given in kcal/mol.

prising in view of the weak nature of the H-bond itself. The surface is punctuated by a large number of small undulations, as the molecules reorient themselves. The fluctuations become steeper for motions that more nearly break the H-bond. While the single, nearly linear, H-bond is most stable and corresponds to the unique minimum on the surface, other geometries which contain multiple bent H-bonding interactions can be rather close in energy, although they correspond to first or higher-order saddle points. The flatness of the potential makes its characterization sensitive to the level of theory. Semiempirical methods seem prone to mistake higher-energy saddle points for minima. Even ab initio results must be viewed with some caution. For example, Smith et al. noted a number of occasions where the order of a given saddle point was altered upon including correlation, even with a flexible basis set. At about the same time, Vos et al.27 published another investigation of various conformers of the water dimer using a polarized double- basis set. MP2 correlation was compared directly with the CEPA-1 approach, and the binding energies were found nearly identical, to within about 0.1 kcal/mol or better. The cyclic geometry was found less stable than the linear minimum by about 1 kcal/mol. The authors attributed most of this difference to the deformation energy at the SCF level (non-Heitler-London effects). The bifurcated structure is further destabilized with respect to this SCF deformation energy and also contains less dispersion energy as a result of the greater R(OO) separation. On the other hand, the bifurcated arrangement does have stronger coulombic attraction due to the better orientation of the two molecular dipole moments. As a result of these competing effects, the bifurcated geometry is about 0.75 kcal/mol less stable than the cyclic. Since their results indicate that distortion energies from equilibrium are dominated by Heitler-London effects, the authors were optimistic that SCF calculations can provide reasonable equilibrium geometries for H-bonded complexes. Further examination by Marsden et al.28 of the nature of the bifurcated structure applied a basis set containing three sets of diffuse functions, along with a doubly polarized quadruple-Jj valence set, amounting to 146 contracted Gaussians. These workers concluded that

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Hydrogen Bonding

the bifurcated structure is certainly a transition state on the SCF surface with an imaginary frequency in the neighborhood of 200i c m - 1 . The authors expect this value to increase with correlation, strengthening their contention of a saddle point. The energy barrier between the true minimum and this transition state is estimated as 1.3 kcal/mol, increasing to perhaps 1.7 kcal/mol after inclusion of correlation. 4.4.2 Components of the Interaction Energy The shape of the potential energy surface in the general vicinity of the minimum of the water dimer has been addressed by examining the individual contributors to the total interaction energy. Singh and Kollman21 examined how the various components of a Morokuma decomposition vary as the proton acceptor molecule of the linear dimer "wags" as a function of in Fig. 4.9. As described in Section 4.4.1. the authors combined electrostatic and exchange into a single term, ESX, along with the "mixing" energy. There is little change in ESX as varies between 180° and 90°, with a shallow minimum at approximately 120°. The behavior of the total interaction energy is very nearly parallel, although lower in energy as a result of other stabilizing factors. The charge transfer, polarization, and dispersion were all computed to be attractive and of smaller magnitude, in the order indicated. All three become monotonically more attractive as decreases from 180° to 0°. Similar calculations were carried out for the cyclic structure. Again, the ESX term closely parallels the full interaction energy, as a function of angular orientation. In this case, the minimum is located at 50°. The charge transfer, polarization, and dispersion are all maximized in magnitude for smaller angles. 4.4.2.1 Electrostatic Contribution Cybulski and Scheiner29 also considered the factors that contribute to the distortion energy that must be overcome when the H-bond in the water dimer is bent. Both the proton donor and acceptor molecules were subjected to 40° distortions from their optimal alignment, and the energetics monitored. The authors were able to draw a strong parallel between these distortion energies and the change in the electrostatic component of the interaction energy. These two quantities are nearly identical when R(OO) = 3.25 A but less coincident for R(OO) = 2.75 A, shorter than the equilibrium H-bond. In contrast, the other contributors to the interaction, namely, exchange repulsion, polarization, charge transfer, and a catch-all "MIX" term, are much smaller in magnitude and do not change very much with misorientation. Indeed, the distortions considered brought the amounts of these contributions down close to zero. Because of its importance in the water dimer, as well as in a number of other H-bonded systems, the electrostatic interaction was partitioned into a multipole series in powers of 1/R, consisting of terms corresponding to interactions between dipole, quadrupole, and so

Figure 4.9 Geometry of the water dimer equilibrium structure.

Extended Regions of Potential Energy Surface

221

on moments29. Truncation of this series at the R-5 term led to excellent reproduction of the full electrostatic interaction, particularly for the longer intermolecular separation, even though the last term was not necessarily small in magnitude. Near cancellation between the R-4 and R-5 terms was observed, leaving R-3 as a good indicator of the full electrostatic energy for most modes of distortion examined. It is thus possible for the simple dipole-dipole interaction, constituting the R-3 term, to predict quite well the behavior of the full energetics with respect to angular distortions of this H-bond. 4.4.2.2 Contributions from Electron Correlation Szczesniak et al.30 considered the factors leading to the degree of linearity of the H-bond in the water dimer and the pyramidalization of the proton acceptor oxygen. The dependence of the Hartree-Fock interaction energy was calculated as a function of both a and (3 (see earlier), as were the dispersion energy, disp (20) ,and second-order M011er-Plesset correlation energy, EMP(2). It is primarily the SCF components that lead to the equilibrium geometry of the dimer. But particularly interesting was a comparison of the behavior of disp (20) and EMP(2) which includes the former. The dispersion energy favors a linear H-bond, and amounts to nearly 2 kcal/mol in this geometry. A 60° distortion of a reduces the dispersion attraction by about half. The more complete second-order correlation energy behaves in an opposite fashion, with EMP(2) favoring a nonlinear H-bond. The two quantities behave more similarly when the proton acceptor molecule is rotated. Both disp(20) and EMP(2) tend to push (3 away from 180°, that is, they favor a pyramidal arrangement. The authors were able to rationalize the behavior of the dispersion energy based on a model where this attractive component is built up from the interactions between occupied molecular orbitals on the two subunits. The two lowest MOs of the water molecule, la1 and 2a1, are not very polarizable and so contribute relatively little to the dispersion. The third MO is largely of O—H bonding character, while the fourth and fifth are the oxygen lone pairs. Fig. 4.10 illustrates schematically the angular dependence of several of the more important interorbital dispersion interactions. For example, the interaction between the O—H bond (orbital 3) of the donor and the a lone pair of the acceptor (orbital 4) is strongest due to the maximal overlap when a = 0°, as pictured in the figure. Rotation of the donor turns the O—H bond away so this term rapidly becomes less negative. Similar behavior, although less dramatic, is observed in the 3-5 interaction involving the lone pair of the acceptor. The bilobal character of the a lone pair leads to a substantial 4-4 interaction at a = 0°. A maximum occurs when the two orbitals are perpendicular to one another, as indicated when a is approximately 50°, after which the 4-4 interaction again becomes more negative. This sort of analysis lends hope that the anisotropy of dispersion can be predicted from first principles, based on knowledge of the general MO structure and polarizabilities, in the same way that electrostatics can be decomposed into interactions between multipole moments of the monomers. An alternate approach, and one which is more commonly taken, is to partition the dispersion into interactions between atoms on the two subunits. Probably the simplest example assumes an inverse sixth power dependence upon interatomic separation.

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Hydrogen Bonding

Figure 4.10 Angular dependence of various interorbital pairwise contributors to dispersion energy in the water diraer30. (3 was held fixed at 180°.

The CXY are parameters fit to each pair of atoms X and Y on molecules A and B, and rXY is the distance separating these atoms. Szczesniak et al.30 fit these parameters by a leastsquares method so as to reproduce their calculated dispersion as closely as possible. The reason for the discrepancy between the angular dependence of disp (20) and EMP(2) can be traced by examining the distance-dependence of these two terms. At long distance the former remains negative while the latter becomes repulsive. This repulsive character results from another term that appears in the second-order correlation. Since the multipole moments of the water monomers are lower in the correlated wave function than SCF, the attractive electrostatic interaction becomes weaker when correlation is included. Hence, the correlation correction to the electrostatic interaction, es(12), shows up as a repulsive term. Since electrostatics dies off more slowly than dispersion, it is this repulsive correlation correction that remains at long distance. The authors left their results as a warning against attempts to simulate a true correlated intermolecular potential by supplementing the SCF interaction by dispersion alone. Another point is that there is a definite anisotropy to dispersion, as there is to other correlation components, that should not be ignored in construction of empirical potentials to model the interaction. 4.4.2.3 Natural Bond Orbitals Most recently, Glendening and Streitwicser31 have decomposed the interaction energy of the water dimer using natural bond orbitals. Their natural energy decomposition analysis (NEDA) combines the normal electrostatic and exchange energies into a single ES term,

Extended Regions of Potential Energy Surface

223

and the DEF term refers to the energy required to deform the wave functions of the isolated subsystems to those they will assume within the complex. In the case of a fully linear Hbond, with a = 0°, the total interaction energy has a minimum (or at least within several degrees of 0°), as do the ES and charge transfer terms; DEF is at its most positive here. Distorting the H-bond by rotating the proton donor molecule causes both ES and CT to rise rather sharply. These two terms each become less negative by about 5 kcal/mol when a = 60°. But the total interaction energy increases by only 2 or 3 kcal/mol because this bending motion reduces the (electronic) deformation energy DEF by about 8 kcal/mol.

4.5 Carbonyl Group

The complexity of the PBS for the pair of H2CO with H2O offers a rich field of candidates for minima and stationary points. Kumpf and Damewood32 explored an extensive region of the potential energy surface of the H2CO...HOH complex using a polarized basis set of the 6-31G** variety. Correlation was added, as well as zero-point vibrations, once stationary points were identified. Thirteen different configurations were considered. Some of these had formaldehyde as proton acceptor and some tested its capability as donor. Linear H-bonds were compared with bifurcated arrangements and with a sort of cyclic geometry in which both molecules can act simultaneously as donor and acceptor. The authors also checked on the ability of the bond of H2CO to accept a proton. Some of the more important of the configurations examined by Kumpf and Damewood are illustrated in Fig. 4.11 where the same nomenclature has been used as in their original article for purposes of consistency. The interaction energies, — Eelec, computed for each structure at the MP2/6-311 +G**//SCF/6-31G** level are indicated along with the letter designation, followed after the comma by the same quantity after correction by zero-point vibrational energies (but all with the 6-31G** basis set). The reader is cautioned at the outset that the authors of this paper were not entirely clear as to when they had identified true minima or stationary points on their surface and when the structures were the result of optimization under some sort of constraint. As pointed out in Chapter 2, the energetically preferred conformation is the "ringlike" geometry (b). The energetic cost of the nonlinearity of the H-bond is presumably compensated by the contact between the water oxygen and one of the hydrogens of H2CO. Also of note is the nearly antiparallel alignment of the molecular dipole moments of the two subunits in (b). The more conventional single linear H-bond in (a) is slightly less stable. Structure (c) is very much like (a) except that the proton donor approaches the formaldehyde along its C=O axis. The lesser stability of (c) is an indication that a lone-pair approach, as in (a), is preferred by about 1 kcal/mol. The bifurcated geometry in (d) is favored by the head-to-tail alignment of the two molecular dipoles. Nevertheless, the lack of a strong H-bond makes this structure less stable than any of the aforementioned geometries. One can disrupt the latter partial H-bonds by rotating one of the two molecules by 90° about the O..O axis, rotating the hydrogens out of the plane of the carbonyl oxygen lone pairs. Such a rotation leaves intact the favorable alignment of the molecular dipoles. The highest level MP2/6-311+G** calculations predict an energetic cost of 0.5 kcal/mol arising from this rotation. Structure (f) permits both hydrogens of the water to interact with the carbonyl oxygen, as well as allowing the water oxygen to approach one of the CH2 hydrogens. This geometry contains an antiparallel arrangement of the molecular dipoles. It was found to be only

224

Hydrogen Bonding

Figure 4.1 1 Interaction energies (in kcal/mol) of various complexes of H2CO with H2O, computed at MP2/6-311 +G**//SCF/6-31G** level. The first number refers to electronic part of E; the second includes ZPVE correction32.

slightly less stable than the other bifurcated structure (d). A 90° rotation of the H2CO molecule around its C=O axis removes the interaction between O of water and the H of CH2. The rotation simultaneously displaces one of the C=O lone pairs from the water hydrogens. Nevertheless, the ensuing destabilization relative to (f) is only about 0.4 kcal/mol. The arrangement in (h) permits the testing of the ability of H2CO to donate a proton to the water. The authors found (h) to be a true minimum on the PES, but not surprisingly, less stable than the H-bonds where HOH acts as the donor. An analogous reversal between the roles as donor and acceptor, but probing a bifurcated type of H-bond, results in structure (j). Like (d), the molecular dipoles are arranged head-to-tail, but the two bridging hydrogens are both associated with the H2CO molecule. As indicated in Fig. 4.11, this structure is only about 0.8 kcal/mol less stable than the more conventional bifurcated geometry (d). The authors tested the possibility that the water might donate a proton to the bond of H2CO, but were unable to locate corresponding minima. They conclude that the PES is relatively shallow, consistent with recent findings for the simpler water dimer. Dimitrova and Peyerimhoff33 ignored structure (b), and instead focused their attention on (a), (d), and (j') (where the prime indicates a 90° rotation to make a fully planar complex). They optimized the geometries and added BSSE corrections, after which they obtained binding energies, Eclcc, of 4.0, 2.4, and 2.0 kcal/mol, respectively, at the MP2/6311 + +G(2d,2p) level. Following ZPVE correction, these values become 2.2, 1.6, and 1.5 kcal/mol.

Extended Regions of Potential Energy Surface

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Later work by Ramelot et al.34, incorporating full geometry optimizations at SCF and correlated levels, verified the characterization of (a), (b), and (f) as stationary points on the PES of H2CO...HOH. While (b) is a true minimum, (a) is a transition state. The rotation of the water is associated with barrierless collapse to (b). Like (a), (f) is also a transition state and not a true minimum. Rotation of the water again leads back to (b) without an energy barrier. Its energy yields the conclusion that interchange of the two water hydrogens in (b) passes through an energy barrier of 1.2 kcal/mol.

4.6 Amines

A surface of a different type was generated for the complexes of amines with hydrogen halides. Brciz et al.35 considered the four possible pairs of HC1 and HBr with NH3 and CH3NH2 with an eye toward determining if any would prefer a proton-transferred ion-pair structure. Their surfaces were hence functions of the H—X distance, r, and the distance between the two heavy atoms, R(N .. X). For three of the systems, there is only one minimum on the surface, corresponding to the neutral dimer. However, a sort of low-energy corridor appears in each PES that would take the system toward the ion pair, even though the energy rises monotonically along this pathway. It is only for the complex pairing HBr with CH3NH2 that a second minimum appears, which corresponds to Br ...+HNH2CH3. Indeed, the latter well is deeper than that for BrH...NH2CH3 by about 1.2 kcal/mol. The barrier which must be surmounted to pass from one minimum to the next is 1.2 kcal/mol higher in energy than the neutral complex. Correlation was added to this picture several years later by Jasien and Stevens36 who also employed gradient optimization techniques. These authors found the appearance of a secondary, ion-pair minimum on the potential energy surface for C1H...NH3 and BrH ... NH 3 at the SCF level which disappears with correlation. In the case of IH...NH3, both the neutral and ion-pair minima survive the inclusion of electron correlation and are within 1 kcal/mol of each other. The authors located a transition state for conversion that lies 1.4 kcal/mol higher in energy than the neutral pair. As an important point, the authors found that when they added zero-point vibrations, the ion-pair disappears as a minimum from the surface. Latajka et al.37 followed up this line of reasoning by considering complexes of HBr and HI with ammonia and mono-, di-, and trimethylamine. They learned that the earlier finding that correlation can change the character of the potential energy surface was indeed rather general. While two minima were encountered in the SCF surfaces, most of the correlated analogues contained only a single minimum. BrH ... NH 3 is present as the indicated neutral pair while both neutral and ion pairs are present in the correlated surface of IH...NH3. Like Brciz et al., the authors generated a potential energy surface for the latter complex in terms of the two pertinent distances. The two minima are very close in energy, and an interconversion pathway involves changes in R as well as the I—H distance. The ionic minimum is not very highly developed and the system must surmount a barrier of only about 0.1 kcal/mol to climb out of it at the MP2 level. On the other hand, raising the level of correlation to MP4 heightens the barrier to perhaps 0.6 kcal/mol, making the ion-pair more likely to be observed. As methyl groups are added to the amine, it becomes more basic and hence leads to a stronger tendency for an ion pair. Only a single minimum is identified in the MP2 potential energy surface of CH3NH2 with either HBr or HI. While the ion pair structure is clear for HI, the proton is approximately midway between the nitrogen and halogen atoms in Br .. H .. NH 2 CH 3 . Both dimethylation and trimethylation lead to only single, ion-pair min-

226

Hydrogen Bonding

ima in the complexes of the amines with HBr and HI. This sort of analysis has been extended by Bacskay and Craw38 who have illustrated that trimethylamine is a strong enough base to extract a proton from HC1; the proton transfer potential of this complex contains a single well, corresponding to the ion pair.

4.7 Summary

The potential energy surface of the ammonia dimer is unquestionably very flat. The calculations indicate that there is no clearly defined minimum. Rather, a very shallow trough exists on the surface connecting a configuration containing a C linear N—H--N arrangement with an equivalent geometry in which the two NH3 molecules exchange places. This path passes through a cyclic structure, which is comparable in energy to the two linear configurations. Complexes pairing water with a hydrogen halide molecule were used as a testbed to examine the energetic cost of large-scale nonlinearity in the H-bond. In the case of H2O...HC1, rotation of the HC1 molecule by 90° completely eliminates the stability of the complex, raising the energy by 8 kcal/mol from equilibrium. A secondary minimum occurs for HBr, where H 2 O ... BrH is less stable than the H-bonded H2O...HBr by 4 kcal/mol. Transition from one minimum to the other must overcome a barrier of 6 kcal/mol. The two minima are much closer in energy for HI: the energy of H2O...IH is less than 1 kcal/mol different from that of H2O...HI. One mode of interconversion of proton donor and acceptor roles in HP...HP to FH...FH passes through a transition state with C2h cyclic character. The energy barrier to this interconversion is estimated as approximately 1 kcal/mol. A similar barrier is encountered in the transition path which moves hydrogens from one side of the H-bond axis to the other. The transition state consists of a fully linear HP...HF configuration of all four atoms. The two transition states have interfluorine distances that differ by some 0.25 A. The head-to-head F—H . . . H—F and tail-to-tail H—F . . . F—H linear configurations are maxima on the surface. The former is particularly high in energy, 40 kcal/mol higher than the minimum whereas the latter lies only about 7 kcal/mol above this point. Calculations showed that the dispersion part of the interaction energy is more anisotropic than is usually thought, particularly large in magnitude for the H—F . . . F—H configuration. In this case, the attractive dispersion energy is approximately canceled by other correlation effects, most notably the correlation-correction to the electrostatic interaction which is repulsive. When added to the SCF interaction, the latter two terms can reproduce the fully correlated potential energy surface rather well. As for (HF)2, the cyclic C2h structure of (HC1)2 is not a minimum, but rather a transition state for interconversion between the two equilibrium geometries. In this case, however, the cyclic geometry lies only about 0.2 kcal/mol higher in energy than the minima, rather than the 1 kcal/mol in (HF)2. Also analogous to (HF)2, the fully linear C1H...C1H geometry represents a transition state on the surface of (HC1)2; here, this structure lies about 1.5 kcal/mol higher in energy than the minimum, as compared to 1 kcal/mol in (HF)2. In general, the PES of (HC1)2 is somewhat flatter than that for (HF)2. Despite some notions in the literature that the PES of the water dimer contains minima other than the equilibrium geometry, with a linear H-bond, no other geometries appear to represent true minima. The bifurcated arrangement is a transition state for the interchange of the two protons in the donor molecule, and a trifurcated structure is a stationary point of second order. Although not true minima, some of these geometries arc not much higher in energy than the linear H-bond, typically within 1 or 2 kcal/mol.

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227

The total interaction energy of the equilibrium water dimer was dissected into its components to understand the nature of its anisotropy. The first-order component, prior to dimerization-induced charge rearrangement, is fairly insensitive to wags of the proton acceptor, whereas the higher-order terms favor a large [3 angle, that is, planar versus pyramidal. Further partitioning reveals that the electrostatic interaction is largely responsible for the observed energetics of bending either the donor or acceptor away from the equilibrium geometry. In certain cases, it is useful to represent the electrostatic contribution by its multipole series, and then focus on the first few terms, each of which has a simple physical interpretation; for example, a dipole-dipole term. The anisotropy of correlated terms is smaller but reveals an interesting trend. Whereas the dispersion energy favors a linear H-bond, that is, a tends toward zero, a more complete treatment of correlation would tend toward nonlinearity. The behavior of the former can be rationalized on the basis of interactions between individual MOs on each subunit. The trend toward nonlinearity on the part of the full correlation component arises from the influence of the correlation upon the multipole moments of the individual subunits. This effect can be embodied in es(12) which represents the effect of correlation upon the electrostatics of the interaction. The potential energy surface of the H2CO,H2O pair contains a number of possible minima. Most stable is the geometry wherein HOH acts as the donor to the C=O group of H2CO in the primary H-bond, but a weaker interaction occurs between a C—H group of H2CO and the water oxygen. This structure has been characterized as a true minimum on the surface. Slightly higher in energy is a strictly linear H2CO...HOH H-bond, with no secondary interaction, but this geometry is a transition state. Forcing the proton donor HOH to lie along the C=O axis, rather than a carbonyl lone pair, destabilizes the system by about 1 kcal/mol. Slightly higher in energy is a bifurcated H-bond wherein both HOH hydrogens approach the carbonyl oxygen. A number of other geometries appear bound relative to the isolated monomers, including a reverse C—H ... OH 2 interaction, where a C—H group acts as proton donor. Another sort of motion that a full PES must include is the transfer of the proton from the donor molecule to the acceptor. Such a transfer would require a particularly strong acid, coupled with a strong base, as it would yield an ion pair with a high degree of charge separation. The hydrogen halides are strong acids, just as amines are strong bases. It appears that this proton transfer can take place for HBr or HI paired with alkylated amines. This topic is explored in more detail in Chapter 6. References \. Latajka, Z., and Scheiner, S., The potential energy surface of (NH3)2, J. Chem. Phys 84, 341-347 (1986). 2. Tao, F.-M., and Klemperer, W., Ab initio search for the equilibrium structure of the ammonia dimer, J. Chem. Phys. 99, 5976-5982 (1993). 3. Liu, S.-Y., Dykstra, C. E., Kolenbrander, K., and Lisy, J. M., Electrical properties of ammonia and the structure of the ammonia dimer, J. Chem. Phys. 85, 2077-2083 (1986). 4. Sagarik, K. P., Ahlrichs, R., and Erode, S., Intermolecular potentials for ammonia based on the test particle model and the coupled pair functional method, Mol. Phys. 57, 1247-1264 (1986). 5. Hassett, D. M., Marsden, C. J., and Smith, B. J., The ammonia dimer potential energy surface: resolution of the apparent discrepancy between theory and experiment?, Chem. Phys. Lett. 183, 449-456(1991). 6. Hannachi, Y., Silvi, B., Perchard, J. P., and Bouteiller, Y., Ab initio study of the infrared photoconversion in the water-hydrogen iodide system, Chem. Phys. 154, 23-32 (1991).

228

Hydrogen Bonding

7. Curtiss, L. A., and Pople, J. A., Ab initio calculation of the force field of the hydrogen fluoride dimer, J. Mol. Spectrosc. 61, 1-10 (1976). 8. Michael, D. W., Dykstra, C. E., and Lisy, J. M., Changes in the electronic structure and vibrational potential of hydrogen fluoride upon dimerization: A well-correlated (HF)2 potential energy surface, J. Chem. Phys. 81, 5998-6006 (1984). 9. Redmon, M. J., and Binkley, J. S., Global potential energy hypersurface for dynamical studies of energy transfer in HF—HF collisions, J. Chem. Phys. 87, 969-982 (1987). 10. Szczesniak, M. M., and Chalasinski, G., Anisotropy of correlation effects in hydrogen-bonded systems: The HF dimer, Chem. Phys. Lett. 161, 532-538 (1989). 11. Bunker, P. R., Kofranek, M., Lischka, H., and Karpfen, A., An analytical six-dimensional potential energy surface for (HF)2 from ab initio calculations, J. Chem. Phys. 89, 3002-3007 (1988). 12. Karpfen, A., Bunker, P. R., and Jensen, P., An ab initio study of the hydrogen chloride dimer: the potential energy surface and the characterization of the stationary points, Chem. Phys. 149, 299-309(1991). 13. Gdanitz, R. J., and Ahlrichs, R., The averaged coupled-pair functional (ACPF): A size-extensive modification of MR CI(SD), Chem. Phys. Lett. 143, 413-420 (1988). 14. Tao, F.-M., and Klemperer, W., Ab initio potential energy surface for the HCl dimer, J. Chem. Phys. 103,950-956(1995). 15. Elrod, M. J., and Saykally, R. J., Determination of the intermolecular potential energy surface for (HCl) 2 from vibration-rotation-tunneling spectra, J. Chem. Phys. 103, 933-949 (1995). 16. Latajka, Z., and Scheiner, S., Structure, energetics and vibrational spectra H-bonded systems. Dimers and trimers of HF and HCl, Chem. Phys. 122, 413-430 (1988). 17. Diercksen, G. H. F., SCF-MO-LCGO studies on hydrogen bonding. The water dimer, Theor. Chim. Acta 21, 335-367 (1971). 18. Matsuoka, O., Clementi, E., and Yoshimine, M., CI study of the water dimer potential surface, J. Chem. Phys. 64, (1976). 19. Baum, J. O., and Finney, J. L., An SCF-CI study of the water dimer potential surface and the effects of including the correlation energy, the basis set superposition error, and the Davidson correction, Mol. Phys. 55, 1097-1108 (1985). 20. Frisch, M. J., Pople, J. A., and Del Bene, J. E., Molecular orbital study of the dimers (AHn)2 formed from NH3 ,OH2, FH, PH3 ,SH 2 , and CIH, J. Phys. Chem. 89, 3664-3669 (1985). 21. Singh, U. C., and Kollman, P. A., A water dimer potential based on ab initio calculations using Morokuma component analyses, J. Chem. Phys. 83, 4033-4040 (1985). 22. Douketis, C., Scoles, G., Marchetti, S., Zen, M., and Thakker, A. J., Intermolecular forces via hybrid Hartree-Fock-SCF plus damped dispersion (HFD) energy calculation. An improved spherical model, J. Chem. Phys. 76, 3057-3063 (1982). 23. Hobza, P., Mehlhorn, A., Carsky, P., and Zahradnik, R., Stacking interactions: Ab initio SCF and MP2 study on (H2O)2,
Extended Regions of Potential Energy Surface

229

30. Szczesniak, M. M., Brenstein, R. J., Cybulski, S. M., and Scheiner, S., Potential energy surface for dispersion interaction in (H2O)2 and (HF)2, J. Phys. Chem. 94, 1781-1788 (1990). 31. Glendening, E. D., and Streitwieser, A., Natural energy decomposition analysis: An energy partitioning procedure for molecular interactions with application to weak hydrogen bonding, strong ionic, and moderate donor-acceptor interactions, J. Chem. Phys. 100, 2900-2909 (1994). 32. Kumpf, R. A., and Daraewood, J. R., Jr., Interaction of formaldehyde with water, J. Phys. Chem. 93,4478-4486(1989). 33. Dimitrova, Y, and Peyerimhoff, S. D., Theoretical study of hydrogen-bonded formaldehyde-water complexes, J. Phys. Chem. 97, 12731-12736 (1993). 34. Ramelot, T. A., Hu, C.-H., Fowler, J. E., DeLeeuw, B. J., and Schaefer, H. P., Carbonyl-water hydrogen bonding: The H2CO H2O prototype, J. Chem. Phys. 100, 4347-4354 (1994). 35. Brciz, A., Karpfen, A., Lischka, H., and Schuster, P., A candidate for an ion pair in the vapor phase: Proton transfer in complexes R3N HX, Chem. Phys. 89, 337-343 (1984). 36. Jasien, P. G., and Stevens, W. J., Theoretical studies of potential gas-phase charge-transfer complexes: NH3 + HX(X=Cl,Br,I), Chem. Phys. Lett. 130, 127-131 (1986). 37. Latajka, Z., Scheiner, S., and Ratajczak, H., The proton position in amine-HX (X=Br,I) complexes, Chem. Phys. 166, 85-96 (1992). 38. Bacskay, G. B., and Craw, J. S., Quantum chemical study of the trimethylamine-hydrogen chloride complex, Chem. Phys. Lett. 221, 167-174 (1994).

5

Cooperative Phenomena

he formation of a H-bond causes certain changes in the internal geometries of each monomer. For example, the bridging hydrogen moves a little farther away from the donating atom. In addition to the deformations of the nuclear positions, the H-bond formation is accompanied by real redistributions in the electronic structure of each subunit. These polarizations are readily apparent from experimental spectra, as well as from molecular orbital calculations. Taking into account the above perturbations of the two molecules involved in a H-bonded dimer, it is reasonable to presume that the ability of either of these two molecules to form another H-bond is altered by their participation in the first bond1. Consider, for example, the pair of molecules AH and BH, each of which has a proton to donate in a H-bond, and each of which contains one or more lone electron pairs appropriate to accept a proton. If they form a H-bond of the type AH-BH, the proton of BH is still available to form a H-bond to another molecule, CH. But the CH molecule will encounter two different situations depending on whether the BH molecule is involved in the aforementioned dimer, or is a single isolated BH molecule. In fact, formation of the AH---BH complex will remove electron density from the BH subunit, density which is transferred across to AH. This loss of negative charge will make BH a more powerful proton donor so that one can expect the BH---CH interaction in the AH--BH--CH trimer to be stronger than in the simpler BH- CH dimer. Analogous reasoning would make the AH molecule in AH---BH a better proton acceptor, in comparison to the isolated AH molecule. These effects that make the "whole larger than the sum of its parts" wherein a chain of H-bonds is more strongly bound together than any of the individual links would be in the absence of the others, is an expression of the "cooperative" nature of H-bonds. It is this cooperativity that leads to the common occurrence of long strings of H-bonds. It has been noted from surveys of crystal studies, for example, that H-bonds that occur as parts of such strings tend to be considerably shorter, and presumably stronger, than isolated H-bonds2. Cooperativity is a powerful enough phenomenon that it is responsible for the formation in liquid formamide of six-membered rings3. Rigorous calculations have confirmed this rea-

T

230

tCooperative Phenomena 231 soning, indicating that it is polarizability of this sort that is largely responsible for the nonadditivity effects described in greater detail below for H-bond chains4. But it should be emphasized that multiple H-bonds are not always cooperative in a positive sense. Again referring to the AH ... BH dimer, the electron density removal from BH makes this molecule not only a better proton donor, but also a poorer proton acceptor. So BH would now be less inclined to accept a proton from another molecule like CH. For this reason, one would expect the total interaction energy in a trimer like that in Fig. 5.1 to be weaker than in the pair of dimers AH ... BH and CH...BH. This sort of general weakening is sometimes referred to oxymoronically as "negative cooperativity." It is usually energetically unfavorable for a molecule to act as a double proton acceptor as BH would in Fig. 5.1. For similar reasons, cooperativity is typically negative also when a molecule acts as double proton donor. Of course, even in the case of negative cooperativity, formation of the second H-bond is usually energetically favorable when compared to the complete absence of a second H-bond. That is, even though the CH ... BH interaction above is weaker than it would be in the absence of the other proton donor, AH, this interaction energy is still negative, and so will form spontaneously. In other words, two H-bonds are always better than one (or usually so). For the sake of consistency of terminology, triads of molecules in which the central unit acts simultaneously as both proton donor and acceptor will be termed "sequential" to distinguish such configurations from those in which the central molecule acts as double proton donor or double acceptor. A perhaps more quantitative expression of cooperativity is referred to in the literature as "nonadditivity." The latter term is commonly taken as the difference between the total interaction energy of an aggregation of molecules on one hand and the sum of all the pairwise interactions on the other. If adding a third molecule to a H-bonded dimer in a sequential fashion strengthens all interactions, it is natural to wonder if the same will occur for a fourth and fifth and so on. Calculations can address the behavior of the energetics as the chain grows. It is also possible to consider the effects of multiple H-bonding upon the geometries of each element in the chain and the charge distributions within each. Another point of interest concerns the vibrational modes of the growing chain which reflect upon the bonding characteristics. The first section in this chapter focuses on HCN as an element in a chain of H-bonded units. The presence in this molecule of only one proton and one lone electron pair provides a simple testing ground for ideas about cooperativity. The linearity of the complex also simplifies analysis of electron density redistributions caused by multiple bonding. HCCH is similar in some ways, but its absence of a lone electron pair causes significant distinctions.

Figure 5.1 Example of negative cooperativity where B serves as proton acceptor to both AH and CH.

232

Hydrogen Bonding

Since the equilibrium dimer is T-shaped, there is little propensity of the oligomers to consist of linear chains. HX molecules form strong H-bonds as dimers and are good candidates to manifest cooperativity. The bent geometry of the dimer suggests that a ring can be formed even by oligomers as small as the trimer. These complexes also provide a fertile ground to examine many-body effects upon the energetics, their basis set sensitivity, and the relative importance of SCF versus correlation contributions. Of greatest importance from a practical point of view, perhaps, is the cooperativity in liquid water. For this reason, this phenomenon has been the subject of the highest-level calculations, leading to the most well founded conclusions and the deepest insights. Mixed systems, in which the molecules involved in the oligomers are not identical to one another, are discussed last.

5.1 HCN Chains We begin our discussion of past consideration of cooperativity with the homogeneous (HCN)n series because the linearity of this molecule and the single lone pair of the HCN molecule make a linear geometry likely for a chain of any length. The analysis is hence unclouded by more complex angular distortions that accompany most other chains. The presence of only a single proton and a single lone pair on each monomer also simplifies our understanding as one need not be overly concerned with any molecule acting as double proton donor or acceptor. Motivated by gas phase spectroscopic studies of (HCN)35, Kofranek et al.6 used a variety of fairly large basis sets to examine oligomers of HCN of up to five molecules. They optimized the geometries of the complexes by gradient procedures. 5.1.1 Geometries Considering first fully linear arrangements, Fig. 5.2 illustrates how the C—H and C=N bonds change their length as the chain is built up, one subunit at a time. In the jump from the monomer to dimer, Fig. 5.2a illustrates the expected elongation of the C—H bond in the proton donor molecule, and the concomitant smaller increase in the C—H bond length of the proton acceptor molecule. As the chain gets longer, the donor designation indicates the leftmost molecule of NCH .. (NCH) n-2 .. NCH, and the furthest right molecule is considered the acceptor. Fig. 5.2a indicates that these elongation trends continue with each additional molecule added in the middle of the chain, although these bond lengths slowly approach asymptotes as n continues to increase. For odd n, we also consider the central molecule which has an equal number of molecules on its left as on its right. The data points in Fig. 5.2a show how the elongating effects of being a donor or acceptor reinforce one another in the middle molecule which simultaneously plays both roles. Hence, the C—H bonds are longest in the central molecule. Analogous data are presented in Fig. 5.2b for the CN bonds in the HCN linear chains. In this case, the act of donating a proton makes the CN bond longer while it becomes shorter if the molecule acts as acceptor. Again, the bond lengths approach asymptotes for the terminal molecules as the chain grows longer. As before, the central molecule acts both as proton donor and acceptor, but in this case the middle molecule shows little modification as compared to the monomer, since these two roles produce opposite effects on the CN bond length.

Figure 5.2 Optimized lengths of (a) C—H and (b) C N bonds in linear arrangements of (HCN)n, n = 1-5. Donor refers to the leftmost molecule and acceptor to the rightmost in NCH .. (NCH) n _ 2 .. NCH. The middle designation indicates the central molecule for n = 3 and n = 5. Data are taken from SCF optimizations with a [53/3] basis set6.

234

Hydrogen Bonding

5.1.2 Energetics The binding energies of the various linear oligomers were also computed by Kofranek et al.6 and the data are listed in Table 5.1. The first two columns report the energetics of assembling each complex from n isolated monomers to represent the total binding energy and enthalpy, respectively. For example, at the particular SCF/[53/3] level used, the electronic contribution to the binding energy of the dimer is 5.6 kcal/mol. After inclusion of zero point vibrational and other effects, the authors obtain a binding enthalpy of 4.7 kcal/mol. — Eelec for assembly of the full trimer is 12.5 kcal/mol. Since this total is more than twice that of — Eelec for the dimer, the two H-bonds in the trimer are stronger in sum than two isolated H-bonds, as would occur in a pair of dimers. This difference is reported in the next two columns of Table 5.1 as the cooperativity. Hence, for either Eelec or H°, the binding energy of the trimer exceeds that of two dimers by 1.3 kcal/mol. The series progresses to n = 4 in the next row where total binding energies are nearly 20 kcal/mol. These totals exceed the sum of three dimers by 3 kcal/mol. The latter amount is divided by two in the next two columns of Table 5.1 to facilitate comparison with the trimer in the preceding row. That is, there are two "trimers" present within the tetramer so the authors divide by 2 for a fair comparison. Note that these cooperativities of 1.5 kcal/mol are larger than the values for the trimer. The cooperativity increases further in the pentamer. One can imagine that these cooperativities continue to increase while approaching an asymptote as n . The convergence of the binding energy with chain length can be gleaned from the last two columns of Table 5.1. — Qn/(n-l) refers to the average H-bond energy of a given oligomer where the full interaction energy of the n monomers is divided by the number of adjacent pairs in the oligomer. For example, while - Eelec is 5.6 kcal/mol for the dimer, the average interaction energy of the four H-bonds present in (HCN)5 is approaching 7 kcal/mol. This sort of analysis was later extended to the correlated MP2 level, using a 6-31+G** basis set, with counterpoise corrections introduced7. Calculations up to the heptamer indicated the anticipated approach toward an asymptote of the average binding energy. This asymptote is somewhat smaller than in Table 5.1, tending toward about 5.8 kcal/mol at the SCF level for — Eelec/(n-l). This asymptote represents an approximate enhancement of 40%, as compared to the dimer.

Table 5.1 Energetics of binding (kcal/mol) in linear oligomers (HCN)n, as calculated at the SCF level6. coopa n

2 3 4 5

- Eelec 5.60 12.52 19.89 27.47

-H

4.72 10.75 17.21 23.87

- Eelec

1.32 1.55 1.69

— Qn/(n — 1) -

H°

1.31 1.53 1.66

- E.,.,,.

5.60 6.26 6.63 6.87

- H°

4.72 5.38 5.74 5.97

a Cooperativity is defined here as follows: if Qn is the property of interest for (HCN)n, coop is evaluated as [ Qn (n- 1) Q 2 ] /(n- 2).

Cooperative Phenomena

235

5.1.3 DipoleMoments It was stated earlier that formation of a first H-bond in a dimer will polarize both molecules. One can hence expect a dipole moment in the dimer that is larger than the vector sum of the individual moments of the isolated monomers. This supposition is confirmed by the SCF moments of these oligomers which are reported in the second column of Table 5.2. The third column lists the cooperativity, measured in a manner analogous to that for the energies. This property is approximately 1 D and increases with n. The average dipole moment per molecule in the chain is exhibited in the last column of Table 5.2 and echoes the cooperative nature of the electronic rearrangements within the chain. Later experimental measurements by Ruoff et al.8 indicated that the dipole moment of the linear trimer exceeds the vector sum of the three monomers by 1.8 D, quite close to the estimation of 2.0 D in Table 5.2 (i.e., twice the value listed for n = 3 in column three). 5.1.4 Vibrational Spectra Kofranek et al.6 carried out vibrational analyses of their various complexes. It must be stated at the outset that the frequencies are not so clearly identified with any particular single bond as are geometrical features. Nevertheless, the authors were able to identify various modes as largely C—H or C N stretches, or intramolecular bends, even in the longer chains. With respect to the C—H stretches, the longer oligomers exhibit a spread of frequencies. The highest of these is fairly clearly identified with the proton acceptor molecule and the others are within a narrow range of each other. We denote the lowest frequency as that of the terminal donor molecule with the caveat that this designation is somewhat arbitrary. Analogous reasoning was applied to the CN stretches. The calculated frequencies exhibit behavior very much like the bond lengths in some ways. For example, the CH stretching frequencies in Fig. 5.3a suffer decreases as the chain grows, with this red shift being more pronounced for the proton donor end of the chain. These frequency drops are consistent with the bond stretches described in Fig. 5.2a. In fact, the two plots are nearly perfect mirrors of one another. The CN stretching frequency of the donor end of the chain, illustrated in Fig. 5.3b, diminishes with larger n, also consistent with the analogous bond stretches of Fig. 5.2b; the behavior of the acceptor is again opposite in sign. Note the approach to an asymptote for all frequencies in Fig. 5.3. Another important aspect is the magnitude of changes in bond length and frequency. The CH bonds stretch by nearly 0.01 A upon forming the serial H-bonds; the red shift of the associated frequency is Table 5.2 Calculated dipole moments (D) in linear oligomers (HCN)n, as calculated at the SCF level6. n

n

coopa

/n

1

3.20 7.29 11.63 16.08 20.57

0.89 1.02 1.09 1.14

3.20 /n 3.65 3.88 4.02 4.11

2 3 4

5 a

Cooperativity is defined here as [

n

— n 1 ] / (n — 1).

Figure 5.3 Calculated stretching frequencies of (a) C—H and (b) C N bonds in linear arrangements of (HCN)n, n = 1-5. Donor refers to the leftmost molecule (assumed to be the lowest frequency) and acceptor to the rightmost (highest v) in NCH .. (NCH) n _ 2 .. NCH. Data derived at SCF/[53/3] level in6.

