Computational Chemistry Reviews of Current Trends

Computational Chemistry Reviews of Current Trends Volume 9

Computational Chemistry: Reviews of Current Trends Editor-in-Charge:

Jerzy Leszczynski, Dept. of Chemistry, Jackson State University, USA

Published Vol. 1: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 2: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 3: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 4: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 5: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 6: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 7: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 8: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski

Computational Chemistry Reviews of Current Trends Volume 9

editor

Jerzy Leszczynski Department of Chemistry Jackson State University USA

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COMPUTATIONAL CHEMISTRY: REVIEWS OF CURRENT TRENDS Volume 9 Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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PREFACE There is no doubt that many of the readers of this book series wonder how the research/computational facilities will look like in the near future. It seems that the efficiency and compactness of the silicon-based computers are almost exhausted, and for significant progress one has to look for alternative solutions. A very promising future for computing is based on molecular computers. Among the molecular elements required for such computers, the developments rely on the existence of nanostorage devices. A recent Science article (Z. M. Liu, A. A. Yasseri, J. S. Lindsey, and D. F. Bocian; Science, 302, 1543, 2003) reveals silicon-tethered porphyrins that are resistant to heat (up to 400° C) and harsh conditions. Since porphyrins possess well known data storage characteristics, the unique properties of these new derivatives reveal the possibility for their commercial production. It is assumed that porphyrin-based memory chips can be assembled using facilities devoted to the traditional silicon devices, opening the way toward nano components of molecular computers. The research in this area is extensive. To assist the readers, we have decided to collect more chapters for this book that are devoted to the reviews of current advances in nano science. As a result, this volume includes three chapters related to this important research area. The first chapter, written by J. J. Palacios, A. J. Perez-Jimenez, E. Louis, E. San Fabian, J. A. Verges, and Y. Garcia, reviews fundamental issues underlying first-principle quantum theory in atomic and molecular systems. The authors have recently developed a new approach called Gaussian Embedded Cluster Method that is based on the GAUSSIAN98 and GAUSSIAN03 codes. The review provides numerous examples that illustrate the applicability and reliability of this method. This is supported by the study and interpretation of various experiments in the field of molecular electronics. Apparently, an ultimate goal of nano sciences is the development of tools for spatial and chemical control of single molecules. Among the experimental techniques that have emerged in recent years are single molecule fluorescence, optical and magnetic tweezers, and atomic force spectroscopy. All of these techniques allow for mechanical manipulations of single molecules. The proper comprehension of the experimental results requires adequate theory. W. Nowak and P. E. Marszalek present an overview of

V

vi Preface

recent advances and trends in the applications of computer simulations aimed at understanding atomic force microscopic experiments. Those variants of classical molecular dynamics (MD) simulations which are particularly helpful in the interpretation of experimental data obtained for single biopolymer molecules, such as steered MD or biased MD, are reviewed and discussed in detail. In the third chapter, J. Seminario, P. Derosa, L. Cordova, and B. Bozard discuss the application of combined molecular dynamic simulations based on ab initio force fields and signal processing techniques to analyze the dynamic properties of nanocells. The analysis includes several characteristic modes at different temperatures. The authors consider the applications of such techniques for the characterization of new, molecular electronics. The prediction of the excited state properties of molecular systems are among the most challenging tasks of computational chemistry. They are assisted by the development of new methods designed for such tasks. The most accurate approaches are based on the coupled cluster theory. D. Mukherjee and his group (S. Chattopadhyay, D. Pahari, and U. S. Mahapatra) review formulations of linear response theory for excited state potential energy surfaces which are based on multi-reference coupled electronpair approximation categories of methods. This review includes a thorough discussion of the genesis of the developed theory. In addition, applications to the potential energy surfaces of low-lying excited states of the model P4 and Li2 molecules are also provided. Recently progress has been achieved in the theoretical prediction of the magnetic exchange coupling constants in compounds of rare earth transition metals. In the fifth chapter, M. Atanasov, C. Daul, and H. U. Giidel describe a method developed for such tasks. After a comprehensive discussion of the calculation procedure, the authors present examples of its application to various dimers of rare earth metals. My recent web search of the phrase "hydrogen bonding" resulted in over 50,000 entries. This demonstrates the popularity of this topic which was formally recognized more than 80 years ago. Over the years new and intriguing aspects of this phenomenon have been studied. Such studies reveal new classes of hydrogen bonds. One of such new categories, dihydrogen bonds (DHB), are discussed in the last chapter of this volume. S. J. Grabowski and J. Leszczynski present the examples of such interactions existing in crystal structures. These interactions are compared with the conventional hydrogen bonds. A comparison shows that DHBs are often similar to typical H-bonds. The discussion includes the results of ab initio calculations,

