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