Cooperative Phenomena

237

some 150 c m - 1 . In contrast, the CN bonds change their length by less than 0.002 A, with shifts of only 20 cm-1 or so. The intermolecular frequencies were also computed and are presented in Fig. 5.4 as a function of chain length. For the dimer, the stretching frequency, v , is 122 cm - 1 , bracketed by the two intermolecular bending modes. For each of these three modes, growth of the chain leads to a drop in one frequency and an increase in the other. The proportional effects of chain length on these frequencies are enormous. For example, v nearly doubles upon going from dimer to pentamer; the lower-frequency bend drops by 80%. The same authors followed up their earlier work with an elaboration of the oligomers up to trimer which computed the intensities, as well as frequencies, of the vibrational transitions9. This work included electron correlation through the coupled pair functional (CPF) technique, using polarized basis sets. Their results for the trimer are summarized in Table 5.3. Correlation appears to dampen the red shift of the three CH stretching frequencies while making all of the CN frequencies more positive. The changes in the bending frequencies are not quite as regular. But correlation does not appear to produce major changes in any of these shifts. The band intensifications, listed in the last two columns of Table 5.3, also show slight correlation-induced increases in the CH stretches, with virtually no changes occurring at all in the bending intensities. However, correlation produces marked changes in the CN stretches, raising the intensification by an order of magnitude or more in each case. This large discrepancy between SCF and correlated calculations is consistent with that noted above for the dimer.

Figure 5.4 Calculated intermolecular frequencies in linear arrangements of (HCN)n n = 2-5. Highest and lowest frequency of each type are illustrated. Data derived at SCF/[53/3] level in6.

238

Hydrogen Bonding

Table 5.3 Calculated frequency shifts and intensification ratios in linear trimers of HCN, as compared to the monomer9. vCcm-1) A

tri/Amon

Mode

SCF

CPF

SCF

CH stretch

-8 -76 -86 10 -4 -14 85 74 20

-3 -68 -76 15 3 -7 96 90 10

0.9

1.0

0.01

0.02 11.1

CN stretch

HCN bend

8.6 5 9.8 3.0 0.5 1.4 0.9

CPF

59 258 33 0.5 1.5 0.9

Just as one can consider the effects of dimerization upon the frequencies or intensities of the monomer modes, it is useful to ponder how juxtaposing two "dinners" within a trimer affects the vibrational spectra of the dimer. Kofranek et al.9 provided such information, reported in Table 5.4, which suggests that the SCF data graphed in Fig. 5.4 are quite representative of the correlated level as well. All of the trends in the SCF frequencies are mirrored in the CPF data. The similarity of the last two columns in the table indicates no major perturbations of the intermolecular band intensifications caused by electron correlation. Anex et al.10 later repeated some of this work using a 6-31G* basis set and including electron correlation via MP2. Their data verified the earlier calculations of (HCN)n and went further by describing the various normal modes that correspond to the ab initio force field. The highest frequency mode was found to correspond to independent C—H stretching motion of the terminal proton acceptor molecule (with 8% C N stretching mixed in). The C—H stretches of the other two molecules mix with each other in the next two modes. Karpfen 11 was able to consider an infinitely extended linear chain of HCN molecules using an ab initio crystal-orbital approach, with basis sets ranging from STO-3G to [641/41], albeit without any consideration of electron correlation. Polynomials of second and third order were fit to the results, computed for a grid of values of r(CH), r(CN), and

Table 5.4 Calculated frequency shifts and intensification ratios in the intermolecular modes of the linear trimers of HCN, as compared to the dimer9. Atri/Adim

v3-v2 (cm-1)

Mode

SCF

CPF

SCF

CPF

H-bond stretch

36 -24 31 -17 11 -23

38 -27 27 -13 4 -27

0 1.5 1.5 0 2.6

0 0.7 2.0 0 2.0

0.05

0.06

H-bond bend

Cooperative Phenomena

239

Table 5.5 Quadrupole coupling constants (in MHz) in linear dimers of DCN, computed at SCF/6 31+G* level12. n

1 2 3

a d a d-a d

4

a

5

d-a d a d-a d

6

a d-a d

D

14N

0.225 0.223 0.209 0.223 0.204 0.205 0.222 0.200 0.204 0.222 0.199 0.204 0.222 0.197 0.204

-4.758 -4.426 -4.660 -4.366 -4.345 -4.638 -4.347 -4.317 -4.629 -4.340 -4.252 -4.625 -4.336 -4.241 -4.623

r(H .. N). He found with his best basis set that the formation of such an infinite chain diminishes the CH stretching frequency, relative to the monomer, by 138 cm - 1 , while that of the CN stretch is constant to within 1 cm - 1 . In comparison to a calculated binding energy of 4.4 kcal/mol for the single H-bond in the dimer, this same basis set yielded an average of 6.1 kcal/mol in the chain. 5.1.5 Quadrupole Coupling Constants The effects of cooperativity can also extend to other properties such as quadrupole coupling constants. These quantities were computed for the D and 14N nuclei in linear clusters of (HCN)n, with n varying up to 6, at the SCF level with a 6-31+G* basis set12. The calculated values are listed in Table 5.5 where d refers to the proton donor molecule on one end of the chain, a to the acceptor on the other end, and d-a to a combined donor-acceptor molecule in the middle of each chain. The coupling constants of both the D and 14N atoms of the monomer are lowered in magnitude when the H-bond is formed in the dimer. The value of D is not affected much in the proton acceptor molecule; conversely that of 14N is lowered much more in the proton acceptor. As the chain becomes longer, the quadrupole coupling constants of the D atoms of the molecules on the two ends of the chain change very little, indicating little cooperativity. On the other hand, the value for a molecule in the middle of the chain, serving as both donor and acceptor, is slightly smaller than for either terminal molecule. Cooperativity is more obvious in the 14N coupling constants. The values for the terminal molecules continue to diminish as the chain grows beyond the dimer level. As in the case of the D atom, the coupling constants are smaller for the molecules in the middle of the chain. The authors noted a close linear correlation between the 14N quadrupole coupling constants and the lengths of the pertinent H-bonds. This same property was found to correlate also with the VCH stretching frequencies.

240

Hydrogen Bonding

5.1.6 Cyclic Chains The terminal HCN molecule on one end of a linear chain contains a nitrogen lone electron pair that is not involved in a H-bond. Nor is there any involvement by the proton of the HCN on the other end of the chain. If these two entities could be paired up, an additional H-bond might be formed, presumably stabilizing the entire system. However, in order to bring these two groups into proximity to one another, by forming a ring, the deformation from nonlinearity would destabilize the existing H-bonds. Hence, the relative stability of a cyclic oligomer rests on two competing effects: formation of the last H-bond versus deformation of all the others. In the case of a long chain (large n), the latter deformation can be rather small since each H-bond must bend by only a small amount. But when there are only a small number of molecules, the angular deformations are considerable. Hence, one can expect linear chains to be preferred for small n, but the cycles will become more favorable as n increases past some threshold value. Of course, the foregoing analysis is based only upon energetics; the higher constraints in the cyclic geometry would disfavor this arrangement from entropic considerations. In the case of (HCN)n, Kofranek et al.6 found the threshold of conversion of linear to cyclic, on energetic grounds, to occur at approximately n = 4. The linear trimer is favored over the cyclic by 2.1 kcal/mol, but the latter tetramer is more stable than the linear by 0.1 kcal/mol, again at the SCF/[53/3] level. Kurnig et al.13 reexamined the relative stabilities of the linear and cyclic trimers, motivated by experimental indications that two conformers coexist8,10,14. They included correlation via the ACPF method15 which they consider equivalent in this case to CPF, along with a basis set of [321/31] quality. Their results in Table 5.6 show that the linear trimer is more stable than the cyclic by 2 kcal/mol, with respect to binding energy at the SCF level, but this difference is reduced to 0.7 with correlation included. Including zero-point vibrations and other corrections yields the AH° data in the last row which nearly mirror the trends in E. Given the errors which remain at this level of calculation, the calculated preference for the linear trimer by 0.5 kcal/mol in AH° was not considered by the authors as definitive, and the smallness of this quantity accounted for the presence of both geometries in the experiments.

5.2 HCCH Aggregates The C—H group of HCCH is of comparable strength as a proton donor to that in HCN, albeit somewhat less acidic, but the former molecule does not contain a lone electron pair.

Table 5.6 Comparative stabilities of linear and cyclic trimers of HCN13. Data shown represent binding relative to three isolated monomers (kcal/mol) SCF

E - Eelec elec - H° -

ACPF

linear

cyclic

linear

cyclic

12.5 10.7

10.4

10.1

9.0

8.4

9.4 7.9

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241

Hence, the interaction between a pair of acetylene molecules places the proton in some proximity to the alternate source of electron density, the cloud between the two C atoms. The weakness of the interaction, with — Eelec less than 2 kcal/mol, and the absence of appreciable perturbations in the monomer geometries, makes questionable the characterization of the interaction in (HCCH)2 as a H-bond. It is legitimate to wonder if perhaps the interaction might be strengthened by the effects of cooperativity. That is, since the interaction between any pair of HCN molecules in a chain is enhanced relative to that in the dimer, the same phenomenon might also occur for oligomers of HCCH, such that the pairwise interaction might be unambiguously classified as a H-bond. With this in mind, we peruse the literature. 5.2.1 Trimers Alberts et al.16 were the first to consider (HCCH)3 applying a DZP approach at the SCF level. Keeping in mind that the absence of a lone pair on either end of the HCCH molecule makes any linear arrangement unlikely, it is not surprising that this system tends toward a cyclic arrangement. Optimization led to the C3h structure pictured in Fig. 5.5. One can envision how this geometry permits one H atom of each molecule to interact with the cloud of its neighbor. This configuration is stable by 2.6 kcal/mol, relative to three monomers, or 0.9 kcal/mol for each pairwise interaction. The latter quantity is nearly identical to — Eelec calculated at the same level for the dimer (T-shaped), so the evidence of cooperativity here is weak. On the other hand, this equality is indicative of a certain degree of cooperativity since each of the pairwise interactions is distorted from a true T-shape. Moreover, this is only the SCF result and earlier work with the dimer illustrated that a good deal of the interaction takes place in the context of electron correlation effects. With the inadequacy of SCF treatments for a system of this type in mind, the trimer was reexamined shortly thereafter17, employing the MP2 treatment of correlation. A number of different configurations were first optimized at the SCF level so as to identify stationary points. In addition to the minimum identified previously and illustrated in Fig. 5.5, a number of other configurations were identified as minima, albeit of lesser stability. Fig. 5.6 illustrates the other two minima wherein the protons of two different molecules attack the same cloud (a) and the conjugate situation where the two hydrogens of a single central molecule approach the clouds of separate molecules (b). In comparison to the cyclic arrangement, with a binding energy of 2.3 kcal/mol, (a) and (b) have — Eelec values of 1.4 and 1.5 kcal/mol, respectively. Reoptimization of the cyclic structure in Fig. 5.5 incorpo-

Figure 5.5 C3h geometry of the HCCH trimer 16 .

242

Hydrogen Bonding

Figure 5.6 Alternate geometries of the HCCH trimer17. The central molecule acts as a double proton acceptor in (a) and as a double donor in (b).

rating correlation through MP2, enhanced the binding energy to 3.0 kcal/mol; enlarging the basis set to TZ2P raised this quantity still more, to 3.8 kcal/mol, again at the MP2 level. There is evidence of weak cooperativity in the geometries. The MP2/DZP distance between centers of mass is 4.310 A in the dimer, but only smaller by 0.02 A in the cyclic trimer. The binding energy of this trimer is 3.0 kcal/mol at the MP2/DZP level, so that each of the three pairwise interactions contributes 1.0. Since this value is equal to the binding energy of the dimer at the same level of theory, we again conclude that the evidence of cooperativity in the cyclic trimer is present but weak. One may consider a comparison between dimer and cyclic trimer as unfair in that the pairwise geometries are different, distorted T-shapes in the latter. A better comparison might involve the other trimer minima in Fig. 5.6 which contain true T-shaped interactions. The interaction energies of the trimers (a) and (b) are both less than twice the binding energy of the dimer, indicating a negative cooperativity. This result is consistent with the idea that a single molecule cannot act effectively as either double proton acceptor, as in (a), or donor, (b). 5.2.2 Tetramers and Pentamers Yu et al.18 took matters a step further and computed optimized geometries of acetylene oligomers up to n = 5. Taking as a cue the fact that monomer geometries are virtually unchanged within the dimer, they held internal geometries fixed during the course of their optimizations. Trimer and tetramer geometries were optimized at the SCF and MP2 levels with a 3-21G basis set, and then MP2/6-311G** was applied to obtain the interaction energy al

Cooperative Phenomena

243

these geometries; pentamers were considered only at the SCF/3-21G level. The tetramer was interesting in that experimental work19 had indicated a cyclic S4 complex very much like the trimer in Fig. 5.5 except that each molecule is tilted by nearly 30° with respect to the overall molecular plane. This configuration is illustrated in Fig. 5.7a while a fully planar C4h equivalent is shown in Fig. 5.7b. Calculations revealed no discernible difference in energy between these two configurations at any level of theory. The MP2/3-21G interaction energy of the entire complex is 9.3 kcal/mol, which works out to 2.3 kcal/mol for each "H-bond," that is, pairwise interaction, as indicated in the second column of data in Table 5.720. This value is a little larger than 2.0 kcal/mol in the trimer or the T-shaped dimer. The SCF values in the preceding column of the table indicate only small fluctuations in the Hbond energy as the size of the complex increases from n = 2 to n = 5. The pentamers studied by Yu et al. were comparable to the tetramers in Fig. 5.7 in that both were cyclic, one fully planar and the other staggered. The 8° distortions in the latter stabilized the system slightly, by around 0.15 kcal/mol. Bone et al.20 also considered the higher oligomers, n = 4, 5. They improved on the earlier methodology by employing a DZP basis set which contains polarization functions, and optimized some geometries at the MP2 level, including internal parameters. Their optimized tetramer was planar, as pictured in Fig. 5.7b, with the monomer centers of mass defining the vertices of a square. However, closer inspection of the SCF frequencies revealed this structure to be a saddle point. The small values of a number of imaginary frequencies (less than 1i cm-1) prevented a determination of the order of this saddle point. The authors hence searched for a S4 structure which is probably the true minimum, but the flatness of the surface prevented an unambiguous identification of such a minimum. They concluded the planar C4h geometry is in reality the global minimum, but the surface is extremely flat in this region, such that large-amplitude excursions from planarity will be experienced by the cluster. In the case of the pentamer, Bone et al.20 considered two possible geometries: a fivemembered version of the planar ring in Fig. 5.7, as pictured in Fig. 5.8a, or a different arrangement which contains two three-membered rings, clustered around a central molecule, illustrated in Fig. 5.8b. Both were found at the SCF/DZP level to be stationary points on the PES. The double ring (b) was bound relative to five monomers by 4.7 kcal/mol, as compared to 4.5 kcal/mol for (a). The authors attributed the greater stability of (b) to the

Figure 5.7 Configurations of the HCCH tetramer18. The complexes in (a) and (b) are nonplanar and planar, respectively.

244

Hydrogen Bonding

Table 5.7 Interaction energies per H-bond (— Eelec in kcal/mol) in oligomers of HCCH18,20. 3-21G

dimer trimer tetramer pentamer

DZP

SCF

MP2

SCF

MP2

1.6 1.5 1.8 1.7

2.0 2.0 2.3

0.78 0.78 0.91 0.89

0.97 1.01 1.15 1.13

presence of six pairwise interactions, as compared to only five in (a). The six pairwise contacts in (b) average 0.8 kcal/mol each, as compared to precisely the same amount in the T dimer at this level of theory, indicating little cooperativity. Frequency calculations revealed both (a) and (b) to represent transition states on the surface, with the true minima probably nonplanar and puckered, analogous to the tetramers. The last two columns of Table 5.7 illustrate the average binding energies of the various cyclic structures of the HCCH oligomers, calculated with the DZP basis set, and with BSSE corrections included. These values are considerably smaller than the 3-21G data of Yu et al., due in large part to the poor performance of the latter set for many molecular interactions, coupled with its large BSSE (left uncorrected by the authors). Nonetheless, both sets of data evidence similar trends. There is little cooperativity and what there is seems to peak at n = 4. This may be due to the perfect perpendicular arrangement of each pair in the tetramer. The antisymmetric C—H stretching frequencies also provide little evidence of cooperativity. The two such frequencies in the T-dimer are 3572 and 3576 cm-1. Forming the cyclic trimer in Fig. 5.5 yields three frequencies of 3567 c m - 1 , which is reduced only slightly to 3564 in the cyclic tetramer and 3565 in Fig. 5.8a.

Figure 5.8 Possible geometries of the HCCH pentamer20.

Cooperative Phenomena

245

5.3 Hydrogen Halides HX molecules contain a single proton but three lone electron pairs on the halogen atom. Since none of these lone pairs is collinear with the HX bond, and the HX dimer does not contain all four atoms on a single axis, it is unlikely that a chain of such molecules will be linear. 5.3.1 Open versus Cyclic Trimers One likely geometry for a trimer would contain a pair of dimer structures as in Fig. 5.9a, where each H-bond is linear, or nearly so, and the H . . X—H angle is in the 100-130° range. As in some of the cases already described, a third H-bond can be formed if the proton on the last molecule is brought near the X atom of the first, forming a cyclic structure as in Fig. 5.9b. Of course the price paid for this third H-bond is some angular distortion of all three. Some early studies of the comparative stability of these two arrangements of the HF trimer with small basis sets had yielded ambiguous answers. In 1983, Karpfen et al.21 applied more extended basis sets and optimized the geometries with gradient procedures. They found the cyclic trimer more stable than the open one by 2.2 kcal/mol. Their results confirm the greater strain in each of the three H-bonds of the trimer: the interaction energy per H-bond is 3.9 kcal/mol in the cyclic structure, as compared to the 4.7 kcal/mol in each of the two bonds of the open form. Soon thereafter, another study of the same system focused on the vibrational frequencies of the cyclic structure22. Each of the three H-bonds was calculated to contribute 5.0 kcal/mol, slightly more than computed earlier by Karpfen et al. Calculations in 198623 confirmed the greater stability of the cyclic geometry, as well as its planar nature, and suggested that the open form could convert to this geometry with little or no energy barrier. (Indeed, other calculations in the same year dispensed with the notion that the open form of the HF trimer represents a real minimum on the potential energy surface24.) It was also pointed out that all experimental measurements were indicative of the existence of the cyclic structure25. It was found here that the H-bond energy per bond

Figure 5.9 Open and cyclic arrangements of the HX trimer.

246

Hydrogen Bonding

in the trimer (4.3 kcal/mol) is in fact slightly greater than the interaction energy of the dimer (4.1) despite the strain in the former. One may presume that the cooperativity effect around the ring (each molecule is both donor and acceptor) is compensating for the unfavorable strain. This same study considered the ramifications of electron correlation and found little qualitative change. All H-bonds, whether in dimer or trimer are strengthened by a small amount. The correlated per-bond H-bond energy in the trimer is 4.7 kcal/mol, compared to 4.6 in the dimer. Interestingly, the authors pointed out a sharp increase in the correlated Hbond energy per bond to 6.4 kcal/mol in the tetramer. This increase is probably the result of a drop in the strain energy. Latajka and Scheiner26 considered the trimer of HF, along with (HC1)3, with extended basis sets designed to minimize basis set superposition error. The geometry optimizations led to C3h minima for both trimers, conforming to experiment27-30. Electron correlation induced a contraction in the interhalogen distances, more noticeable in (HC1)3. With correlation included and BSSE corrected by the counterpoise procedure, the binding energy in (HF)3 is 4.2 kcal/mol per H-bond, considerably stronger than the 1.2 kcal/mol for each Hbond in (HC1)3. Confirming earlier calculations, the binding energy in the HF trimer indicates a compensation between the strengthening caused by cooperativity, and the weakening that accompanies the angular distortions necessary to form the triangular trimer. The interchlorine distance computed for (HC1)3 is 3.78 A, in surprisingly good coincidence with an experimental measurement of 3.69 A between centers of mass29, in light of the fairly small basis set, 6-31G**, used in the calculation. 5.3.2 Three-Body Interaction Energies The question of cooperativity was further probed by computation of the three-body interaction energy, E326. This quantity is defined as the difference between the total interaction energy in the entire trimer, and three times the two-body interaction, E2 (one for each pair of subunits), all computed as Eelcc. This property differs from the prior means of considering cooperativity chiefly in that the geometry of all species are frozen in that of the optimized trimer. Thus, the two-body interaction is not computed in the optimized structure of the dimer, but that of the trimer, so the effects of angular strain are present here, too. The resulting quantities are reported in Table 5.8 where the "cc" superscript indicates counterpoise correction of basis set superposition error. The two-body terms in the first row underscore the greater H-bond interactions in oligomers of HF, as compared to HC1. When corrected for superposition error, MP2 adds little to the interactions between HF monomers, but is responsible for a 25% enhancement in HC1. It is important to mention that erroneous

Table 5.8 Two and three-body interaction energies ( Eelec in kcal/mol) in the trimers of HF and HC1, computed with +VPS basis set26. (HF)3

E2 CC E2

E3 ECC 3

(HC1)3

SCF

MP2

SCF

MP2

-3.67 -3.45 -1.47 -1.66

-4.04 -3.35 -1.39 -1.73

-0.94 -0.90 0.31 -0.23

-1.58 -1.13 -0.36 -0.22

Cooperative Phenomena

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conclusions of an exaggerated MP2 component would be drawn if counterpoise corrections were not included (compare with first row of data). This exaggeration is even more profound with other basis sets, particularly the "standard" ones such as 6-31G*26. The threebody terms are sizable: E3CC is roughly half the magnitude of E2CC for (HF)3 at the MP2 level, and E3CC/ E2CC = 20% for (HC1)3. Quite similar ratios pertain to the SCF calculations. It is worth stressing that correlation contributes very little to the three-body terms. This conclusion is confirmed by more rigorous partitioning of interaction energies31, which suggest that SCF computations are adequate to study most aspects of 3-body interactions involving H-bonds4. The nonadditivity of the cyclic HX trimer was partitioned into various contributions by intermolecular perturbation theory, combined with M011er-Plesset treatment of correlation, in 198932. After demonstrating the importance of correcting basis set superposition error of all terms, the authors stress that the three-body effect is dominated by its SCF component, as demonstrated by the data in Table 5.9. The latter is, in turn, primarily composed of the term attributed to electron density deformation, as the HeitlerLondon exchange nonadditivity amounts to only 5% of the total. The SCF portion of the three-body term is rather insensitive to improvements of the basis set beyond 6-31G* *. Correlation adds very little to the nonadditivity, making up only 5% of the total. The three-body terms arising from each level of correlation alternate in sign, making for cancellation. That is, the MP2 and MP4 terms are both negative (attractive), while MP3 contributes a positive quantity. There is surprisingly little basis set sensitivity of any of these terms. The manner in which the two- and three-body terms change as the three molecules are pulled apart is illustrated graphically in Fig. 5.10. In the vicinity of the equilibrium geometry (R = 2.73 A), the SCF terms clearly dwarf MP2 contributions. It is interesting to note the crossing of the two- and three-body MP2 terms. The three-body SCF term diminishes toward zero more quickly than does the two-body term, which approaches its asymptote gradually. Most of the nonadditivity can be traced to the deformation terms within the SCF approximation. The two- and three-body MP2 contributions become comparable in magnitude for interfluorine distances of 4 A or so.

Table 5.9 Two and three-body interaction energies (kcal/mol) in the trimers of HF and HC1. Data calculated with mediumpolarized [7s5p2d/5s3p2d/3s2p] basis set for [Cl/F/H]32. (HF), SCF Heitler-London Deformation MP2 MP3 SCF + MP2 + MP3 SCF Heitler-London Deformation MP2 MP3 SCF + MP2 + MP3

2-body terms -3.20 -1.89 -1.31 -0.20 -0.07 -3.47 3-body terms -1.59 -0.11 -1.48 -0.07 0.05 -1.62

(HC1)3 -0.70 -0.41 -1.24 -0.66 0.17 -1.20 -0.25 -0.00 -0.24 0.00 0.02 -0.23

248

Hydrogen Bonding

Figure 5. \ 0 Variation of SCF and MP2 two-body (solid lines) and three-body (broken lines) terms for cyclic (HF)332.

Variations in these quantities as the H atoms are pivoted around the fluorines are exhibited in Fig. 5.11. The two-body SCF term is clearly the most anisotropic. Its minimum in the vicinity of a = 30° dominates the determination of the equilibrium geometry of the trimer. Both the SCF and MP2 three-body terms favor smaller angles, with the SCF component both of larger magnitude and more sensitive to a. More recent work using a somewhat different formalism33 has largely supported the angular sensitivity reported here although there are certain points of lingering discrepancy. 5.3.3 Larger Oligomers The cyclic complexes of HP investigated were extended to the hexamer34 in 1990. Karpfen applied a polarized basis set, and included correlation via the averaged coupled pair functional method15. He restricted his investigations to C h planar symmetries; application of correlation prohibited gradient optimizations. Later work by the same group35 applied a much larger basis set, albeit without explicit correlation included. Nonetheless, similar trends are observed. Fig. 5.12 illustrates the progressive stretching of the HF bond as the ring builds up to the hexamer level, in concert with the continued contraction of the distance between F atoms. The correlated data, indicated by the solid curves in Fig. 5.12, show more of a sensitivity to ring size than do the SCF values, reported as the broken curves. The data in Table 5.1034,35 reveal the progressively smaller deviation from linearity of each Hbond as the ring is enlarged. This better arrangement, when coupled to the cooperativity. leads to the increase in H-bond energy (per H-bond) such that this quantity in the hexamer is 66% larger than the interaction energy of the fully optimized dimer.

Cooperative Phenomena

249

Figure 5.11 Angular dependence of SCF and MP2 two-body (solid lines) and three-body (broken lines) terms for (HF)3, with R(FF) held at 2.73 A. a is defined as the deviation of the HF bond from the F..F axis, as indicated32.

As in the case of linear, open oligomers, the stretching frequencies of the HF bonds undergo progressive red shift as the number of monomers increases. These shifts are illustrated in Fig. 5.13 where they may be seen to surpass 400 cm-1 in the tetramer. The reader should understand that in a cyclic structure (i.e., n > 2), each molecule acts as both donor and acceptor, and the symmetry is such that vibrations cannot be distinguished as belonging to one particular molecule. It is also worth pointing out that a third v(FH) stretch is present in the tetramer, which at 3571 c m - 1 , amounts to a red shift of 611 c m - 1 from the monomer. Just as the aforementioned HF bond stretches are magnified when correlation is included, so too are the vibrational frequencies more sensitive to ring enlargement when computed at a correlated level. The MP2 red shifts of the HF stretching frequency in the Cnh HF oligomers, resulting from each addition of another HF molecule, are approximately double the SCF values36. Three normal modes were investigated in this work, using a rather large 6-311 + +G(3df,3pd) basis set. vl refers to the symmetric HF stretching mode. It differs from those reported in Fig. 5.13 in that it represents the totally symmetric combination of all HF stretching motions, the lowest of such HF stretching frequencies. v2 corresponds to motion of all atoms toward the center of the ring, and a symmetric bend of all molecules is characterized by v3. The behavior of these three frequencies as the ring is enlarged is illustrated in Fig. 5.14. v1 undergoes the characteristic sharp drop as n is increased, like the other HF stretches in Fig. 5.13. The enhancement in this sensitivity at the correlated level is evident by a com-

250

Hydrogen Bonding

Figure 5.12 Calculated internal and intermolecular distances in cyclic (HF)n. Solid curves refer to correlated values with polarized basis set34; dashed lines to SCF data with 6-311 + +G(2d,p)35.

parison of the solid and broken curves. In contrast to this reduction, v3, the bending mode, is increased as the ring is enlarged. This increase is enhanced when correlation is included. The mode corresponding roughly to a "swelling" of the ring, v2, is rather small in magnitude and is not strongly affected by the number of molecules in the ring. As the size of the cyclic oligomers increases, there is progressively greater overlap between the experimental bands originating from various values of n. This spectral congestion makes it difficult to identify those of a given size oligomer with certainty. This complication fueled speculation that the pentamer might not be found in the gas phase, or might

Table 5.10 Angular deformation of each H-bond and average interaction energy in cyclic (HF)n (SCF data with 6-311 + +G(2d,p) basis set35, correlated values34). (FFH) (degs) n 2 3 4 5 6 a

SCF

55.8 26.5 13.2 7.8 4.0

Optimized dimer, not cyclic structure.

— Eelec/n (kcal/mol)

ACPF a

6.0 23.6 11.6 2.4

SCF

ACPF

3.2 4.2 5.6 6.2 6.4

5.0a 5.1 7.2 8.3

Cooperative Phenomena 25 I

Figure 5.13 Computed harmonic vibrational frequencies in cyclic (HF)n34. Solid and dashed lines refer to smallest and second smallest red shifts, respectively, relative to monomer.

exist in only very low proportions, being replaced by tetramers and hexamers. However, the pentamer has recently been identified by FTIR spectra of continuous and pulsed super38 sonic jet expansions of HF37,38 at low temperature. An analysis of the surface generated from a large number of points calculated at the MP2/DZP level led to the conclusion that sizable anharmonicities are present in the HF stretching fundamental frequency shifts of such oligomers but that these changes can be canceled by other errors (BSSE, zero-point bond-weakening, etc.) in certain cases38. Although perhaps not global minima, the open, chain-like oligomers were reexamined in 1994 using large basis sets by Karpfen and Yanovitskii35,39, who considered how the length of the chain influences the properties of the subunits within. As the chain elongates, it is found that the internal bond lengths and intermolecular separations of most of the "internal" molecules, that is, those far from the ends, are fairly uniform, with sharp dropoffs noted in the last three or four molecules at each end. The r(HF) bonds in the internal molecules are longer than those of the end molecules by 0.01-0.02 A; the F..F separations are smaller in the middle of the chain by about 0.1 A. As the chain elongates with additional HF molecules, there are progressively more HF stretching frequencies. These frequencies are all red-shifted relative to the monomer, and these shifts increase as the chain becomes longer. There is always one which is changed by only a small amount, less than 100 cm - 1 , even for a long chain. The largest shift was noted for the longest chain, n = 19, which had one HF stretching frequency smaller by 800 c m - 1 , relative to the monomer. This compares to a shift of only 42 cm-1 for the highest frequency when n = 19. Intensities were also computed for the various HF stretching modes. Some

252

Hydrogen Bonding

Figure 5.14 Frequencies of three fully symmetric normal modes of Cnh (HF)n, computed with a 6-311 + +G(3df,3pd) basis set36. Solid curves refer to MP2 values and dashed lines to SCF.

salient findings are as follows: The intensity of the lowest-frequency mode, which corresponds to the simultaneous, in-phase motion of all hydrogens, increases the most as n is incremented. The intensity of this mode, for n = 19, is 200 times larger than for the monomer. Even dividing this quantity by 19 to obtain an enhancement per molecule, one finds an increase of 10 or so. The behavior of the other modes can be modeled by a vibrating string. That is, the intensities of modes with an odd number of nodes in the phase relation are zero. For example, when n = 18, the intensity vanishes for modes containing 1, 3, and 9 nodes.

5.4 Water The cooperativity in aggregates of water is particularly important for attempts to understand the behavior of the liquid and aqueous solvation. It is in part responsible for the ability of water to maintain H-bonds up to very high temperatures, probably even above 800 K in supercritical water40-42. This importance has motivated numerous attempts to incorporate nonadditivity into water-water potentials43-51 and, more recently, related liquids such as alcohols52,53. The question of optimal orientation in oligomers of water differs from the (HX)n case since each water molecule contains two lone pairs and two protons. Based upon the general principles of cooperativity, it is reasonable to presume that if each water molecule has two neighbors, it will prefer to act as donor to one and acceptor to the other, as compared to serving as double donor or double acceptor. A full complement of four neighbors would permit the central water to serve as donor in two H-bonds and acceptor in two others.

Cooperative Phenomena

253

After some earlier studies had yielded uncertainties as to the magnitude of the cooperativity, Clementi et al.54 considered the sensitivity of this quantity in the water trimer to the type of basis set used. Rather than perform a geometry optimization, the authors assumed certain particular configurations. Despite the earliness of this work, the authors corrected some of their results for basis set superposition error. The nonadditivity in most configurations examined was rather small, in most cases less than 0.5 kcal/mol. The authors concluded that minimal basis sets were capable of providing nonadditivities comparable in accuracy to more extended sets, provided the former are well balanced and BSSE is corrected. As a harbinger of later work, these authors attributed a fraction of the nonadditivity to induction energy. Calculation of the latter quantity, based upon atomic point charges and bond polarizabilities of the individual molecules, was able to reproduce the ab initio data with surprising accuracy. 5.4.1 Extended Open Chains The charge redistributions that accompany formation of a H-bond, and the consequences for a string of water molecules, were investigated with two rather small basis sets55. By extrapolating the results up to the pentamer level, Scheiner and Nagle found the component of the dipole moment, in the direction of the chain, is increased by 40-60% as compared to the moment in the water monomer. Shortly thereafter, Karpfen and Schuster used a different formalism wherein infinite chains could be considered explicitly to examine the same question56. Their means of analysis yielded a reduced enhancement of the moment of only 16%; they attributed the discrepancy to edge effects. A later work57 continued the investigation of extended chains of water molecules, incorporating the effects of electron correlation. As in the oligomers of HF, the length of the H-bond contracts as the chain enlarges. The small nonlinearity present in the dimer vanishes as well. Crystal orbital techniques were employed to consider infinitely extended chains. Some of the more interesting features of the infinite chain are listed in Table 5.11

Table 5.11 Computed features of infinite chain of water molecules57. Basis set

- Eeleca

(kcal/mol)

r(OHb) (A)

r(OH)

2.672 2.837 2.877 2.882 2.880 2.881

0.973 0.956 0.953 0.967 0.959 0.958

0.950 0.944 0.942 0.941 0.939 0.938

2.619 2.728 2.734 2.731 2.742 2.729

1.008 1.010 1.010 1.004 1.003 1.001

0.976 0.963 0.960 0.959 0.956 0.954

R(O .. O) SCF

DZ DZ(d,p) DZ(2d,p) TZ(2df,p) TZ(2df,2pd) TZ(3d2f,3p2d)

10.74 6.72 6.24 5.82 5.61 5.48

DZ DZ(d,p) DZ(2d,p) TZ(2df,p) TZ(2df,2pd) TZ(3d2f,3p2d)

11.07 7.89 7.51 7.16 6.98 6.84

MP2

a b

Average for each H-bond, counterpoise corrected. Bridging hydrogen.

254

Hydrogen Bonding

as a function of basis set. There appears to be a clear convergence toward a H-bond energy of 5.5 kcal/mol, at the SCF level and after counterpoise correction. This value represents an enhancement of 1.7 kcal/mol or 45%, relative to the dimer, which can be attributed to the cooperativity in the chain. The interoxygen distance is rapidly approaching an asymptote of 2.88 A. The O—H bond of the hydrogen involved in the H-bond is longer by 0.02 A than the other O—H bond. The correlated asymptote for the H-bond energy is somewhat larger, close to 7.0 kcal/mol. The MP2 interoxygen distance is shorter by 0.15 A; the correlated R(O..O) of 2.73 A is rather close to the interoxygen separation of 2.74 A in ice. Raising the level to MP4 would likely increase the calculated distance by 0.01 A, making for even better agreement. This accord is surprisingly good in light of the difference between the onedimensional chain considered here and the three-dimensional lattice of ice which includes not only sequential chains in which each molecule is both donor and acceptor, but the monomers in ice are involved in more than two H-bonds each. The differential between the two O—H bond lengths in the water molecule is 0.05 A after correlation. Another work considered finite chains of varying length58. The water molecules were connected in a uniform sequential fashion as illustrated in Fig. 5.15. All interoxygen distances were taken as 2.84 A, chosen to mimic the liquid. The internal geometry of each molecule was frozen in its experimental structure. The basis set employed was DZP, with MP2 evaluation of correlation. Since the geometries were not optimized, normal modes could not be calculated. Instead, the workers focused on what they termed "uncoupled OH stretching frequencies," derived from pointwise computation of the energy for the system with different OH bond lengths, holding fixed the remainder of the geometry of the chain. The harmonic frequency was then evaluated as the second derivative of this potential curve. A variational solution of the Schrodinger equation for OH oscillator motion enabled a calculation of the anharmonic frequency. Some of the more interesting computations of the multibody interaction energies in the chain are reported in Table 5.12. With regard first to the two-body interactions in the first five rows, these terms die off fairly quickly with distance. The 1-4 interaction is mildly repulsive. Moving on to trimers in the next five rows, the entries in the two-body column of Table 5.12 refer to the sum of all three pairwise interaction energies. The 3-body interaction represents less than 10% of the total binding energy in the consecutive 1-2-3 trimer. It is much smaller in those trimers containing a "gap," for example, 1-2-4 or 1-2-5, amounting to about —0.04 kcal/mol in these cases. Turning now to the tetramers, the fourbody interaction energies are much smaller still, the largest of them being only —0.06 kcal/ mol for the consecutive tetramer with no gaps; higher-order terms such as 5-body terms can be safely ignored. The highest nonadditivity, defined by these authors as the difference between the total interaction energy and that computed by summing pairwise interactions, is observed for the full heptamer, where it amounts to 16% of the total binding energy. Fig. 5.16 illustrates

Figure 5.15 Arrangement of water molecules in sequential chain58.

Table 5.12 Multibody interaction energies (kcal/mol) in the sequential open oligomers of water. Calculated at MP2/DZP level58. Molecules"

2-body

3-body

4-body

5-body

1-2 1-3 1-4 1-5 1-6

-4.62 -0.81 0.03 -0.12 -0.01 -10.16 -5.41 -4.72 -5.50 -4.81 -15.66 -11.06 -10.22 -24.77

-1.03 -0.05 -0.03 -0.04 -0.04 -2.15 -1.11 -0.17 -21.28

-0.06 -0.00 -0.01 -3.34

-0.00

1-2-3 1-2-4 1-2-5 1-3-4 1-4-5 1-2-3-4 1-2-3-5 1-2-4-5 1-2-3-4-5 a

SeeFig. 5.15 for numbering scheme.

Figure 5.16 Percentage contribution of nonadditivity to total binding energies of linear and ring oligomers of water58.

256

Hydrogen Bonding

the percentage contribution of nonadditivity to the total binding energy of the linear chains of water. The solid line denotes a chain like 1-2-3-4 where there are no gaps present, as compared to the broken line which corresponds to a chain containing one gap, for example, 1-2-4 or 1-2-3-5. Note that the greatest nonadditivity is present in the consecutive chains and that the gap significantly reduces the amount of nonadditivity present. The percentage contribution of nonadditivity appears to level off for the longer chains, and one might project it to asymptotically approach something just less than 20% for an infinitely long chain. For ring oligomers, on the other hand, the degree of nonadditivity shows no sign of leveling off, for n as large as 5. So cooperativity may be said to be appreciably greater in the rings than in the chains for n > 3. The effects upon the computed OH stretching frequencies of lengthening the chain are illustrated in Fig. 5.17. The harmonic frequencies are illustrated by the broken lines and anharmonic v by solid. Molecule "1" refers to the first in the chain. It shows a red shift of 100-150 c m - 1 for the dimer, and a large further shift of about 70 c m - 1 when a third molecule is added. Chain elongation beyond this point results in only small further red shifts. Molecules ensconsed in the middle of the chain show the largest red shifts of all. Just as for the terminal molecule, these shifts continue to climb, albeit slowly, as the chain elongates. It is important to stress the very parallel behavior of the harmonic and anharmonic data, suggesting trends can be accurately deduced from study of the harmonic data. Effects are even stronger in the pentamer ring. The red shift of the anharmonic frequency is computed to be 377 cm - 1 , with a corresponding harmonic shift of 321 c m - 1 .

Figure 5.17 Calculated frequency shifts of OH stretches, involving the bridging hydrogen. Solid lines refer to anharmonic and broken to harmonic frequencies. The first molecule in the chain is designated "1," and the central molecule corresponds to that with largest frequency near center of chain 5 8 .