Preface vii

decomposition energy, topological parameters, and experimental neutron diffraction results. I would like to thank all authors for the excellent contributions and fine collaborations. The very efficient technical assistance of Dr. Manoj K. Shukla in putting together this volume is greatly appreciated. As always, your feedback is very important to me, please feel free to e-mail your suggestions to [email protected]. Jerzy Leszczynski Jackson, MS December 2003

CONTENTS Preface

v

1. Molecular Electronics with Gaussian98/03 J. J. Palacios, A. J. Perez-Jimenez, E. Louis, E. SanFabidn, J. A. Verges and Y. Garcia

1

2.

Molecular Dynamics Simulations of Single Molecule Atomic Force Microscope Experiments W. Nowak and P. E. Marszalek

47

3.

Molecular Dynamics Simulations of a Molecular Electronics Device: The NanoCell J. Seminario, P. Derosa, L. Cordova and B. Bozard

85

4.

Computation of Excited State Potential Energy Surfaces via Linear Response Theories Based on State Specific Multi-Reference Coupled Electron-Pair Approximation Like Methods S. Chattopadhyay, D. Pahari, U. Mahapatra, D. Mukherjee

121

5.

Modelling of Anisotropic Exchange Coupling in Rare-Earth-Transition-Metal Pairs: Applications to Yb 3 +-Mn 2+ and Yb 3 +-Cr 3 + Halide Clusters and Implications to the Light Up-Conversion M. Atanasov, C. Daul and H. U. Giidel

153

6.

Is a Dihydrogen Bond a Unique Phenomenon? S. J. Grabowski and J. Leszczynski

195

Index

237

Content Index

243

ix

Chapter 1: Molecular Electronics with Gaussian98/03

J. J. Palacios ab , A. J. Perez-Jimenez0, E. Louisa'b, E. SanFabian"'0, J. A. Verges"" and Y. Garcia" *Departamento de Fisica Aplicada, Universidad de Alicante San Vicente del Raspeig, Alicante 03690, Spain b Unidad Asociada del Consejo Superior de Investigations Cientificas, Universidad de Alicante San Vicente del Raspeig, Alicante 03690, Spain

c Departamento

de Quimica-Fisica, Universidad de Alicante San Vicente del Raspeig, Alicante 03690, Spain

d Departamento de Teoria de la Materia Condensada Institute de Ciencia de Materiales de Madrid (CSIC) Cantoblanco, Madrid 28049, Spain

Abstract In this review we discuss the fundamental issues underlying first-principles quantum transport theory in molecular- and atomic-scale systems, making emphasis on the actual numerical implementation of them. For this purpose we focus on the ab initio method named Gaussian Embedded Cluster Method, recently developed by the authors, which is based on the popular quantum chemistry code GAUSSIAN98/03. Various examples that illustrate the applicability and reliability of this method in the study and interpretation of a large range of experiments in the field of molecular electronics are also presented.