Cooperative Phenomena

257

5.4.2 Branching Clusters The observation that water molecules are about 0.2 A closer together in ice than in the gasphase dimer has motivated a number of theoretical inquiries. Yoon et al.59 considered how the R(O .. O) shrinkage might be tied together with stretches in r(OH). Their polarized basis set was satisfactory in yielding both a low BSSE and a good moment for the monomer. The three-body terms computed here were considerably larger than those obtained earlier by Clementi et al.54, albeit for different configurations: Yoon et al. limited their analysis to geometries corresponding to the structure of ice. An optimization of the water cluster geometry using the nonadditivity term approached within 0.06 A of the interoxygen distance in ice; the shrinkage relative to the dimer was thus attributed mainly to the three-body term. The cooperativity involved in a single water molecule, surrounded by four others in a tetrahedral arrangement, was investigated at the SCF level60, with all interoxygen distances set to 2.90 A. The central molecule in the pentamer acts therefore as simultaneously double proton donor and double acceptor. There are various subset trimers of this full complex: one with the central molecule as double donor, one as double acceptor, and several where the central molecule is both donor and acceptor; that is to say, "sequential." As expected, the three-body interaction energies of the double-donor and double-acceptor trimers are positive (destabilizing), but negative values are obtained for the sequential types. The magnitudes of these terms vary between 0.5 and 1.0 kcal/mol, as compared to two-body terms between 3 and 4 kcal/mol; four-body terms are all much smaller, less than 0.03 kcal/mol. Because of large-scale cancellation between the various three-body terms, which are both positive and negative, the total of all these terms accounts for only about 5% of the sum of two-body interaction energies (most of which are negative). The author60 also considered the charge shifts occurring as a result of pairwise and higher-order interactions. In comparison to changes in Mulliken charges of the central molecule caused by dimerization which are of the order of 0.005-0.030 e, the (3-body) nonadditivity in the shifts amounts to less than 0.010, more commonly 0.002; four-body nonadditivities are typically less than 0.001. The authors conclude that the additive contributions to the electron redistributions are responsible for the bulk of the nonadditivity in the energy. 5.4.3 Cyclic Oligomers The importance of electron correlation to the cooperativity in a cyclic tetramer was investigated61 in 1987. The choice of system was based on its experimental detection62,63. The geometry was fully optimized via gradient algorithms, and superposition error was corrected by the Boys-Bernardi counterpoise procedure. The formation of the cyclic tetramer stretches the donor OH bonds by three times more than the stretch in the dimer, and R(O..O) is shorter by 0.15 A. With regard to energetics, the per-bond H-bond energy of the tetramer is about 27% larger than the dimer interaction energy. This enhancement is virtually unchanged when correlation is included. The authors attribute the majority of this cooperative effect to three-body interactions, with the remainder due to two-body terms such as mutual polarization. 5.4.3.1 Vibrational Spectra The vibrational modes of clusters of water molecules were emphasized in a study up to the tetramer level64, motivated by IR and coherent anti-Stokes Raman spectroscopic data. Cal-

258

Hydrogen Bonding

culations were restricted to the SCF level, using primarily the 4-31G and 6-31G* basis sets. Optimization of the cyclic trimer with the larger basis set yielded an isosceles triangle of the three O atoms: one of the R(O..O) is slightly longer than the other two. Its cyclic character is consistent with experimental observation62. A normal mode analysis was carried out within the harmonic approximation and yields what the authors refer to as "delocalization of intramolecular modes" wherein all three molecules participate in each normal mode. The v1 and V2 modes of the monomer can be recognized within the trimer, but each of these monomer modes appears as a set of three vibrations, each with participation from all three molecules. The exception is v3 which remains localized in the trimer. The bending modes are closely related to v2 in H2O while the vibrational amplitudes of the bridging and free O—H stretching coordinates are quite unequal in the stretching modes of the trimer. The stretching modes which place more emphasis on the H-bonding hydrogens are red-shifted versus the monomer by 82-130 c m - 1 , while the shift to lower frequency in the other set of stretches is 37-42 c m - 1 . In contrast, the bending modes in the trimer are higher than that in the monomer by 15-37 c m - 1 . The tetramer is more symmetric than the trimer, being very close to S4. All of its intramolecular modes are highly delocalized, in contrast to the trimer which had varying degrees of this phenomenon. This work was continued several years later with a study of the effects of deuterium substitution on the vibrational modes of (D2O)365. The size of the ring was extended to the pentamer, again limited to the basis sets mentioned above66. The ring was found to be very close to planar, except for the nonbridging hydrogens. Two of these lie on one side of the ring, and three on the other, with a nonplanarity of roughly 45°. Each H-bond is very nearly linear, at least within 2-3°. The pentamer is "floppier" than the trimer and tetramer, with a very low-frequency 21 c m - 1 librational mode. The same study considered also (H2O)8, and found a quasicubical D2d structure. 5.4.3.2 Energy Components The size of the cyclic cluster under investigation was increased to six by White and Davidson67. Because of their underlying interest in ice, the authors took the O..O distance as 2.75 A, rather than the larger separation in the gas phase; r(OH) was fixed at 0.99 A. The structure examined was not of the classic type where all six molecules act as both donor and acceptor: one molecule (#5) serves as double donor and another (#4) as double acceptor, as illustrated in Fig. 5.18. The two- and three-body interaction energies in the water hexamer were decomposed via the Morokuma procedure, without counterpoise correction, and some of the results are listed in Table 5.13. Beginning with the two-body terms, the results for all adjacent molecules are identical to the data for the 1-2 pair in the first row of the table. This similarity is explained by the fact that all adjacent pairs constitute a single H-bond; the concept of dou-

Figure 5.18 Proton donor and acceptor characteristics in water hexamer67.

Cooperative Phenomena

259

Table 5.13 Morokuma components (kcal/mol) of two and three-body interactions computed for the water hexamer illustrated in Fig. 5.1867. E elec total in hexamer

-23.4

1-2 1-3 2-4 3-5 4-6 1-5 1-4 2-5 3-6 SUM (

-2.8 -1.2 -0.6 0.8 0.4 -0.6 0.2 0.2 -0.4 -19.2

1-2-3 2-3-4 3-4-5 4-5-6 1-3-5 2-4-6 1-2-4 1-3-4 1-4-5 2-5-6 SUM ( Z2 E+ a

E2)

) 3E E3

-1.4 -1.0 0.8 1.2 0.0 0.0 -0.1 -0.1 0.2 -0.2 -3.8 -23.0

ES

-81.0 2-body terms -13.1 -1.1 -0.6 0.8 0.4 -0.6 0.2 0.2 -0.4 -81.0 3 -body termsa 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -81.0

EX

POL

CT

86.9

-10.9

-18.5

14.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 87.0

-1.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -8.4

-2.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -16.8

0.0 0.0 -0.1 0.2 0.0 0.0 0.0 0.0 0.0 0.0 -0.1 86.9

-0.8 -0.6 0.5 0.4 0.0 0.0 0.0 -0.1 0.1 -0.1 -2.2

-0.6 -0.4 0.4 0.5 0.0 0.0 0.0 0.0 0.1 0.0 -1.4 -18.3

-10.6

Not all terms are listed in the table.

ble donor or acceptor is only meaningful within the context of three or more molecules. The interaction energy amounts to —2.8 kcal/mol. The attractive electrostatic energy is canceled by the exchange repulsion; polarization and charge transfer energies are both attractive. For all nonadjacent pairs, the EX, POL, and CT terms are quite small, leaving only electrostatic energy in the pairwise interactions. Because of the long-range character of the ES term, there are significant contributions even from molecules on opposite ends of the ring; for example, 1-4 or 3-6. The signs of these nonadjacent pairwise electrostatic energies can be understood on the basis of the orientations of the particular molecules. For example, molecules 3 and 5 have H atoms pointed at one another, leading to the repulsive 3-5 term. When summed together, the two-body terms amount to —19.2 kcal/mol, less attractive than the full interaction energy in the hexamer by 4 kcal/mol. With respect to the individual components, the electrostatic term is by nature fully additive, so the sum of two-body terms is equal to the full ES energy of the hexamer. The exchange is very nearly additive, with a discrepancy of only 0.1 kcal/mol. The sum of two-body polarization and charge transfer components are each about 2 kcal/mol less attractive than the full components in the hexamer. The three-body terms are listed in the lower part of Table 5.13. The first several entries represent triplets of consecutive molecules around the ring. It is notable that these interactions can be of either sign. Repulsive terms are associated with triplets like 3-4-5 and

260

Hydrogen Bonding

4-5-6 that contain either a double-donor or double-acceptor; others are attractive. The ES contributions are identically zero and the exchange components are quite small. The threebody terms are composed of similar amounts of polarization and charge transfer components. It is worth noting that some of the three-body terms, for instance, 1-2-3, are of larger magnitude than certain pairwise interactions, particularly those between nonadjacent pairs. Nonconsecutive triplets may contain no adjacent pairs, as in 2-4-6, or one adjacent pair, as in 1-2-4. In the former case, the three-body energies are less than 0.1 kcal/mol; the latter are all less than 0.2 kcal/mol. The total of all three-body interactions is —3.8 kcal/mol, as compared to —19.2 kcal/mol for the sum of all two-body interactions. When added together, the total of all pairwise and three-body interactions comes within 0.4 kcal/mol of the total interaction energy of —23.4 kcal/mol in the hexamer. With respect to the individual components, there is very little nonadditivity in ES or EX. The total nonadditivity of some 4 kcal/mol is approximately equally divided between POL and CT. 5.4.3.3 Anisotropy of Energy Components The nature of the cooperativity in the water trimer was dissected by perturbation theory, coupled with extended basis sets, in 199168. Rather than a full geometry optimization, the authors assumed an equilateral triangle C3h arrangement, with all atoms in a common plane, as illustrated in Fig. 5.19. A basis set, especially designed for molecular interactions was employed, with a [5s3pld/3slp] contraction, and counterpoise corrections were applied at all stages. The equilibrium geometry of this cyclic C3h arrangement has interoxygen separations of 3.0 A, with an orientation angle a of 75°. Fig. 5.20 emphasizes that the optimal angle for this trimer is controlled by pairwise forces present in the SCF wave function. The anisotropies of three-body effects, at either SCF or MP2 levels, or of the two-body MP2 interactions, are dwarfed by the strong angular dependence of two-body SCF terms. The sum of SCF pairwise energies is also much larger in magnitude than the latter terms which are all less than 2 kcal/mol. We focus on the three-body forces in Fig. 5.21 where the SCF portion is compared with its components extracted before (Heitler-London, HL) and after (deformation, def) the wave functions of the monomers are perturbed by one another. It is apparent that the latter SCFdef forces are largely responsible for the anisotropy of the SCF three-body terms. The Heitler-London component is weaker, and resembles a mirror of SCF-def, with a maximum where SCF-def contains a minimum. The extremum at 20° corresponds to a configuration

Figure 5.19 Planar C 3h geometry of water trimer 68 . a refers to the angle between O..O axis and HOH bisector.

Cooperative Phenomena

261

Figure 5.20 Anisotropy of two and three-body interaction energies in the water trimer, at SCF and MP2 levels68. See Fig. 5.19 for definition of a.

where one H atom of each water molecule is pointing toward the center of the equilateral triangle, a so-called H-to-H geometry. It is thus not surprising to see a maximum in exchange repulsion-type forces at this angle. One can conclude that HL acts to ameliorate the deformation effects at the SCF level. It is instructive to compare the SCF three-body interaction, represented by the solid curve in Fig. 5.21, with the induction energy, with which there is a tendency to approximate it in the literature. In this context, it should be noted that the induction curve is far too attractive, by a factor of more than 2 in the vicinity of 20°. Other characteristics of its shape differ from the full SCF curve or deformation energy in Fig. 5.21 as well. The three-body forces at the correlated MP2 level are very small in magnitude, and insensitive to angular characteristics of the trimer. Most of these conclusions have been verified by later calculations69 and by symmetry-adapted perturbation theory calculations, although there were a number of discepancies as well33. The issue is not entirely closed. 5.4.3.4 Comparison with Open Trimers In addition to the closed cyclic trimer, the open trimers displayed in Fig. 5.22 in which the central molecule acts as (a) simultaneous donor-acceptor, d-a, (b) double donor, d-d, and (c) double acceptor, a-a were considered as well68. In the first case, one would expect positive cooperativity, which is confirmed by an attractive total three-body interaction energy equal to about 10% of the two-body contribution. Most of this three-body term is due to the SCF-deformation, as in the cyclic structure. The double-donor is bound by only about 1/10

262

Hydrogen Bonding

Figure 5.21 Anisotropy of various contributions to the 3-body forces for the cyclic water trimer68. See Fig. 5.19 for definition of . HL refers to Heitler-London term which prohibits modification of the charge clouds of each molecule in the presence of the others, and SCF-def to the result of such deformation, both at the SCF level. Three-body induction is computed directly via perturbation theory.

the strength of the donor-acceptor trimer. This weaker total interaction is due in large measure to the pairwise interaction between the two terminal molecules, which are oriented with their O atoms pointing directly toward one another. The three-body term is attractive but only very slightly. The double-acceptor trimer is more strongly bound than the double donor, with about half the total binding energy of d-a. The three-body term here is a repulsive one, and is roughly 10% of the magnitude of the cumulative two-body terms. This repulsive character is due chiefly to the deformation of the wave functions. It is concluded that total nonadditive effects are dominated by SCF terms, which are directly attributed to electric polarization. However, since the polarization is constrained by the Pauli exclusion principle, classical models which neglect exchange phenomena may incur certain errors, especially in regions of strong overlap between electron clouds of the monomers. Correlation contributions to nonadditivity do appear small enough that they can be safely ignored, with efforts better concentrated on an accurate portrayal of the SCF phenomena. 5.4.4 Identification of True Minima The structures investigated for the water trimer up to this point have been idealized, with no guarantee that any are true minima on the full potential energy surface (PES). There are

Cooperative Phenomena

263

Figure 5.22 Three open trimer geometries68.

many more possibilities to be considered. Since each water molecule contains two hydrogens and two O lone pairs, it is possible in principle for it to participate in up to four interactions: as a double proton donor and/or double acceptor, all simultaneously. A diverse collection of trimer configurations, taking into account these various possibilities, was examined in 199270, using extended polarized basis sets, up to 6-311 + + G(2df,2p). Geometries were optimized and harmonic frequencies obtained. While counterpoise corrections were introduced, the authors decided that since these were of the approximate magnitude 0.5 — 0.7 kcal/mol (1.2 — 1.4 at the MP4 level) for all configurations examined, they were unimportant for purposes of comparative stability. The most stable geometry found is a cyclic one like that in Fig. 5.19, except that the nonbridging hydrogens do not lie in the plane. The cyclic character is consistent with observations in the gas phase62,71 and in Ar and Kr matrices72. (Indeed, the hydroxyl groups in the related phenol molecules will also form a cyclic trimer in gas-phase supersonic jets73.) This complex is bound, relative to three isolated water molecules, by 12.0 kcal/mol at the SCF/6311 + +G(2df,2p) level, which is increased by 3.7 kcal/mol for MP2. After including MP4 and zero-point vibrational corrections, the total binding energy becomes 10.8 kcal/mol. This represents an increase of 1.1 kcal/mol, relative to three times the interaction energy of an optimized dimer. A sequential dimer of the d-a type in Fig. 5.22 is not a true minimum on the full PES of the trimer. Optimizations beginning with such a structure bring the two terminal molecules closer together until the trimer closes up to the cyclic arrangement. The d-d structure in Fig. 5.22 does in fact represent a minimum on the PES. Its total binding energy is less than that of the cyclic trimer by 5-6 kcal/mol. In contrast, the double acceptor geometry a-a in Fig. 5.22 is not a minimum at all, but rather a second-order saddle point. On the other hand, Ihe

264

Hydrogen Bonding

double-acceptor does represent a minimum when the two terminal molecules approach one another to form a cyclic trimer as indicated in Fig. 5.23. This geometry is predicted to be very slightly (1 kcal/mol) more stable than the open double-donor chain. The structure of the cyclic water trimer was optimized also using correlation-consistent polarized basis sets, augmented with additional diffuse functions74. Whereas the three R(O..O) distances are all different at the SCF level, MP2 correlation brings them within 0.002 A of being equal to one another. The three H-bonds are all nonlinear by about 30°, that is, (OH..O)~150°. Replacement of one of the water molecules by phenol causes no fundamental changes in the geometry of the trimer75. Of course, the three R(O..O) distances are now all clearly different, varying between 2.85 and 2.98 A at the SCF/6-31 + +G(d,p) level. The shortest, and presumably strongest, H-bond is that where the phenol acts as the proton donor molecule. Deviations from nonlinearity are also affected by the phenyl substitution, but to only a minor degree. The phenyl ring is rotated about 120° from the plane of the three O atoms. Restricting calculations to the SCF level, and with counterpoise correction, the 6-31G(d,p) interaction energy (- Eelec) of the entire trimer is 16.3 kcal/mol, as compared to 15.5 kcal/mol for the water trimer at the same level of theory. The enhanced binding introduced by the phenol molecule carries over when ZPE corrections are added, yielding a DO value of 11.5, as compared to 9.7 kcal/mol for the water trimer. The positions of the three nonbridging hydrogens of the water trimer were the focus of more recent work. A potential energy surface was constructed as a function of the out-ofplane bending angles of these three atoms76. The energetics arise from MP2 level computations, with counterpoise corrections, using a singly polarized basis set, that includes functions centered on each H-bond. The structure with all of the nonbridging hydrogens on one side of the plane appears to be a minimum in the surface, albeit higher in energy than the global minimum in which one hydrogen is on the side of the plane opposite from the other two76. An all-planar structure, with all nonbridging hydrogens in the plane of the ring, is described as a third-order stationary point on the PES. After fitting the ab initio data to an analytical functional form, it was found77 that the lowest torsional levels of the non-bridging hydrogens lie well above the torsional barrier of 0.28 kcal/mol. This observation leads to the conclusion that the corresponding nuclear wave functions are fully delocalized amongst the six symmetry-related minima. The tetramer is also cyclic74, as illustrated in Fig. 5.24, and with a greater degree of symmetry than the trimer, belonging to the S4 point group. The ring is sequential in the sense that each molecule acts as simultaneous donor and acceptor. This type of structure has been confirmed by analysis of the far-IR vibration-rotation tunneling spectrum78. The correlated R(O .. O) distances are all equal to 2.743 A, about 0.06 A shorter than in the trimer. (The ex-

Figure 5.23 Cyclic water trimer containing a doubleacceptor molecule70.

Cooperative Phenomena

265

Figure 5.24 Cyclic water tetramer with S4. symmetry.

perimental estimate78 is 2.79 A in the tetramer.) The larger ring permits less nonlinearity in the four H-bonds as 6 (OH..O) rises to 168°. A later work79 found that inclusion of electron correlation is mandatory if one wishes to study the torsional motions of the hydrogens in the tetramer. Fortunately, the results indicated that correlation effects beyond second-order are unlikely to make significant contributions to the tetramer, consonant with findings for the dimer and trimer. The global minimum, with two hydrogens above the molecular plane and two below, is more "classical" than the trimer, in the sense that the torsional zero-point vibrational level lies below the saddle points for interconversion79. The barriers to torsional motions of the nonbridging hydrogens are about 1.2 kcal/mol. These results were largely confirmed by later optimizations of the trimer and tetramer with a variety of polarized basis sets80 where the 0.16 A reduction in H-bond length on going from trimer to tetramer was stressed. The VRT measurements78,81 support the notions advanced here concerning tunneling from one minimum to a symmetrically related one. Just as in the case of the trimer75, replacement of one of the water molecules of the cyclic tetramer by the hydroxyl-bearing phenol molecule yielded much the same structure82. Of course the S4 symmetry classification no longer applies. The tendency of the phenyl ring to make its hydroxyl more acidic is reflected in the four R(O..O) distances. The shortest (2.784 A at the SCF/6-31G** level) corresponds to that in which the phenol acts as proton donor, and the longest (2.900 A) when phenol is the acceptor. The combination of phenol as a better proton donor, but poorer acceptor than water, leads to little net effect on the binding energy of this complex. With correlation included along with BSSE corrections, the binding energy De is computed to be quite similar to that of the water tetramer (0.4 kcal/mol stronger), although this difference increases to 1.7 kcal/mol when zero-point vibrations are considered. It was not possible74 to optimize the larger oligomers at the correlated level, so Xantheas and Dunning reported SCF optimizations instead. Both the pentamer and hexamer were found to be cyclic, sequential rings, in the same spirit as (H2O)3 and (H2O)4. There is some puckering of the ring of the five O atoms in the pentamer, with two of them lying within about 20° of the plane of the other three. This sort of geometry in the pentamer is supported by recent VRT tunneling spectral data83. (In an interesting aside, crystal diffraction data suggest that water molecules may assemble in cyclic pentamers in hydrophobic clefts of proteins84.) The hexamer belongs to the S6 point group, and has the general structure of chair cyclohexane. There is virtually no angular distortion of the H-bonds which are within 2.5° of linearity.

266

Hydrogen Bonding

When the number of water molecules grows beyond six, the situation begins to change. In terms of energy alone, a di-bipyramid shape of the heptamer is computed with a 6311G* * basis set to be some 4 kcal/mol more stable than a cyclic geometry85. Consequently, it is the former structure that would be observed at 0 K. However, the entropy of the dibipyramid is quite a bit less than that of the seven-membered cycle. As a result, the cyclic geometry is favored with respect to free energy by 2.6 kcal/mol at room temperature. This situation is echoed in the octamer where the cyclic structure is superceded in stability by a geometry in which the oxygen atoms occupy the corners of a cube86. This sort of arrangement permits each molecule to be involved in three, rather than two, separate Hbonds. Calculations at the SCF/6-311G** level find the cube to be more stable than an eightmembered ring by 13 kcal/mol. This result confirms an earlier finding with the smaller 431G basis set66. On the other hand, the latter ring has a very high entropy so that its free energy is within about 0.2 kcal/mol of the cube at 298 K86. This trend is confirmed when correlation is added to the calculations by the MP2 method and the results are corrected for superposition error87. In fact, this particular set of correlated computations indicates the cyclic structure is unquestionably of lower free energy at room temperature. Other geometries that lie within about 10 kcal/mol of the global minimum cube can be characterized as five- (or six-) membered clusters, with three (or two) additional molecules, and a pair of four-membered rings, connected side to face. Altogether this series of calculations identified 29 separate minima on the PES of the water octamer86. 5.4.4.1 Vibrational Spectra Xantheas and Dunning74 computed the vibrational spectra of these water clusters in some detail, providing not only the frequencies at both the SCF and MP2 levels, but also IR intensities for each and Raman activities. Table 5.14 lists the shifts that occur in the intramolecular frequencies, relative to the monomer, and all computed at the same correlated MP2 level with the same basis set. Considering first the bending modes, the increase in this frequency with each progressive enlargement of the oligomer is apparent. Smaller blue shifts are observed in the stretch of the free O—H. More dramatic than these are the red shifts in the stretching frequencies of the bridging O—H group, which diminish by several hundred cm-1 as each new water molecule is added. The relationship between the latter shifts and the optimized length of the corresponding O—H bonds is very close to linear, whether at the correlated or SCF levels, conforming to the Badger rule88,89. The trends in the computed data were later confirmed by measurements of the spectra in the gas phase90. The harmonic intermolecular modes are reported in Table 5.15. As for the intramolecular modes, the MP2 frequencies are uniformly higher than the SCF values. In the case of Table 5.14 Shifts in harmonic intramolecular vibrational frequencies (cm - 1 ) of cyclic oligomers of water, relative to the monomera, computed at MP2/aug-cc-pVDZ level74. n

Bends

Free O—H stretch

Bridging O—H stretch

2 3 4

1,28 9, 12, 37

57,35 26, 24, 20 15, 15, 15, 14

-73, -160 -231, -240, -299 -350, -388, -388,

a

14,30,30,60

-481

Frequencies in monomer are 1623, 3807, 3936, respectively. O—H stretches are referenced to average of two O — H stretching frequencies in monomer.

Cooperative Phenomena

267

Table 5.15 Harmonic intermolecular vibrational frequencies of cyclic oligomers of water (cm - 1 ) computed at MP2/aug-cc-pVDZ level74. n

2 3

4

141 158 51 451

147 173 79 754

155 185 200 826

185 193 211 826

342

218 237 996

632 235 237

343 255

351 255

444 261

571 291

667 403

863 435

451

the dimer, the two higher frequencies correspond to bending motions of the monomers, followed by the intermolecular stretch at 185 cm - 1 . The authors were not clear about the types of motions that correspond to the frequencies listed for the trimer or tetramer. A later set of computations69 was more explicit about identification of the various modes in the trimer. Frequencies of "flipping" motions of the three nonbridging hydrogens were in the 200-270 cm-1 range. Intermolecular stretches were somewhat lower, 170-220 c m - 1 . The remaining intermolecular modes correspond to librations and vary from 350 to 830 c m - 1 , with the highest of in-plane type. 5.4.4.2 Many-Body Contributions An update of this work91 emphasized the many-body contributions to the binding energies. MP4 results are very close to MP2, as evident in Table 5.16. The three-body term contributes some 12% of the total binding energy of the trimer at the SCF level, which increases to 17% at MP4. The results for the tetramer indicate the four-body term is quite small, on the order of —0.5 kcal/mol. One would hence introduce only a 2% error by ignoring this term completely. The combined three-body terms make up some 25% of the total binding energy of the tetramer. Correlation seems to nearly double the three and four-body terms, but has a much smaller enhancing effect upon the pairwise interaction. The total binding energy of the tetramer is 23.8 kcal/mol, the first reported correlated value for this quantity. Another manner of viewing the cooperativity is through the progressive increase in the average binding energy of the oligomers. Table 5.17 reports the average H-bond energy as

Table 5.16 Total multibody energetics (kcal/mol) for cyclic water oligomers with aug-cc-pVDZ basis set91.

dimer

trimer

tetramer

SCF MP2 MP4 SCF MP2 MP4

SCF MP2 MP4

2-body

3-body

4-body

-3.71 -4.42 -4.36 -9.75 -11.80 -11.57 -16.04 -18.55 -17.97

-1.38 -2.45 -2.40 -3.52 -6.24 -6.16

-0.22 -0.54 -0.56

268

Hydrogen Bonding Table 5.17 Average binding energy per molecule, — Eelec/n, (kcal/mol) in water oligomers with aug-cc-pVDZ basis set91.

dimer trimer tetramer pentamer hexamer

SCF

MP2

MP4

1.86 3.67 4.88 5.19 5.40

2.21 4.62 6.08

2.18 4.55 5.95

(- Eelec/n) at various levels. The data clearly indicate the cooperativity as the binding energy rises from 1.9 kcal/mol (at the SCF level) for the dimer up toward 5.4 as the number of molecules reaches six. (However, the data for the dimer may be misleading as the complex is not cyclic and so contains only a single H-bond.) This study concludes that two and three-body interactions can provide most of the total interaction. Correlation is recommended, but the MP2 level appears satisfactory. 5.4.4.3 Higher Level Searches for Minima The theoretical treatment of the cyclic trimer was improved when BSSE corrections were added to the geometry optimizations and to the harmonic vibrational frequencies92. The work was not truly a full optimization in that each molecule was frozen in its experimental monomer geometry. The three O atoms were presumed to occupy the vertices of an equilateral triangle, with the bridging hydrogens lying in this O—O—O plane. In this manner, the optimization was reduced to only three parameters: R(O..O), (HO..O), and the dihedral angle of rotation of the nonbonding hydrogens out of the plane, . The optimal value for the latter out-of-plane angle was found to be 48°. This value differs by only 2° from what it would be in the absence of cooperativity. It deviates from a previously optimized value of 30°70 which may be a result of the counterpoise corrections included here. A 0.07-0.09 A shortening of the interoxygen separation was attributed to cooperativity, but another factor is in operation. Specifically, "bent" H-bonds tend to have shorter intermolecular distances, so some of the contraction can be associated with the nonlinearity imposed on each H-bond by formation of a cyclic trimer. The energy of interaction in the trimer was found in this work to be —14.7 kcal/mol, 0.6 kcal/mol stronger than three times the binding energy of the dimer. It should again be emphasized that this small net enhancement is really indicative of a larger cooperativity since each of the three H-bonds in the trimer is significantly distorted from linearity, in this case by 20°. The nonadditivity in the trimer amounts to —2.0 kcal/mol, perhaps a better indicator of cooperativity here. When zero-point vibrations were included in the analysis, the dissociation energy of the trimer, Do, came to 10.2 kcal/mol, larger than three times that of the dimer by 2.4 kcal/mol. In comparison to the magnitude of the effects of nonadditivity upon the energetics in the trimer, much larger changes are observed in the shifts of the O—H stretching frequencies, vOH, which are 70% larger than the dimer value 92 . The major component in the nonadditivity was traced to the SCF deformation, consistent with prior calculations. Whereas this effect is a relatively minor contributor to the total interaction energy, it dominates the fre-

Cooperative Phenomena

269

quency shift. Another important finding is the high sensitivity of frequency shift to the intermolecular distance (and out-of-plane angle as well). The authors reason that the differences in calculated frequency shifts from one theoretical study to another may be largely due to differences in R(O..O) used in these works. The reader is thus cautioned to carefully consider intermolecular distances when comparing theoretical calculations of such frequency shifts. One of the highest level searches of the potential energy surface of the water trimer to date added diffuse functions to a DZP basis set, and included correlation via single and double excitation coupled cluster (CCSD)93. At this level of theory, the structure is that pictured in Fig. 5.25, with two non-bridging hydrogen atoms above the plane of the oxygens and one below. The three R(O..O) distances are not quite equivalent, being equal to 2.825, 2.828, and 2.837 A. Each of the H-bonds is significantly distorted from linearity, with (OH..O) in the range 147-150°. The non-bridging hydrogens lie out of the plane by some 40-50°. The geometry obtained by the CCSD treatment of correlation is very similar indeed to the MP2 results earlier74. Forcing full planarity of the cyclic trimer, and a C3h geometry, results in a stationary point of index 3, that is, it is not a minimum and there are three imaginary frequencies. The harmonic vibrational frequencies computed with this basis set are listed in Table 5.18, along with the change resulting from adding correlation effects. There is a clear pattern in that the nine higher frequencies, representing the intramolecular modes, above 1000 cm-1 are all diminished by correlation, and the twelve intermolecular frequencies are raised. The lowering of the intramolecular stretching frequencies are of the order of 6-9%, with slightly smaller increases in the three intramolecular bending modes, 3-5%. The increments in the intermolecular frequencies caused by correlation are larger, generally 10-20%, but varying all the way up to 39% for the lowest frequency. The lowest of these intermolecular frequencies is some 170 c m - 1 , about double that of a band that has recently been observed at 87 cm-1 94. Another recent set of calculations69 confirmed the structure of the minimum, using a modification of the standard MP2 correlation procedure, and with counterpoise corrections applied. Stationary points were located at the MP2/6-311 + +G(d,p) level, and vibrational frequencies determined. While the global minimum has two of the hydrogens above and one below the plane of the three O atoms, a secondary minimum occurs when all three hydrogens are on the same side of the plane. This so-called "crown" configuration is 0.8 kcal/mol less stable than the global minimum. A barrier of 0.2 kcal/mol must be crossed to transit from the global to the secondary minimum. A configuration with all nonbridging hydrogens lying in the molecular plane lies 1.2 kcal/mol above the global minimum.

Figure 5.2.5 Equilibrium geometry of water trimer, belonging to C1 point group93.

270

Hydrogen Bonding

Table 5.18 Harmonic vibrational frequencies (cm - 1 ) of the water trimer, computed at SCF and correlated level, with DZP+diff basis set, along with percentage change93. SCF

CCSD

CCSD-SCF (%)

4234 4233 4229 4072

3957 3953 3951 3769

-6.5 -6.6 -6.5 -7.4

4068

3762

-7.5

4038 1776

3692 1720 1671 1667 941 664 536 443

-8.6 -3.2 -4.7

1754

1750 741 614 481 392 314 305 207 184 166 151 144 122

356 331 264

208 195 186 177 170

-4.7

27 8 11 13 13 9 28 13 17 23 23 39

The complex is bound by 16.3 kcal/mol, relative to three isolated water molecules. This value is higher by 1.3 kcal/mol than three times the 5.0 kcal/mol binding energy of the dimer at the same level. The three-body component is negligible at the MP2 level, and more than 2 kcal/mol at the SCF level. This very small correlation component is in agreement with a previous work68 but at odds with a later calculation91. The authors believe the large MP2 three-body term identified by Xantheas is due to a change in geometry rather than a true cooperative effect. The authors went on to compute the energies of seventy points on the PES, as a function of the torsional angles of the three non-bridging hydrogen atoms. These data were then fit to an analytical function95 so that the dynamics of these hydrogen motions could be examined. It appears that these three atoms undergo large-amplitude flipping and torsional motions, even in the lowest vibrational states. 5.4.5 Substituent Effects Some of the effects of replacement of a nonbridging hydrogen by a phenyl group have already been described earlier in Section 5.4.4 in the context of interactions between water and phenol in various oligomers. The effect of a progressively larger alkyl group replacing one of the hydrogens of water was probed by considering a series of alcohols96. Study was restricted to cyclic trimers, at the SCF level with a 6-3 1G basis set, and with no correction for superposition error. The

Cooperative Phenomena

271

Table 5.19 Binding energies (kcal/mol) of cyclic trimers of mixed water (W) and alcohols, computed at SCF/6-31G level. Data computed at SCF/6-31G level96. ROH CH3OH C2H5OH C3H7OH(1) C3H7OH(2)

W3 24.3 24.3 24.3 24.3

W2 ROH 23.9 23.6 23.6 23.5

W (ROH)2 23.3 22.9 22.8 22.7

(ROH)3 22.8 22.1 22.0 22.0

cyclic character of these trimers, and indeed of higher oligomers as well, is consistent with molecular beam electric deflection results71. The binding energetics in Table 5.19 indicate the nature of the alcohol, whether it is methanol, ethanol, or 1- or 2-isopropanol, has little effect on the strength of the interactions. The number of alcohol molecules is a minor concern as well: As each water molecule of the trimer is replaced by an alcohol, there is a very minor decrement in the binding energy, on the order of 0.5 kcal/mol. Unlike the energetics, there is an elongating effect of alkyl substitution upon the interoxygen distances. Since the complexes are not fully symmetrical, and there are three different values of R(O..O), it is the average values that are listed in Table 5.20. Reading across a given row illustrates that each replacement of a water molecule by an alcohol stretches the average H-bond by some 0.01 A. The size of the alkyl group on the alcohol has a small effect as well, with the larger groups elongating the H-bonds by slightly more. There appears to be a linear correlation between energetics and geometry in the sense that the strongest total binding energy is associated with the shortest mean interoxygen separation. The similarity of energetic and geometric data for ethanol and 1-propanol suggests the 2carbon chain is sufficient to model a much longer one. Comparison of the cyclic homotrimers with the corresponding dimers reveals that only in the case of water is the binding energy of the trimer larger than three times that in the dimer. The three alcohol trimers all show a slightly negative cooperativity. Nevertheless, the important result is that in all cases, the energetics of distortion of the three H-bonds are approximately compensated by the cooperativity factor. A far better 6-3 11 + + G(2d,2p) basis set was applied to the cyclic methanol trimer97, although the calculations were limited to the SCF level. At this level of calculation, the R(O .. O) distances are shorter by 0.004 A than in the water trimer, in contrast to the study at the lower level where the contraction is five times greater. In both trimers, the stretches of the OH bonds, relative to the monomer, are 0.007 A. The energetic features of cooperativity in the methanol trimer are very similar to those for water.

Table 5.20 Average interoxygen distances (A) in cyclic trimers. Data computed at SCF/6-31G level96. W3 ROH CH3OH C2H5OH C 3 H 7 OH(1) C3H7OH(2)

2.702 W2 ROH 2.711 2.717 2.716 2.720

W (ROH)2 2.718 2.728 2.728 2.734

(ROH)3 2.724 2.738 2.738 2.755

272

Hydrogen Bonding

Table 5.21 Shifts in harmonic intramolecular vibrational frequencies ( c m - 1 ) of dimer and trimer of memanol, relative to the monomer, computed at SCF/6-311 + +G(2d,2p) level97. n

O—H bends

O—H stretch

C — O stretch

2 3

2,45 34, 42, 82

-5, -69 -98, -103, -128

-14, 10 -2,0, 10

Table 5.21 reports the shifts that arise in the intramolecular vibrational frequencies, relative to the methanol monomer. It is possible to draw inferences by comparison with the data for water in Table 5.14, even though the latter are correlated and obtained with a different basis set. In either case, the bending frequencies are blue shifted in the dimer, more so in the trimer. A red shift occurs in the O—H stretch which is magnified in the trimer. The authors pointed out that these shifts are significantly smaller than experiment, and attribute the discrepancy to a lack of correlation. This contention is in part confirmed by the much larger red shifts in the corresponding modes of the water trimer, following correlation. Unlike the blue shifts noted in the O—H stretches for the free (nonbridging) hydrogen of water, there is much less change in the C—O stretching frequencies of methanol dimer or trimer.

5.5 Mixed Systems

There has been a good deal of work on the cooperativity of H-bonds involving mixed systems. As an example, a pair of HX (X=F,C1) molecules were added to NH3 in order to examine the effects of multiple H-bonds on the charge distributions within each monomer98. Rather than optimize the geometry, the NH3 molecule was retained in its experimental monomer structure. The symmetry of each complex was taken as C3v, presuming that both HX molecules are collinear with the N lone electron pair. Calculations were limited to the SCF level. Comparison of the first two columns of Table 5.22, representing the charges in the isolated monomers, with those for the 1:1 XH ... NH 3 complexes, illustrates that formation of the initial H-bond causes a polarization of the two molecules. The N atom of NH3 increases its negative charge while the hydrogen becomes more positive. An even stronger

Table 5.22 Mulliken atomic charges in complexes of NH3 with HF and HC198. XH...NH 3 X

XH+NH3

NH3 HXi

HX()

XHo...XHi...NH3

X=

F

Cl

F

Cl

F

Cl

H N H X H X

0.243 -0.727 0.380 -0.380 0.380 -0.380

0.243 -0.727 0.197 -0.197 0.197 -0.197

0.269 -0.793 0.436 -0.449

0.264 -0.776 0.274 -0.291

0.274 -0.796 0.461 -0.481 0.411 -0.420

0.267 -0.781 0.292 -0.298 0.210 -0.211

Cooperative Phenomena

273

polarization occurs in HF and HCl The last two columns show the continuation of these trends as the second HX molecule is added to the complex. The changes in NH3 upon adding the second molecule are quite small, 5 me or less. The redistributions in the HXi molecule, now the central of the three molecules, are substantially larger: The H atom loses about 25 me of electron density while F picks up an additional 30 me. The inner HX molecule is quite a bit more polarized than is the outer one. In the case of HF, the inner H atom has 50 rne less density than does Ho, with a comparably larger negative charge on Fi. 5.5.1 Geometries These systems were examined in further detail when a 6-31G** basis set was applied to H3Z...HF...HF, Z=N,P99. The geometries of all complexes were fully optimized at the SCF level. Unlike the earlier assumption98 that the trimer would contain fully linear H-bonds and belong to the C3v point group, the equilibrium structure is highly bent, as illustrated in Fig. 5.26. This structure is sensible in light of the two dimers contained within. That is, one can consider the complex from the standpoint of taking H3Z...HF as a starting point. This dimer would contain a linear H-bond. The HF molecule would act as proton acceptor to an additional HF molecule, which would orient itself toward one of the inner F (Fi) atom's lone pairs. Were there no other interactions present, then, the (Z..Fi..Fo) angle would tend toward the 100° or so, characteristic of HF..HF. However, when this angle decreases, it is possible for the Fo atom to form a favorable interaction with a proton of H3Z, which can be aided by nonlinearities developing in each of the two H-bonds already present. These nonlinearities are denoted by the a angles in Fig. 5.26. Table 5.23 reports the salient features of the geometries of the various complexes. It is immediately clear that addition of a second HF molecule to H3Z...HF contracts the H-bond in the latter 1:1 complex. When Z==N, this contraction amounts to 0.13 A; it is 0.17 A in the case of P. The interfluorine distance in either trimer is shorter than in the simple HF dimer. This is particularly true in the case of H3N, which is a better proton acceptor than is H3P. The effects of cooperativity are also apparent in the internal r(HF) bond lengths. The bond which winds up in the central molecule, denoted ri, stretches by 0.03 A when H3N is added to HP..HF. The addition of the H3Z molecule also lengthens the outer HF bond, albeit by a smaller amount. With regard to the angular features, it might be noted that addition of the second HF molecule to H 3 Z .. HF causes a 12-21° nonlinearity. Coupled with this effect is a 17° nonlinearity in the new H-bond to (HF)o, and a reduction in the 0 angle from 102° in HP..HF to only about 70° in the full trimer. All of these angular changes operate to allow the (HF)o molecule to more closely approach H3Z.

Figure 5.26 Illustration of geometry of H 3 Z ... HF ... HF complex", and definition of

Ebend.

274

Hydrogen Bonding

Table 5.23 Geometric features of H3Z...HF...HF, Z=N,P. See Fig. 5.26 for definition of parameters. Data in A and degs99. R(Z..F) ..

H 3 N HF H3P..HF HF..HF H3N..HF..HF H3P..HF..HF

R(F..F)

ri

2.725 2.605 2.662

0.918 0.906 0.904 0.935 0.913

2.756 3.479 2.623 3.307

r

o

i

%

0.0 0.0

0.905 0.912 0.908

12.0 21.0

14.3 16.1 17.4

101.8 72.1 66.8

5.5.2 Energetics The above geometrical changes are all indicative of a positive reinforcement in the strengths of the various H-bonds. Further evidence of this cooperativity arises from consideration of the energetics. The binding energies of complexes are reported in Table 5.24. The data is organized into two separate paths for building the 1:2 trimer. Route 1 first combines a H3Z molecule with HF, then adds the second HP molecule. The same complex is attained in Route 2 by first combining the two HF molecules, then adding H3Z. The total energy of the complex is of course independent of the route chosen. Comparison of reaction 2a with 1b in Table 5.24 shows that HF binds more strongly to the HF end of the preformed H 3 Z ... HF complex than it does to the isolated HF molecule. This enhanced binding is equal to 5.70 kcal/mol when Z=N and about half that amount for Z=P, as indicated by the entry in Table 5.24 for Ecoop. This effect may be understood on the basis of the H3Z molecule transferring some electron density to HF in the H 3 Z ... HF complex, making the HF a better proton acceptor. The stronger cooperativity associated with H3N relative to H3P is due to the better proton-accepting capability of the former. Of course there are other factors, such as polarization of HF, which contribute to the cooperativity. An analogous comparison between la

Table 5.24 Complexation energies ( Eelec) and measures of cooperativity in H3Z...HF...HF. Data in kcal/mol99.

route 1 la: H3Z + HF H3Z..HF 1b: H 3 Z .. HF + HF H3Z..HF..HF route 2 2a: HF + HF HF..HF 2b: H3Z + HF..HF H3Z..HF..HF totala Ecoopb Eabcc Ebend

Z=N

Z=P

-11.81 -11.67

-4.13 -8.78

-5.97 -17.51 -23.48 -5.70 -4.09 -1.73

-5.97 -6.94 -12.91 -2.81 -1.42 -1.06

"la - l b ( = 2a + 2b) lb - 2a (=-- 2b - la) °Eabc = K(ahc) - |K(ab) + E(ac) I E(bc)l + [E(a) + E(b) + E(c)] d Ebend = E(Cs) - E(C3v), see Fig. 5.26. b

Cooperative Phenomena

275

and 2b illustrates that the H3Z molecule binds more strongly to the preformed HF dimer than to a single HF molecule. The reasons are quite similar, with the dimerization making the proton more accessible to the approaching lone electron pair of H3Z. The above definition of cooperativity ignores any interaction between the terminal HF molecule and H3Z, folding any attraction into the cooperativity term, Ecoop. One can remove this interaction from the effect by computing this interaction energy directly. Specifically, the three-body term is defined in the usual manner: the total binding energy of the complex, minus the interaction energy of each pair of subunits:

Another important point of distinction with Ecoop is that the three-body term is evaluated with all interactions computed in the precise geometry, internal as well as intermolecular, of the trimer. In contrast, Ecoop permits relaxation of the geometry of each entity, monomer, dimer, or trimer. The appropriate row of Table 5.24 reports these three-body terms for the 1:2 complexes, which are each reduced relative to E coop . This reduction is understandable since Eabc removes the interaction between the two terminal molecules. This formulation suggests the cooperativity associated with H3N is nearly three times larger than that of H3P. One can obtain a measure of this interaction between terminal molecules by forcing the geometry to adopt a fully linear, C3v, geometry, as illustrated in Fig. 5.26. Of course, the bending energies, reported as Ebend in Table 5.24, reflect also the strain that full linearity imposes on the HP..HF interaction, which prefers a 9 angle of 102°, much smaller than the 180° in the C3v structure. Note that the sum of Ebend and Eabc is roughly equql to Ecoop.

5.5.3 Vibrational Spectra The same systems were probed also for the effects of cooperativity upon the vibrational frequencies and intensities100. The data in Table 5.25 focus on the stretching vibrations of the HF molecule. The results are organized along the same two reaction schemes presented in Table 5.24 so as to parallel the energetics. The first row illustrates the red shift induced in the HF stretching frequency when it donates a proton to H3Z. This shift is much larger for H3N due to its stronger basicity. The formation of the H-bond also substantially increases the intensity of this band, by a factor of seven for H3N...HF, as indicated in the last two

Table 5.25 Frequency shifts and intensity enhancements in HF stretch of central subunit of H 3 Z ... HF ... HF 100 . a

v (cm--1)

Ap /Ara

Z=N

Z=P

Z=N

Z==P

-432 -389

-134 -143

7.1 1.0

4.0 1.1

-41 -780

-41 -236

1.4 5.2

1.4 3.1

route 1

la: H3Z + HF H3Z..HF Ib: H3Z..HF + HF H3Z..HF..HF route 2

2a: HF + HF HF..HF 2b: H3Z + HF..HF H 3 Z .. HF .. HF a

Ratio of intensity of product to that of reactant.

276

Hydrogen Bonding

columns of Table 5.25. Addition of the second HF molecule has little effect upon the intensity of the stretching band of the HF molecule already bound to H3Z. On the other hand, this second H-bond introduces a further red shift, comparable in magnitude to that occurring when HF forms its first H-bond with H3Z. Comparison of the 1b row with the data in row 2a illustrates the cooperativity in this complex. That is, the binding of one HF to another only red shifts the HF stretching frequency of the proton acceptor by 41 cm-1. But if the latter HF molecule is already bound to H3Z, the formation of the bond with HF lowers the frequency by several times that amount. Another measure of the cooperativity emerges from comparison between la and 2b. As indicated above, bonding of H3Z to HF lowers the frequency of the latter molecule's stretch by a substantial amount. But the last row of Table 5.25 illustrates that this shift is magnified by a factor of about two if the HF molecule has already attached itself to another HF, prior to the approach of H3Z. The intermolecular modes also provide some insights into the nature of cooperativity in these complexes. The computed harmonic frequencies are reported in Table 5.26, using the nomenclature developed by Bertie and Falk101 wherein v has its usual meaning of the Hbond stretch. The bending motions of the proton donor molecule are denoted vb and vt, whereas v 1 and v 2 refer to the bends of the acceptor. Because of the angular characteristics of the H-bond, and the low effective mass for a proton motion, the proton donor bending motions have the highest frequencies of the intermolecular modes in the binary complexes. The weaker nature of the H-bond in H3P...HF, as compared to H3N...HF, explains the fact that the frequencies in the former complex are roughly half those in the latter. Of particular relevance here is a comparison of the 1:1 complexes in the first two columns with the 1:2 complexes in the last two columns. In each case, addition of the second HF molecule acts to increase the frequency of the intermolecular mode within H3Z...HF. This increment can be as small as 27 cm-1 in the V band of H3P...HF, or as much as 300 cm-1 in a number of cases. The intensities of the intermolecular modes are compiled in a similar organizational scheme in Table 5.27. The intensities of the proton donor bending modes of the 1:1 complexes are by far the strongest, with those corresponding to H 3 N ... HF consistently larger than those of H3P...HF. Comparison of the first two columns with the last two again brings out the cooperative effects in the trimer. Adding the second HF molecule intensifies all the bands by a substantial amount; the only exception is the v 2 band of H3P...HF...HF.

Table 5.26 Frequencies ( c m - 1 ) of intermolecular vibrational modesa in 1:1 and 1:2 complexes of H3Z with HF100. H3Z...HF...HF

.. H3Z HF

Z=N

v

v V 1

V a

...

Z=N

Z=P

113

290 1162

140 800

877

467 464

1024

603

236 238

108 117

417 307

247

240 876

V

Z=P

175

v refers to Z F stretch, vb and v to bending motions of proton donor: bonds of acceptor arc denoted by v and v 101.

Cooperative Phenomena

277

Table 5.27 Intensities (km/mol) of intermolecular vibrational modes in 1:1 and 1:2 complexes of H3Z with HF100. H3Z...HF

v vb v

V V

H3Z...HF...•HF

Z=N

Z=P

3 208 208 10 10

Z=N

Z=P

1

13

134 134 6 6

264 245 49 13

5 247 275

10 0.1

5.5.3.1 Analysis of Intensities The intensity of a given vibrational band is directly related to the change in the molecular dipole moment that is associated with the motions of the atoms corresponding to that particular normal mode. Hence, the intensities offer a point of contact between the computed electronic distributions and experimental observation. Just as atomic charges can be defined by integrating the computed electron density in a given region surrounding a particular center, another means of assigning a charge is via the dipole moment change associated with the atom's motion 102-105. Following a scheme designed some years ago106, an atomic polar tensor (APT) is defined for a given atom a as

where i represents one of the three components of the dipole moment and qj is the x,y, or z coordinate of atom a. For example, Pxz refers to the change in the x-component of the molecular dipole moment when atom a is moved a certain distance along the z-axis. An effective charge for an atom a is defined as the sum of the squares of all nine elements of Pij. Beginning with the HF bond, when this bond direction is taken as the z-axis, the PZZH element describes the change occurring in the component of the dipole moment in this direction, when the H atom is stretched along the HF bond. In the isolated HF molecule, this element is equal to 0.36100. It increases to 0.96 in H3N...HF, and then to 1.14 when the second HF molecule is added, forming H3N...HP...HF. The intensities calculated for the HF stretch are approximately proportional to these values of PzzH. One can conclude that as the H-bond is strengthened through the cooperativity, the charge cloud around HF changes such that the molecular dipole is more easily altered by displacement of the proton along the HF axis. Perhaps a better way to view this process is along the following lines: when the bridging proton of a H-bond moves in one direction, electron density shifts in the opposite direction107. Thus the formation of a H-bond causes the motion of this proton to drastically increase its "net" positive charge, resulting in a very large change in the moment, and thus to a stronger intensity. The intensity changes occurring in the proton acceptor molecules H3N and H3P offer some real insights into the effects of multiple H-bonding upon electron distributions100. Whereas the intensity of the symmetric stretch in the H3N monomer is quite weak, it is intensified by a factor of 35 when complexed with HF, and then by a further factor of 40 when

278

Hydrogen Bonding

the second HF is added to the complex. In concert with these magnifications in the intensity of this mode are large increases in the PZZH element of the atomic polar tensor of the H3N protons. It is possible to further partition a dipole moment derivative like z/ z into separable contributions106,108.109. Of utility here is a fractional charge q and a "charge flux," CF. The latter term accounts for the loss or gain of electron density as the atom is displaced along a given axis. Within this framework, the intensity patterns were rationalized along the following lines. Beginning with the H3N monomer, the NH stretching modes are of low intensity due to a cancellation between two factors. Each H atom has a fractional positive charge, so its displacement away from N would cause the molecular moment to change. But as the H atom moves, electron density accumulates on it, lowering its positive charge, so the moment is changed very little, resulting in the low intensity. When the H3N molecule is complexed with HF, each of the three H atoms becomes more positively charged. Working in concert with this effect is the tendency of the H-bond to "restrict" the electron cloud, making it less able to accompany the H atoms as they stretch away from N. Together, these effects account for the large dipole moment change associated with the N—H stretch, and the consequently intense band. The further intensity enhancement resulting from the addition of the second HF molecule is due to a continuation of the above two effects: greater positive charge on N—H protons, and less deformable electron cloud. A comparison of the above patterns of H3N with those observed when H3P acts as proton acceptor is enlightening as well100. Whereas the N—H stretching intensities increased manyfold when the HF molecules were added, the analogous P—H stretches show relatively small perturbations. In fact, addition of the first HF molecule reduces the intensity of the symmetric stretch by a factor of 2.5. The first point of contrast between the two proton acceptor molecules is that the H atoms of H3P are negatively charged. As for H3N, the electron cloud follows these H atoms, enhancing their negative charge. As a result, the symmetric stretch in isolated H 3 Phas substantial intensity, 500 times larger than in H3N. When the H-bond is formed with HF, the withdrawal of density from the H3P molecule now acts to diminish the negative charge of the hydrogens. At the same time, the H-bond causes the electron cloud to become less deformable, reducing its ability to follow the H atoms as they stretch away from P. Together, these two effects act to lower the intensity of the symmetric stretch. In summary, the profoundly different behavior of the intensities of the H3N and H3P stretches can be simply explained on the basis of the opposite charges of the H atoms in the two molecules. The H-bond has similar consequences for both the charges on the atoms and the deformability of the electron cloud. 5.5.4 Effects of Electron Correlation This style of analysis was followed up by another group of coworkers who replaced the ZH3 molecule by water and also included the effects of electron correlation110. The 6-31G** set was used, as was the larger + (VP)s(2d)s containing two sets of d-functions, plus a diffuse sp-set111. As in the earlier cases, the absolute minimum is of cyclic type, with a weak Hbond between the H2O and the second HF molecule, as illustrated in Fig. 5.27. Since H2O is a poorer proton acceptor than is NH3, one would expect the H-bond to HF to be weaker, and the cooperativity within the trimer to be less extensive. On the other hand, the better proton-donating ability of H2O should strengthen the last H-bond that completes the cyclic character of the trimer.

Cooperative Phenomena

279

Figure 5.27 Illustration of geometry of H2O...HF...HF complex, including definition of geometrical parameters110.

The geometric features of the binary complexes are listed in Table 5.28, along with the 1:2 complex in the last two rows. The expected contraction of the O...F distance as the second HF molecule is added is immediately apparent. This contraction amounts to 0.10 A at the SCF level, less than that in the H 3 N ... HF ... HF complex where the shrinkage was 0.13 A. Correlation has little influence on the H-bond reduction; the MP2 contraction is 0.11 A. From the perspective of adding the proton acceptor molecule to the HF dimer, R(F..F) is reduced by 0.12 A for both H2O and NH3, at the SCF level. The distinction between the latter two molecules is perhaps most clearly seen in the bond length of the inner HF molecule, ri This bond stretches by 0.017 A when the outer HF molecule is added to H3N...HF, but by only 0.009 A if H3N is replaced by H2O. Note, however, that when correlation is added, this stretching doubles. Table 5.29 reports the energetics of step-by-step construction of the H2O...HP...HF complex, using the same scheme established earlier for H3N...HF...HF99. In addition to considering the effects of electron correlation, the investigation of H2O...HF...HF added counterpoise corrections for the complexation energies, reported as the "cc" data in Table 5.29. The comparison with H3N...HF...HF is most consistent if the first column of this table is placed alongside the first column of Table 5.24, all results at the SCF level with a 6-31G** basis set. The first rows verify the better proton accepting ability of NH3, as the binary complex with HF is stronger by some 2.7 kcal/mol. The cooperativity fostered by NH3 and H2O are apparently similar, based on the second rows. Addition of the second HF molecule to H2O...HF accounts for 12.1 kcal/mol; replacement of H2O by NH3 makes this process exergonic by 11.7 kcal/mol. Another means of assessing this cooperativity is Ecoop , the enhancement of the binding energy of the second HF molecule after the proton acceptor has attached to the first HF. This quantity is comparable for H2O and NH3 in Tables 5.24 and 5.29, as are the three-body terms Eabc.

Table 5.28 Geometric features of H2O...HF...HF. See Fig. 5.27 for definition of parameters. Data calculated with +(VP)s(2d)s basis set, in A and degs110. R(O..F) ..

H2O HF

SCF MP2

HF..HF

SCF

H 2 O .. HF .. HF

MP2 SCF MP2

R(F..F)

r

2.822 2.764 2.696 2.622

0.910 0.939 0.901 0.924 0.919 0.956

2.723 2.661

2.623 2.550

i

r

o

a

0

0

7.5 6.2 20.2 18.1

116.5 111.2 70.2 68.6

1.7 1.5

0.903 0.927 0.908 0.937

12.7 11.2

280

Hydrogen Bonding

The remaining data in Table 5.29 provide an opportunity to examine the effects of BSSE, basis set, and correlation upon the cooperativity of energetics. The 6-31G** basis set is subject to a fairly large superposition error, accounting for the quite large discrepancies between unconnected and corrected interaction energies in the first two columns of the table. For example, 4.5 kcal/mol of the total of 12.1 SCF binding energy of the second HF molecule to H2O...HF is due to this error. The superposition error is much reduced with the +(VP)s(2d)s basis set. Once the counterpoise corrections are added, the results with the two basis sets are comparable (within about 1 kcal/mol of one another), if not identical. Consistent with expectations, correlation adds to the binding energy of each H-bond. The correlation is apparently responsible for an increase in the cooperativity effect as well. The counterpoise-corrected values of Ecoop are —2.9 and —4.1 kcal/mol at the SCF and MP2 levels, respectively, and a similarly larger value of Eabc is obtained with MP2. 5.5.5 Other Mixed Trimers The H2O...HF...HF complex was reexamined more recently in conjunction with a comparison with H 2 O ... H 2 O ... HF 112 . The focus of this work was an exploration of the potential energy surface to identify all the local minima. In addition to the cyclic structure of Fig. 5.27, a bifurcated geometry, in which the water serves as double proton acceptor, was also located as a minimum of the PES of H2O...HP...HF. In the complex containing a pair of water molecules and one HF, the only minimum located was of cyclic type. Both of these structures are illustrated in Fig. 5.28. The binding energy for the cyclic H2O...HF...HF complex in Fig. 5.27 and the analogous H 2 O ... H 2 O ... HF in Fig. 5.28 are calculated to be quite similar to one another. As illustrated in Table 5.30, the SCF-level calculations suggest the former is marginally more strongly bound, whereas including correlation shifts the balance toward the latter. In any case, it is clear that there is little difference between the two. Computation of the cooperativity within

Table 5.29 Complexation energies and measures of cooperativity in H2O...HP...HF. Data in kcal/mol110. +(VP)s(2d)s

6-31G** SCF

route 1 la: H20 + HF H2O..HF 1b: H2O..HF + HF H2O..HF..HF route 2 2a: HF + HF HF..HF 2b: H2O + HF..HF H2O..HF..HF totala Ecoopb E

abc a

MP2

no cc

cc

no cc

cc

no cc

cc

-9.06 -12.09

-8.00 -7.54

-7.36 -6.76

-7.23 -6.61

-8.86 -9.04

-7.83 -8.07

-5.97 -15.18 -21.15 -6.12 -3.16

-4.23 -11.31 -15.54 -3.31 -3.08

-3.83 -10.29 -14.12 -2.93 -2.18

-3.75 -10.09 -13.84 -2.86 -2.28

-4.60 -13.30 -17.90 -4.44 -3.94

-4.02 -11.88 -15.90 -4.05 -4.12

la + l b ( = 2a + 2b) Mb - 2a (= 2b - la) E abc = E(abc) [E(ab) + E(ac) + H(bc)] + [K(a) - li(b) i K(c)]

b

c

SCF

Cooperative Phenomena 281

Figure 5.28 Secondary minimum of H2O...HF...HF complex, and single minimum for H2O...H2O...HF112.

the two complexes, analogous to Ecoop in Table 5.29, leads to MP2 values of —1.38 and - 2.41 kcal/mol for H 2 O ... HF ... HF and H2O...H2O...HF, respectively. Another set of calculations replaced the HF molecule of H2O...H2O...HF by HC1 and obtained the optimized geometry of H2O...H2O...HC1 at the MP2 level with a series of three different polarized basis sets80. The structure is analogous to the C1 geometry of H2O...H2O...HF in Fig. 5.28. The interoxygen separation is in the range 2.75-2.81 A and R(C1...O) is 2.99-3.06 A (depending upon particular basis set). The H-bonds are within about 10° of linearity. A harmonic frequency analysis of the MP2 data revealed that the red shift of the water molecule's OH stretch (relative to the monomer) increases from 98 cm-1 in the water dimer to 210 cm-1 when HC1 is added to form H2O...H2O...HC1. The former refers to the OH ... O H-bond which is apparently strengthened by addition of HC1; the red shift in OH...C1 is 48 c m - 1 , verifying the expected weaker nature of this bond These values are in good agreement with experimental measurements of 172 and 52 cm-1 for this complex in Ar matrix 113 . The energetics of formation of the H2O....H2O...HCl complex are reported in Table 5.31, in a format comparable to that for H2O...HF...HF in Table 5.29. All data were obtained at the MP2 level and are corrected for BSSE by the counterpoise procedure. The three basis sets all contain polarization functions and yield comparable results. The binding of HC1 to the pre-formed water dimer (step 1 b) is considerably more favorable than its binding to a water monomer, as in step 2a. This enhancement is referred to as ECOOp in Table 5.31, and amounts to between 2 and 3 kcal/mol. (The same quantity is identically equal to the enhancement of binding of water to H2O...HC1 as compared to HC1.) It is worth noting that whereas the total binding energy in the appropriate row of Table 5.31 is rather insensitive

Table 5.30 Computed binding energies of cyclic H2O...HF...HF and H2O...H2O...HF. Data corrected for superposition error, in kcal/mol112.

6-31G**

6-31 + +G** 6-31 + + G(2d,2p)

MP2/6-31 + +G**

H2O...HF...HF

H2O...H20...HF

-16.02 -16.49 -14.41 -17.25

-15.95 -15.46 -13.97 -17.58

282

Hydrogen Bonding

Table 5.31 Complexation energies ( Eelec) and measures of cooperativity in H2O...H2O...HCl. Data computed at MP2 level and corrected for BSSE, in kcal/mol80.

route 1 la: H2O + H2O

H2O..H2O 1b: H2O..H2O + HCl H2O..H2O..HC1 route 2 2a: H2O + HC1 H2O..HC1 2b: H2O + H2O..HC1 H2O..H2O..HC1 total" Ecoopb E coop,vib

DZP

6-31G(2d,p)

Poll

-5.04 -7.32

-4.58 -7.67

-4.32 -7.86

-5.09 -7.26 -12.36 -2.23 -1.52

-5.25 -7.00 -12.25 -2.42 -2.12

-4.92 -7.26 -12.18 -2.94 -2.96

a

la + 1b (= 2a + 2h) lb - 2 a ( = 2 b - la) c Same as Ecoop, with zero-point vibrational corrections. b

to details of the basis set, there is a clear and steady increase in the cooperativity as the basis set becomes more flexible. It is intriguing that this 2-3 kcal range of Ecoop is quite similar to the 2.4 kcal/mol computed for H2O...H2O...HF, also at the MP2 level and with BSSE correction112, despite the substitution of HF by HC1 in the present system. Perhaps Ecoop would have been larger in the H2O...H2O...HF system had a basis set more flexible than 6-31 ++G** been used. At any rate, this cooperativity is significantly smaller than the 4 kcal/mol reported in the last column of Table 5.29 for H 2 O ... HP ... F where there are two HF molecules present, again obtained at the MP2 level with BSSE corrections. The last row of Table 5.31 refers again to the energy of cooperativity, except that all binding energies have been corrected for zero-point vibrations. Ecoop,vib pretty much duplicates the results without such corrections, except that the sensitivity to basis set flexibility is heightened. So much so that there is a doubling in this quantity between the "standard" DZP basis set and the Pol 1 type. One might conclude that even though it is possible to obtain good estimates of the binding energies of H-bonded clusters, accurate assessment of the degree of cooperativity in multiply H-bonded systems can be particularly demanding in terms of a flexible basis set. Such a rule is commensurate with the notion that the cooperative effects originate largely from polarizations of electron clouds induced by H-bond formation.

5.6 Summary

The C—H bonds of HCN become progressively longer as the number of molecules in (HCN)n increases. This elongation is greater for the proton donor molecule on the end and least for the acceptor. Longest of all are the C—H bonds of molecules in the center of the chain. The C N bond of the donor is elongated while that of the acceptor is shortened as the chain grows; those in the middle exhibit little change. The energetic consequences of cooperativity grow as the chain is enlarged, in that the total interaction energy in an n-mer is more than n — 1 times larger than the interaction in a simple dimer. The average H-bond

Cooperative Phenomena

283

energy within an infinitely long chain is projected to be about 25% larger than the interaction energy in the dimer. Cooperativity also makes for a dipole moment of an assembly of HCN molecules that is greater than the sum of moments of each monomer. The stretching frequencies behave much like the bond lengths in that bond stretches correspond to red shifts and contractions to frequency increments. The longest C—H stretches of the terminal proton donor molecule correlate with progressively larger red shifts as the chain grows. The frequency shifts and bond length changes in the CN bonds are considerably smaller than for CH. Cooperativity exhibits particularly dramatic effects upon the intermolecular frequencies. For example, there are two H-bond stretching frequencies, v , in the trimer. One of these is much larger than the value of V in the dimer, while the other is much smaller. These trends continue as the chain elongates. Although each NCH ... NCH connection prefers full linearity, a bending of each such connection in an oligomer would enable the chain to bend around on itself, eventually allowing the terminal lone pair of the first molecule to pair up with the proton of the last. The advantage would be the formation of an additional H-bond, but this would come at the expense of angular distortion of each of the other H-bonds in the oligomer. Calculations indicate that for a chain containing four or more HCN molecules, the former outweighs the latter and a cyclic complex will be favored over a linear arrangement. There is some evidence for a small amount of Cooperativity in the HCCH trimer, albeit of a lesser degree due to the absence of a true H-bond. The interaction energy of the cyclic trimer is very nearly equal to three times that of a single dimer, even though each pairwise geometrical relationship is distorted from the true T-shape favored by the dimer. The tetramer and pentamer are also likely to be of cyclic type, but again little variance is observed in the H-bond energy per pairwise interaction. The HF trimer adopts a cyclic geometry wherein the geometric distortion of each H-bond is compensated by the Cooperativity as each molecule serves as both proton donor and acceptor. The lesser degree of deformation in the cyclic tetramer leads to a significantly greater apparent Cooperativity. Similar trends were observed in the trimer of HC1. Threebody interactions, another measure of Cooperativity phenomena, are well modeled at the SCF level. The primary contributor arises from the deformation of electron density, akin to polarization effects. The two-body SCF interaction is primarily responsible for the radial and angular anisotropy of the energetics of (HF)3. Continued enlargement of the ring leads to a progressive elongation of the HF bonds, coupled with reduction in the interfluorine distances. Also in evidence is a progressively larger red shift of the HF stretches as the ring includes more members. If maintained in an open chain structure, the molecules in the middle of the chain exhibit the longer HF bonds and the shortest F...F contacts. Calculations of an infinitely long open chain of water molecules indicate the interoxygen separation is 2.73 A, and the bridging hydrogen is stretched away from the O atom by 0.05 A. The H-bond energy (per bond) is enhanced by nearly 50% relative to the isolated dimer. Four-body terms are considerably smaller than three-body and can probably be ignored, along with higher-order terms. The sum of two-body interactions in a long chain, with idealized geometry, is projected to account for about 80% of the total interaction energy, the rest arising from terms of higher order. The cumulative sum of three-body interactions can be much smaller in a cluster wherein some triads of molecules contain a double proton donor or acceptor (destabilizing) and other triads are sequential (stabilizing). Red shifts of the OH bond involving the bridging hydrogen in the chain become progressively larger as the chain elongates; patterns for anharmonic frequencies bear a striking resemblance to harmonic data.

284

Hydrogen Bonding

Experimental data indicate that small oligomers of water prefer a cyclic geometry. As in the dimer, it becomes difficult to assign a particular band as the H-bond stretching mode. Moreover, most of the normal modes involve significant motions of atoms on more than one molecule. The sum of pairwise interactions in a cyclic hexamer accounts for only about 80% of the total interaction energy. The electrostatic energy is rigorously additive and the exchange repulsion is nearly so. The majority of the nonadditivity arises in the charge transfer and polarization terms. Consistent with prior findings on simpler systems, the three-body term tends to be attractive for sequential triads (where the central molecule is simultaneously donor and acceptor) and repulsive for a triad containing a double donor or double acceptor. The cyclic trimer contains H-bonds that are all significantly distorted from linearity. The orientation of each water molecule is controlled largely by two-body terms present at the SCF level. The anisotropy of the three-body term is due primarily to forces that arise from deformation of the electron clouds; Heitler-London forces are opposite in sign and smaller in magnitude. Induction energy per se is not a good substitute for the full SCF three-body term. While the shapes of the two are not too dissimilar, with respect to angular distortion, the former is far too attractive. As in the closed, cyclic trimer, the three-body terms in the open trimers are dominated by SCF deformation forces. The three-body term in the double proton donor configuration is weakly attractive, but this force is counteracted by a strongly repulsive pairwise two-body interaction between the two terminal molecules. Correlation effects are minimal in examining three-body terms. Geometry optimization leads to the conclusion that the cyclic trimer is indeed the global minimum on the PES of the water trimer. Despite the distortion of each of the three H-bonds, the total interaction energy is greater than three times that of a dimer. The 3-body term contributes 17% to the total binding energy of the trimer and 25% for the tetramer; 4-body terms in the tetramer are nearly negligible. The three R(OO) distances are not quite equivalent, but all are about 2.83 A. Another minimum on the PES, albeit much less stable than the sequential cyclic one, is one in which the central molecule acts as double donor. However, even this structure takes on cyclic character as the two terminal molecules approach one another to form a third H-bond. Enlarging the system to the tetramer brings about a cyclic structure, as is true as well for the pentamer and hexamer. The latter is large enough that there is little nonlinearity that must be suffered by the six H-bonds. There are large red shifts in the O—H stretching frequencies, amounting to several hundred cm-1 as each new water molecule is added. This change is apparently due to SCF deformation effects. The average H-bond energy in the oligomer rises with the number of molecules, surpassing 5.4 kcal/mol at the SCF level, more than 6 kcal/mol when correlated. Alkylation of the oxygen causes only small perturbations of these results. A system combining NH3 with a pair of HX (X=F,CI) molecules illustrates the sorts of charge rearrangements that accompany the cooperativity. After the redistributions that occur in the proton donor NH3 molecule upon formation of XH ... NH 3 , the further changes upon elongation to XH ... XH ... NH 3 are rather minimal. However, the central HX molecule undergoes significant redistribution when the second HX molecule is added. Optimization shows this complex to be bent, resembling a cyclic structure, although the interaction between the terminal ZH3 (Z=N,P) and HF molecules is a weak, distant one which would not fit the description of a true H-bond. Addition of the second HF molecule induces changes in the geometry of FH ... ZH 3 that are characteristic of positive cooperativity: The H-bond contracts by more than 0.1 A. From the other perspective, that is, the influence of ZH3 upon the geometry of FH ... FH, there is a similar reduction in the F...F distance. Addi-

Cooperative Phenomena

285

tion of the second H-bond also yields further stretches in the internal F—H bonds. The cooperativity emerges from comparison of binding energies as well. HF binds more strongly to the H 3 Z ... HF complex than it does to HF. Similarly, H3Z forms a stronger interaction with the HF dimer than to a single HF molecule. Using another measure of cooperativity, the three-body term in the complex is negative, signaling a stabilizing force. The vibrational spectra provide additional insights into the cooperativity in these trimers. While addition of a second HF molecule to FH...ZH3 has little effect upon the intensity of the F—H stretch, the frequency of this mode is red shifted by an amount comparable to the shift occurring when the first H-bond was formed to ZH3. It is worth stressing that this red shift of the HF stretch in FH ... ZH 3 , is many times larger than would occur were this FH molecule not first bound to ZH3. Similarly, the red shift occurring when ZH3 adds to FH ... FH is considerably larger than when it adds to the FH monomer. In other words, a preexisting H-bond is more amenable to a change in frequency than is the unbound monomer. In terms of intermolecular modes, addition of the second HF molecule raises the frequency of the intermolecular stretch within H3Z...HF. This increment can be as small as 30 cm-1 in the v band of H3P...HF, or as much as 300 cm-1 for some of the bending motions. Adding the second HF molecule intensifies most bands by a substantial amount. Analysis of the changes in the molecular dipole moment reveals that when the H-bond is strengthened by cooperativity, the charge cloud around HF changes such that the molecular dipole is more sensitive to displacement of the proton along the HF axis. Any motion of the proton is accompanied by a displacement of electronic charge in the opposite direction. The intensity enhancements in the internal NH stretches of NH3 when engaged in a Hbond are due to a combination of more positively charged hydrogens and a charge cloud less able to follow the protons. The opposite effect in PH3 is due to the negative charges on these hydrogens. Similar sorts of trends are witnessed in complexes such as FH...FH...OH2, H2O...H2O...HF, and H 2 O ... H 2 O ... HC1 that contain OH2 instead of NH3.

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10. Anex, D. S., Davidson, E. R., Douketis, C., and Ewing, G. E., Vibrational spectroscopy of hydrogen cyanide clusters, J. Phys. Chem. 92, (1988). 11. Karpfen, A., Ab initio studies on hydrogen-bonded chains. III. The linear, infinite chain of hydrogen cyanide molecules, Chem. Phys. 79, 211-218 (1983). 12. King, B. F., Farrar, T. C., and Weinhold, R, Quadrupole coupling constants in linear (HCN)n clusters: Theoretical and experimental evidence for cooperative effects in C—H . . . N hydrogen bonding, J. Chem. Phys. 103, 348-352 (1995). 13. Kurnig, I. J., Lischka, H., and Karpfen, A., Linear versus cyclic (HCN) 3 . An ab initio study on structure, vibrational spectra, and infrared intensities, J. Chem. Phys. 92, 2469-2477 (1990). 14. Jucks, K. W., and Miller, R. E., Near infrared spectroscopic observation of the linear and cyclic isomers of the hydrogen cyanide trimer, J. Chem. Phys. 88, 2196-2204 (1988). 15. Gdanitz, R. J., and Ahlrichs, R., The averaged coupled-pair functional (ACPF): A size-extensive modification of MR CI(SD), Chem. Phys. Lett. 143, 413-420 (1988). 16. Alberts, I. L., Rowlands, T. W., and Handy, N. C., Stationary points on the potential energy surfaces of (C2H2)2, (C2H2)3, and (C 2 H 4 ) 2 ,J. Chem. Phys. 88, 3811-3816 (1988). 17. Bone, R. G. A., Murray, C. W., Amos, R. D., and Handy, N. C., Stationary points on the potential energy surface of (C 2 H 2 ) 3 , Chem. Phys. Lett. 161, 166-174 (1989). 18. Yu, J., Su, S., and Bloor, J. E., Ab initio calculations on the geometries and stabilities of acetylene complexes, J. Phys. Chem. 94, 5589-5592 (1990). 19. Bryant, G. W., Eggers, D. F., and Watts, R. O., High resolution infrared spectrum of acetylene tetramers, Chem. Phys. Lett. 151, 309-314 (1988). 20. Bone, R. G. A., Amos, R. D., and Handy, N. C., Ab initio studies of acetylene tetramer and pentamer, J. Chem. Soc., Faraday Trans. 86, 1931-1941 (1990). 21. Karpfen, A., Beyer, A., and Schuster, P., Ab initio studies on clusters of polar molecules. Stability of cyclic versus open-chain trimers of hydrogen fluoride, Chem. Phys. Lett. 102, 289—291 (1983). 22. Gaw, J. R, Yamaguchi, Y, Vincent, M. A., and Schaefer, H. F., Vibrational frequency shifts in hydrogen-bonded systems: The hydrogen fluoride dimer and trimer, J. Am. Chem. Soc. 106, 3133-3138(1984). 23. Liu, S.-Y, Michael, D. W., Dykstra, C. E., and Lisy, J. M., The stabilities of the hydrogen fluoride trimer and tetramer, J. Chem. Phys. 84, 5032-5036 (1986). 24. Scuseria, G. E., and Schaefer, H. F., Vibrational frequencies and geometries for the open HF trimer, Chem. Phys. 107, 33-38 (1986). 25. Kolenbrander, K. D., Dykstra, C. E., and Lisy, J. M., Torsional vibrational modes of (HF)3: IR—IR double resonance spectroscopy and electrical interaction theory, J. Chem. Phys. 88, 5995-6012(1988). 26. Latajka, Z., and Scheiner, S., Structure, energetics and vibrational spectra of H-bonded systems. Dimers and trimers of HF and HCl, Chem. Phys. 122, 413-430 (1988). 27. Michael, D. W., and Lisy, J. M., Vibrational predissociation spectroscopy of (HF)3, J. Chem. Phys. 85, 2528-2537(1986). 28. Andrews, L., and Bohn, R. B., Infrared spectra of isotopic (HCl) 3 clusters in solid neon, J. Chem. Phys. 90, 5205-5207 (1989). 29. Han, J., Wang, Z., Mclntosh, A. L., Lucchese, R. R., and Bevan, J. W., Investigation of the ground vibrational state structure of H35Cl trimer based on the resolved K, J substructure of the v5 vibrational band, J. Chem. Phys. 100, 7101-7108 (1994). 30. Suhm, M. A., and Nesbitt, D. J., Potential surfaces and dynamics of weakly bound trimers: Perspectives from high resolution IR spectroscopy, Chem. Soc. Rev. 45—54 (1995). 31. Moszynski, R., Wormer, P. E. S., Jeziorski, B., and van der Avoird, A., Symmetry-adapted perturbation theory of nonadditive three-body interactions in van der Waals molecules. I. General theory, J. Chem. Phys. 103, 8058-8074 (1995). 32. Chalasinski, G., Cybulski, S. M., Szczesniak, M. M., and Scheiner, S., Nonadditive effects in HF and HCl trimers, J. Chem. Phys. 91, 7048-7056 (1989).

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54. Clementi, E., Kolos, W., Lie, G. C., and Ranghino, G., Nonadditivity of interaction in water trimers, Int. J. Quantum Chem. 17, 377-398 (1980). 55. Scheiner, S., and Nagle, J. P., Ab initio molecular orbital estimates of charge partitioning between Bjerrum and ionic defects in ice, J. Phys. Chem. 87, 4267-4272 (1983). 56. Karpfen, A., and Schuster, P., Ab initio studies on hydrogen bonded chains. V. The structure of infinite chains of methanol and water molecules, Can. J. Chem. 63, 809-815 (1985). 57. Suhai, S., Cooperative effects in hydrogen bonding: Fourth-order many-body perturbation theory studies of water oligomers and of an infinite water chain as a model for ice, J. Chem. Phys. 101,9766-9782(1994). 58. Ojamae, L., and Hermansson, K., Ab initio study of cooperativity in water chains: Binding energies and anharmonic frequencies, J. Phys. Chem. 98, 4271-4282 (1994). 59. Yoon, B. J., Morokuma, K., and Davidson, E. R., Structure of ice Ih. Ab initio two- and threebody water-water potentials and geometry optimization, J. Chem. Phys. 83, 1223—1231 (1985). 60. Hermansson, K., Many-body effects in tetrahedral water clusters, J. Chem. Phys. 89,2149-2159 (1988). 61. Koehler, J. E. H., Saenger, W., and Lesyng, B., Cooperative effects in extended hydrogen bonded systems involving O—H groups. Ab initio studies of the cyclic S4 water tetramer, J. Comput. Chem. 8, 1090-1098(1987). 62. Dyke, T. R., and Muenter, ]. S., Molecular beam electric deflection studies of water polymers, J. Chem. Phys. 57, 5011-5012 (1972). 63. Vernon, M. E, Krajnovich, D. J., Kwok, H. S., Lisy, J. M., Shen, Y. R., and Lee, Y. T., Infrared vibrational predissociation spectroscopy of water clusters by the crossed laser-molecular beam technique, J. Chem. Phys. 77, 47-57 (1982). 64. Honegger, E., and Leutwyler, S., Intramolecular vibrations of small water clusters, J. Chem. Phys. 88, 2582-2595 (1988). 65. Schiitz, M., Burgi, T., Leutwyler, S., and Burgi, H. B., Fluxionality and low-lying transition structures of the water trimer; J. Chem. Phys. 99, 5228-5238 (1993). 66. Knochenmuss, R., and Leutwyler, S., Structures and vibrational spectra of water clusters in the self-consistent-field approximation, J. Chem. Phys. 96, 5233-5244 (1992). 67. White, J. C., and Davidson, E. R., An analysis of the hydrogen bond in ice, J. Chem. Phys. 93, 8029-8035(1990). 68. Chalasinski, G., Szczesniak, M. M., Cieplak, P., and Scheiner, S., Ab initio study of intermolecular potential of H2O trimer, J. Chem. Phys. 94, 2873-2883 (1991). 69. Klopper, W., Schiitz, M., Luthi, H. P., and Leutwyler, S., An ab initio derived torsionalpotential energy surface for (H 2 O) 3 . H. Benchmark studies and interaction energies, J. Chem. Phys. 103, 1085-1098(1995). 70. Mo, O., Yanez, M., and Elguero, J., Cooperative (nonpairwise) effects in water trimers: An ab initio molecular orbital study, J. Chem. Phys. 97, 6628-6638 (1992). 71. Kay, B. D., and Castleman, Jr., A. W., Molecular beam electric deflection study of the hydrogenbonded clusters (H 2 O) N , (CH 3 OH) N , and(C 2 H 5 OH) N , J. Phys. Chem. 89, 4867-4868 (1985). 72. Engdahl, A., and Nelander, B., On the structure of the water trimer. A matrix isolation study, J. Chem. Phys. 86, 4831-1837 (1987). 73. Ebata, T., Watanabe, T., and Mikami, N., Evidence for the cyclic form of phenol trimer: Vibrational spectroscopy of the OH stretching vibrations of jet-cooled phenol dimer and trimer, J. Phys. Chem. 99, 5761-5764 (1995). 74. Xantheas, S. S., and Dunning, T. H. J., Ab initio studies of cyclic water clusters (H2O)n, n = 1-6. 1. Optimal structures and vibrational spectra, J. Chem. Phys. 99, 8774-8792 (1993). 75. Gerhards, M., and Kleinermanns, K., Structure and vibrations of phenol(H2O)2, J. Chem. Phys. 103,7392-7400(1995). 76. van Duijneveldt-van dc Rijdt, J. G. C. M., and van Duijneveldt, F. B., Ab initio potential energy surface for the low-frequency out-of-plane bending motions of the water trimer, Chem. Phys. Lett. 237, 560-567(1995).

Cooperative Phenomena 289 77. Sabo, D., Bacic, Z., Burgi, T., and Leutwyler, S., Three-dimensional model calculation of torsional levels of (H2O)3 and (D2O)3, Chera. Phys. Lett. 244, 283-294 (1995). 78. Cruzan, J. D., Braly, L. B., Liu, K., Brown, M. G., Loeser, J. G., and Saykally, R. L, Quantifying hydrogen bond cooperativity in water: VRT spectroscopy of the water tetramer, Science 271, 59-62 (1996). 79. Schiitz, M., Klopper, W., Luthi, H.-P., and Leutwyler, S., Low-lying stationaiy points and torsional interconversiom of cyclic (H2O)4: An ab initio study, J. Chem. Phys. 103, 6114-6126 (1995). 80. Packer, M. J., and Clary, D. C., Interaction of HCl with water clusters: (H2O)nHCl, n = 1-3, J. Phys. Chem. 99, 14323-14333 (1995). 81. Liu, K., Cruzan, J. D., and Saykally, R. J., Water clusters, Science 271, 929-933 (1996). 82. Burgi, T., Schiitz, M., and Leutwyler, S., Intermolecular vibrations of phenol-(H 2 O) 3 and d 1 -phenol-(D 2 O) 3 in the So and S1 states, J. Chem. Phys. 103, 6350-6361 (1995). 83. Liu, K., Brown, M. G., Cruzan, J. D., and Saykally, R. J., Vibration-rotation tunneling spectra of the water pentamer: Structure and dynamics, Science 271, 62-64 (1996). 84. Teeter, M. M., Water structure of a hydrophobic protein at atomic resolution: Pentagon rings of water molecules in crystals of crambin, Proc. Nat. Acad. Sci., USA 81, 6014-6018 (1984). 85. Jensen, J. O., Krishnan, P. N., and Burke, L. A., Theoretical study of water clusters: heptamers, Chem. Phys. Lett. 241, 253-260 (1995). 86. Jensen, J. O., Krishnan, P. N., and Burke, L. A.., Theoretical study of water clusters: octamer, Chem. Phys. Lett. 246, 13-19 (1995). 87. Kim, J., Mhin, B. J., Lee, S. J., and Kim, K. S., Entropy-driven structures of the water octamer, Chem. Phys. Lett. 219, 243-246 (1994). 88. Badger, R. M., The relation between the internuclear distance and force constants of molecules and its application to polyatomic molecules, J. Chem. Phys. 3, 710-714 (1935). 89. Badger, R. M., A relation between internuclear distance and bond force constants, J. Chem. Phys. 2, 128-131 (1934). 90. Huisken, R, Kaloudis, M., and Kulcke, A., Infrared spectroscopy of small size-selected water clusters, J. Chem. Phys. 104, 17-25 (1996). 91. Xantheas, S. S., Ab initio studies of cyclic water clusters (H2O)n, n = 1-6. II. Analysis of manybody interactions, J. Chem. Phys. 100, 7523-7534 (1994). 92. van Duijneveldt-van de Rijdt, J. G. C. M., and van Duijneveldt, F. B., Ab initio calculations on the geometry and OH vibrational frequency shift of cyclic water trimer, Chem. Phys. 175, 271-281 (1993). 93. Fowler, J. E., and Schaefer, H. R, Detailed study of the water trimer potential energy surface, J. Am. Chem. Soc. 117,446-452 (1995). 94. Liu, K., Loeser, J. G., Elrod, M. J., Host, B. C., Rzepiela, J. A., Pugliano, N., and Saykally, R. J., Dynamics of structural rearrangements in the water trimer, J. Am. Chem. Soc. 116, 3507-3512(1994), 95. Burgi, T., Graf, S., Leutwyler, S., and Klopper, W., An ab initio derived torsional potential energy surface for (H2O)3. I. Analytical representation and stationaiy points, J. Chem. Phys. 103, 1077-1084(1995). 96. Peelers, D., and Leroy, G., Small clusters between water and alcohols, J. Mol. Struct. (Theochem) 314, 39-47 (1994). 97. Mo, O., Yanez, M., and Elguero, J., Cooperative effects in the cyclic trimer of methanol. An ab initio molecular orbital study, J. Mol. Struct. (Theochem) 314, 73-81 (1994). 98. Hinchliffe, A., Ab initio study of the hydrogen-bonded complexes NH 3 . . . HX, PH 3 . . . HX and NH 3 . . . (HX) 2 , where X=F,Cl, J. Mol. Struct. (Theochem) 105, 335-341 (1983). 99. Kurnig, I. J., Szczesniak, M. M., and Scheiner, S., Ab initio study of structure and cooperativity in H 3 N . . . HF . . . HF and H 3 P . . HF . . . HF, J. Phys. Chem. 90, 4253-4258 (1986). 100. Kurnig, I. J., Szczesniak, M. M.. and Scheiner, S., Vibrational frequencies and intensities of Hbonded systems. 1:1 and 1:2 complexes of NH3 and PH3 with HF, J. Chem. Phys. 87, 2214-2224 (1987).