1

2 J. J. Palacios et al.

1

The basics

For the past forty years, computers and, more generally, electronic devices have grown more powerful as their basic subunits have shrunk. Investigators in the field of next-generation electronics forsee that during the next 10 to 15 years, as the smallest features of mass-produced transistors shrink below 100 nanometers, the devices will become more difficult and costly to fabricate. In addition, they may no longer function effectively since the laws of quantum mechanics may prevent transistors from working under the same principles as today's. In order to continue the miniaturization of circuit elements down to the nanometer scale new paradigms need to be proposed. For instance, unlike today's field effect transistors, which operate based on the movement of masses of electrons in bulk matter, the new devices should take advantage of quantum mechanical phenomena that emerge on the nanometer scale such as the quantum coherence and the discreteness of the electron charge. There are, essentially, two roads leading towards nanometer-scale ultra-dense circuits: • Solid-state quantum-effect devices (silicon-based devices). • Molecular electronic devices (carbon-based devices). It is not the purpose of this review to analyze which alternative is more likely to succeed in the long run. While the first alternative seems the logical continuation of present technologies, we believe, in our modest and biased view, that molecular electronics may have a better chance. Even if this is not the case, basic research in molecular electronics may contribute to advances in multidisciplinary areas such as biotechnology and medicine. Behind the words "molecular electronics" lies the promising idea that functional units for electronics can be built out of very stable and wellcharacterized carbon-based molecules such as fullerenes, carbon nanotubes, or polyphenylene chains, to name a few[l]. The advantage of using carbonbased molecules resides in that they are fabricated identical by nature, can be obtained in any desired quantity, and can be easily functionalized to meet any required property by means of standard and cheap chemistry techniques. Regardless of the final road taken by electronics, there is little doubt that the functionality of future devices will rely on the electrical and mechanical properties of nanoscopic regions composed of a number of atoms that could range from several thousands down to a single one. Hereby the necessity of quantum chemistry or first-principles computational support to which these notes are devoted.

3

Molecular Electronics with Gaussian98/03

2

Electronic transport at the nanoscale: A quick view of Landauer formalism

The first proposal of using molecules for electronic applications was put forward by Aviram and Ratner in 1974[2]. Since then various ideas to use molecules for electronic applications have been proposed [1]. How to put into practice these ideas, however, remains the main issue. It is important to realize that the coexistence of molecules and metallic electrodes will probably form the basis of future molecular devices since it is very unlikely that electronics will be entirely based on carbon. Keeping this in mind, the simplest molecular electronic unit that can be envisioned is a two-terminal device consisting of two large metallic electrodes, several nanometers apart, joined by a molecule or molecules anchored to them. This proposal was actually realized for the first time in the laboratory by Reed's group using a Mechanically Controlable Break Junction (MCBJ)[3], although early measurements go back to the work of Mann and Kuhn[4]. This system is generically known as a molecular bridge or molecular junction and plays a central role in molecular electronics. Conventional lithography, on the other hand, had already been successful in contacting large molecules like carbon nanutubes[5]. Before Reed's work, various groups had already managed to make nanoscale junctions connecting the two metallic electrodes of the MCBJ by just a chain or even a single atom of the same metallic element [6]. These systems are known as atomic contacts or metallic nanocontacts and, although they are not expected to be of any practical application, constitute an excellent test bed to learn about the world of electrical transport at the atomic scale. It is fair to say that other groups had already been working on molecular and atomic junctions since the early nineties from a different perpective using the scanning tunneling microscope (STM)[6, 7, 8]. The actual fabrication of these electronic devices at the molecular and atomic scale poses new challenges for theoretical physicists and chemists who must develop new theories and, most importantly, implement existing ones to address the present and upcoming questions. The basics to calculate the zero-bias, zero-temperature conductance Q of a molecular bridge or metallic nanocontact were established by Landauer long before the concept of molecular electronics was commonplace. In Landauer's formalism Q is simply related to the quantum mechanical transmission probability T for electrons at the Fermi level to go from one electrode to the other [9] (we will assume spin degeneracy throughout):

g = goT(EF) = —T(EF) = -^YlT^)i

(!)