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6

Weak Interactions, Ionic H-Bonds, and Ion Pairs

he classic picture of a H-bond involves the approach of a pair of neutral molecules. One contains a hydrogen atom covalently attached to a highly electronegative atom like O or F. The other molecule also contains an electronegative atom, associated with which is at least one lone pair of nonbonding electrons. It is perhaps oversimplistic to expect that a specific interaction either does or does not represent a true H-bond,with no grey area between. For example, one can imagine a scenario where a genuine H-bond, as in FH...FH, becomes weaker and weaker as one or the other F atom is replaced by less electronegative halogen atoms. The HI dimer is clearly not associated via a H-bond, whereas C1H...C1H arguably contains a H-bond, albeit a weak one. Another example replaces the proton-accepting FH molecule by a much less polar one, such as FF. Does the absence of a molecular dipole moment in the proton acceptor preclude the existence of a H-bond, even if the latter does contain a lone electron pair? Still another case in point takes as a starting point the equilibrium geometry of a classic H-bond such as FH...FH. As the two molecules are pulled apart, the interaction clearly becomes weaker and weaker. At what point would one cease to categorize this interaction as a H-bond? It would probably be best to think of interactions as spanning a continuum. In the middle of this continuum are the classic H-bonding interactions. On one end are those that are much weaker and clearly not of the H-bonded variety. On the other are those that are considerably stronger, typically dominated by electrostatic factors. It then becomes somewhat arbitrary as to where on this continuum one draws the line between a true H-bond and a different sort of interaction. Interaction energy can be used as one measure, and one can assign a minimum and maximum strength to the definition of a H-bond. Or the magnitude of the red shift of the A—H stretching frequency can be used as an indicator, with an arbitrary threshold assigned. But regardless of how carefully one designs the criteria, it must be understood from the outset that one person's H-bond is another person's van der Waals complex, or another's Coulombic interaction.

T

291

292

Hydrogen Bonding

Some of the foregoing chapters have included discussions of borderline H-bonds. The ammonia dimer, for example, is very weakly bound and its potential energy surface is so flat that there is no clearly defined equilibrium geometry. The definition as H-bonds of some of the systems that contain second or third-row atoms is also questionable. This chapter considers a number of other types of interactions that are somewhere near the boundaries of a true H-bond. We discuss the details of these interactions and the magnitudes of some of the indicators. One issue discussed is the proton-accepting ability of an electronegative atom when involved in a bond to another electronegative atom, leaving the bond of low or zero polarity. Hydrogen atoms bonded to carbon are typically of low acidity, so their ability to participate in H-bonds is questionable as well. On the opposite extreme are H-bonds wherein one of the two partners bears an electric charge. These interactions are severalfold stronger than H-bonds involving a pair of neutral molecules. They are dominated by electrostatics and the bridging proton tends to drift so far from the proton donor atom that it becomes questionable as to which unit is the donor and which the acceptor. The strong role of electrostatics can call into question their categorization as H-bonds. For example, the fundamental nature of the interaction in (H 3 NH +... NH 3 ) is very similar indeed to that in (K +... NH 3 ) where there is clearly no Hbond present1. In addition to the so-called ionic H-bonds in which one of the two partners bears an electric charge, there is the further possibility that both of the subunits might be charged. An ion pair, represented in general as A - . . . + H B , is also dubbed a "salt bridge" on occasion. The opposite charges of the two ions have the potential of producing a very strong attractive force between them; such forces have been implicated in stabilizing certain peptide conformations, for example 2-4 . There is some question as to whether a system of this type is truly a H-bond or would be better characterized as a simple electrostatic interaction. What makes them particularly fascinating is the fact that the entire nature of this system can be changed to the more conventional H-bond between neutral molecules, AH ... B, by a simple proton transfer from the cation to the anion. It becomes an interesting and relevant question as to what conditions would cause a neutral pair to convert to an ion pair.

6.1 Weak Acceptors When two HX molecules are paired together, it is clear that a H-bond can form between the proton of one molecule and the X atom of the other. But the situation is less clear cut if the H atom of the proton acceptor molecule is replaced by another halogen atom. In such a case, the dihalogen will not be polarized much at all, lessening the electrostatic part of the interaction. 6.1.1 Dihalogens Ab initio calculations5 considered the interaction of HF with C1F and with C12. Two minima were found in each case. The first corresponds to the standard sort of arrangement which can be denoted as H-bonded, at least from a geometric perspective. The other minimum is clearly not H-bonding as it is the F atom of HF that approaches the halogen atom. As indicated in Fig. 6.1, the latter geometry is designated as "F-bonding." The authors used 4-31G to probe the general character of the potential energy surface. They then used a larger basis set, representing H by [3slp], F by [5s3pld], and Cl by [7s5pld], basically a TZP set. Correlation was included via a coupled pair functional (CPF)

Weak Interactions, Ionic H-Bonds, and Ion Pairs

293

Figure 6.1 Two types of geometries for HF + C1X (X = F,C1).

formalism6, which considers single and double excitations and adds the effects of quadruple and higher excitations in an approximate way. It was found that there is a delicate balance between the stabilities of the two geometries. Failure to include counterpoise corrections, correlation, or zero-point vibrational effects can lead to the wrong conclusion as to which configuration is preferred. Indeed, earlier calculations at the SCF level7,8 had computed comparable stabilities of these two structures, and been unable to account for the experimental observation of only one of them9. The sensitivity is illustrated in Table 6.1 which shows that at the SCF level the two geometries are within 0.1 kcal/mol of each other, but that the H-bonded structure is favored after correlation is included. Once counterpoise corrections are added, however, the Fbonded structure is preferred by 0.1 kcal/mol at either level. This preference is amplified by zero-point vibrational energies, resulting in a greater stability of the F-bond by 0.6 kcal/mol. Later computations using a larger basis set10 confirmed the preference for the Fbonded structure at the correlated level, in this case by 0.3 kcal/mol. This same preference is confirmed by experimental observation9 that only the F-bonded geometry occurs in the gas phase. One caution arises from earlier computations that noted that despite a computed electronic contribution to the binding energy of these complexes by as much as —3 kcal/mol, addition of vibrational and entropic factors leads to a much smaller free energy of complexation at 100° K; G becomes positive at higher temperatures, reaching +3 kcal/mol at 298° K7. Calculations5 lead to a similar conclusion for the HF + C12 pair: both of these structures have been subsequently observed by their infrared spectra in solid Ar and Ne11. Due to the questionable existence of the H-bonded geometry, as well as the very weak interaction energy of only 2 kcal/mol or less, one can conclude that the dihalogen molecule does not act as a proton acceptor in a H-bond. It is interesting to note that dihalogens can interact also with a base like NH3, in a complex of the H 3 N ... XX' type, even though there is no proton to act as bridge12-15. In fact, the

Table 6.1 Interaction energies of HF + GIF complex, computed with TZP basis set. Data in kcal/mol5. C PF

SCF

F-bonded E elec

elec E

+ CCa

e E lec + CCa + ZPVE

Counterpoise correction.

-2.44 -2.01

-133

H-bonded -2.52 -1.90 -0.72

F-bonded

H-bonded

-2.97 -2.08 -1.43

-3.31 -2.00 -0.83

294

Hydrogen Bonding

experimental results suggest that the binding energy of H3N...C1F may even be comparable to that in the clearly H-bonded H 3 N ... HF complex16.

6.1.2 CO If HF will not form a H-bond to a nonpolar dihalogen molecule, what sort of interaction might form with a molecule that has slightly more polar bonds? Calculations to answer this question were carried out 17-19 incorporating electron correlation. It was found that CO participates in two different minima with HF, both of which have the general structure of a Hbond. They are both linear with the proton pointing toward the O or C atom: OC ... HF and CO...HF. As listed in Table 6.2, the former is favored over the latter by some 1.8 kcal/mol17. In fact, the latter is barely bound at all after zero-point vibrational energies are included. The greater stability of OC ... HF is consistent with the idea that the dipole moment of CO is negative on the C side, so it is this atom that can better attract the partially positively charged hydrogen of HF. Continuing this qualitative reasoning, UV photofragmentation measurements of this complex lead to an enthalpic preference for OC ... HF by about 0.1 kcal/mol20. The C...F distance is 2.99 A for the more stable structure, and R(O ... F) = 3.00 A for the other, both on the long end of the range of H-bond distances. Arguing for the presence of a H-bond is a red-shift in the HF stretch of 154 cm-1 in OC...HF, but this shift is only 20 cm-1 in the weaker complex. Also notable is the stretch in the HF bond brought about by complexation. This stretch is 0.006 A in OC...HF, but practically zero for the other. It appears that OC...HF has many of the characteristics expected of a H-bond, but CO ... HF clearly does not. When CO is paired with water, one again sees the former molecule acting as a formal proton acceptor. SCF calculations had suggested that the role of acceptor atom can be served equally well by both the C and O atoms21, although inclusion of correlation led to a marked preference for OC...HOH. The binding energy was computed to be weaker than that in OC ... HF by 1 or 2 kcal/mol. The details of the complex in the gas phase22 confirm the presence of OC...HOH and would lead one to doubt that it is a true H-bond. The O and C atoms are separated by some 3.37 A and the bridging hydrogen is located 11.5° from the H-bond axis. Improved calculations utilizing a polarized basis set designed to accurately mimic molecular properties of the monomer23 were able to reproduce the experimental tilt angle of 11° and reaffirmed the preference for the OC ... HOH geometry. This nonlinearity was traced to a balance between electrostatic attraction and exchange repulsion. Later calculations demon-

Table 6.2 Characteristics of various complexes, computed at MP2/TZP level, and corrected by counterpoise procedure17.

OC ... HF CO ... HF OCO ... HF NNO ... HF ONN ... HF

- Eelec (kcal/mol)

- E + ZPVE (kcal/mol)

R(A...F)

r(HF)

(A)

(A)

3.59 1.14 2.42 2.99 1.93

1.83 0.03 1.12 1.59 0.67

2.99 3.00 2.86 2.87 2.91

0.006 0.001 0.003 0.004 0.004

v(HF) (cm -1 ) -154 -20 -47 -85 -87

Weak Interactions, Ionic H-Bonds, and Ion Pairs

295

strated that the nonlinearity does not require a particularly large basis set for quantitative reproduction; diffuse functions are generally sufficient24. The electronic contribution to the binding energy was computed to be 1.9 kcal/mol at the MP4 level with a basis set including f functions. This interaction is due largely to the electrostatic and dispersion energies; terms corresponding to deformation of the electron density (e.g., induction) contribute little. Coupled-cluster computation of the binding energy, CCSD(T), using a 6-31 + +. G(d,p) basis set and correcting for BSSE, yielded an electronic binding energy of 1.4 kcal/mol24. The weakness of this interaction casts further doubt upon its categorization as a H-bond. Other minima present on the surface correspond to CO...HOH, bound by less than 1 kcal/mol, and a higher-energy T-shape wherein the O atom of water approaches the C—O midpoint. However, a later work classified the T-structure as a saddle point on the PES25. The possible categorization of OO...HOH as a H-bond is discounted by the r(OH) stretch which is only 0.0007 A (although some other flexible basis sets predict stretches of as much as 0.002 A24). The OH stretching frequencies of water were computed at the MP2 level to be red-shifted by only 15 to 18 c m - 1 , further arguments against the presence of a H-bond25. The intermolecular stretching frequency, V , was calculated to be 101 c m - 1 . Other computations comparing a range of basis sets24 found the red shift of the asymmetric OH stretch to be anywhere from 5 to 22 c m - 1 . The shifts for the higher-energy CO...HOH minimum are to the blue. In summary, while one might convincingly argue for a H-bond in the complex between CO and HF, such an interaction in OC...HOH is more dubious. 6.1.3 C02

When CO is replaced by CO2, the proton acceptor no longer has a net dipole moment at all. On the other hand, the C=O bond's polarity might be expected to favor a H-bond, even if a weak one. Calculations reported in Table 6.2 revealed a linear OCO...HF structure as the only minimum on the potential energy surface17, conforming to experimental observation [26] (although the bending potential is extraordinarily flat27). Indeed, the insensitivity of the energy to bending is underscored by later studies where MP4 calculations confirmed a linear structure28 whereas others obtained a bent geometry, with (F..OC) = 146°29. The interaction energy of this structure is intermediate between OC ... HF and CO...HF, but the H-bond length is shorter than in either of the others. The red shift of the HF stretching mode is rather small, only 47 c m - 1 , and the HF stretch is 0.003 A. This arrangement would probably be considered as a marginal sort of H-bond at best. The structure of the complex appears to retain its linear character when HF is replaced by HC1, on the basis of SCF computations, coupled with observation of IR and Raman spectra in Ar matrices30 or IR absorption31 and microwave spectroscopy in the gas phase32. When CO2 is paired with HBr, the geometry loses its H-bond character: the H atom approaches the C atom, with the HBr axis perpendicular to OCO31,33,34. (Another recent microwave investigation suggested that it is the Br atom that more closely approaches the C of OCO35, with the HBr molecule undergoing large-amplitude oscillations about the C .. Br axis). A similar T-shaped geometry occurs as well when HF is replaced by HCN: in this case, it is the N atom which approaches the C from above36. Calculations37 find this structure less stable than the linear configuration but only by about 0.2 kcal/mol, easily within the margin of error. Whereas HF appears to be a strong enough proton donor that it forms what has some of the characteristics of a H-bonded complex with OCO, the same is apparently not true of wa-

296

Hydrogen Bonding

ter. When paired with OCO, the water molecule approaches OCO O-atom first to form the complex indicated in Fig. 6.238, not resembling a H-bond in any way. The structure can be described in terms of a T shape, permitting the negative end of the H2O dipole to approach the partially positively charged C atom. This finding confirmed experimental measurements39,40 and earlier calculations41 concerning this structure, but the latter computations had noted another minimum on the surface, also illustrated in Fig. 6.2, as containing what appears to be a H-bond. At the MP2 level, with a 6-311+G* basis set, the H-bonding geometry is bound by 2.0 kcal/mol (without counterpoise correction), as compared to 3.4 kcal/mol for the non-H-bonded structure. Due to the weak nature of the interaction, the stretch of only 0.001 A in the bridging O—H bond, and the long distance between O atoms (more than 3.25 A), it would not be appropriate to refer to this interaction as a true H-bond. A slightly larger basis set (6-31+G(2d,2p)) obtained similar binding energies, De, of 3.0 and 2.2, respectively40, for the two geometries in Fig. 6.2, but another work could not locate the less stable of the two42. Later computations of this same system43 were improved in the sense that counterpoise corrections were added and a larger basis set was employed (D95 + +(3d,2p)). All parameters were fully optimized and minima verified by frequency analysis. This work verified that the second structure in Fig. 6.2, which has certain characteristics of a H-bond, is a minimum on the surface. The electronic contributions to the binding energies of the T-shape and secondary minima are 2.2 and 1.3 kcal/mol, respectively, at the MP2 level. Despite the very weak binding energy of the secondary minimum, the OH bond of the water is stretched by 0.005 A as a result of the complexation. On the other hand, this indication of a possible H-bond is belied by the observation that the stretching frequencies of the water molecule are virtually unchanged in the complex. Hence, in complexes involving both CO and OCO, HF appears to be a strong enough proton donor to form a H-bond (albeit a debatable one) whereas there is no such interaction present in complexes with water.

6.1.4 NNO NNO is isoelectronic with OCO. There are several possible sites where a proton might be accepted were a H-bond to form. Calculations of HF + NNO17 locate two minima reported in Table 6.2. When HF adds to the end N site, the entire complex is linear as in OCO...HF. On the other hand, interaction with the O atom leads to a bent structure as indicated in Fig. 6.3. Both are found to be minima on the PES, but the interaction with the oxygen atom is significantly stronger, 3.0 kcal/mol prior to addition of zero-point vibrational energies. The presence of both minima, and their shapes, conform to experimental observations44-46. The red shift of the HF stretch in the more stable geometry is 85 c m - 1 , and 87 cm-1 in the less

Figure 6.2 Structures of complexes of H2O with CO2.

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297

Figure 6.3 Two minima in PES of HF + NNO.

stable linear structure, lending credence to the presence of a H-bond here. This contention is supported by the 0.004 A stretches of the HF bond length. The study of NNO + HF was later extended to NNO + HC147 in which case two minima were identified roughly corresponding to those in Fig. 6.3, in which the HC1 molecule donates a proton to either the N or O end of the NNO molecule (although the C1H...ONN complex is nearly linear). But a third minimum was also located, which does not have the appearance of a H-bond. As illustrated in Fig. 6.4, the two molecules are nearly parallel, and it is this structure which has been observed experimentally48,49. The lesser proton donating ability of HC1 versus HF is thus responsible for the loss of any H-bonding in the complex with NNO. Reproduction of the preference for this structure required surprisingly high levels of theory, as most calculations erroneously predicted one of the H-bonded geometries to be more stable. Hence, the complex between NNO and HC1 does not contain a H-bond but lower-level calculations of this sort of system could easily mislead the unwary researcher.

6.1.5 SO 2 The SO2 molecule is like NNO in that it contains several potential sites, including O atoms, that might accept a proton. HF forms a H-bonded complex with the oxygen atoms of the SO2 molecule, with a nearly linear F—H . . . O arrangement50. This complex can either be of cis or trans type, but there is little difference in energy between the two. At the MP2/TZP level, and with appropriate counterpoise correction of the BSSE, the electronic contributions to the binding energy are —4.44 and —4.35 kcal/mol for the cis and trans geometries, respectively. (In light of the later results described below for HCN, it is important to note that the geometry optimizations assumed a plane of symmetry in this complex.) HCN is comparable in strength as a proton donor to HF so it too can serve to test the proton accepting ability of certain molecules. Calculations that pair HCN with SO251 were unable to locate as a minimum on the potential energy surface a geometry that would correspond to proton donation by HCN to any site on the SO2 molecule. The only stable geometry identified has the line of the HCN molecule lying nearly perpendicular to the plane of

Figure 6.4 Preferred geometry of complex of NNO with HC147.

298

Hydrogen Bonding

SO2, with the N of the former approaching the S atom of the latter, consistent with spectroscopic indications52,53. One can conclude that the electron pairs of the S and O atoms are not competitive to attract the HCN proton, when compared with the factors leading to the geometry observed. SO2 can be paired with another proton donor, H2O, which is a weaker base than the N of HCN so one would not expect a complex of the type formed with HCN. Two different structures appear to represent minima on the H2O/SO2 surface54. The more stable of the two has the planes of the two molecules roughly parallel, with no H-bond present, and is consistent with microwave data55. The secondary minimum has some of the geometric features of a H-bond between a proton of H2O and one of the SO2 oxygens. However, this complex has only half the binding energy of the primary minimum. And one might question the existence of a true H-bond here as the OH bond is stretched by only 0.002 A and this stretching frequency undergoes a red shift of only 8 cm-1 upon forming the complex. Furthermore, the bridging proton is 2.144 A distant from the acceptor oxygen atom. Nor is a H-bond present in the complex formed when SO2 is paired with the weaker proton donor H2S, neither on the basis of calculations54, nor from microwave measurements56. Taking this into consideration, it is hence not surprising that the weak proton donor, HCCH, also does not form a H-bonding interaction with SO257, nor does CH3OH58.

6.1.6 CC12 Carbenes, wherein a central carbon atom is bound to only two other atoms via single bonds, present the possibility of a singlet and triplet electronic state that are close in energy. In the case of CC12, it appears that the unusual electronic structure reverses the normally expected electronegativities. More specifically, the carbon seems to act as a better proton acceptor than do the chlorine atoms. When paired with water as a proton donor, the only minimum on the SCF/DZP surface has a classical H-bonding arrangement, wherein the water proton acts as a bridge to the carbon59. The H-bond is longer than is typical: R(O..C) is equal to 3.32 A. However, the strength of the interaction is in the normal H-bonding range. After inclusion of correlation via MP2, with appropriate BSSE correction, the electronic contribution to the binding energy is —3.9 kcal/mol. Addition of ZPVE leads to a best estimate of the dissociation energy of 2.4 kcal/mol.

6.2 C—H as Proton Donor The matter of whether the carbon atom can act as a proton donor in a H-bond has been discussed for some time60,63. A systematic analysis of crystallographic data for a large collection of molecules had shown a statistically meaningful tendency for C—H hydrogens to approach oxygen, as compared to C or H atoms64. In these cases, the C—H hydrogen lies within about 30° of the plane containing the lone electron pairs of the O atom and approaches within the sum of van der Waals contact radii. The data also suggest that N and Cl can act as proton acceptors. The authors were reluctant to classify these interactions as true H-bonds, based as they are on purely geometric considerations. Examining H-bonds within the context of a crystal also subjects the results to crystal packing forces which might be misleading65. Implementation of neutron diffraction data allowed much better refinement of the position of the hydrogens, and thus of the putative H-bonds themselves. Of particular interest

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due to their thorough study, and the prevalence of C—H donor groups and O acceptors, are the carbohydrates66,67. A thorough study of 26 carbohydrate structures68 found the shortest C—H . . . O contact to have r(H ... O) = 2.27 A, considerably longer than is usually considered a H-bond; most of the contacts identified are even longer. The C—H bond stretches no more than 0.004 A in even the shortest contact, at the limit of experimental accuracy. A later extension examined structures in which water serves as the proton acceptor69. Again, there were no contacts found with r(H ... O) < 2.3 A. Instead of arguing for the existence of true H-bonds, the authors point out that the proton acceptor molecules may resort to a C—H donor rather than leaving a lone pair with no interaction at all. A survey of crystal structures of organometallics70 provides additional indications that C—H protons might form a Hibond to oxygen. The former can approach the latter from many directions, with a peak in the neighborhood of 0 (CO..H) = 140°. On the other hand, any such H-bond is rather long with C..O distances typically greater than 3.3 A, peaking at around 3.4-3.5 A. 6.2.1 Alkynes It is well known that carbon can act as a proton donor within the context of a molecule like HCN where the triply bonded C behaves like a more electronegative atom. The bridging hydrogen and the proton-accepting N atom of the N=CH ... NH 3 complex, for example, are separated by 2.33 A71. Like HCN, acetylene also contains a hydrogen atom which is acidic due to its placement on a carbon which is involved in a triple bond. When paired with a potential proton acceptor like water, HCCH will indeed use its proton to act as a bridge72,73, as it will when alkynes are paired with other oxygen proton acceptors74 or with N-bases75. In fact, there is even evidence that alkynes will donate a proton to bases containing second or third-row acceptor atoms76. The H-bond length between acetylene and water is only slightly longer than in the N CH...OH2 complex. On the other hand, the rotational spectra of complexes pairing O-bases oxirane and formaldehyde with HCCH and HCN indicate a much stronger contrast between the proton-donating abilities of the latter two molecules. The proton of HCN approaches within 2.21 A of the oxygen of formaldehyde, 1.99 A for oxirane. These distances are considerably longer when HCN is replaced by HCCH; 2.48 and 2.40 A, respectively77. The dubious nature of a H-bond in the acetylene complexes is further underscored by a 30-40° CH ... O nonlinearity and nearly perpendicular direction of approach toward the oxgyen atom. Not surprisingly, when paired with a strong proton donor like HC1, HC CH acts as proton acceptor, via its density78. Unlike the situation in HCN, HC CH does not contain an atom with an available lone electron pair which can act as the proton acceptor in a H-bond. The next best pool of electrons is the cloud between the C atoms in the triple bond. While this is certainly a rich source of electron density, it is natural to wonder if a bonding set of electrons can serve the same purpose as a nonbonding pair. For this reason, it is of fundamental interest to enquire as to whether acetylene molecules can form H-bonds with one another. One would expect the sort of bonding described above to lead to a T-shaped complex, as indicated in Fig. 6.5a. It is also conceivable that this same attraction of the H atom to the cloud could be represented by a sort of "cyclic" structure in Fig. 6.5b wherein there are two such interactions possible. This arrangement, a kind of staggered parallel geometry, has also been referred to as "S-shaped." There has been some controversy in the literature as to which is actually observed in the gas phase, or whether both coexist79-87. The theoretical literature offers a chronological picture that illustrates the pitfalls encountered when applying low level or incomplete theoretical treatments to weak inlerac-

300

Hydrogen Bonding

Figure 6.5 (a) T and (b) S-shaped complexes of HCCH dimer.

tions. For example, Sakai et al.80 applied an empirical intermolecular potential, based upon dispersion and exchange, through a 6-12 function, and the electrostatic interaction between molecular quadrupole moments. Their calculations found the staggered parallel geometry of Fig. 6.5b to be the most stable, with an interaction energy nearly double that of the Tshape. Ab initio calculations which followed soon thereafter88 contradicted this conclusion when the 6-31G calculations predicted a more stable T-geometry. Alberts et al.89 later applied an enhanced (DZP) basis set, and included MP2 treatment of correlation. Another major improvement was their ability to identify stationary points on the surface and characterize them as true minima or saddle points. They found both the T and S shapes to be stationary points on the SCF surface. While T does indeed represent a minimum, the S is a transition state; its single imaginary frequency corresponds to displacement toward the T. The latter is stable by 0.9 kcal/mol, with respect to dissociation to two monomers, including zero-point vibrations and a counterpoise correction. Reoptimization of the T structure at the correlated level leads to a contraction of the intermolecular separation by 0.4 A and a binding energy of 1.6 kcal/mol. In fact, the T structure would appear to be that which occurs in the gas phase 84-86 , and the S conformers lie along the path for interconversion of one T to the next. MP2 calculations confirm the earlier findings of Alberts et al. that the T is more stable than S90. The calculated distance from the bridging hydrogen to the C^C midpoint of 2.677 A is only slightly shorter than the experimental estimate of 2.743 A85. After computing harmonic frequencies of the dimer, Bone et al. estimate a barrier of 20 cm-1 for interconversion of various T conformers. The results cast doubts as to whether a true H-bond exists between the acetylene molecules. The T-shape is precisely what one would expect based solely upon electrostatic considerations91,92. The symmetry of HCCH yields a zero dipole moment, so the moment of lowest order is a quadrupole. A T-arrangement would best allow the approach of the two quadrupole moments. The interaction, probably less than 2 kcal/mol, is less than normally expected for a H-bond. Finally, the ease of rotation of one molecule around the other contrasts with the directionality of most H-bonds. As it appears that the pool of electrons in the triple bond is not adequate to accept a proton from acetylene, perhaps a more conventional acceptor might be successful. Various N and O acceptors were paired with an acetylene derivative in an argon matrix and monitored

Weak Interactions, Ionic H-Bonds, and Ion Pairs

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by IR spectroscopy93. Table 6.3 lists the various bases in increasing order of basicity, as measured by gas-phase proton affinity, in the first row. Starting with the weakest base, CH3CN in the first column, all of the shifts are less than 100 cm - 1 , so none of these would probably be classified as a H-bond. On the other end of the spectrum is the very basic (CH3)3N which induces red shifts of at least 100 cm - 1 , approaching 300 c m - 1 in some cases. It would hence be easy to argue that the alkynic H is being donated to this base and that a legitimate H-bond has formed. There are gradations in between these two extremes. Depending on what minimum threshold red shift one wishes to establish for the presence of a H-bond, one can argue that there are or are not H-bonds present for the intermediate situations. In any case, it seems clear that the alkynic C—H group is indeed capable of forming a H-bond under certain circumstances. Perhaps another litmus test of the ability of the alkynic C—H group to donate a proton in a H-bond arises when a molecule of this type is paired with a hydrogen halide, HX. One then has two distinct possibilities. The X atom, although a weak proton acceptor by nature, can form a complex of the C—H . . . XH type. An alternative would have the XH acting as the proton donor, with the electron-rich alkyne triple bond acting as the acceptor. Experimental measurements94,95 indicate the latter is the more stable of the two alternatives. Indeed, a similar sort of geometry is adopted when HF approaches the system of ethylene96, even though the electron source in this double bond is less rich than in the triple bond of an alkyne. An experimental study of diacetylene and HF in solid argon97 suggested both sorts of complexes (a and b in Fig. 6.6) were present and that they are of comparable stability. Correlated (MP2) calculations with a 6-31 + +G(d,p) basis set98 found the perpendicular complex (a in Fig. 6.6), wherein FH approaches one of the two triple bonds of diacetylene, is more stable than is complex b wherein C—H acts as proton donor. The electronic contributions to the binding energies of complexes a and b are calculated to be —3.8 and —2.6 kcal/rnol, respectively. However, these values are surely inflated by the failure to correct them for BSSE. One can conclude that the triple bond is a better proton acceptor than the alkynic C—H is a donor, at least when paired with the rather strong acid HF. The preference for this sort of geometry is confirmed by gas-phase measurements, and are valid also when HF is replaced by HC199. The importance of using a satisfactory level of theory for such complexes is reinforced by comparison with earlier SCF-level calculations100 which predicted a structure like b to be most stable.

Table 6.3 Experimentally measured red shifts (cm - 1 ) of H—C band of substituted alkynes (R—C=CH), when paired with various O and N bases in Ar matrix93. R

proton affinitya -Cl -CH3 -H -CH2C1 -COCH3 -CF3 a

Kcal/mol.

CH3CN

(CH3)2O

(CH3)2CO

NH,

(CH3)3N

188 69 49 57 63 82 82

192 103 69 73 111 131 125

197 95 64 54 103 121 118

204

225 138 156 160 200 241 288

94 115 125 146 170

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Hydrogen Bonding

Figure 6.6 Two complexes pairing diacetylene with HF.

There was some discussion in an earlier chapter of the propensity of the C—H group of molecules in which the C atom is involved in a double bond (e.g., HCOOH or HCONH2) to act as a proton donor in a H-bond. While there was certainly evidence presented of a stabilizing interaction between this CH group and the electronegative O atom of the partner molecule, it was not entirely clear whether this interaction truly constitutes a H-bond. 6.2.2 Alkanes We turn our attention now to C—H groups in which the C atom participates in single bonds only. While the H atom of CH4 is clearly not capable of forming a H-bond under most circumstances, it can be made more acidic by replacing some of the H atoms by more electronegative substituents. An early theoretical study in the mid-1970s101 predicted that CHF3 would donate a proton to NH3, for example. The presence of such a H-bond in this complex was later confirmed by experimental work in the gas phase102, where its strength was found comparable to that in HCCH...NH3. Water was taken as a potential proton acceptor and paired up with both CF3H and CC13H, and studied at the SCF level with a 4-31G basis set103. These complexes were found to be bound by a H-bond nearly as strong as that in the water dimer. Similarly, these two proton donors form H-bonds with formamide of strength comparable to that of the formamide dimer104. This line of inquiry was later extended to the series CHmCln, m+n = 4105. The calculations were performed with the 4-31G basis set and at the SCF level and, therefore, are crude by modern standards. On the other hand, the data were improved by removing the BSSE and adding in the effects of dispersion by an atom-atom empirical expression. The data listed in Table 6.4 indicate that, as expected, there is only a very weak interaction between CH4 and OH2. However, the replacement of one of the methane H atoms by chlorine immediately imparts sufficient acidity to the other H atoms that a reasonably strong H-bond can be formed with water. The C...O distance drops precipitously, another indication of a much strengthened interaction. (The nonlinearity of the H-bond is likely due to the electrostatic interaction between the dipoles of CC1H3 and OH2.) As additional H atoms are replaced by Cl, the interaction undergoes additional strengthening and the H-bond becomes progressively shorter. Of course, there is no H-bond when all four H atoms are replaced. The qualitative aspects of the above study with the 4-31G basis have recently been confirmed at a higher level and including correlation106. The interaction of a series of fluorosubstituted methanes with water was computed at the MP2/6-31G** level. The results are

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Table 6.4 Computed properties of complexes pairing H2O with CHmCln. Data computed with 4-31G basis set, with counterpoise correction, and dispersion term added105. -

Eelec (kcal/mol)

CH4..OH2 CClH3..OH2 CC12H2..OH2 CC13H..OH2 CC14..OH2 a

0.7 5.6 6.4 8.1 1.1

R(C ... O) (A)

a (degs)a

3.79 3.22 3.16 3.07 4.88

0.0 29.7 2.6 0.0 0.0

Nonlinearity of H-bond, (O..CH).

exhibited in Table 6.5, from which it may be seen that the interaction energy is 1.4 kcal/mol for CF3H. This value is considerably smaller than the 5.6 kcal/mol computed in the earlier work for CC13H, a discrepancy due in part to the difference between F and Cl, but probably due more to the difference in theoretical approach. Comparison of the data in Tables 6.4 and 6.5 suggests that the earlier results for the chlorosubstituted methanes likely exaggerated the binding. But there are nonetheless clear similarities apparent. In either case, the replacement of each H by a halogen atom adds an increment to the binding energy and shortens the intermolecular distance. (It should be noted that the latter distances refer to H...O contacts in Table 6.5.) These changes are approximately 1 kcal/mol and 0.1 A, respectively, for the fluoromethanes. The interaction energy of monofluoromethane with water lies on the lower end of the energy spectrum of what are usually considered H-bonds, but the diand trifluoromethanes are more strongly bound. When the various chloromethanes are paired with HF rather than water, there is an obvious tendency for the strong HF acid to act as the proton donor. For this reason, the minima identified at the correlated MP2 level with a 6-31 +G(d,p) basis set contained F—H ... C1 H-bonds for the most part107. The mono-, di-, and trichloromethanes did form "cyclic" complexes which contain elements of a strongly bent C—H . . . F H-bond along with a distorted F—H . . . Cl bond. But it is difficult to separate the properties of the former from those of the latter, so one cannot ascertain whether there is a true C—H . . . F H-bond present in these structures. An analogous study replaced HF by the weaker proton donor HC1180. The results were similar in that cyclic geometries were obtained for the mono-, di-, and trichloromethanes. Again, in no case was an unambiguous C—H ... Cl H—bond identified in any of the minima present in the surface.

Table 6.5 Computed properties of complexes pairing H2O with CHmFn. Data computed at MP2/6-31G** level, with counterpoise correction106.

CFH3..OH2 CF2H2..OH2 CF3H..OH,

- Eelec (kcal/mol)

R(H ... O) (A)

1.41 2.13 3.16

2.51 2.39 2.28

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Hydrogen Bonding

Electron-withdrawing substituents other than halogens are also capable of making methane a stronger proton donor. When paired with ammonia, nitromethane forms a complex which contains what can be described as a C—H . . . N hydrogen bond, in addition to auxiliary interactions between the O atoms and the hydrogens of ammonia109. The electronic contribution to the binding energy of this complex is computed at the MP2 level with a large polarized basis set to be —4.4 kcal/mol, after correction for BSSE. After addition of vibrational and other corrections, H° is —3.0 kcal/mol. The absence of a true H-bond between unsubstituted CH4 and OH2 was confirmed by higher level calculations with much larger basis sets, and with correlation included explicitly 110-112 . In the optimum geometry of this complex111, one of the methane hydrogens is pointing directly toward the O atom of water as in Fig. 6.7a, consistent with the earlier 431G results105. The binding energy is only 0.5 kcal/mol, remarkably similar to the cruder earlier value, as is the C... distance of 3.75 A. It is interesting that when OH2 is replaced by SH2, the type of complex reverses and it is now SH2 which is the nominal proton donor, as illustrated in Fig. 6.7b. Nonetheless, there is no true H-bond present as the interaction energy here is still only 0.5 kcal/mol. Further verification of the absence of a H-bond came from a basis set near the HartreeFock limit113. The geometry of CH4 + OH2 illustrated in Fig. 6.7a was found there to be the only minimum on the PES but its binding energy was computed to be only 0.6 kcal/mol, a value which is largely confirmed by later calculations as well 112,114 . Only 34% of this interaction energy is present at the SCF level, another indication that it is not a H-bond. In partial contrast, a later careful study of a wider swath of the PES112 suggested that while Fig. 6.7a does indeed represent a local minimum on the surface, the global minimum of the methane-water complex is in fact akin to that in Fig. 6.7b. That is, HOH is the nominal proton donor rather than CH4. This structure was computed to be more stable than Fig. 6.7a by some 0.2 kcal/mol. The total binding energy of this particular complex, with C as proton acceptor, is 0.8 kcal/mol. Just as changing the O of water to its second-row analog, S, does not lead to a H-bond, the same is true when the C of CH4 is replaced by Si. The optimized geometry in this situation is illustrated in Fig. 6.7c115. Van Mourik and van Duijneveldt argue114 that the relatively short C...O distances sometimes encountered in organic crystals are a result of the "softness" of the carbon atom, as opposed to any H-bonding character in the interaction itself. Amplifying on evidence that C—H can act as a proton donor in certain circumstances, a recent work116 reports spectroscopic evidence that this group can donate a proton to a second-row atom. Unusually short distances were found in the C—H . . . Se interaction in the crystal structure of diselenocin; the proton was located 2.92 A from the Se center. IR data indicate a 53 cm-1 shift in the C—H stretch. Arguing against this interpretation is the large deviation from linearity, closer to a right angle. When combined with water, methane may act as a proton donor, even if only in a geometric sense. Suppose that OH2 is replaced by a molecule which is a superior proton donor:

Figure 6.7 Optimala geometries of complexes of CH4 and SiH4 with OH2 and SH2.

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305

will the geometry reverse such that methane acts as proton acceptor? Rotational spectroscopy furnishes an answer to this question. When combined with HC1 or HCN, one does indeed see a geometry indicative of proton donation to the C atom of CH4117,118, as illustrated in Fig. 6.8a. However, it is questionable if this structure is indicative of a true H-bond any more than the complex of CH4 with SH2 in Fig. 6.7b, particularly because the distance between C and Cl is nearly 4 A. Calculations with a 6-31G** basis set at the MP2 level119 indicate the H of HC1 approaches the C of CH4 along a C3 symmetry axis. The red shift of the HC1 bond is computed to be 16 c m - 1 . When HF replaces HC1, the structure loses all semblance of a H-bond: a C—H of CH4 points toward the F atom119. The structure is as shown in Fig. 6.8b. This geometry has certain features of a cyclic structure with two weak H-bonds. However, calculations carried out of CH4 + HF at the SCF/6-31G** level117 revealed the total binding energy to be only 1.0 kcal/mol, 2.1 after correlation is included via MP2. Moreover, these results are likely to be reduced substantially were BSSE removed. The absence of a true H-bond in this complex is confirmed by the long distance between C and F of 3.60 A. Spectral data of the complex between HF and CH4 in an inert matrix120 appear to confirm the absence of an H-bond geometry. That is to say, the structure likely resembles Fig. 6.8b. A structure similar to that in Fig. 6.8 is obtained when ethane is combined with HCN121. Again, the structure has certain angular features of a H-bond but R(C .. C) is very long, estimated to be 3.76 A. Infrared matrix isolation studies support the possibility that CH4 can act as a proton acceptor when complexed with HNO3122. The presence of a H-bond is questionable, however, since the red shift of the bridging OH stretch is only 43 c m - 1 . 6.2.2.1Anionic Acceptors Whereas the hydrogen atoms of methane are certainly unable to form H-bonds in general, the situation changes when the proton acceptor is an anion. One then has a greatly amplified electrostatic component, in addition to the ability of the ion to polarize the neutral molecule, increasing the polarization energy as well. Methane was paired up with the series of halides and the geometries optimized at the MP2 level using a 6-31 + +G(d,p) basis set; energies were then obtained using larger sets approaching the Hartree-Fock limit123. The types of structures examined are illustrated in Fig. 6.9. In the lexicon of H-bonds, the first might be denoted linear, the second bifurcated, and a trifurcated geometry is represented in Fig. 6.9c. The interaction energies reported in Table 6.6 make it immediately evident that the linear structure is most stable for each anion. Indeed, geometries b and c are not true minima on the PES. The binding energies of the linear structure diminish as the anion becomes larger, varying from 5.8 kcal/mol for F~ down to less than 2 for the larger anions. Using

Figure 6.8 Experimental geometries of complexes of CH4 with HCI and HF.

306

Hydrogen Bonding

Figure 6.2 Possible geometries of CH4 I

energy as a criterion, one might consider methane to be capable of forming a H-bond to F and probably also to Cl- as well. Applying a basis set approaching the Hartree-Fock limit, and removing counterpoise errors, the binding energy of the linear structure is computed to be 5.6 kcal/mol for CH4 + F-, and 2.5 kcal/mol when F- is replaced by C l - , quite close to the data reported in Table 6.6. What about other signs of a H-bond? The R(C..F) distance in the first complex is 3.05 A, rather long for a H-bond but not entirely unreasonable. The C—H bond involving the bridging hydrogen is stretched relative to the other bonds by 0.011 A, another sign of a true H-bond. These same properties would argue against a H-bond for C l - , however. R(C..C1) is 3.79 A, and the bridging C—H bond is stretched by only 0.001 A. The harmonic frequency corresponding to the H-bond stretch v is computed to be 159 cm-1 for CH4 + F-, but less than 70 cm-1 for the other complexes. The C—H stretching frequency of CH4 undergoes a red shift of 105 c m - 1 in CH4 + F- but 20 c m - 1 or less for the other anions. In summary, one could argue that methane forms a H-bond with the fluoride anion. The geometries are similar for the other anions, but the interaction is considerably weaker as are the other indicators of a H-bond. 6.2.3 Metal Atoms as Acceptors Another sort of unconventional type of H-bond that has been proposed in the literature involves electron-rich transition metals as proton acceptor124-128. These ideas are derived from spectroscopic data but the most explicit information arises from diffraction studies of crystals, where the evidence is largely geometrical. For example, an NH group approaches unexpectedly close to a Pt atom in a square-planar geometry, with R(Pt .. H) = 2.262 A129. The arrangement is nearly linear as the 9 (N—H ... Pt) angle is 167°. The authors point to prior crystal structures in the literature where N—H and even C—H groups are positioned

Table 6.6 Computed binding energies (— Eelec in kcal/mol) of complexes pairing CH4 with halide anions. Data computed at MP2 level with 6-31 + +G(d,p) basis set, uncorrected for BSSE123.

F-

C1BrI-

Linear

Bifurcated

Trifurcated

5.8 2.7 2.0 1.3

2.7 1.3 0.7 0.2

1.8 1.2 0.5 0.3

Weak Interactions, Ionic H-Bonds, and Ion Pairs

307

directly above the plane of a square complex with the metal in a d8 configuration. Mention is made also of possible interactions with d10 metals such as Co and Ni130,131. For example, theR(Co .. H) distance is 2.613 A, less than the sum of the van der Waals radii, and the (N—H ... Co) arrangement is very close to linear132. Moreover, the N—H distance is measured to be surprisingly long, 1.054 A, an indication of possible H-bonding activity. However, as the authors point out, there is a fine line between a true H-bond and a simple electrostatic interaction between the cation and anion. 6.2.4 Hydride as Proton Acceptor A different sort of interaction involving hydrogens has been recently noted in crystal studies of certain systems. It has been observed by X-ray diffraction and NMR techniques that two hydrogens can approach one another unexpectedly closely if one of them is covalently bonded to an electronegative atom such as N, and the other to a metal atom like Ir. It is supposed that the Ir-H hydride acts as a sort of "base" to accept the proton from N—H. An example of this type arises in the iridium complex with trans PCy3 groups and a cis 1 -SC5H4NH ligand133 where the two hydrogens in question are separated by 1.75 A. NMR data indicate this interhydrogen distance persists in solution as well. A comparably short Ir-H .. HO interaction has been observed when the OH is part of an iminol group134; the interhydrogen distance here may be as short as 1.58 A. The two hydrogens are separated by some 1.77 A when an alcohol is paired with a W—H hydride in a tungsten derivative in an intermolecular complex135. Estimates of the strength of this type of interaction, based upon experimental quantities, vary from 2.5 to as much as 6.9 kcal/mol in certain cases135-137. Geometry optimizations carried out at the SCF level, using a moderate-sized basis set including polarization functions136, found the Ir-H and N—H hydrogens could approach within 1.96 A in a model system. The authors attributed the source of the attraction to the opposite charges calculated on the two hydrogens, +0.22 on one and —0.26 on the other. Another contributing factor is the ease of polarization of the Ir-H bond. Although a positive bond order was computed between the two hydrogens in question, the small value of only 0.012 might lead one to question whether the interaction contains much covalent character. An analysis of various energetic quantities yielded a theoretical estimate of the interhydrogen interaction energy of 5.65 kcal/mol. The fundamental nature of this interaction as primarily electrostatic attraction between the opposite charges of the two hydrogens was supported by later calculations138. When the geometry of the complex of FH with HMn(CO)5 was optimized at the SCF/3-21 G(*) level, the H atoms of the two subunits were separated by 1.68 A. This distance is likely too short due to the large BSSE common to this sort of basis set. Likewise, the computed interaction energy of 6.55 kcal/mol is probably exaggerated by the same phenomenon. Nonetheless, the results do support the notion of a short interhydrogen contact when one H is covalently bonded to an electronegative atom and the other to a metal. Similar sorts of interactions have recently been proposed as well when the metal is replaced by a boron atom. Crystallographic data yields a number of NH ... HB interhydrogen separations of less than 2.2 A139. The nitrogenic hydrogen seems to prefer to approach the other hydrogen from a direction nearly perpendicular to the BH bond. The authors attributed this significantly nonlinear geometry to the unfavorable alignment of bond dipoles that would result in the case of a linear N — H . . . H — B arrangement.

308

Hydrogen Bonding

6.3 Symmetric Ionic Hydrogen Bonds

On one end of the spectrum of H-bonds are those that are very weak, as in the case of numerous C—H . . . X interactions. On the other end of this continuum are very strong interactions. HX molecules are quite acidic. If paired with a proton acceptor that has a negative charge, the interaction can become very strong indeed. In fact, the H-bond can become so short that the proton loses its association with one molecule versus the other. That is, the proton can be drawn into the midpoint of the X...X axis. Such a H-bond is referred to as centrosymmetric.

6.3.1 Hydrogen Bihalides Simple examples are the hydrogen bihalide anions FH..F- and C1H .. C1 - . Polarized basis sets were used to examine this pair of systems, along with CI treatment of correlation140. At the SCF level, their binding energies were computed to be 39 and 15 kcal/mol, respectively. These values increase to 41 and 21 kcal/mol at a level which approximates a full CI treatment. The computed results mimic experimental data rather well. The binding energy in FH..F- had been measured to be 42 kcal/mol 141,142 ; the range obtained for C1H..C1- is 13-19 kcal/mol143. In both cases, the optimized position of the proton is midway between the two halogen centers. The interfluorine distance is calculated to be 2.26 A, one of the shortest contacts known (2.278 A in the gas phase144); R(Cl .. Cl) is 3.13 A, also very short. The hydrogen bifluoride anion was considered at higher levels of theory, focusing upon its infrared spectrum145. CISDT was applied, in conjunction with a TZ3P basis set, augmented by very diffuse Rydberg(R)-type functions. Anharmonic effects were considered by evaluating the full quartic force field of this triatomic. At all levels, SCF through CISDT, and DZP through TZ3P+R+d, the H-bond was found to be centrosymmetric, and the linear (FHF)- anion belongs to the D h point group. Interfluorine distances lie in the narrow range of 2.24-2.28 A. The harmonic frequencies of the symmetric (H-bond stretch) and antisymmetric (proton motion) stretches are computed to be 668 and 1195 cm - 1 , respectively. Because of the large dipole moment change associated with the proton motion, the intensity of the latter mode is very high indeed, 3724 km/mol. Anharmonicity lowers the first frequency by 36 cm - 1 , but the asymmetric stretching mode is much more sensitive, increasing by 265 cm^'. With regard to sensitivity to level of quantum mechanical treatment, the symmetric stretch does not change much as the basis set is altered, nor as correlation is added, remaining in the 650-700 cm-1 range at all levels of theory. The antisymmetric stretch, on the other hand, is highly sensitive, with frequencies varying between 627 and 1538 cm-1 depending upon level of theory. One may conclude that computation of the vibrational characteristics of the mode involving the proton transfer between the two F centers is very demanding, whereas the other modes of the complex may be treated with only reasonable levels of theory. The sensitivity of the antisymmetric stretch to particular theoretical treatment is paralleled by a similar sensitivity to experimental conditions: v3 has been measured to lie anywhere between 1284 and 1740 cm - 1 , depending upon the particular host146; the frequency in the gas phase is 1331 cm - 1 . 1 4 4 Later work evaluated the two-dimensional potential energy surface using various correlation treatments including many-body perturbation theory and coupled cluster techniques 147 . Evaluation of the vibrational spectrum was explicitly anharmonic in nature, mak-

Weak Interactions, Ionic H-Bonds, and Ion Pairs

309

ing use of a highly flexible analytic function to fit the calculated point-wise surface. The results agreed with earlier work in that the symmetric stretch is relatively insensitive to the variety of correlation treatment, in the range of 570-600 c m - 1 for all methods examined. The asymmetric stretch, too, is relatively stable with respect to the order of MP or CCSD treatment, lying in the range between 1380 and 1450 c m - 1 . It was mentioned that the use of a standard quartic potential leads to poor reproduction of experimental frequencies. Another treatment of the vibrational modes in FH..F- computed the energetics of 710 points at the CID level with a [3s2pld/2slp] basis set148. These results were then fit to an analytical function (a superposition of Morse potentials plus other terms), which provided clear evidence of the nonharmonic nature of the potential surface, and illustrated the transition from a centrosymmetric single minimum to a pair of equivalent minima as the interfluorine distance was increased. This transition occurs at approximately R(F..F) = 2.4 A on this surface. The authors' results agreed with the other study in that the surface is more complex than that presuming a quartic force field. Ahigh level correlated study of the FH..F- complex149 yielded a MP4/6-311 +G(2d,2p) interaction energy — Eelec of 44.3 kcal/mol, although this value was not corrected for superposition error; a computation with an even more extended basis set resulted in a H-bond energy of 45.6 kcal/mol150. Comparable calculations, including counterpoise corrections 151 , reduced the binding energy to 39 kcal/mol, in good agreement with an experimental measurement of 39152. The (Cl—H—Cl) - analog represents a particularly interesting system as crystal structures are inconclusive regarding the nature of the proton transfer potential. The C1..C1 distance is observed to vary between 3.14 and 3.22 A in various different crystalline environments153. When in its shorter range, a centrosymmetric H-bond is observed whereas the proton is displaced as much as 0.24 A from the C1..C1 midpoint for longer distances. The interchlorine separation is measured to be 3.15 A in the gas phase154. The data suggest a double-minimum potential for this anion, but the lowest vibrational level is close enough to the top of the barrier that a centrosymmetric H-bond is observed154,155. An early SCF study treated the (Cl—H—Cl) - system with a DZP basis set156. R(Cl..Cl) was optimized to be 3.24 A. Consistent with experimental data for longer interchlorine distances, the proton transfer potential contained two separate minima in each of which the proton is offset from the bond midpoint by 0.23 A. The barrier separating the two wells is rather low, only 0.6 kcal/mol. The H-bond energy of the (Cl—H—Cl) - complex was computed to be 20 kcal/mol, in reasonable agreement with an experimental measurement of 24 kcal/mol157. When correlation is added, the barrier appears to vanish and the H-bond takes on centrosymmetric character151,158-161. An extensive basis set of [11s9p2dlf] character for Cl, with H represented by [5s2pld], was applied to this system, and correlation applied via MP2-4159. The results in Table 6.7 illustrate the change in character from C v to D h that occurs upon application of correlation. That is, the proton moves to the center of the C1..C1 bond. Along with this change in character comes a shortening of the H-bond by some 0.2 A. As in the earlier case of (FHF) - , the symmetric stretching frequency is rather insensitive to the level of theory while the asymmetric stretch shows a good deal of variation. This result is consistent with the flatness of the potential for proton transfer, and the change in character upon adding correlation. After adding in zero-point vibrational corrections, the enthalpy of association in the (Cl—H—Cl) - complex was computed to be —23.5 kcal/mol at the MP4 level, in nice agreement with experimental estimates of —23.1 and — 23.7 152,162,163 . This interaction is quite a bit weaker than in the FH..F- analogue. The data emphasize the sensitivity of the nature of the H-bond in this complex to level of the-

3 10

Hydrogen Bonding

Table 6.7 Calculated characteristics of optimized (Cl—H—Cl) - complex159. Basis set

Level

DZP [Ils9p2dlf/5s2pld] DZP DZP

SCF SCF MP2 MP4

Symmetry C C D D

V v h h

R(Cl .. Cl) (A)

ra (A)

3.343 3.331 3.126 3.126

0.32 0.32 0.0 0.0

a

Displace of proton from C1..C1 midpoint.

ory, or to any environmental effects that might tend to shorten or lengthen the H-bond. It is also important to note that due to the change from a noncentrosymmetric to centrosymmetric H-bond that occurs upon inclusion of electron correlation, the SCF and correlated vibrational structures are quite different160, so one would be ill-advised to add SCF zeropoint vibrational corrections to the correlated energetics. Correlated computations have also been performed on the next in the series, (Br .. H .. Br )-164 . Using a DZP basis set (more specifically [9s7p2d/3slp]) and a multireference CI approach, this anion was also found to be centrosymmetric, with R(B . . Br) = 3.429 A. This distance is estimated to be about 0.27 A shorter than the sum of the van der Waals radii. The dissociation energy to HBr and Br- is 15.4 kcal/mol, bracketed by experimental estimates of the same quantity152,165. Note that the computed value was not corrected for superposition error. The zero-point vibrational correction to the dissociation energy was calculated to be negligible. Electron correlation is absolutely critical to proper treatment of this system. 6.3.2 Comparison with Other Anionic H-bonds A systematic comparison of various symmetric anionic complexes provides a solid basis for comparison151. The data listed in Table 6.8 illustrate the rapid decline in binding energy as the electronegativity of the atoms diminishes, or as one passes from first to second-row atoms. Concomitant with this weakening of the interaction is the lengthening of the H-bond. In a number of cases, there is a fine distinction as to whether the H-bond is centrosymmet-

Table 6.8 Binding energies, H-bond lengths, and proton displacements of anions, computed at MP4/6-311+G(d,p) level151. Complex (FH..F)(HOH..OH)(H2NH..NH2)(ClH..C1)(HSH..SH)(H2PH..PH2)a

— Eeleca (kcal/mol)

R(A)

38.8 23.2 10.2 18.0 7.1 unbound

2.30 2.44 2.91 3.10 3.28

BSSF and ZPE corrections added. Displacement of proton from H-bond midpoint. c Barrier lies below first vibrational level. b

rb (A) 0.0

0.oc 0.40 0.0C 0.0c

Weak Interactions, Ionic H-Bonds, and Ion Pairs

3II

RIc or not. It is not uncommon to find a proton transfer potential that contains two wells, but the barrier is quite low. 6.3.2.1 (HOH..OH)The (HOH..OH)- anion serves as a classic case of the latter situation. For example, calculations with basis sets ranging from 4-31G to polarized such as 6-311G* and larger166-170 showed how correlation and/or zero-point vibrations can change a double to a single-well potential, and how the results might differ depending upon the particular means of treating the correlation. But all surfaces were flat, regardless of whether there were two minima present or one. The highest barriers obtained were less than 1 kcal/mol, so one can conclude that this barrier would lie below the first vibrational level. Consequently, the complex would behave for all intents and purposes as though it contained a centrosymmetric bond. Of course, the amplitude for proton motion would be quite large. In cases such as these, it is probably more appropriate to compute the binding energy of the centrosymmetric geometry, even though this configuration represents a first-order saddle point on the purely electronic potential energy surface. The fine balance between a centrosymmetric and noncentrosymmetric H-bond is confirmed by crystal studies. Lithium hydrogen phthalate monohydrate, for example, contains H-bonds of the OH..O type about 2.4 A in length 171 . While one of these has the bridging hydrogen closer to one O atom than to the other, the other H-bond is centrosymmetric (or very nearly so). Another centrosymmetric H-bond is observed in the crystal structure of methyl ammonium hydrogen succinate monohydrate172, this one being 2.44 A long. (The authors do not rule out the possibility of a double-well potential for proton transfer, albeit one with a low barrier.) Perhaps the shortest such H-bond observed occurs in the crystalline complex of l,8-bis(dimethylamino)naphthalene with 1,2-dichloromaleic acid173; the 2.38 A H-bond is apparently centrosymmetric. The authors provide a graphic of measured r(OH) bond lengths in similar complexes that illustrates the transition from noncentrosymmetric to centrosymmetric as R(O..O) contracts down to 2.4 A. Minor external effects can also influence the shape of the proton transfer potential between carboxylate anions. Whereas there are apparently two minima present in complexes of this type in aqueous solution, the H-bond takes on centrosymmetric character when immersed in nonpolar solvent174. Calculations have suggested the change from single to double-well character is due not to the surrounding dielectric but rather to interactions with discrete molecules175. The binding energy computed for (HOH..OH)- at a relatively high level, 23 kcal/mol151, is in reasonably good coincidence with experimental measurements of the enthalpy of binding of 27 kcal/mol176,177. Computed enthalpies, which include rotational and translational corrections160,178 were even closer at 28.6 kcal/mol, although neither study removed BSSE. Recent NMR measurements of a series of complexes in the solid state179 indicate that many of the same principles apply to H-bonds of the FH..F type. Centrosymmetric H-bonds occur only for short interfluorine distances, about 2.3 A or less. The gas-phase bond length of 2.2,78 A is therefore consistent with the ab initio finding of a centrosymmetric H-borid in (F...H...F)-. The crystalline environment can alter the proton transfer potential to a noncentrosymmetric type by just a small H-bond elongation179. Methyl substitution apparently induces little change in this behavior. MP2 calculations with a 6-31+G* basis set180 yield a binding enthalpy of 26.3 kcal/mol for

312

Hydrogen Bonding

(CH 3 OH ... OCH 3 )- . The small barrier present in the proton transfer potential at the SCF level (2.2 kcal/mol) is eliminated by the inclusion of zero-point vibrational energies and/or by consideration of electron correlation. Experimental measurements indicate the interaction energy is increased by 1-2 kcal/mol by methyl substitution. Tandem flowing afterglowselected ion flow tube181 and variable-temperature pulsed high-pressure mass spectrometric measurements182 of (CH3OH...OCH3)- have led to a dissociation energy of 28-29 kcal/mol. This change in binding energy accompanying methyl substitution is rather small, considering that the deprotonation energy of CH3OH is lower than that of HOH by 9.6 kcal/mol182. And like the simpler (HOH .. OH) - , the proton transfer potential in (CH3OH...OCH3)- in the gas phase contains either a single central minimum or a small barrier, as determined by ion-cyclotron resonance spectroscopic measurements183. Even a more major change in the character of the oxygen bases yields a relatively small perturbation in the H-bond energy. For example, replacement of the two methoxide ions in (CH3OH...OCH3)- by a pair of acetate anions increases the enthalpy of binding by only 0.5 kcal/mol184. 6.3.2.2 Anions with Triple Bonds The C=N - anion is of particular interest as it is capable of forming a H-bond from either end. Table 6.9 reports the properties of the H-bonds formed between this anion and HCN (or HNC), as well as the analogous (HC=CH .. C=CH) - which also contains a triple bond185. The first column reflects the greater acidity of HNC as compared to HCN, as the former forms stronger H-bonds when it acts as proton donor. The weaker acidity of HCCH is apparent from the relatively low interaction energy in (HCCH .. CCH) - . The binding enthalpies at 300 K in the next column are similar in magnitude to the electronic data in the preceding column. The stronger H-bonds are also associated with shorter intermolecular separations. The noncentrosymmetries of the H-bonds are illustrated in the r column; all are predicted to be noncentrosymmetric at the SCF level. The degree of asymmetry is inversely related to bond strength. The values of E+ listed in the next columns refer to the barriers in the proton transfer potential. There is a clear correlation between strong H-bonds on one hand and low barriers on the other. It is important to stress that inclusion of electron con-elation lowers all barriers, to the point where the NH—N interaction collapses to a centrosymmetric (CN .. H .. NC) - H-bond.

Table 6.9 Properties of anionic complexes of triply bonded species, computed with 6-31 + G* basis set185. - Eelec

a

(kcal/mol)

- Hb (kcal/mol)

R (A)

rc (A)

Et,d (kcal/mol)

Complex

MP2

MP2

SCF

SCF

SCF

MP2

(CNH .. NC) (NCH..CN)(HCCH..CCH)-

27.5 19.5 10.8

26.3 18.5 9.7

2.75 3.20 3.35

0.34 0.51 0.60

3.4 9.5 13.1

0.0 5.3 7.6

a

Corrected for BSSE. Evaluatedat 300 K. Displacement of proton from H-bond midpoint. d Proton transfer barrier. b

e

Weak Interactions, Ionic H-Bonds, and Ion Pairs

3 I3

6.3.2.3 Anions with Double Bonds An equivalent study was carried out for the analogous systems containing double instead of triple bonds186. Because the doubly-bonded C or N atoms are less acidic, the H-bond energies in Table 6.10 are smaller than those in Table 6.9. The H-bonds are also uniformly longer and more noncentrosymmetric. Other systematic trends remain consistent. The N atom is more acidic than C, and the C of HN=CH 2 is more acidic than H2C=CH2. The Hbonds lengthen as the interaction weakens. The noncentrosymmetry of the proton position grows as the barrier for proton transfer becomes higher. Correlation lowers these barriers but not to the point where any of them vanish entirely. (However, the lowest barrier, that in (H 2 CNH .. NCH 2 )- , does indeed vanish when zero-point vibrational energies are added.) 6.3.2.4 Carboxylate When bound together by a proton, a pair of formate anions (HCOO - ) can in principle adopt a number of different conformations since each O atom has associated with it a syn and anti lone electron pair. Using a fairly large basis set containing diffuse functions on O, along with polarization functions, the geometry obtained for this complex is illustrated in Fig. 6.10a187. The proton donor is designated as anti because the OH proton is opposite the carbonyl oxygen; the formate is referred to as syn since the proton which it is accepting approaches a syn lone pair of that O atom. This particular arrangement also permits the CH proton of the HCOOH to interact favorably with the oxygen of the HCOO- which is not involved in the H-bond with —OH. The principal R(O .. O) H-bond length is 2.592 A, while the distance separating the C—H hydrogen from the O of the acceptor group is 2.51 A. Since this is much longer than would normally be expected for a H-bond, this secondary interaction would probably be better characterized as an electrostatic interaction. The binding energy (- Eelec) of this structure is calculated to be 28.8 kcal/mol at the SCF level, increasing to 33.0 when MP2 correlation is added. These results are smaller than the gas-phase measurement of H° of 36.8 kcal/mol184. The disagreement is amplified by the fact that BSSE and ZPE corrections have not been added to obtain a theoretical enthalpy. Like several of the systems mentioned above, the noncentrosymmetric H-bond illustrated in Fig. 6. l0a, changes to a more symmetric structure, and the proton transfer potential converts from double to single-well character when correlation effects are accounted for. At the

Table 6.10 Properties of anionic complexes of doubly bonded species, computed with 6-31+G** basis set186. Etd (kcal/mol)

- Eelec (kcal/mol)

- Hb (kcal/mol)

R (A)

rc (A)

Complex

MP2

MP2

SCF

SCF

SCF

MP2

(H2CNH..NCH2)(HNCH2..CHNH)(H2CCH2..CHCH2)-

15.4 10.3 5.6

13.8 9.6 4.8

3.03 3.52 3.74

0.50 0.68 0.78

8.6 18.2 19.2

1.7 11.7 12.8

a

Corrected for BSSE. Evaluated at 300 K. ^Displacement of proton from H-bond midpoint. d Proton transfer barrier. b

314

Hydrogen Bonding

Figure 6.10 Geometrical configurations of HCOOH ...-- OOCH 187 .

approximate bottom of the correlated transfer potential, the R(O..O) distance is 2.47 A. The authors estimate that the secondary interaction in the anti-syn configuration in Fig. 6.10a accounts for perhaps an additional stabilization of 2 kcal/mol. For purposes of comparison, the anti-anti and syn-syn geometries were also studied by the authors187. The configurations pictured in Fig. 6.10b and 6.10c were optimized. The anti-anti geometry was found to be more stable than syn-syn by about 3 kcal/mol. At the SCF level, the anti-anti is 4 kcal/mol higher in energy than anti-syn geometry, but this difference is reduced to only 1 kcal/mol at MP2. A later calculation disputed these results in that anti-anti was computed to be the most stable conformer188. This work, however, was limited to the SCF level, using a 631 + +G** basis set. (If consulting the original paper, it is important to note that this set of workers reversed the standard nomenclature for syn and anti.) Anti-syn was computed to be somewhat less stable, followed by syn-anti and syn-syn. These results were compared to a statistical survey of such interactions in proteins where a marked propensity was observed for the syn-syn structure, the least stable in the gas phase. In contrast, the anti-anti configuration, most stable in the gas phase, is observed rarely in proteins. The authors attributed these discrepancies to crystal packing forces. There also seems to be a preference in the crystals for centrosymmetric (or nearly so) H-bonds between carboxyl and carboxylate, although noncentrosymmetric H-bonds are also present in large numbers. Centrosym-

Weak Interactions, Ionic H-Bonds, and Ion Pairs

315

metric H-bonds in the crystal tend toward R(O--O) distances of about 2.45 A; the noncentrosymmetric bonds are a little longer, in the general range of 2.5-2.6 A. In either case, the crystal seems to exert a mild lengthening effect upon the H-bonds. 6.3.2.5 Environmental Effects The effects of the forces present in the crystal upon the H-bond connecting a pair of formate units was examined explicitly by placing positive point charges in the vicinity of the complex, and then employing a self-consistent procedure to evaluate point charges of the remainder of the crystal189. Rather than carry out a full geometry optimization, the positions of all atoms were taken from a neutron diffraction study of potassium hydrogen diformate, with the exception of the bridging hydrogen atom. The study was hence restricted to an R(O-O) distance of 2.437 A, with a geometry corresponding roughly to anti-anti in Fig. 6.10. Prior to inclusion of the crystal forces, the SCF/DZP proton transfer profile contains a pair of minima, separated by an energy barrier of 1 kcal/mol; the MP2 H-bond is centrosymmetric. The noncentrosymmetric nature of the crystal forces induces a displacement of the position of the proton in the MP2 potential, moving the minimum 0.053 A from the O--O midpoint, in nice agreement with the experimental shift of 0.052 A. For this particular case, then, it is apparent that asymmetric forces present within a crystal can shift the equilibrium position of the proton away from the midpoint of the H-bond, while maintaining its single-well character. Another crystal structure, that of sodium hydrogen bis(formate), seems to contain all noncentrosymmetric H-bonds between formate anions190. In aqueous solution, on the other hand, a large observed primary isotope effect suggests the proton moves within a double-minimum potential191. The ability of the environment to influence the nature of the proton transfer potential has been underscored by work192 that differentiates between a centrosymmetric single-well potential and one containing two wells, by isotopic perturbation of the NMR spectrum. It was found that when certain H-bonded systems that have a single well in the crystal are dissolved in aqueous media, the potential changes its character to a double-well. One explanation is that solvation can better stabilize a localized charge as would occur when the proton is shifted toward one subunit or another, as compared to a centrosymmetric geometry where the charge is delocalized over the entire complex. This contention is confirmed to the extent that systems of this type revert to a single-well proton transfer potential when placed in less polar solvents. The disorder of water can be invoked as a second and supplementary explanation, in that it would be unlikely for the two ends of the H-bond to be solvated to the same degree, thereby preferentially stabilizing one side versus the other. 6.3.2.6

HSO4

Perhaps the strongest of H-bonds might be expected when H2SO4 combines with its conjugate base, H S O 4 , since three separate and distinct H-bonds can occur. SCF optimizations of the geometry of this complex193 with a 6-31G* basis set led to a structure of this complex as illustrated in Fig. 6.11. Applying the MP2/6-31+G* level of theory to the optimized geometry, the total binding enthalpy was computed to be 47 kcal/mol. While this result probably suffers from some inflation due to BSSE, it is certainly quite a strong interaction. The potential for proton motion from the neutral to the anion contains a pair of equivalent minima, separated by a barrier of 2 kcal/mol at the SCF level. It is likely that this low barrier would vanish entirely were correlation applied to the transfer potential.

3 16

Hydrogen Bonding

Figure 6.11 Geometry of complex between H2SO4 and HSO4- l93 .

6.3.3 Cationic H-bonds In addition to complexes pairing a neutral molecule like HF with its anion, F--, one can construct a cationic complex where HF is combined with the protonated H2F+. Such complexes are also expected to be quite strong. Following earlier work at the SCF level194, a high-level computation of this particular system195 obtained a binding energy of 31.9 kcal/mol. This result was obtained at the fourth-order M011er-Plesset level with a polarized basis set. As illustrated in Table 6.11, the interfluorine distance is quite short, only 2.29 A, similar to the H-bond length in the anion but not quite as strong. As in the anion, the H-bond is centrosymmetric, with the proton located in the center of the F-F axis. 6.3.3.1 (H 2 OH .. OH 2 ) + The cationic analogue of (HOH..OH)-- is (H 2 OH .. OH 2 ) + , in which two water molecules are held together by a fifth proton. The nature of the geometry in this complex, like the anion, has generated a good deal of theoretical study. Early studies at the SCF level, and with small basis sets, had indicated the complex contains a short, centrosymmetric Hbond196-200. This question has been probed by much higher levels of calculation recently201, 202 and it appears that the previous lower-level work was largely correct. More precisely, the bottom of the proton transfer potential appears to be only very slightly asymmetric if at all; the barrier, if it exists, is less than 0.5 kcal/mol. As in the (HOH..OH)-- anion, the character of the potential alternates from single to double well upon small changes in level of theory, the former being favored by electron correlation. The calculations, employing CCSD(T) treat-

Table 6.11 Properties of cations, computed at MP4SDQ/6-31 +G(d,p) level195. Complex (HFH..FH)+ (H2OH..OH2)+ (H3NH-NH3)+ (HC1H-C1H)+ (H 2 SH--SH 2 ) h (H3PH-PH.,) ' a

- Ea (kcal/mol)

R(A)

rb (A)

31.9 33.0 23.6 12.2 10.5 7.3

2.29 2.48 2.85 3.16 3.75 4.18

0.0 0.0 0.37 0.0 0.52 0.70

Includes ZPVE correction. Displacement of proton from H-bond midpoint.

b

Weak Interactions, Ionic H-Bonds, and Ion Pairs

3 17

ment of correlation, in conjunction with a TZ2Pf basis set, predict R(O..O) to be some 2.40 A; the H-bond is close to linear, but not quite, with a (O--H--O) angle of about 173°201. The binding enthalpy is calculated to be 33.4 kcal/mol, agreeing rather nicely with experimental estimates in the 32-33 kcal/mol range203"205. An MP2 treatment of correlation178, with a 6-311 + +G(d,p) basis set yielded a binding enthalpy of 35.8 kcal/mol; this overestimate is likely due to the failure to remove BSSE. It is notable that this particular calculation indicated a centrosymmetric H-bond, as did other comparable calculations202'206, further evidence of the sensitivity of this question to the particular theoretical approach. A later study of the anharmonicity of the OH vibration involving the bridging proton202 indicates that this effect would reduce the computed binding enthalpy by some 0.5 kcal/mol. Other calculations have tested whether the complex pairing H3O+ with H2O contains a single H-bond or more than one207. Whereas the single bond does appear to be preferred in the neutral water dimer, the question is different when one of the two species is charged. It is conceivable that the ion-dipole interaction might favor the closer approach that is possible in the double (D) or triple (T) H-bonds illustrated in Fig. 6.12 as the intervening hydrogen gets "out of the way" of the two oxygens. (These interactions can also be described as bifurcated and trifurcated, respectively.) At the MP4/6-31G* level, the single H-bond is in fact preferred by 8.0 kcal/mol over the D configuration, which is in turn favored by 4.8 kcal/mol over T. Moreover, R(O-O) elongates from 2.456 A in S to 2.527 A in D and 2.633 A in T, consistent with the progressive weakening. In fact, the D geometry is not a minimum on the potential energy surface, but represents instead a transition state on the path of interchange of one bridging hydrogen in S to another. 6.3.3.2 (H3NH..NH3)+ The nitrogen-containing cation, (H3NH--NH3)+, unlike the others, does appear to have a legitimate double-well potential. Early studies at the SCF level208'209 indicated that R(N-N) should be about 2.81 A, and that the potential for proton transfer contains a pair of equivalent wells, separated by a barrier of 3.8 kcal/mol. The minimum in the potential has the proton displaced by 0.28 A from the N..N midpoint. A preliminary study210 was inconclusive as to whether electron correlation would raise or lower this barrier: A generalized valence bond approach increased the barrier while a POL—CI treatment yielded a lowering, although both approaches suggested the H-bond is lengthened by correlation. M011er-Plesset treatments yielded a barrier of less than 1 kcal/mol, suggesting it might lie below the first vibrational level211. H-bond strengths in the 22-31 kcal/mol range were obtained from these calculations. MP4 calculations195 confirmed that the barrier in this complex is probably in the neighborhood of 1 kcal/mol. After zero-point vibrational energies were added to the interaction energy, a value of —23.6 kcal/mol was obtained, in nice agreement with an experimental enthalpy of —25 kcal/mol212. The effect of correlation was made more explicit

Figure 6.12 Candidate geometries pairing H 3 O + with H2O, S, D, and T refer to single, double, and triple H-bonds, respectively.

318

Hydrogen Bonding

in a later study213 where a contraction of R(N .. N) by 0.085 A was observed; the MP2/631G* value is 2.731 A. The noncentrosymmetry of the complex, as measured by the deviation of the proton from the N..N midpoint, decreased from 0.35 to 0.25 A. Applying MP4SDQ to this geometry, along with a larger 6-311 + +G(d,p) basis set, so as to obtain vibrational frequencies, the binding enthalpy of the (H 3 NH .. NH 3 ) + complex was calculated to be —24.9 kcal/mol at 298 K (no BSSE correction was applied)213. Another study computed a AH of —27.1 kcal/mol with the same basis set at the MP2 level178, with this value becoming slightly less attractive, —25.5 kcal/mol, when the level was raised to MP4/6311 + +G(2d,2p). As in the case of H5O2+, H 7 N 2 + also prefers a single linear Hbond207,214. The double (or bifurcated) H-bond, analogous to D in Fig. 6.12, is higher in energy than the single bond by 7.4 kcal/mol. 6.3.3.3 Comparison with Second-Row Analogs Calculations have addressed the second-row analogs of the aforementioned cations. An early study of (H2SH..SH2)+ indicated a noncentrosymmetric H-bond. At the SCF level the intersulfur distance was computed to be 3.48 A215, and the proton is displaced 0.28 A from the S..S midpoint. A barrier of 0.6 kcal/mol was predicted to separate this minimum from its equivalent following proton transfer. Without BSSE correction or correlation effects, the binding energy was estimated to be 17 kcal/mol. Correlation was later introduced into this complex, as well as the other second-row cations216 and resulted in an improved binding energy. The enthalpy computed at the MP4SDQ/DZP level is 12.9 kcal/mol, in the range of experiment of 12.8 — 15.4 kcal/mol203, 217. The noncentrosymmetry of the proton position diminishes from 0.51 to 0.36 A upon adding correlation. The binding enthalpy of the (HQH..C1H)+ complex was computed to be about 2 kcal/mol stronger, while that of (H3PH..PH3)+ was found weaker by 4-5 kcal/mol. In each case, correlation tends to bring the bridging hydrogen atom closer to the H-bond midpoint. It might be noted parenthetically, however, that the single H-bond studied for (H3PH..PH3)+ may not in fact be the most stable. Calculations have suggested that a trifurcated geometry, similar to the T structure in Fig. 6.12, may be preferred to the single Hbond by as much as 2 kcal/mol207. This result suggests that the interaction should perhaps be considered less of a H-bond and more of a simple ion-dipole interaction. The R(P..P) distance is only 3.64 A in the trifurcated arrangement, as compared to 4.17 A in S where the intervening hydrogen holds the two P atoms apart. 6.3.4 Comparisons between Cations and Anions Table 6.11 presents the results of a systematic study of the proton-bound cationic dimers, all calculated at a fairly high level. These data may be compared with Table 6.8 to obtain some insights about the cations and their corresponding anions. There are some clear patterns that emerge. Considering first the H-bond lengths, as the electronegativity of the atoms diminishes, F > O > N, the bond elongates rather quickly. The two F atoms can come as close as 2.3 A to one another, while R(O..O) is approximately 2.45 A, and the internitrogen distance is just under 3 A. The F and O ions contain centrosymmetric H-bonds, whereas the NH .. N bond is noncentrosymmetric (although the barrier separating NH .. N from N .. HN is probably less than 1 kcal/mol). In the case of second-row atoms, the C1..H..C1 bond is centrosymmetric. The anionic (HSH .. SH) complex probably is as well, while the cationic (H 2 SH .. SH 2 ) + apparently contains a double-well potential. It is emphasized that the spe-

Weak Interactions, Ionic H-Bonds, and Ion Pairs

319

cific nature of the proton transfer potential, that is, single or double-well, height of energy barrier, and so on, is very sensitive to particular level of theory so results can vary widely from one study to the next, even with nominally similar approaches. At any rate, most of these systems appear to have a potential which is quite broad at its bottom, and a harmonic treatment is prone to errors. With regard to the strengths of these ionic H-bonds, (FH..F)--- contains the strongest, nearly 40 kcal/mol. Its cationic analog is considerably weaker, with a binding energy of 32 kcal/mol. The same is true for the second-row analogs where (C1H--C1)" is more strongly bound than is (HC1H..QH)+. The situation reverses for the oxygen-containing systems where it is the cationic (H2OH-OH2)+ that is more strongly bound than is (HOH .. OH) -- , by some 10 kcal/mol (as well as for the S analogs). The greater strength of cationic versus anionic H-bonds containing O-bases has been observed in comprehensive experimental surveys177, 184. This latter pattern holds for N as well where (H3NH..NH3)+ is much more tightly bound than (H 2 NH .. NH 2 ) -- . The P-containing anion is probably not bound at all, as noted in Table 6.8. It is emphasized that the H-bonds in the second-row ions are considerably weaker than the first-row species. Nonetheless, even in such cases, the ionic character of the complexes leads to interaction energies that tend to be larger than many of the strongest neutral H-bonds. The comparisons in bond strength between cation and anion are highlighted in Table 6.12 which illustrates that the aforementioned patterns are true at SCF as well as correlated levels218. Correlation tends to add to the binding energy, particularly in the case of secondrow atoms. MP2 appears to provide results close to MP4 in most cases. MP4 results are illustrated in Table 6.13 to nearly coincide with coupled cluster including triples219. (FH-F)~ is apparently the strongest H-bond, with an interaction energy of some 44 kcal/mol. Many of the earlier calculations, up to 1987, have been compiled along with experimental H-bond enthalpies220. Although somewhat dated, as most of the calculations are limited to SCF level with fairly small basis sets like 4-31G, many of the trends are illustrative of what is seen at higher levels of theory. 6.3.5 Alkyl Substituents Methyl substitution appears to have little influence upon the cationic complexes. For example, R(O..O) in (CH3OH)2H+ is longer by only 0.01 A than in (HOH)2H+221. In both cases, the H-bond is centrosymmetric. This insensitivity to methyl substitution has been confirmed by experimental measurements222: The binding enthalpy of (CH3OH)2H+ is within experimental error of that of (HOH)2H+. A small decrease is noted when both hy-

Table 6.12 Comparison of binding energies (kcal/mol) of cations and anions. Data uncorrected for BSSE218. Complex

SCF

MP2

MP4

(FH..F)(HFH..FH)+ (C1H..C1)(HC1H .. CIH) 1 (HOH-OH) (H2OH..OH2) '

42.5 29.9 15.0 8.2 23.9 31.2

45.3 33.2 21.8 15.5 28.3 35.9

44.6 33.1 20.4 14.4 27.7 35.2

320

Hydrogen Bonding

Table 6.13 Comparison of correlated binding energies (kcal/mol) of cations and anions. Data, uncorrected for BSSE, calculated with aug'-cc-pVTZ basis set219. Complex

MP4

CCSD+T(CCSD)

(FH-F)(HFH..FH)+ (HOH..OH)(H2OH..OH2)+ (H2NH..NH2)(H 3 NH .. NH 3 )+

43.3 33.1 26.2 33.8 13.9 26.4

44.2 33.1 26.9 33.8 14.1 26.3

drogens of HOH are replaced by methyl groups as in (CH3)2O)2H+. Similar results have been noted with the NH +.. N interaction. As (H3N)2H+ is changed to proton-bound methylamine, dimethylamine, and trimethylamine dimers, the binding enthalpy decreases slowly from 26 kcal/mol for (H3N)2H+ to 22 kcal/mol for ((CH3)3N)2H+222. When various symmetric substitutions are made on the pyridinium-pyridine complex, there is little change observed in the internitrogen separation (2.5 A), even though these substitutions account for a span of over four pK units in the monomers223. Nor do these substitutions have much effect upon the proton transfer barrier calculated at the HF/6-31G level, which lies between 18 and 20 kcal/mol for all complexes. 6.3.6 Other Considerations Strong ionic H-bonds can be formed also when the O atom is engaged in a double bond. The proton-bound acetaldehyde dimer, for example, is computed to be bound by 28 kcal/mol at the SCF/4-31G* level224. R(O..O) is equal to 2.52 A, and the H-bond is noncentrosymmetric at this level, but the two wells in the proton transfer potential are separated by a barrier of only 1.7 kcal/mol. This barrier will likely vanish entirely when correlation is added. Similar results were obtained for proton-bound formaldehyde dimer225, with R(O .. O) = 2.51 A and a transfer barrier of 1.4 kcal/mol226. Replacing one hydrogen of each formaldehyde by F leads to a slightly deeper pair of wells, separated by a barrier of 2.4 kcal/mol226. But again, these barriers are likely to disappear when correlation is included. The combination of a pair of carboxylate anions and a proton also results in a potential that has two minima at the SCF level, but a centrosymmetric H-bond when correlation is included187. The cationic analog on the other hand, pairing two neutral formic acids and a bridging proton, appears to retain a double-minimum potential even in its MP2 transfer potential, for which R(O..O) is shorter by 0.05 A than in the SCF-optimized structure227. The strengthening of H-bonds that occurs when one of the two partners is charged can be sufficient to form H-bonds between C atoms. For example, a recent crystal structure228 revealed that a proton can bind together the C atoms of a pair of imidazole species. The Hbond is apparently noncentrosymmetric with r(C—H) distances of 2.03 and 1.16 A; the bond is within 8° of being linear. There are certain manifestations of H-bonds which are either centrosymmetric or in which the transfer potential contains a broad minimum, or a pair of minima separated by a low barrier. In such cases, the proton position can be easily shifted by external influences and its vibration extends over a wide amplitude. Consequently, the infrared spectrum con-

Weak Interactions, Ionic H-Bonds, and Ion Pairs

321

tains a very broad, intense band. The "proton polarizability" can be two orders of magnitude larger than would be the case in more traditional H-bonds223,229 233. One reason for the large changes in moment associated with displacement of the proton along the H-bond is that the electron density tends to shift in the direction opposite to proton motion, thus amplifying the entire effect234,235.