The fact that the zero-bias conductance is proportional to an intrinsically coherent quantity such as the elastic transmission probability should not

4

J. J. Palacios et al.

Figure 1: Schematic drawing of the scattering of conduction channels in a constriction in a semiclassical picture. Quantum conduction channels are typically associated to specific quantum numbers of the transversal part of the wavefunctions occupied at the Fermi level when the Schrodinger equation is separable in bulk. come to us as a surprise at zero temperature since inelastic scattering events due to vibrations or many-body interactions are strictly prohibited. The fact that the concept of elastic transmission probability or elastic scattering can still be determinant even at finite bias and temperature is what makes Landauer's formula extremely useful. This is esentially due to the fact that inelastic scattering lengths are typically much larger than the extent of the nanoscopic region that determines the conductance of the whole device. The wide success of Landauer's formula lies in its simplicity. However, this simplicity is misleading and still is the source of some debate (for a recent discussion see Ref. [10]). Landauer's formula has been discussed on numerous occasions[9] and we do not intend to do it again in these notes. Instead, we will simply give some plausible arguments in favor of it. The sum in the last expression of Eq. 1 runs over all possible conduction channels deep in the bulk of the metallic electrodes and not only over the conducting channels in the actual nanoscopic region of interest. It is customary to associate quantum conduction channels to specific quantum numbers m, n of the transversal part of the bulk wavefunctions occupied at the Fermi level when the Schrodinger equation is separable in bulk; in other words, \&(f) = ijjm(x)
Molecular Electronics with Gaussian98/03

Figure 2: Schematic drawing of the alignement between the molecular orbitals of the molecule and the Fermi energy of the metallic electrodes for two extreme cases as discussed in the text. R = A is still finite due to the electrons in all the other conduction channels that fully backscatter (see Fig. 1). It is also implicit in Landauer's formalism that the Joule heating associated to this resistance takes place away from the nanoscopic region. The factor QQ in Eq. (1) can be easily understood making use of the Heisenberg's uncertainty principle. We can define the current per channel as I = e/At, where At is a measure of the time an electron takes to cross the sample and e is the electron charge. On the other hand we can define a typical energy scale for the electron, AE, as V = AE/e, where V is the bias voltage. Knowing that Q = I/V, Heisenberg's uncertainty principle leads us to Q = e2/(AtAE) « e2/h. Unfortunately, the simple expression in Eq. (1) can seldom be used as it is since channels cannot always be defined deep in the electrodes[ll, 12]. This is only the case when translational symmetry is present and the three-dimensional Schrodinger equation can be separated into transversal and longitudinal solutions (actual electrodes do not fall into this category). Instead it is convenient to use a different, but completely equivalent expression (see below) where the sum does not run over conduction channels, but over localized basis functions. Still, the concept of channel can be rescued as will become clear after a few examples have been seen. Finally, we refer the reader to Refe. [9, 13] for other issues regarding inelastic effects and many-body corrections to Landauer's formalism. The overall transmission probability T(Ep) can be easily estimated on generic considerations for metallic nanocontacts[14, 15] either by estimating the section of the nanocontact or by counting the valence orbitals of the metallic element. It is much more difficult, however, to do so for molecular bridges. The main reason is the following: One does not know a priori where the electrode Fermi energy E-p lies with respect to the molecular lev-