6.4 Asymmetric Ionic Systems

Now let us consider how the H-bond energy might be expected to vary when the two partners are chemically different species. This sort of bond, where A and B of AH...B are different, is referred to as an "asymmetric" H-bond. The reader should be wary of the literature, however, where the use of this term can be ambiguous as it often refers to a "noncentrosymmetric" H-bond as well. Taking the proton bound ammonia dimer as an example, this H-bond would be considered symmetric in that the two partners are identical, and the proton transfer potential is similarly symmetric. The proton transfer potential contains a pair of minima, corresponding to (H 3 NH +... NH 3 ) and (H 3 N ...+ HNH 3 ), in each of which the bridging hydrogen is closer to one N atom than to the other. This noncentrosymmetric geometry is sometimes given the label "asymmetric" in the literature. To reword this description, the proton transfer potential in any symmetric H-bond, containing a pair of identical partners, must by definition also be symmetric (provided there are no geometric restraints that impose an asymmetry into the system). The H-bond can be centrosymmetric if the profile contains a single, central minimum, or noncentrosymmetric when there are two equivalent minima present, separated by a local maximum when the proton is equidistant between the two partners. 6.4.1 General Principles As a starting point for our discussion, consider the proton-bound dimer between A and B0. If these two species have the same proton affinity, the AH+ + BO pair would have the same energy as A + H+BO, as depicted at the top of Fig. 6.13. The stabilization achieved when the two interact to form a H-bond, A...H+BO, is indicated by the arrow labeled Eo in Fig. 6.13. (We assume for the moment that the H-bond would be noncentrosymmetric with the proton closer to B0 than A.) Now we change BO to a more basic molecule, B 1 .Because the latter has been defined to be more basic, H + B 1 is more stable than H + B 0 , so the energy of the A + H + B 1 pair would be lower than that of the original A + H+BO. As illustrated in Fig. 6.13, this greater stability is manifested also in the complex, with A ... H + B 1 more stable than A ... H + B 0 . The net result is that the more basic B1, forms a stronger H-bond with AH + , as expected. That is, E1 is greater than Eo. The same trend continues as B becomes progressively more basic as B2, and so forth. In summary, the H-bond formed between AH+ and B becomes progressively stronger as the basicity of B increases. The reader may have noted that the formation of the H-bonds in Fig. 6.13 has involved also the transfer (or at least partial transfer) of the proton from AH+ to the base within the complex. That is, since B is a stronger base than A, A ... H + B is preferred over AH +... B. It is thus of interest also to consider the H-bond energy of each complex from the perspective of the "other" reactants: A + H + B. The arrows labeled E' in Fig. 6.13 refer to the reaction A + H+B A...H + B where B serves as proton donor rather than A. It is evident that the stabilization of the complex arising from the increasing basicity of B is not as large in

322

Hydrogen Bonding

Figure 6.13 Schematic diagram of the H-bond energies arising from formation of proton-bound dimers of A with Bi.

magnitude as the stabilization of H+B itself. As a result, the arrows shorten as B becomes more basic; that is, less acidic: E'2 < E' l < E'O. In other words, the H-bond energy diminishes as the acidity of H+B drops. This trend is fully consistent with the pattern discussed above for AH+ + B where A donates the proton to the H-bond. To encapsulate the qualitative trends illustrated in Fig. 6.13, the H-bond will be energetically strengthened if either (i) the proton acceptor becomes more basic, or (ii) the acidity of the donor increases. These rules will be in force whether or not the formation of the H-bond incorporates the partial transfer of the proton from one group to the other within the context of the complex. Finally, it is worth stressing that one must be careful about the precise meaning of the H-bond energy of a complex such as A ... H + B. It is clear from Fig. 6.13 that E'1 differs from E1 ,for example. In other words, the H-bond energy is quite different, depending on whether one takes as reactants AH+ + B1 or A + H + B 1 .(In fact, the discrepancy between these two measures of the H-bond strength is equal to the difference in proton affinity between A and B1.) 6.4.1.1 Quantitative Relationships At this point, it is useful to inquire into any quantitative relationship between the H-bond energies in Fig. 6.13 and the proton affinities of the two partners. Numerous experimental studies have indicated a strong correlation between these two quantities157,184,236 238. The relationship usually tested is a linear one where the H-bond enthalpy varies as

where the slope m is less than unity, typically in the range 0.2 — 0.4239 244. The proton affinity difference, PA, which obeys this relationship is fairly wide, extending over a range of perhaps 60 kcal/mol239. H-bond types considered include ionic interactions between N, 0,C, and S bases184,239 242.

Weak Interactions, Ionic H-Bonds, and Ion Pairs

323

A number of concepts have been proposed over the years to explain the linearity of this relationship, and deviations therefrom. Probably the most popular is due to R. A. Marcus whose original ideas were developed for electron transfer and then extended to proton transfer reactions245 249. These ideas apply primarily to the height of the barrier for these reactions and how it is influenced by the exothermicity of the overall reaction. But one can certainly imagine how "flipping the energy profile upside down" could allow the same notions to apply to the depth of energy minima. In fact, the idea of applying Marcus theory to systematize the well depths of H-bonded complexes has been described in the literature on a number of occasions250 253. The concepts behind this approach are summarized in Fig. 6.14 which "fills out" the energy profiles for some of the reactions in Fig. 6.13. The symmetric system, that in which the proton affinities of A and Bo are identical, is taken as a base point, and the depth of the energy well, Eo, is denoted an "intrinsic" well depth. As Bo is changed to a stronger base B 1 ,the increase in its proton affinity is described by PA and the new proton transfer profile is represented by the dashed curve in Fig. 6.14. The Marcus formulation relates the new well depth ( E1) to the depth in the "unperturbed," (that is, symmetric) system, Eo, and the change in basicity of B, PA:

This equation is derived from a number of simplifying assumptions254. In summary, it is presumed that each profile in Fig. 6.14 can be constructed by a pair of inverted parabolas. The energy of the total system is taken to be that of the first parabola, but then switches

Figure 6.14 Profiles for the transfer of a proton from A to B0 and from A to B1.

324

Hydrogen Bonding

abruptly to the second parabola at the point that they intersect. The increased basicity of the base B is incorporated into the model by lowering the second parabola relative to the first, which in turn lowers their point of intersection and increases E1 relative to EO. One of the prime objections to this prescription is that it would lead to an incorrect shape for the transfer profile, namely a sharp "cusp" rather than the rounded bottom. Nonetheless, the approach has seen some surprising successes. 6.4.2 Test of Quantitative Relationships The accuracy of Eq (6.2) was rigorously tested using ab initio calculations in 1986 by taking as a starting point the proton-bound water dimer255 where A and Bo in Fig. 6.14 are both modeled by H2O. Eo then becomes the computed energetics of Reaction (6.3):

The proton affinity of the base B was enhanced by replacing H2O by CH3OH, then by C2H5OH, and by (CH3)2O. With the use of a 4-31G basis set, each of these substitutions leads to respective increases in proton affinity of the O atom, relative to H2O, by 16.8,21.3, and 26.4 kcal/mol256. The binding energies computed for each of these bases with H 3 O + , also at the SCF/4-31G level, are reported in Fig. 6.15 as the three data points. It is evident that these binding energies are predicted quite well by the Marcus Eq (6.2), represented by the solid curve in Fig. 6.15. Similar computations were carried out with the nitrogen analogs

Figure 6.15 Energetics of binding of H3O + to various alkylated O bases. The data points were computed explicitly and the solid curve is derived from Equation (6.2)255.

Weak Interactions, Ionic H-Bonds, and Ion Pairs

325

of Reaction (6.3), pairing NH4+ with various alkylated amines, and the results again confirmed the accuracy of Eq (6.2). It might be noted that the solid curve in Fig. 6.15 is nearly a straight line, even though the proton affinity difference extends over a 30 kcal/mol range. This approximate linearity is consistent with the experimental gas-phase findings mentioned above. Making use of Eq (6.2), the linearity can be traced to the fairly large value of the binding energy of the symmetric system, Eo, which reduces the impact of the third (quadratic) term in the equation. As an example, even at the far right of Fig. 6.15, where PA is equal to 30 kcal/mol, the third term is equal to only 1.3 kcal/mol, as compared to 15 kcal/mol for the second, linear term. This approach would lead one to expect a much less linear relationship between Hbond energy and the basicity difference of the two partners when the former quantity is smaller in magnitude, that is, for weaker H-bonds. It should be stated parenthetically that the Marcus formulation is not the only one that could in principle reproduce the patterns of H-bond energy changes precipitated by proton affinity changes. There are other extant theories257 261 that differ from the Marcus equation chiefly in the last term, a rather unimportant one in many cases as the relationship is very nearly linear anyway. Later work attempted to extend the applicability of this sort of analysis to systems which are more asymmetric. For example, instead of altering the degree of alkyl substitution on two O-bases in a H-bond, the atoms directly involved were different, for example, O and N, or O and C262. The agreement was not as good, a fact which might be attributed to the distinction between single and double-well character in the various systems, resulting in geometry mismatches. It might also be noted that computations at a higher level, and extending the interaction energies from the electronic contribution to E to the full enthalpy AH, with allowances made for zero-point vibrational energies, confirm the original conclusions. The addition of a methyl group to one end of the nitrogen system, namely CH3NH4+...NH3, reduces AH by 3.5 kcal/mol at the MP2/6-311 + +G(d,p) level178, relative to NH 4 +... NH 3 , as compared to a reduction in Eelec of 3.7 kcal/mol for SCF/4-31G256. We finally address a fundamentally interesting question that may best be understood by way of example. The transfer potential of the proton-bound ammonia dimer contains a pair of minima, (H 3 NH +... NH 3 ) and (H3N...+HNH3), separated by an energy barrier255. As one of the nitrogens, say the one on the left, is alkylated so as to make it more basic, the lefthand minimum becomes more stable than the one on the right. As the process continues and the left minimum is progressively lowered, the right minimum becomes merely a shoulder in the potential and eventually disappears entirely, leaving only a single-well potential: (R 3 NH +... NH 3 ). So beginning with a symmetric system that contains a pair of minima, the potential is transformed into an asymmetric single-well potential when the proton affinity of one base exceeds that of the other by a certain amount. Chu and Ho have attempted to quantify this idea via a parameter, , which is directly related to the difference in protonation energies (PEs) of the two bases A and B:

where B is the less basic of the two. The authors found that when exceeds about 0.08, the collapse from double- to single-well proton transfer potential occurs. In other words, one might expect only a single minimum when the difference in proton affinity is greater than

326

Hydrogen Bonding

8% of the proton affinity of the subunits themselves. However, their data sample was rather small, restricted to carbonyl oxygen atoms as bases. And it is clear that in some classes of systems, one would not expect a double-well transfer potential regardless of the value of . An example is provided by the (R 2 OH +... OR' 2 ) family of H-bonded hydroxyl groups, where even the unsubstituted, symmetric (H2O..H+..OH2) contains a single centrosymmetric minimum.

6.5 Syn-Anti Competition in Carboxylate The carboxylate anion is of particular importance to enzymatic activity as it serves as the functional portion of the Asp and Glu residues in proteins. As described above, there are a number of geometrical dispositions in which the carboxylate can form H-bonds with other groups. Of particular interest is the question as to whether this group will form a stronger H-bond on its syn or anti side, with these terms defined in Fig. 6.16, with respect to the two O lone pairs. From the point of view of an incoming proton donor, with a partial positive charge, the O atoms carry the bulk of the negative charge of the anion. (Indeed, one study places a negative charge in excess of — 1 on each of these atoms in formate263.) The donor might be expected to prefer the syn approach so that it could foster an interaction with both of these atoms. Indeed, the difference in basicity of these two sides of the O atom has been considered to play an important role in general base catalysis264,265, so is potentially an important issue. 6.5.1 Ab Initio Calculations The water molecule has been the most commonly studied proton donor with regard to interactions with the carboxylate anion. A number of modes of H-bonding that one might envision are illustrated in Fig. 6.17. In addition to the aforementioned H-bonds between the two O atoms in the "pure" syn and anti configurations, there is the possibility of the "linear" (lin) geometry, in which the water molecule lies directly along the C—O axis rather than on either side of it. Another candidate is the "bifurcated" structure (bif) in which there are two H-bonds present simultaneously. Although both of these bonds would be bent to some degree, this geometry might be stabilized by the excellent alignment between the dipole moments of water and HCOO . It is important to note the possibility of a smooth transition from one configuration to the next, with only small motions of the water molecule necessary. In other words, the sys-

Figure 6.16 Definition of syn and anti directions in the carboxylate group.

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327

Figure 6.17 Idealized geometries for interaction between a water molecule and the carboxylate anion.

tern can smoothly transition from and to bif, passing through first lin and then syn along the way. It is also worth stressing that each of these categorizations refers not so much to a single structure as to a class of them. That is, the anti designation can be used for any geometry which places the water on that side of the pertinent O atom. One would not expect the C—O . . . O angle to necessarily be precisely 180.00° in lin so the distinction between this configuration and anti or syn is an arbitrary one. An early study of this problem was limited to the SCF level with relatively small basis sets266. The geometries were not fully optimized but were restricted instead to adjustment of the intermolecular distance only. The bifurcated geometry was found most stable, with an interaction energy in the 19-29 kcal/mol range. The lin geometry was considerably less stable; syn and anti were not considered. A later work267 confirmed the stability of the bifurcated geometry, in comparison with syn and anti. Their data suggested that one might reach an erroneous conclusion were one to use a basis set that is too small, like STO-3G. At the SCF/6-31G** level, the bifurcated structure is favored over syn by 2.2 kcal/mol; the preference is increased to 2.7 kcal/mol with MP2 corrections. The electronic contribution to the binding energy of bif is 23.6 kcal/mol at the latter level. Geometries of syn and anti were fully optimized soon thereafter268, within the framework of the 6-31G* basis set (syn and lin were not considered here); bif was favored by 4.6 kcal/mol over anti. In 1988, Hermansson et al.269 computed the energies of nearly 600 points on the PES of this complex at the SCF/DZP level so as to derive an analytical form of the surface. The internal geometries of the HCOO and HOH species were frozen, so the surface was concerned with intermolecular variables only. BSSE was corrected by the counterpoise procedure at each point. Their data confirmed the greater stability of the bif geometry. The geometries of bif and anti were fully optimized and compared to syn, using a 4-31+G* basis set270. Bif was preferred with a binding energy of 20.5 kcal/mol at the MP2 level. The syn and anti geometries were virtually indistinguishable on energetic grounds, both with binding energies of 17 kcal/mol. (The nonbridging water hydrogen was rotated 180° from that pictured in Fig. 6.17 to prevent the collapse of syn into the bif structure.) A more recent MP2 optimization, employing a 6-31 + +G**(d,p) basis set178 confirmed the bif structure as preferred on energetic grounds, favored by 2.8 kcal/mol over anti. This energy difference remains nearly the same when elevated to AH by including zero-point vibrational terms. The Gibbs free energy also favors bif over anti by 1.3 kcal/mol, after factoring in entropic factors, although this difference diminishes to only 0.3 kcal/mol when the temperature is raised to 500° K. When paired with ammonia as proton donor instead of water, the syn and anti structures are equal in energy to within 1-2 kcal/mol 271 .

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Hydrogen Bonding

The conclusion from these various ab initio calculations is that a water molecule would prefer to approach the carboxylate anion so as to form the bifurcated geometry. If this structure is distorted by stretching one of the two H-bonds and pulling the water toward a lone pair of the other oxygen, so as to form the syn geometry, this deformation would represent an energetic cost of perhaps 3 kcal/mol. The syn structure per se, (not the bifurcated arrangement) is very similar in energy to the anti geometry in Fig. 6.17, in which the water molecule approaches the O atom from the other direction. 6.5.2 Experimental Findings The theoretical conclusions are counterpointed by experimental data. There is some indication that a linear complex versus the bifurcated structure is favored in the gas phase184 although this preference is based on entropic rather than energetic factors. A survey of crystal structures in which a carboxylate group interacts with various metals272 finds 63% of the interactions are with the syn lone pair, as compared to 23% with anti (the remainder fall into the bif category). However, the interactions between carboxylate and metal ions like potassium and copper would be expected to be dominated by electrostatics in contrast to the H-bonds when water acts as the partner. Moreover, it is difficult to isolate the intrinsic preferences from the perturbing influence of crystal packing forces. Indeed, a survey of protein structures which focused on true H-bonds273 reached the conclusion of a weakened preference for syn. While the ratio of syn to anti H-bonds of the Glu carboxylate group was significantly greater than 1 (57:43), the same quantity was within statistical error of unity for Asp. Together, the syn:anti ratio for all groups examined is 53:47. Another survey, this time focusing on H-bonds to small-molecule carboxylate groups274, was able to account for and remove crystal packing forces. The authors found no statistical preference for syn over anti H-bonds (in the absence of steric interference) over a database spanning 876 interactions. It would hence appear that crystal studies confirm the computational finding that there is little energetic distinction between H-bonds to the syn and anti sides of the carboxylate oxygen atom. 6.5.3 Carboxylic Group If there is no strong distinction between the syn and anti sides of the carboxylate oxygen atom with regard to formation of a H-bond, one might wonder what would happen when the hydrogen is much closer, within range to form a covalent bond. Calculations have shown that syn carboxylic acids (the Z-rotamers) are more stable than anti (E), by some 5-7 kcal/mol in the case of HCOOH270,275,276, consistent with microwave data that indicate an energy difference of 4 kcal/mol277. It is in this context that the syn lone pair of HCOO may be said to be more basic. This greater stability may be due to a favorable alignment of the C=O bond dipole with that of the O—H bond in the acid, since the Z-rotamer has a much smaller molecular dipole moment than E276,270. Similar arguments relating to alignments of bond dipoles have been invoked in explaining anomalous acidities of esters278,279. 6.5.4 Solvent Effects In contrast to these calculations and microwave information, there have been numerous studies of model systems that question the magnitude of the preferential stabilization of the syn geometry280 282, which does not appear to exceed 1 pKa unit 283 285. It was conjectured that perhaps the resolution of the discrepancy resides in solvent effects282 since the equi-

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librium ratio of E to Z isomers of primary amides is known to be strongly solvent dependent286,287. This premise was confirmed by later calculations that indicated that the difference in energy between the syn and anti geometries of acetic acid is reduced from 6 to 1 kcal/mol when the molecule is taken from the gas phase into aqueous solvent288,289. This result can be taken as a simple consequence of the much larger dipole moment of the anti geometry as compared to syn (about threefold), which is subject to proportionately larger stabilization in solvent. Thus, while there is a clear energetic preference for the syn (Z) conformer of carboxylic acids in the gas phase, this preference is reduced or eliminated altogether when in aqueous solvent. 6.5.5 Resolution of the Question Let us return now to the gas phase where the syn conformer of the carboxylic acid is favored. Is this observation in contradiction to the finding that there is no real preference for syn or anti H-bonding of the carboxylate? The answer to this question is negative because there is a difference between the two phenomena274. That is, just because a proton has a preference for the syn side when it has formed a covalent bond with the O atom in question does not necessarily indicate that the same side will be preferred when the proton is more distant, as in HCOO ...HA. The distinction can perhaps best be explained for a simpler carbonyl group, as in formaldehyde. A proton would be added to the O atom so as to yield a C — O — H angle in the neighborhood of 110° due to the hybridization of the O atom. However, as the proton is pulled away, there would be a tendency for the molecule to rotate so that the C=O bond dipole can point toward the positive charge developing on the proton acceptor, leading to a C—O . . . H angle closer to 180°290 292. An example of how this principle would apply to the carboxyl group is illustrated in Fig. 6.18270,271. While the syn geometry of a carboxylic acid is intrinsically more stable than the anti, if the proton extracted by carboxylate comes from a neutral molecule, the resulting —COOH acid will be paired with an anion. As indicated in Fig. 6.18, a destabilizing electrostatic interaction can occur with the partially negatively charged O atom when the anion is in the syn position, but not if it is anti. This destabilization can dampen, or negate entirely, the intrinsic energetic preference of syn versus anti carboxylic acids. This concept

Figure 6.18 Illustration of the potential destabilizing electrostatic interaction between the carbonyl O atom of the carboxylic acid and the anion in the syn position (S). No such destabilization occurs if the anion is anti (A).

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Hydrogen Bonding

was further elaborated and shown to be in operation in some form regardless of whether the carboxylic acid interacts with an anion or a neutral molecule293. In summary, there is little distinction between the syn and anti lone pairs of the carboxylate oxygen atom with regard to forming H-bonds with proton donors. When a proton has approached closely enough to form a covalent bond, the syn position is favored, but only in the gas phase. The syn and anti conformers of the carboxylic acid are close in energy in aqueous solution. Even in the gas phase, the preference for the syn configuration of the isolated carboxylic acid can be eliminated when it forms a H-bond, due to more favorable electrostatic interactions between the partner and the anti geometry of the carboxyl.

6.6 Neutral Versus Ion Pairs As the acidity of the proton donor AH increases, and there is a concomitant enhancement in the basicity of the acceptor B, the H-bond will become stronger. But it also stands to reason that for a strong enough pair of acid and base, the proton can simply transfer across from A to B, converting the system into an ion pair:

An important force working against such a transfer is the energetic difficulty of separating charge, so such a transfer will occur only for the strongest acid-base combination. On the other side of the equation, once formed, the ion pair will be held together by a powerful electrostatic ion-ion attraction, a strong consideration in its favor. In addition, moving the complex from complete isolation in the gas phase into an environment where the medium can interact with the system to help stabilize the charges will clearly favor the ion pair. This bias will become stronger as the polarity of the surrounding molecules or solvent increases. 6.6.1 Amine-Hydrogen Halide The question as to what it takes to generate an ion pair (sometimes referred to as a "salt bridge"294) has generated a good deal of inquiry in the literature2. Much of this work has focused on the hydrogen halides as the strong acids, and the amines as the requisite strong base. Indeed, a very early ab initio study of a proton transfer reaction was directed at the possibility that H 3 N ... HC1 might spontaneously undergo the reaction to form H3NH+... C1295, although a somewhat later calculation disputed this prediction296. A later and more thorough examination as to how the nature of the complex is affected by the level of theory297 illustrated that certain "mid-range" basis sets are prone to error in this matter. The minimal basis sets tested, with and without polarization functions, indicated it is only the neutral pair that is present. This result is confirmed by more flexible sets, containing multiple sets of polarization functions. In contrast, application of the 3-21G basis set suggested it is the ion pair that is favored in the gas phase. Addition of diffuse functions leads to the same conclusion, whereas inclusion of polarization functions on non-hydrogen atoms points instead to the neutral pair. Another split-valence set, MIDI-1, shows the same pattern; still another, 4-31G, indicates two separate minima in the potential energy surface, one corresponding to the neutral pair, the other to the ion pair. This set of calculations297,298 underscored the necessity to use an appropriately flexible basis set in examining the relative stabilities of the neutral and ion pairs. Interestingly, inclusion of electron correlation via MP2 does not change the central conclusion of a neutral pair for this complex298.

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33 I

Since H3N...HC1 exists only as a neutral pair and HF is less acidic than HC1, it is not surprising that H3N...HF, too, does not form an ion pair299. Similarly for H3P...HC1 and H3P...HF, as H3P is less basic than H3N300. The latter set of calculations is particularly significant as it included the effects of electron correlation. The list of complexes found to be neutral-pair only at the correlated level was later extended to H3P...HBr, H3As...HCl, and H3As...HBr301. Other systems examined at the SCF level, and failing to show evidence of an ion pair, were H 3 N ... HBr and H3As...HF302. If HC1 is not acidic enough and NH3 not basic enough to form an ion pair, it was thought that perhaps HBr might be sufficiently acidic and/or CH3NH2 basic enough. The potential energy surfaces of NH3 and CH3NH2 paired with HC1 and HBr were scanned with a DZP type basis set at the SCF level303. The data confirmed the lack of an ion pair for H3N...HC1. The surface for CH3NH2...HC1 is somewhat flatter, but again contains only a single minimum corresponding to the neutral pair. This result contradicted an earlier SCF/4-31G finding of a CH3NH2H+... C1 ion pair304, reemphasizing the sensitivity to basis set. The surface for H3N...HBr303 was similar, but did contain a hint that there might perhaps be a second minimum for the ion pair. However, if it did, this minimum would probably not be deep enough to sustain a vibrational level, and would lie some 4 kcal/mol higher in energy than the neutral pair. In the case of CH3NH2...HBr, on the other hand, the surface contained two clear minima. The neutral and ion pairs are of comparable energy (the ion pair is favored by 1.2 kcal/mol) and both appear deep enough to contain at least one vibrational level. In contrast to the neutral pair for HC1 + methylamine, combining the same acid with the more basic trimethylamine does appear to produce an ion pair at the MP2 level305. The acidity was turned up one notch when NH3 was paired with HI306. This study included the effects of electron correlation. In agreement with prior results, the surfaces of both H3N...HC1 and H 3 N ... HBr contain only a single neutral-pair minimum. H 3 N ... HI, on the other hand, exists both as the neutral and ion pairs. Whereas the neutral is favored by 5 kcal/mol at the SCF level, the two types of complex are nearly equal in energy when correlation is added. On the other hand, zero-point vibrational contributions to the energy are quite distinct for the two minima. Whereas this effect destabilizes the neutral pair by 1.6 kcal/mol, the ion pair is raised in energy by 5.6 kcal/mol, leading to the near disappearance of the minimum in the potential. It is concluded unlikely that H3NH... I would actually be observed. Because of the possible occurrence of two minima in CH3NH2...HBr, this complex was the subject of further scrutiny, this time including electron correlation307. The calculations confirmed the possibility of two minima at the SCF level; moreover, the earlier hint of a possible second minimum for H 3 N ... HBr was verified here. The data suggested further that the relative energies of the neutral and ion pairs are really quite sensitive to the details of the basis set. But the most important finding of this set of calculations is that correlation can cause the collapse of a double-well potential to one with only a single minimum. The MP2 proton transfer potential in either case contains a single broad minimum. Changing the base from H3N to CH3NH2 shifts the bottom of this well slightly away from the Br and toward the N. This trend opened up the possibility that it is perhaps incorrect to look for either the classical neutral or ion pairs in all cases; the transition may in fact be a gradual one. In other words, as the base and acid are strengthened, there may be some point where the structure would be better described as A ..+ H--B, with the proton somewhere near the middle of the bond. The calculations have demonstrated that the correlated potential energy surface of this sort of system typically contains a single minimum, and that the position of this minimum

332

Hydrogen Bonding

shifts progressively from acid to base as the strength of the two increases. A comprehensive examination of the strongest HX acids was completed, with the full spectrum of methylated amines from H3N to (CH3)3N308. HI forms what can best be characterized as an ion pair with all of the amines. The only exception is with the weakest base, H3N, where the surface contains not a well-defined minimum, but rather a long shallow valley connecting H3NH+... I with H3N...HI. In other words, the proton will undergo very large amplitude vibrations between these two configurations and the complex is some mixture of the two. HBr is a weaker acid than HI. Nevertheless, it too forms ion pairs with the more basic amines. In the case of its complex with methylamine, the proton is very nearly equally shared between the N and Br atoms. The less basic H3N is not capable of extracting the proton from HBr, and so this complex is best described as a neutral pair. Table 6.14 contains a current summary of the nature of the complexes of the various amines, coupled with each of the HX acids, as determined by ab initio calculations. HF does not form ion pairs at all, whereas HI forms ion pairs with all (with the exception of the shallow valley in the potential with NH3). In the case of HBr and HC1, one can see the transition from neutral to ion pair as the amine becomes more basic. The aforementioned trend of a gradual shift of the proton from acid to base, taking the complex in stages from the neutral toward the ion pair, can even be observed with a minimal basis set at the SCF level, given enough care. Complexes pairing HF, HC1, and HBr with H3N and its mono-, di-, and trimethyl derivatives were constructed and studied first in the gas phase309. Despite the use of a minimal basis set, the computed basicities of the amines were in surprisingly good agreement with experimental protonation energies. Each substitution of a hydrogen atom of the amine by a methyl group adds an increment of 5-9 kcal/mol to the protonation energy; similar steps were calculated for the deprotonation energy of the series HBr, HC1, HF. Each increment in acidity of HX or basicity of the amine pushes the bridging proton a little closer to the base in the complex. The progressive shift of the position of the single minimum was reaffirmed in a set of correlated calculations at the MP2/6-31+G(d,p) level310. A single acid, HC1, was paired with a series of nitrogen bases, in this case 4-substituted pyridines. As the pyridine became more basic, the Cl—H bond was progressively elongated, until eventually the proton's equilibrium position was approximately equidistant between the Cl and the pyridine. Further enhancement of the basicity led to a full transfer of the proton, forming an ion pair, as may be seen in Fig. 6.19.

Table 6.14 Character of complexes pairing hydrogen halides with amines, as determined by ab initio calculations.a

NH3 MeNH2 Me2NH Me3N a

HF

HC1

HBr

NP NP NP NP

NP NP ? IP

NP 50% IP IP

HI

NP

IP valley IP IP IP

NP = neutral pair; IP = ion pair; 50% indicates equilibrium proton position about halfway between N and halide atoms. ? indicates the situation is still questionable.

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Figure 6.19 Calculated values of the R(C1—H) bond length (solid line) when HC1 is paired with a series of progressively stronger bases, 4-substituted pyridines. Data310 calculated at the MP2/6-31 +G(d,p) level. Also shown (dashed line) is the stretching frequency involving the central proton.

6.6.1.1 Quantitative Measure of Degree of Proton Transfer None of the complexes pairing an amine with a hydrogen halide were computed to be of the pure ion pair variety with a minimal basis set309. However, there were some that were close. More specifically, since the equilibrium position of the proton need not shift precipitously from one atom to the other, but rather can move gradually as the acidity and basicity increase, a proton-transfer parameter was devised to indicate the degree of transfer of the proton from the acid to the base. The quantity was defined as

where r(NH) is the stretch in the N—H bond, relative to the isolated protonated amine, and r(XH) has a similar meaning for the hydrogen halide HX. A negative value of indicates a smaller stretch for HX than for the amine, and is hence a sign of a neutral pair; in the reverse situation of a greater stretch for HX, becomes positive and one can consider the complex to approach an ion pair. A zero value for p denotes equal sharing of the proton. The values calculated for this proton-transfer parameter are shown in Fig. 6.20 as a function of the acidity of A and the basicity of B. More precisely, the horizontal scale is a "nor-

334 HydrogenBonding

Figure 6.20 Proton transfer parameter, , computed for various methylated amines paired with hydrogen halides. The numerical label n on each data point refers to the number of methyl substitutes on the amine N(CH3)nH3_n. The horizontal scale is a normalized difference in proton affinity between the amine and the halide309.

malized proton affinity difference" (NPAD)311 between the two species competing for the proton, the amine (Am) and X .

Since the proton affinity of the halide is greater than that of the neutral amine, NPAD is of negative sign. The less negative is NPAD, the closer in magnitude will be the intrinsic pulls on the proton by the two species. It might first be noted that is less than zero in all cases, indicating that the optimized complex resembles a neutral pair more than it does an ion pair. There is a clear trend in all cases for to become less negative as the number of methyl groups on the amine increases. In other words, the more basic amine increases its pull on the proton, leading the complex closer toward an ion pair. In the case of HF, this effect is negligible: the complex is clearly a neutral pair for all amines and the number of methyl groups has little effect on p. On the other extreme is the set of complexes with HBr. Whereas < —0.3 for H3N...HBr, this parameter rises quickly, reaching almost zero for (CH3)3N...HBr. In other words, the latter complex can be characterized as about halfway between a neutral and ion pair. HC1 is intermediate in its behavior, with less sensitivity of the proton equilibrium position to the basicity of the amine. The data in Fig. 6.20 illustrate an important point concerning neutral and ion pairs. At first blush, one might expect that should approach zero as the proton affinities of the amine

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and the halide equalize. However, that notion would ignore a very important fundamental difference between the two types of complex. Whereas the neutral pair AH ... B is held together by the "standard" H-bond forces, typically in the range of 5-15 kcal/mol, A ...+ HB has a much stronger attractive force consisting of the electrostatic interaction between the cation and anion. As a consequence, B does not have to have so strong an intrinsic proton affinity as does A, in order to equalize the energies of the neutral and ion pairs. Referring to a projection of the HBr plot in Fig. 6.20, this curve would intersect the = 0 axis at approximately NPAD = —0.21. Making use of Eq (6.7), the proton affinity of the base which would be able to equally share the proton with Br would be 235 kcal/mol, 125 kcal/mol smaller than the proton affinity of Br . The latter quantity, then, represents approximately the enhanced binding energy between the entities in an ion pair, as compared to that in the neutral pair, at least for this particular system. This value agrees surprisingly well with an estimate of 113 kcal/mol, representing the purely Coulombic interaction energy betwen a pair of point charges, separated by the experimentally determined R(N ... Br) in the ion pair formed between HBr and trimethylamine312. The reader is reminded that Fig. 6.20 represents uncorrelated calculations with a minimal basis set. Although quantitatively unreliable, the data illustrate useful trends nonetheless. 6.6.1.2 Environmental Effects As indicated, it is possible to preferentially stabilize the ion-pair side of the proton transfer potential by allowing the complex to interact with a polarizable medium. By incrementally raising the polarizability, one can manually fine-tune the relative stability of the ion pair relative to the neutral pair. This lowering of one end of the proton transfer potential relative to the other is very much akin to an adjustment of the acidity and basicity of the partners. When the systems described above were immersed in a dielectric continuum model of a polarizable medium, the equilibrium position of the proton did indeed shift toward the base309. It was possible to raise to the point where it became positive, indicating the complex had more ion pair character than neutral pair. Just as the sensitivity of p to the basicity of the amine increases in the order HF < HC1 < HBr, so too does its sensitivity to the polarizability of the medium. Whereas complexes with HF show little displacement of the proton's equilibrium position as the polarizability of the medium increases, those in which HBr acts as the acid experience a rapid increase in with more polarizable medium. This treatment was later confirmed at the correlated MP2 level in that the H3N...HC1 neutral pair is superceded in stability by the corresponding ion pair when immersed in a dielectric continuum 298. 6.6.2 Carboxyl/Carboxylate Equilibrium The carboxyl group occupies a prominent place in protein structure and function and is commonly taken to be ionized as a carboxylate, particularly when paired in a salt bridge with a base4,294,313,314. Whereas this may be the case in a protein environment, it is not obvious that a —COOH group will donate its proton to a base to form an ion pair in the gas phase. For example, SCF calculations with a polarized 6-31G basis set indicate that neither methylamine315 nor an arginine model 316 is able to extract a proton from acetic acid so as to form an ion pair. On the other hand, there is some indication that correlation might stabilize the ion pair for formic acid + methylamine 317.

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Hydrogen Bonding

This question was explored in some detail by ab initio calculations pairing formic acid with methyleneimine, in which the N atom participates in a double bond318. As illustrated in Fig. 6.21, the two configurations examined place the N atom either syn or anti with respect to the carboxyl group. The calculations were carried out with a 4-31G basis set at the SCF level. (These results were later confirmed at the correlated MP2 level with a 6-31G* basis set319.) The protonation energies of HCOO~ and NHCH2 at the SCF/4-31G level of theory are 360 and 230 kcal/mol, respectively. When adjusted by vibrational and other terms, the calculated values of AH(300 K) are 352.6 and 221.9 kcal/mol, only a little bit larger than the experimental quantities of 345.2 and 214.3 kcal/mol, respectively. Most important for our purposes, the theoretical overestimation is equal to 7 kcal/mol in both cases; consequently, the calculated protonation enthalpy difference of 130.7 kcal/mol is very close indeed to the experimental value of 130.9 kcal/mol. And it is this proton affinity difference which is the key element in considering the question of neutral versus ion pairs. When the intermolecular R(O..N) distance is taken to be 3.0 A, the neutral pair is favored over the ion pair in both the syn and anti geometries. This preference amounts to some 15 kcal/mol for the former configuration and more than 40 kcal/mol for the latter. The particular instability of the anti ion pair can be understood on the basis of electrostatics. Following proton transfer, one of the negatively charged O atoms of HCOO will be turned away from the positively charged +NH2CH2 in the anti geometry, a particularly unfavorable situation, but not in syn where both oxygens will be oriented toward the cation. Other calculations320 have confirmed the preference for the neutral pair, even in the absence of a restriction on R(O .. N). Relaxing the restriction of the H-bond distance, permitting the geometry to be fully optimized, yields a neutral pair as the only minimum on the potential energy surface. As pointed out earlier, permitting the complex to interact with a polarizable medium would preferentially stabilize the ion pair. Using a self-consistent reaction field formalism to model such interactions, the calculations did indeed witness such preferential effects 318. Increasing the dielectric constant of the medium from unity (gas phase) to 2 lowered the ion pair energy of the syn structure to the point where it became competitive with the neutral pair; further increases in made the ion pair more stable. This reversal is an important one as other studies estimate the average dielectric constant in the protein interior as being in the range between 2 and 4321. These results suggest that even small interactions of the complex with the surroundings can produce strong effects on the neutral/ion pair balance. This idea is confirmed by the influence of an inert matrix, described below. Although differing in quantitative aspects, other ab initio calculations using a different model of solvation, based upon the Born equation, verify that interaction with a medium

Figure 6.21 Syn and anti configurations of the complex pairing formic acid and methyleneimine.

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will preferentially stabilize the ion pair in the imine-carboxyl complex, to the point that the ion pair may be preferred322. The same sort of preference for the ion pair has been noted in the complex of formic acid with trimethylamine when a polarizable medium is simulated 323. Still another means of simulating some of the effects of solvation are to introduce an electric field as a perturbing influence upon the H-bonded solute system324. However, even a field as strong as 5 x 107 V/cm is not capable of inducing a second well in the protontransfer potential of the complex between methanesulfonic acid and dimethyl sulfoxide, which prefers to remain a neutral pair. The preceding calculations were carried out at the MP2 level using a 3-21G* basis set. Another model of solvent effects would be to place a number of discrete solvent molecules around the system of interest. Calculations have suggested that the addition of just two water molecules is sufficient to convert the equilibrium tautomer of glycine to its zwitterionic form, containing the COO and NH3+ groups 325. Other calculations indicate little difference between these two approaches, (continuum versus discrete molecules) at least with respect to the (H 3 N .. H +.. OH 2 ) system 326. 6.6.3 Experimental Confirmation During the period when theorists were addressing the question as to the possible existence of ion pairs, there was a good deal of experimental inquiry as well. The neutral pair nature of H3N...HC1 was confirmed in the gas phase by its rotational spectrum327. The intermolecular distance derived from the spectral data was rioted to be in remarkable agreement with an earlier ab initio computation with MP2 correlation 299. The following year saw the measurement of the microwave spectrum of H 3 N ... HBr 328, and this complex too had the characteristics of a neutral pair. Nor are any of the complexes pairing a hydrogen halide with H3P of the ion-pair type. The stronger base trimethylamine was paired with the acid HCN in the gas phase329. Apparently, HCN is simply not a strong enough acid to form the ion pair, even with trimethylamine. The same is true of HF, which is also unable to donate its proton to trimethylamine330. HC1 is another story, however. When this acid is combined with trimethylamine, there is evidence for an ion pair in the gas phase331, particularly from the nuclear quadrupole coupling constants. The data are consonant with the broad minimum in the proton transfer potential, with a minimum near the Cl atom, suggested by correlated ab initio calculations. It is intriguing that when HC1 is paired with (CH3)3P, a neutral pair is the result 332. This point is notable as the proton affinities of (CH3)3P and (CH3)3N are nearly identical. The difference in behavior may be due to the larger size of the P atom as compared to N. As a result, the two ions cannot get as close together should they be formed, and the diminished Coulombic interaction would mitigate against formation of the ion pair. Perhaps the most comprehensive confirmation of the principles arising from the ab initio calculations comes from the series of complexes pairing mono and trimethylamines with HBr and HC1312. The nuclear quadrupole coupling constants and stretching force constants support the idea that increasing acidity of HX or basicity of the amine leads to a smooth increase in the extent of proton transfer from X to N. Both HBr and HC1 appear to form an ion pair complex with trimethylamine, but not with the monomethyl derivative nor with H3N. Formulating an ad hoc measure of the degree of proton transfer, based upon the coupling constants, the authors concluded that the proton is 62% transferred in (CH3)3N...HC1, and 75% in the Br analog333. In agreement with the ab initio calculations, the pair of the strongest acid, HI, with the strongest base, trimethylamine, leads to a product which is

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clearly an ion pair334. H3N...HI, on the other hand, would appear to remain a neutral pair. Work in solution also offers some verification of this sort of behavior. For example, by pairing 1 -methylimidazole with a series of acids with varying pKa, NMR and IR data indicated a gradual shift of the proton from the acid to the base335. Similar properties have been noted in the complex involving acetic acid and pyridine, in a variety of concentration situations336, and also when substituted pyridines are paired with trifluoroacetate in dichloromethane solvent 337. 6.6.3.1 Matrix Isolation Studies The situation in the gas phase, then, appears to be entirely consistent with results obtained with ab initio calculations, particularly those including electron correlation. Matrix isolation studies are germane as well, although the forces between the H-bonded system and the matrix may influence the results. While the geometries of the structures cannot be worked out by rotational spectroscopy in a matrix, a good deal of information is available through the vibrational absorptions. Of particular relevance to the question of the existence of ion pairs is the stretching frequency of the proton donor, vs. Collecting together the spectra of a number of complexes pairing a hydrogen halide with O and N bases 311, Pimentel and coworkers constructed a "vibrational correlation diagram" to illustrate the transition from neutral to ion pair in N2 matrix. When the red shift of the vs mode (actually the percentage shift vs ,/vs ) is plotted against NPAD between the halide anion and the base (see earlier), a deep minimum is noted in the vicinity of NPAD = —0.23. (Since vs/vs is a negative quantity, this minimum corresponds to a sharp increase in the absolute magnitude of the shift.) The behavior of the vibrational correlation diagram was described in terms of the strengths of the bonds to the bridging hydrogen in question. Considering first the neutral pair AH ... B, it is well understood that increasing the acidity of AH and/or the basicity of B produces a stronger H-bond. The latter is accompanied by a stretch of the A—H bond and a lowering of the stretching frequency of this bond. One would expect this pattern to continue as the bond grows stronger. The same principles apply to the opposite ion pair as a reference point. Starting from A ...+HB, an increase in the basicity of A or in the acidity of +HB should yield a displacement of the proton away from B and towards A, accompanied by a red shift in the vs frequency, now associated with the H—B bond. Since vs drops off from either the AH ... B or the A ...+HB end of the continuum, it is not surprising that this frequency will attain a minimum when the bridging proton's equilibrium position lies somewhere around the middle of the A...B bond. It is particularly notable that when the frequency is plotted as a relative red shift, vs/vs, and the acidity/basicity of A and B is taken as a normalized proton affinity difference, it is possible for a single function to fit the spectral data for a range of different acids and bases. A central conclusion of these results matches nicely with what was found from both ab initio calculations and from gas-phase measurements: the system shifts smoothly from a neutral to an ion pair as the proton affinities of the partners change. When the proton affinities of A and B are precisely equal, NPAD would be zero. This is not where the minimum in the vibrational correlation diagram was found. This displacement from zero is related to an earlier explanation that the ion pair A ...+ HB has a much stronger attractive interaction between the partners than does the neutral pair AH ... B. So the proton affinity of A must be that much larger than that of B in order to compensate for this fact. Consequently, the minimum in the diagram was found to occur at NPAD = -0.23. Note the similarity of this value to that obtained from crude minimal basis set estimates of the complexes pairing HBr with amines above in Fig. 6.20.