5

6

J. J. Palados et al. els when metal and molecule are brought into contact. In a simple-minded picture the positioning of the molecular levels with respect to the electrode Ep will depend on the relative values of the work function of the metal with respect to the electron affinity and the ionization potential of the molecule. In simpler words, if Ep lies above the lowest unoccupied molecular orbital (LUMO), a charge transfer is expected from the metal to the molecule until the charge dipole aligns Ep with the LUMO [see Fig. 2(b)]. Thus, electrons will flow through the LUMO. If, on the other hand, E-p lies below the highest ocuppied molecular orbital (HOMO), charge will transfer from the molecule to the metal until E-p and the HOMO line up [see Fig. 2(a)]. If this is the case, the transport will reflect the character of the HOMO. Unfortunately, the coupling or hybridization of the molecular levels with the free-electron levels in the electrodes compounds significantly this picture. In general, Ep will lie in the middle of the HOMO-LUMO gap (where metalinduced gap states are likely to appear) and the current will have a weak tunnelling component. In summary, the conductance is strongly dependent on the particular molecule, the detailed atomic arrangement of the electrodes where the molecule binds, and the chemical nature of the various elements at play. In order to give an answer to these questions one has to abandon parametrized models and rely on first-principles or ab initio calculations.

3

Numerical implementation of Landauer formula

3.1 Basic caveats Implementing Landauer formalism requires to know the electronic structure of a finite region embedded in an infinite system with no particular symmetry or periodicity and, if a finite bias is applied, out of equilibrium. This is a non-trivial problem to which a lot of effort has been recently devoted. However, the difficulties extend far beyond the usual technical problems in first-principles calculations and, at present, there is no agreement on which the most reliable implementation is. One has to take into account various independent aspects that can affect the final outcome. We enumerate them below in, to our judgement, increasing order of importance: • The self-consistent field approximation. If one sticks to a standard density functional theory (DFT) approach, this will be, in most cases, the ingredient that introduces less uncertainty in the final results. • The basis set. Using finite basis sets always gives a certain degree of uncertainty which can be easily overcome by testing the results against larger basis sets (whenever feasible).

Molecular Electronics with Gaussian98/03 7 • The atomic structure. Calculating the possible atomic arrangements in the electrodes and the possible ways the molecule can bind to them is known to be a major problem in itself. Many standard techniques already implemented in most popular quantum chemistry codes can be applied to this purpose and will not be given further consideration in these notes. Unless especified otherwise, we will assume relaxed atomic structures throughout. • The model for the bulk electrodes. This is related to the previous point and, in our opinion, is an important factor in the uncertainty of any claimed result. There is no obvious way of taking into consideration the bulk of the electrodes in the calculation. • The experimental evidence. Finally, it is crucial to know the details on how the experiment has been done to reduce the phase space of possibilities to consider in the theoretical modelling. Possibly the main advantage of the numerical implementation that we review here with respect to similar proposals that have appeared in the literature[16, 17, 18, 19, 20] is the use of the standard quantum chemistry code GAUSSIAN98/03[21] (see Refs. [22, 23, 24, 25]). The GAUSSIAN98/03 code provides a versatile method to perfom first-principles or ab initio calculations of clusters, incorporating the major advancements in the field in terms of functionals, basis sets, pseudopotentials, etc.. Other groups also make use of GAUSSIAN98/03 at the core of their numerical implementations[26, 27, 28], but these differ in a number of technical details that might deserve attention when a quantitative agreement with experiments is required. We will refer to our method as the Gaussian Embedded Cluster Method (GECM) from now on. 3.2

Methodology: Zero bias, zero temperature

Our procedure goes as follows. A standard electronic structure calculation of the region that includes the molecule and a significant part of the electrodes is performed [see Fig. 3(a)]. Including part of the electrodes in the self-consistent calculation is essential since, as explained above, this will guarantee the proper alignement between the Fermi level of the electrodes and the molecular levels. Here we are faced with the first serious source of uncertainty in the results (not only in ours, by the way). We are forced to describe at the atomic level the structure of the molecular junction or nanocontact. There is, however, a statistical uncertainty in the atomic structure due the many uncontroled factors inherent to most experiments that is impossible to account for by a single model. A statistical analysis is, on the other hand, computationally forbidden. The calculation can be performed within any self-consistent field approach: Hartree-Fock or DFT