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Some of the quantitative aspects of a diagram of this type can be gleaned from inspection of Fig. 6.19 in Section 6.6.1, which illustrates the calculated frequencies in the complex between HC1 and a series of 4-substituted pyridines310. For weak bases, on the left side of the figure, the complex is described as a neutral pair, ClH—pyridine, and it is the C1H stretching frequency that is plotted. This frequency is shifted to the red as the base becomes stronger. For much stronger bases, the proton is fully shifted and the complex is best represented by the Cl •••H+pyridine ion pair, and the frequency of interest refers to the H—N stretch. In the middle region, where the proton is shared between the two species, the harmonic modes are more complex. A single intense mode is replaced by a pattern of several strong ones, all of which are of much lower frequency. (It is the highest of these that is plotted in Fig. 6.19.) One can also construct a diagram of similar character by plotting the isotopic ratio of the vs frequencies (VAH/VAD) against a measure of the shift in this frequency 338. Specifically, the latter quantity is defined as (vs/vs°) where vs° refers to the frequency measured in the uncomplexed monomer. A sharp minimum is observed in this plot when the relative frequency shift is about 0.5. It might also be pointed out that correlation diagrams of this type have relevance in solution work as well, even though the vibrational bands are much broader. The maximal shift in the spectral band for complexes of pyridines with carboxylic acids occurs for a proton affinity difference of 109 kcal/mol; pairing of pyridine N-oxides with these acids peaks at 102 kcal/mol339. When plotted as relative frequency change versus NPAD, the functions have minima at values of the latter parameter somewhat less negative than the —0.23 noted above. It is interesting to compare the extent of proton transfer determined for complexes in. inert matrices with those observed in the gas phase and predicted by ab initio computations. Consistent with the latter, H3N..HF is found to be a neutral pair in Ar matrix340, but the complex is somewhat more polar than it is in the gas phase. In fact, the frequency shift of the HF stretch rises quite noticeably as the polarizability of the matrix increases from Ar to N2. One can hence expect the matrix results to typically indicate greater proton transfer character than observed in the gas phase. Nonetheless, HF does not appear to form an ion pair with even stronger bases such as trimethylamine in Ar341, nor with H3P342. In fact, HF does not release its proton to form an ion pair even in the 1:2 H3N..HF..HF complex where cooperativity would tend to enhance the acidity of the central molecule 343. A higher degree of proton transfer in N2 as compared to Ar was observed also for the similar H3N..HC1 complex, and in complexes with HI344. A systematic examination of a series of related complexes 345 indicated that in Ar matrix, the proton is very nearly equally shared between HC1 and methylamine, unlike the gas phase where this complex is a neutral pair. This same point, that is, equal sharing, occurs for the less basic H3N in N2. The degree of transfer continues to progress to the point where the complex is an ion pair when HC1 is paired with trimethylamine in N2. The latter complex is probably an ion pair even in the gas phase. On the other hand, HC1 does not seem to form an ion pair with H3P in Ar342. Turning to the more acidic HBr346, the point of equal sharing occurs when it is paired with ammonia in Ar. The degree of proton transfer, that is ion-pair character, increases for the alkylated amines or in N2 matrix. 6.6.4 Long Chains The transfer of a proton from the proton donor to the acceptor in an amide-amide dimer would, of course, result in formation of a high energy ion pair, as depicted in Fig. 6.22, and is consequently unlikely.

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Figure 6.22 Opposite charges acquired by two amide molecules as a result of a proton transfer from donor to acceptor.

If one has a large number of amide groups lined up in a long H-bonded chain, the terminal molecules acquire the charges indicated in Fig. 6.22, but the remaining molecules throughout the chain remain electrically neutral. The primary change occurring on each of these non-terminal chain molecules is a tautomerism as indicated in Fig. 6.23. Such largescale tautomerism has been proposed to be the principal component in the mechanism of long-range proton transfer in proteins347 350. Inelastic neutron scattering experiments of Nmethylacetamide and polyglycine have lent support to the possibility of dynamic exchange of the proton between N and O351,352. There are, however, a number of observations which argue against the viability of such a string of transfers. Firstly, the imidic acid resulting from the tautomerism is considerably higher in energy than the amide. Theoretical and experimental data provide an estimate of an 11-12 kcal/mol energy difference in formamide353. The tautomeric equilibrium constant is only some 10 8.192 The proton transfer would be impossibly slow: it would take more than a month to produce an imidic acid intermediate at pH 7. Moreover, neutron diffraction measurements of the acetanilide crystal further refute these notions of massive tautomerism354. There is no evidence found for any significant degree of proton transfer from

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Figure 6.23 Tautomerism in each of the amide units in a chain resulting from a series of proton transfers from one group to the next.

N to O along the amide chain in the temperature range between 15 and 295 K. The authors estimate that any such tautomerization would account for no more than 1% of the total statistical population of the crystal.

6.7 Summary 6.7.1 Low Polarity of Acceptor The typical proton acceptor molecule contains an electronegative atom which is involved in a bond with another atom of much lesser electronegativity. The ensuing bond dipole is an important factor in formation of a H-bond. A number of systems were investigated in which the dipole of the pertinent bond was fairly low.

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Despite the lone pairs on an electronegative atom, a dihalogen molecule like C12 or FC1 does not appear to act as a strong proton acceptor. The very low polarity of the dihalogen bond greatly reduces the electrostatic component of the interaction energy. As a result, the configuration wherein the proton of HF approaches one of the available lone pairs becomes less stable than an entirely different dimer geometry in which the hydrogen plays no direct part in the interaction. Similar considerations apply to C=O, which is a highly nonpolar molecule. The proton of HF prefers to approach the C atom rather than the O, consistent with the direction of the dipole moment of the CO molecule. The red shift of the HF stretch is severalfold larger for OC---HF than for CO"HF; the stretch of this HF bond in the former complex, although small, is consistent with a H-bond, while there is no measurable stretch in CO--HF. The energetics indicate a H-bond may indeed exist in OC'"HF, with an electronic contribution to AE of some —3.6 kcal/mol, as compared to only —1.1 kcal/mol in CO-HE CO2 is completely nonpolar as a molecule, but the quadrupole moment is substantial and consistent with O atoms with a partial negative charge. There is a noticeable red shift of the HF stretch in OCO--HF, and r(FH) stretches a small amount. The interaction energy is small, — 2.4 kcal/mol, but greater than in CO-HE The structure of this complex is fundamentally altered when HF is replaced by HBr, forming a T-shape with the hydrogen approaching the C atom of OCO from above. Interaction of HF with the isoelectronic NNO makes the oxygen a slightly better proton acceptor while the terminal nitrogen is inferior to the O in OCO. It is curious that, despite the different strength of binding to the N or O ends of NNO, the red shifts induced in the HF stretch are virtually identical. When HF is replaced by HC1, the equilibrium geometry is of parallel structure, with little in the way of H-bonding character. Although HF can form what appear to be weak H-bonds with a molecule like OCO, the same is not true of HOH. The complex pairing HOH with OCO has the O atom of the former molecule approaching the C atom of the latter, held together in part by a dipole-quadrupole interaction. OSO is too weak a proton acceptor to form a H-bond with a donor such as HCN. 6.7.2 C-H Donors Geometric data extracted from crystal studies point toward the tendency of C—H bonds to orient themselves in structures consistent with the existence of H-bonds. Careful examination reveals most of these to contain intermolecular contacts considerably longer than usually considered for H-bonds; moreover, the C—H bond stretches by only very small amounts as a result of this contact. An exception arises when the C atom is involved in a triple bond, as in HC=N. The much more electronegative character of this carbon makes for a potent proton donor. Less acidic, but still possessing some proton donating ability, are the alkynes. However, the lack of a lone electron pair in a molecule like HC=CH prevents the formation of a H-bond in the acetylene dimer. The most stable geometry has the hydrogen of one atom approach the rich electron cloud around the central girth of the other, forming a T-shape. One would probably not consider this a H-bond due both to its length and to its weakness, less than 2 kcal/mol. Its stability can be rationalized on purely electrostatic grounds (quadrupolequadrupole), without recourse to other factors. Pairing an alkyne with a more conventional proton acceptor does appear to form true Hbonds in certain cases. Using the red shift of the C—H stretch as a barometer of H-bond strength, there is a continuum that depends upon the strengths of the acid and the base.

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CH3CN is a weak enough base that when paired with R—CCH, the red shift is less than 100 cm-1 regardless of the nature of R. On the other end of the spectrum is the stronger base (CH3)3N, for which the red shifts vary from 140 to nearly 300 cm-1 as the changing character of R makes the alkyne a stronger acid. When the C atom is involved in a double rather than a triple bond, its electronegativity drops, as does its ability to donate a proton. It seems unlikely that the C—H group of HCOOH, for example, acts as proton donor in a true H-bond. The single bonds of the alkyl group make these C—H bonds particularly reluctant to donate protons in a H-bond. Methane, for example, does not engage in H-bonding, neither as proton donor nor as acceptor. Its acidity can be increased the requisite amount by substitution of halogen atoms. CF3H and CC13H, for instance, are both capable of forming a Hbond with a suitable acceptor. In fact, it is probably only necessary that one halogen atom be present, as in CH3C1. Another means to overcome methane's inability to engage in Hbonding is to "cheat" and pair it with an anion. The CH4"-X~ complex adopts a structure with a linear CH---X arrangement. The binding energies vary from 6 kcal/mol when X=F to about 1 kcal/mol for CH4--I". The signs of H-bonding also include a stretch of the C—H bond by 0.01 A for CH4"-F~, whereas the other halides induce very little elongation. Consistent with a picture of H-bonding for the fluoride, and a simpler electrostatic interaction for the other halogens, is the red shift of the C—H frequency of 105 cm-1 in the former case compared to less than 20 cm-1 for the others. 6.7.3 Ionic H-Bonds If one of the partners is charged, the electrostatic attraction will be amplified, as will other factors such as polarization effects. The interaction between HF and a F anion, certainly one of the strongest H-bonds, amounts to more than 40 kcal/mol. Ionic H-bonds can become so strong and so short, and the bridging proton stretched so far from the donor atom, that it approaches the midpoint between the two partners. The (F H F)" system is a case in point, with an interfluorine distance of less than 2.3 A. Other ionic systems, for example, (C1"H"C1)~ and (HO-H-OH)", also contain short, centrosymmetric H-bonds. In some other related systems, the barrier for transfer between the two minima in the proton transfer potential is so low that the wave function corresponding to the first vibrational level would likely correspond to a centrally located proton. It is a common observation that electron correlation lowers the barrier to proton transfer in these ionic H-bonds, or even eliminates it entirely, converting a double-well potential to one with a symmetric single well. The potentials for proton displacement in these systems tend to be very flat, with high-amplitude proton motion. Enlargement of these anionic systems by alkyl substitution has little effect upon the energetics of binding or proton transfer. There are certain features of these systems which differ from conventional H-bonds between neutral molecules. For one thing, the binding energy can be several times larger than even the strongest neutral H-bond. This interaction is dominated by electrostatics, with relatively minor contributions from other attractive forces. And in some cases, a proton donor cation (e.g., NH 4 + ) can be replaced by another cation, one without a proton at all (K + ), and the energetics of complexation are barely altered1. The hydrogen nucleus drifts so far from the original donor atom that the distinction between a donor and acceptor group becomes a murky one. Nonetheless, it would probably be best to retain their categorization as H-bonds based largely upon their equilibrium geometry which maintains a hydrogen nucleus as the glue which cements the entire complex together.

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The triply bonded nitrile and alkyne groups also engage in ionic H-bonds. Unlike the case when the atoms involved are F or O, the H-bonds in which the C or N atoms of these species participate tend to be noncentrosymmetric, that is, the proton transfer potential contains a pair of minima. The exception is (C=NH--N=C) which is strongly bound by some 27 kcal/mol and which appears to contain a centrosymmetric H-bond. The H-bonds are all weaker when the triply bonded species are replaced by their double-bonded analogs. For example, whereas R(C--C) is equal to 3.35 A for (HC=CH"C=CH)~, with a dissociation energy of 11 kcal/mol, the bond stretches to 3.7 A for (H2C=CH2"CH=CH2)~ and the Hbond weakens to 5 kcal/mol. The pairing of a neutral HCOOH molecule with HCOO leads to an interaction of about 33 kcal/mol. The primary interaction, the OH--O H-bond, is centrosymmetric with an interoxygen separation of 2.5 A. There is perhaps an additional, much weaker interaction between the C—H group of the neutral and the other O atom of the anion. It appears from studies of numerous crystals that centrosymmetric H-bonds occur when R(O-O) is less than 2.5 A; the transfer potential contains two minima for longer separations. Crystal forces can influence the single or double-well character of the transfer potential, specifically by stretching the H-bond, and asymmetries of the surroundings can shift the proton's equilibrium position. Immersion of this system in aqueous solvent changes the nature of the proton transfer potential to double-well. One of the strongest interactions occurs between H2SO4 and its conjugate base, HSO4~. The basis of this interaction energy, computed to be nearly 50 kcal/mol, lies in the three OH"O H-bonds which bind the two species together. It appears likely that the proton transfer potential contains a single well. Just as a neutral proton donor can strongly associate with an anionic acceptor, one might expect that a neutral acceptor can accept a proton from a cation to form a strong bond. (HFH--FH)+, for example, is indeed strongly bound, with an interaction energy of 32 kcal/mol. While not quite as strong as its anionic analog, (FH--F)~, the interfluorine distance is comparably short, less than 2.3 A. The cation contains a centrosymmetric H-bond, just as does the anion. Whereas the anionic (HO-H--OH)" complex was considerably less strongly bound than (FH--F)~, the same is not true in the cations where (H2OH"OH2)+ contains a H-bond of comparable strength as is present in (HFH—FH) + . In a related perspective, the anionic interfluorine H-bond is stronger than the cation (also true of chlorine), while the reverse is true for oxygen where the cationic H-bond is stronger then the anion, bound by about 33 kcal/mol. Any barrier in the proton transfer potential of (H2OH--OH2)+ is so small as to be effectively nonexistent. In contrast to the interfluorine and interoxygen ionic H-bonds, the internitrogen distance in (H3NH"NH3)+ is considerably longer, 2.73 A. Its proton transfer potential contains two distinct minima, although the barrier is probably only on the order of 1 kcal/mol. The interaction has the characteristic strength of ionic Hbonds, with a binding energy of 25 kcal/mol. The H-bonds in second-row analogs of the foregoing systems are weaker, but still represent interactions that are stronger than neutral H-bonds. The binding enthalpies of (HC1H-C1H)+, (H2SH-SH2)+, and (H3PH--PH3)+ are about 15, 13, and 8 kcal/mol, respectively. The first system contains a centrosymmetric H-bond whereas the others are characterized by a double-well proton transfer potential. Regardless of whether the ionic H-bond is of the cationic or anionic type, the H-bond elongates as the electronegativity of the atoms involved diminishes. The shortest is the interfluorine distance of 2.3 A in (HFH---FH) + and (FH"-F)~; the binding energy in the latter is in excess of 40 kcal/mol. Ionic H-bonds involving F, Cl, and O are centrosymmetric;

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S is on the borderline, while N and P bonds are characterized by double-well potentials. Methyl substitution has little effect upon the nature of these ionic H-bonds. One can rationalize a simple relationship between the strength of an ionic H-bond (A---H + B) on one hand and the difference in proton affinity between the two partners A and E! on the other. The Marcus formulation provides a convenient framework for predicting the H-bond energy of an arbitrary system based on knowledge of the interaction energy in a symmetric system (A=B), and the difference in proton affinity between the two partners. This model has proven successful in a series of interoxygen and internitrogen H-bonds. One issue that has generated a good deal of interest over the years, in part for its potential importance in enzymatic activity, has been the competition for a proton donor between the two lone pairs (syn and anti) of the O atom of the carboxylate group. When this group is paired with a water molecule as donor, the preferred geometry belongs to the C2v point group, wherein there are two H-bonds present: each hydrogen of the water acts as a bridge to one of the carboxylate O atoms. But there is little energetic difference observed between the approach of the water to the syn and anti sides of a single oxygen. This conclusion, of nearly equal H-bonding potential of the syn and anti lone pairs, is confirmed by crystal studies which attempt to subtract out crystal packing forces and steric interference. In contrast to this equal propensity toward formation of a H-bond, there is a definite preference for a proton to locate on the syn side when coming much closer and forming a covalent bond. That is, the syn conformer of RCOOH is more stable than the anti structure by about 5 kcal/mol in the gas phase. If this is the case, why then is there so little apparent pKa difference between the syn and anti structures? The answer to this question resides in solvation phenomena. The presence of the polarizable medium preferentially stabilizes the anti conformer of the carboxylic acid due to its higher dipole moment. But even if we restrict our attention to the gas phase, one might wonder about a contradiction in that the syn conformer of RCOOH is clearly favored over anti, but that there is no such preference when RCOO" forms a H-bond with a proton donor. This puzzle can be resolved if one recalls that there are different forces in operation for a covalent bond as opposed to a H-bond. That is, the syn structure of RCOOH is indeed more stable than anti, due to internal forces within this molecule. But as the proton acceptor is brought into proximity with the carboxyl group, its partial negative charge will be repelled by the partial negative charge of the other O atom, the one with which it is not interacting directly. This repulsion would act to destabilize the syn H-bond, as compared to the anti which would not have such a force in operation. 6.7.4 Neutral Versus Ion Pairs Starting with a conventional H-bond between a pair of neutral molecules, a proton transfer from donor to acceptor will alter the fundamental character of the interaction to a pair of oppositely charged ions. Such a transfer is disfavored by the energetic cost of generating the high degree of separation of charge in the ion pair, so will occur only for a particularly strong acid paired with a strong base. Hydrogen halides comprise some of the strongest proton donors, and amines, particularly alkylated amines, represent a class of powerful acceptors. Consequently, binary systems containing these species have provided a testbed for notions about neutral/ion pair transitions and proton transfers. Calculations have underscored the dominating importance of electron correlation in understanding the behavior of these systems. Whereas potential energy surfaces computed at the Hartree-Fock level typically indicate the presence of two distinct minima, one corre-

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sponding to the neutral and the other to the ion pair, inclusion of correlation causes the coalescence of these two minima into a single one, in which the proton is located at an intermediate position along the H-bond axis. Correlated calculations lead to the general conclusion that many surfaces contain a single, broad minimum. Increasing the basicity of the acceptor and/or the acidity of the donor shifts the position of the minimum. The transition from neutral to ion pair is best described as a gradual one as the nature of the donor and acceptor are changed. Starting with a highly acidic donor, HI appears to donate a proton to amines that have one or more methyl substituent, forming what can best be characterized as an ion pair. When paired with NH3, however, the surface contains not a well-defined minimum, but rather a long shallow valley connecting H 3 NH + ---~I with H 3 N---HI. HBr, too, forms ion pairs with the more basic amines. The proton is nearly equally shared in a single minimum located between Br~ and CH3NH2, whereas the H 3 N---HBr complex is clearly a neutral pair. Turning next to the still less acidic HC1, neither NH3 nor CH3NH2 is capable of extracting its proton, while (CH3)3N forms the ion pair. (The situation with (CH3)2NH remains not fully resolved.) HF is too weakly acidic to form an ion pair with any of these amines. It is possible to quantify these relationships between the degree of proton transfer from acid to base and the relative proton affinities of the two partners. This treatment illustrates the unlikelihood that HF will form an ion pair with any acceptor, no matter how basic. HC1 is more susceptible to the nature of the acceptor: its proton can be pulled away from the Cl as the basicity of the acceptor is raised. HBr is more sensitive still, as the nature of its complexes with the amines take on more and more ion pair character for the more basic amines. The crossover point, at which the equilibrium position of the proton is about midway between the donor and acceptor, does not occur when the two have equal proton affinities: the proton affinity of the acceptor can be significantly less than that of the donor. The reason for this observation resides in the nature of the force of interaction. Whereas the neutral pair is held together by the "standard" components of H-bonding which amount to no more than 15 kcal/mol, the electrostatic attraction in the ion pair is greatly magnified by the opposite charges of the two subunits. There is thus a natural energetic bias toward the ion pair, lessening the proton affinity of the acceptor that is required for this configuration to form. Another means of shifting the equilibrium position of the proton is through the surroundings. A polarizable medium preferentially stabilizes the ion pair, causing the proton to translate toward the base. The same sensitivity of HX to basicity of the amine (i.e., HBr loses its proton more readily than HC1 or HF as the amine becomes more basic) is noted as the polarizability of the medium is enhanced. The carboxyl group is an oxygen acid that is comparable in acidity to the hydrogen halides. Like most of the latter, RCOOH will not donate a proton to form an ion pair with a nitrogen base in the gas phase. But raising the dielectric constant of the medium does permit this transfer to occur and at only fairly small enhancement of the medium's polarizability over that of a vacuum. Experimental measurements confirm many of the conclusions reached by the calculations. Combination of a hydrogen halide with an amine produces an ion pair in the gas phase for only the strongest acid/base pairs. In concert with spectral information obtained in solid matrix, the transition from neutral to ion pair appears to be a gradual one as the strengths of the acid and base reach threshold values. There are a number of systems examined which do not fall neatly into either the pure neutral or ion pair category, but are better described as having a broad minimum in their proton transfer potential; the bottom of this well occurs

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Index of Complexes

This index is provided so that the reader might locate information about a particular complex of interest. It is organized as follows: The first section contains the neutral binary complexes, followed by charged dimers, and then by larger complexes in the last section. Within each section, the complexes are ordered by the type of molecules contained. The order is as follows: XH, YH2, ZH3, carbonyl, carboxyl, imine, amide, nitrile, alkyne, alkene, alkane, others. (X refers to any halogen: F, Cl, Br, I; Y to O, S, etc.; and Z to N, P, etc.) Any alkylated or similar substitutions follow immediately after their parent group. Thus, any binary complex containing HF is listed first. Within the set of neutral binary complexes containing HF as one partner, the complexes are listed in the order HF, HC1, OH2, O(CH3)2, SH2, SeH2, NH3, NH2CH3, NH(CH 3 ) 2 ,... OCH2, OCHF, SCH2, HCOOH, OCHNH2, NCH,.. . H2CCH2, CH4, CC1H3, and so on. Each complex is listed only once. So, for example, the complex pairing HF with OH2 is listed as FH-OHL, and not as HOH-FH. bending potential 64-66 vibrational spectrum 156-8, 171-3 O(CH3)2 67 SH2 binding energy 62-63 bending potential 64-66 SeH2 62 NH 3 binding energy 54 BSSE 56-59 vibrational spectrum 148-52 anharmonicity 153 proton transfer 331, 334-5, 339 NH2CH3 332,334-5

Neutral Binary Complexes

FHFH 291 decomposition 38 energetics 72-77 geometry 72-74, 209-13 isotope effects 118-9 superposition error 26, 27 vibrational spectrum 143-7 ZPVE 118-9 CIH 74 OH2 binding energy 62-64

365

366

Index of Complexes

FH... (continued) NH(CH3)2 332, 334-5 N(CH3)3 332, 334-5, 337, 339 PH3 binding energy 54 vibrational spectrum 148-52 proton transfer 331, 339 AsH3 54, 331 OCH2 92, 183 OCHF 94 OCHC1 94 SCH2 93 HCOOH 94-96 HCONH2 93, 106 NCH 102, 185-90 NCCH3 102 HCCCCH 301 H2CCH2 301 CH4 305 CC1H3 303 CC12H2 303 CC13H 303 CO 294 CO2 295 N2O 296 SO2 297 C12 292-3 C1F 292-3 HMn(CO)5 307 FD-NH3 153 FLi-NH3 153

C1HC1H energetics 74-77 geometry 74, 213-5 isotope effects 119 vibrational spectrum 146-7 BrH 76-77 IH 76-77 OH2 binding energy 62-63 bending potential 64-66 vibrational spectrum 158 EFG 160 OH(CH3) 67, 159 0(CH3)2 67 SH2 binding energy 62-63 bending potential 64-66, 209 EFG 160

SeH2 62 NH3 binding energy 54 anharmonicity 152 EFG 154 proton transfer 225-6, 330-1, 334-5, 337, 339 NH2CH3152, 225-6, 331-2, 334-5, 337, 339 NH(CH3)2 332, 334-5 N(CH3)3 332, 334-5, 337, 339 PH3 binding energy 54 EFG 154 proton transfer 331, 339 P(CH3)3 337 AsH3 54, 331 OCH2 92-93, 182-4 OC(CH3)2 93 OCHF 94 OCHCI 94 HCOOH 94-96 NCCH3 102, 190-1 HCCH 299 HCCCCH 301 CH4 305 CC1H3 303 CC12H2 303 CC13H 303 CO2 295 N2O 297 2-butanone 93 methyl formate 93 methyl acetate 93 pyridine 155, 332-3, 339

BrH-BrH 76-77 IH 76-77 OH2 binding energy 62-63 bending potential 64-66, 209 vibrational spectrum 158 SH2 62 SeH2 62 NH3 binding energy 54 anharmonicity 152 proton transfer 225-6, 331-2, 334-5, 337, 339 NH2CH3 60-61, 225-6, 331-2, 334-5, 337 NH(CH3)2 60-61, 225-6, 331-2, 334-5 N(CH3)3 60-61, 225-6, 331-2, 334-5, 337

Index of Complexes PH3 54, 331 AsH3 54, 331 CO2 295

IHIH 76-77 OH2 158,209 NH3 225-6, 331-2, 338 NH2CH3 60-61, 225-6, 331-2 NH(CH3)2 60-61, 225-6, 331-2 N(CH3)3 60-61, 225-6, 331-2, 337

HOHOH2 19-22 anharmonicity 15 decomposition 33, 35 energetics 78-79 geometry 77-78, 215-23 isotope effects 119-21 polarizability 162-3 superposition error 25-26, 27 total energy 24 vibrational spectrum 160-8, 173-5 OHCH3 81, 169-70 OHC2H5 81 O(CH3)2 82 OHC6H5 83 OHSiH3 82 OHC1 83-84, 171 SH2 80-81 NH3 geometry 69-70 energetics 69-70 isotope effects 119 vibrational spectrum 175-7 NH2CH3 71 PH3 69-70 OCH2 geometry 89-92, 223-5 isotope effects 119 vibrational spectrum 180-2 HCOOH 96-97 NHCH2 103, 185 NHCHCH3 185 NCH3CH2 185 NCH3CHCH3 185 HCONH2 105-6, 119 CH3CONHCH3 106-9 HCCH 299 CH4 302-4 CFH3 303 CF2H2 303

CF3H 302, 303 CC1H3 302-3 CC12H2 302-3 CC13H 302-3 CC14 302-3 SiH4 304 CO 294-5 CO2 296 SO2 298 CC12 298 CH3OHOHCH3 81-83, 169 OHC6H5 83, 170 OHSiH3 82-83 NHCH2 103, 185 NHCHCH3 185 NCH3CH2 185 NCH3CHCH3 185 SO2 298 C2H5OH-OHC2H5 81 C6H5OH-NH3 71, 177 SiH3OH---OHSiH3 82-83, 169

HSHSH2 79-80, 163-6 NH3 69-70 HCN 102 CH4 304 SO2 298 H3NNH3 292 geometry 84-88, 208-9 vibrational spectrum 177-9 ZPVE 178 PH3 89 HCN 102 CH3NO2 304 2-pyridone 88-89 CIF 294 CH3NH2-HCOOH 97 (CH3)2NH-NH(CH3)2 88 H3P-PH3 89 H3P-HCN 102 HCOOHNH2CH3 335 N(CH3)3 337 HCOOH 99-101

367

368

Index of Complexes

HCOOH- (continued) CH3COOH 101 NHCH2 336 HCONH2 112-3 CH3C1 97-99 CH3COOCH3-HCONH2 113 HCONH2-HCONH2 105, 110-1, 195-7 CH3CONHCH3-CH3CONHCH3 106-9

NCHNH3 191-2 N(CH3)3 337 OCH2 93 NCH 102-3, 119, 192-5 CH4 305 H3CCH3 305 CO2 295 SO2 97-8 HCCH-HCCH 241, 299-300 HCCH--OSO 298 RCCH-O(CH3)2 301 RCCH-NH3 301 RCCH-N(CH3)3 301 RCCH-OC(CH3)2 301 RCCH-NCCH3 301 F3CH-NH3 302 HN03-CH4 305 DNA base pairs dispersion energy 31 geometry 113-8 dipole moments 116 stacked structure 118

(CH3)3NH+-N(CH3)3 320 H3PH+-PH3 316, 318 CH3CHOH+-OCHCH3 320 HCOOH2+-HCOOH 320 HCNH2OH2+-OCHNH2 320 FCNH2OH2+-OCFNH2 320 pyridinium+—pyridine 320 Li+-NH3 59-60 Na+-OCH2 180 K+-NH3 292 Mg+2-OCH2 180 Anionic Complexes

FH-F- 308-11,319 F--CH4 305-6 ClH-Cl- 308-11, 319 C1--CH4 305-6 BrH-Br- 310 Br--CH4 305-6 T-CH 4 305-6 HOH-OH- 120-1, 310-12, 319 HOH-OOCH- 326-8 CH3OH-OCH,- 120, 312 HSH-SH- 310-11 H2NH-NH2- 310-11, 319 H2NH-OOCH- 327 H2PH-PH2- 310-11 HCOOH-OOCH- 313-5, 320 CH3COOH-OOCCH3- 312 H2CNH-NCH2- 313 NCH-CN- 312 HCCH-CCH- 312 H2CCH2-CHCH2- 313 H2SO4-SO4H- 315

Cationic Complexes

HFH+-FH 316 HC1H+-C1H 316, 318-9 H2OH+-OH2 120-1, 316-7, 319, 324-5 H2OH+-OHCH3 324-5 H2OH+-O(CH3)2 324-5 H2OH+-OHC2H5 324-5 H2OH+-NH3 337 H2OH+-OCH2 180 CH3OH2+-OHCH3 121, 319 H2SH+-SH2 316, 318 H3NH+-NH3 33, 38, 292, 316-9, 325 H3NH+-NH2CH3 325 CH3NH3+-NH2CH3 320 (CH 3 ) 2 NH 2 + -NH(CH 3 ) 2 320

Neutral Trimers and Oligomers

(HF)3 245-8 (HF)n 248-52 (HC1)3 245-8 FH-FH-OH2 278-82 FH-OH2-OH2 280-2 FH-FH-NH3 272-80, 339 FH-FH-PH3 273-78 C1H-OH2-OH2 281-2 C1H-CIH-NH3 272-3 (H2O)3 120-1, 260-70 (H2O)n 253-8, 264-6 cyclic clusters 257-62 vibrational spectra 256, 266-7

Index of Complexes

ROH-OH2-OH2 270-2 ROH-ROH-OH2 270-2 (ROH)3 270-2 C6H5OH-OH2-OH2 264 C6H5OH-(OH2)3 265

(HCN)3 232-40 (HCN)n 232-40 (HCONH2)n 340-1 (HCCH)3 241-2 (HCCH)n 242-4

369

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Subject Index

acetic acid 329 anharmonicity 15, 22, 120, 139-40, 169 in ionic complexes 308, 317 in neutral binaries 146, 152-4, 186-90 in oligomers 251, 254, 256 anti-H-bonding 67-68 asymmetric H-bond, definition 321 atomic charge 18 atomic polar tensor 179 definition 150, 162, 277-8 average H-bond energy 234, 244-6, 267-8 Badger-Bauer rule (see relationship between H-bond strength and frequency shift) barrier interconversion 208-9, 212-5, 218-20, 225, 300 proton transfer 225-6, 311, 312, 315-7, 325 rotational 69, 215, 300 torsional 89, 213, 265, 269 basis set 4-7 atomic natural orbital 7 complete 27 correlation-consistent 7, 72, 264 dirner-centered 24 double-zeta 4 minimal 4, 25-26, 84, 217, 330, 332 monomer-centered 24 split-valence 5, 330

triple-zeta 5 well-tempered 7, 38, 59 basis set extension (see basis set superposition error, secondary) basis set superposition error in decomposition terms 37, 39, 260 definition 23-28 in dimers with FH 55-56, 293 in dimers with H2O 216, 224, 296 in H2NH-NH3 84-88 in DNA base pairs 114 in ionic complexes 309, 327 in oligomers 234, 244, 246, 254, 257 effect on EFG 160 effect on NMR spectrum 171 effect on vibrational spectrum 189-90 secondary 24-25, 56-59 spatial aspects 56-59 bifurcated structure 53, 77-80, 215-20, 223-5, 280, 305 in ionic complexes 317, 326-8 bond functions 6, 57, 78, 87, 214 Born equation 336 Born-Oppenheimer approximation 3, 21 branching clusters 257 carbenes 298 carbohydrates 299 CASSCF 10, 88 371

372

Subject Index

centrosymmetric 308-11, 314-5 charge flux 150-1, 278 charge transfer energy 31, 32, 35, 37 chemical Hamiltonian 26-27 CH group as proton donor 99, 113, 191-5, 224, 240-4, 298-307 CISD 8, 75, 141-3, 168, 308 cis/trans competition 216 combination band 190 complexation energy (see interaction energy) configuration interaction 7 contraction 4 convergence 234 cooperativity definition 234, 275 negative 231, 242, 271 correlation (see electron correlation) Coriolis interaction 88 Coulombic energy (see electrostatic energy) counterpoise (see also basis set superposition error) 26 definition 24 coupled cluster 8-9 in (HX)2 72, 75 in H2CO complexes 91-92 in oligomers 269 in ionic complexes 308, 316, 319 in weak complexes 295 vibrational spectra 142-3, 168, 181 coupled pair 8-9 in HX complexes 75, 186, 213 in H2O complexes 219 in NH3 complexes 86,154 in H2CO complexes 93 in oligomers 237, 240, 248 in weak complexes 292-3 coupled perturbed Hartree-Fock 162 crystal orbital method 238, 253 cyclic chains of HX 245-52 of H2O 257-64, 268-72 of HCN 240 of HCCH 241-4 cyclic structure of (HX)2 146, 213-4 of (H2O)2 53, 77-80, 215-9 of (NH3)2 85-88, 177-9, 208-9 of HOH complexes 105 , 223-5 of (CHONH2)2 111 damping factor 26 Davidson correction 75-76

decomposition anisotropy 260—1 definitions 28-39 of dispersion energy 221-2 Kitaura-Morokuma 32—34, 67 natural bond orbitals 34, 222-3 nonadditivity 260 perturbation schemes 37-39 reduced variational space method 34 of dimers with H2O 67-68, 216, 220-3 of H2O clusters 258-60 deformation energy (see also induction energy) 219, 260, 268 degree of polarization 162 density difference map 18 dielectric constant 336 dipole moment alignment 90 average 235, 253 change caused by correlation 64, 222 change caused by H-bond formation 58-59, 155, 235 change caused by vibration 157, 277 change caused by bond stretch 210, 277, 308 directionality 12 dispersion energy 66 anisotropy 210-2, 221-2 definition 31, 37-38 in (HC1)2 74 of dimers with H2O 70, 78, 221-2, 295 in DNA base pairs 117 dissociation energy (see also interaction energy) 17 double proton acceptor 231, 242, 261, 263 double proton donor 231, 242, 261, 263 dynamics 14 effective core potential 7, 62, 76, 158 electric field 337 gradient 154, 159-60 electron correlation 4 contribution to H-bond 54, 63, 72, 221-2 definition 7-10 effect on proton transfer 309-11, 313, 315, 320, 331-2 electrostatic energy correlation correction to 31, 39, 211 as factor in geometry 68, 220-1 electrostatic potential 18 enthalpy 17 entropy 17-18, 21 in dimers containing H2O 71, 78, 218

Subject Index in FH-CIF 293 in (H2O)n 266 in ionic complexes 327 of isotopic substitution 121 equilibrium constant 340 of formation 52, 120 ESCA 14 exchange energy 30, 32, 37 F-bonding (see reverse complex) FG matrix (see GF matrix) force constant 197, 208, 337 formic acid 328 four-body interactions 257, 267-8 fractionation factor 121 functional counterpoise (see counterpoise) Gaussian functions 4, 5, 26 GF matrix 139, 161 ghost orbitals (see also basis set superposition) 24, 39, 56-59 GIAO 19, 171 Gibbs free energy 18, 21 in dimers containing H2O 71, 78, 96 in FH-CIF 293 in (H2O)n 266 in ionic complexes 327 of isotopic substitution 121 gradient algorithms 10, 26, 216, 225, 245

ion pair (see proton transfer) isotopic substitution 21, 118-21, 153, 258, 315, 339 LCAO 4 linear combination of atomic orbitals (see LCAO) localized molecular orbitals 35 London forces (see dispersion) Marcus theory 323-6 matrix, effects of 147, 166-8, 338-9 MCSCF 9-10 minimum global vs. secondary 11, 15, 102 true (see true minimum) mixing term 32 multipole expansion 220-1 nomenclature 52 of vibrational modes 140-1 nonadditivity definition 231 perturbational approach 39 contribution of correlation 262 nuclear quadrupole coupling (see also quadrupole coupling constant) 140 nuclear relaxation energy 16 orbital exponent, choice of 7, 56

Hamiltonian 3, 9, 26, 27 SCRF model 147 harmonic approximation 22, 139 Hartree-Fock approximation 3-4 H-borid energy (see interaction energy) Heisenberg uncertainty principle 3 Heitler-London energy 30, 35, 260 Hessian matrix 10 Hoogsteen geometry 115-8 hybridization 62 hydrogen bond energy (see interaction energy) IGLO 19 independent electron pair approximation 8 induction energy 261, 295 definition 30-31, 37 intensity, vibrational spectrum formulation 139-40 Raman (see Raman intensity) interaction energy 13, 15-17, 322 intrinsic well depth 323

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partition function 16 Pauli blockade 34 Pauli exchange principle 26, 33-34 perturbation theory many-body 9, 308 M011er-Plesset 9, 38, 247 Rayleigh-Schrodinger 37 symmetry-adapted 27, 37, 70, 261 polarizability effects of ghost functions 59 proton 321 polarization energy definition 31, 32, 37 relation to Pauli principle 33 polarization function 5-6 Pople correction 75-76 potential energy surface (see also barrier) shape of 11, 308-9, 311 bending 65-66 primitive function 4, 7

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Subject Index

principal moment of inertia 19, 21 proton affinity 322 difference 322-6, 334, 338-9 proton transfer correlation effects 61 effect of environment 311, 314-5, 328, 335-7 parameter 333-5, 337 potential 14, 61, 225-6, 313, 330-5 in amines 225-6, 330-2, 334-5, 337-339 in arsines and phosphines 331, 337, 339 pseudopotential (see effective core potential) pyramidal geometry 64-66, 221-2 QCISD 8, 87 quadrupole coupling constant 19, 239, 337 Raman intensity 188-9, 191 definition 140, 156-8 redistributions, electronic 13, 18 caused by H-bond formation 57, 67 reference configuration 7-8 relationship between frequency shift and bond stretch 161, 266 between frequency shift and isotopic ratio 339 between H-bond strength and bond stretch 66 between H-bond strength and force constant 161, 197 between H-bond strength and frequency shift 140, 155, 180, 183 between H-bond strength and intensity 155 between H-bond strength and proton affinity difference 322-4 between H-bond strength and proton transfer barrier 161, 197 between H-bond length and quadrupole coupling constant 239 between isotropic and perpendicular shifts 171 between proton transfer potential and proton affinity difference 322-6, 333-6, 338 between stretching frequency and quadrupole coupling constant 239 reverse complex in HF complexes 66, 102, 209-10, 212, 292, 294 in HOH complexes 66, 169-70, 224, 304

rotational energy 16 Rydberg functions 308 salt bridge (see proton transfer) scaling factor 139 scattering activity 163 Schrodinger equation 3, 26, 152, 254 SCRF 147 self-consistent field 3, 336 sequential H-bond chain 231, 339-41 shifts, electronic (see redistributions) size-consistency 8, 9, 75-76, 216 Slater-type orbitals 4, 26 solvent effects 328-9, 336-7 spectrum electron resonance 14 NMR 14, 19, 171, 307, 311, 315 resonant photoacoustic 14 vibrational 13-14 spin-spin coupling constant 14 stacked geometry 216-7 steric repulsion (see exchange energy) STO (see Slater-type orbitals) strength of H-bond (see interaction energy) stretch of AH bond in XH 55, 77, 96 in HCOOH 99 in NCH oligomers 232 stretch of carbonyl bond 90, 93-94, 96, 99, 180 stretch of CN bond 232 substituent effects alkylation 60-61, 66-67, 81, 103, 159, 169, 185, 270-2 effect on proton transfer 311, 319 chloro 83-84, 93, 171 fluoro 93 phenyl 71, 83, 170, 177, 263-5 silyl 82, 169-70 supermolecule approach 32 syn/anti competition 326-30 thermodynamic quantities 15-18 three-body interactions 246-8, 254-62, 267-70, 275, 279 anisotropy 260-1 trans/cis (see cis/trans competition) translational energy 16 trifurcated geometry 209, 217-8, 305 in ionic complex 317, 318 true minimum 10

Subject Index in complexes with H2O 69, 91, 215-6, 224 in (NH3)2 85, 177, 208-9 in complexes with HCONH2 113, 196 in nucleic acid base pairs 115 in water cluster 262-4 tunneling 88

Watson-Crick geometry 115-8 wave function 3, 34, 64 vibrational 65-66, 264

van der Waals force 12 van der Waals radius 12, 310 variation principle 23 vibrational averaging 89

zero-point vibrational energy 16, 17 effect on proton transfer 312, 331 means of calculation 20-22 in neutral dimers 56, 224, 225, 293 in oligomers 234, 240, 282 in ionic complexes 309, 317, 331

correlation diagram 338 levels 65-66

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