8

J. J. Palacios et al.

Figure 3: (a) A molecular bridge model where a short capped nanotube is contacted by two (001) parallel surfaces. The standard self-consistent procedure performed initially by GAUSSIAN98/03 is also shown schematically on the right, (b) The same system where the first phantom atoms of the Bethe lattices are also depicted. The self-consistent procedure when the Bethe lattices are included is also shown schematically on top. (c) A finite section of a Bethe lattice model in two dimensions with coordination three. in any of its multiple approximations. As far as transport is concerned, the self-consistent hamiltonian matrix H (or Fock matrix F) of this central cluster or supermolecule contains the relevant information. We will now make use of the Green's function formalism. The retarded(advanced) Green's functions associated with F are defined in the standard way[29]: [(£±i<5)I-F]G(±)=I.

(2)

In this expression I is the identity matrix and S is an infinitesimal quantity (in practice 10~10 will do it). One of the many advantages of the use of Green functions is that one can incorporate the rest of the infinite electrodes in a very elegant and simple way:

[(SI - F - S ( ± ) (£)] G (±) (E)=I,

(3)

SW(£) = SW(E) + 4 ± ) ( E ) ,

(4)

where and S R ( S ) [ S L ( £ 1 ) ] denotes a self-energy matrix that accounts for the part of the right (left) semi-infinite electrode which has not been included in the

Molecular Electronics with Gaussian98/03

self-consitent calculation. This is another source of uncertainty in the calculations. A correct choice of this quantity or, in other words, an appropriate model for the bulk of the electrodes can avoid spurious results and misleading conclusions. The first problem one encounters is that the self-energy matrices can only be explicitly calculated in ideal situations. This is why a number of groups in the field have chosen to model the bulk electrodes as perfect semi-infinite crystals[30, 18, 26]. In our opinion this choice can only be justified for mathematical convenience. It is, however, the least appropriate one since real electrodes are not monocrystals, but polycrystalline structures with no expected long-range order beyond a few lattice constants, particularly near the contact where the atomic structure cannot possibly be perfect. In these models the perfect crystal band structure necessarily reflects in the final conductance of a molecule and this is not desirable. Alternatively, jellium models are used in methods based on a description in terms of scattering states[16, 20,17]. This is also a choice of convenience not free from controversy [31, 32]. We choose to describe the bulk electrode with a Bethe lattice parametrized tight-binding model with the coordination and parameters appropriate for the chosen electrode (part of which has already been included in the cluster). Figure 3(c) depicts schematically a Bethe lattice of coordination three. The advantage of choosing a Bethe lattice resides in that, although it does not have long-range order, the short-range order is captured and it reproduces fairly well the bulk density of states of most commonly used metallic electrodes. For each atom in the outer planes of the cluster, we choose to add a branch of the Bethe lattice in the direction of any missing bulk atom (including those missing in the same plane). In Fig. 3(b) the directions in which tree branches are added are indicated by bigger atoms which represent the first atom of the branch in that direction. Assuming that the most important structural details of the electrode are included in the central cluster, the Bethe lattices should have no other relevance than that of introducing the most generic bulk electrode for a given metal. In Appendix A we explain how to calculate Bethe lattice self-energies for those interested in the mathematical details. As many other ab initio codes GAUSSIAN98/03 makes use of nonorthogonal localized orbital basis sets. In a non-orthogonal basis the Green's function takes the form (see Appendix B)

G W (E) = S [(£S - F - £ W (£) j - 1 S

(5)

where S is the overlap matrix. As explained in Appendix B this Green's function allows us to obtain the density matrix P through the expression

P = _- f

F lm\s-lGi-+\E)S-1]

1" J-oo

L

dE. J

(6)

9

10

J. J. Palacios et al. We can now obtain the total charge through the text-book expression N = Tr[P • S],

(7)

where the trace runs over all the basis set elements of the cluster. The upper limit of the integral in Eq. (6), Ep, is not the Fermi energy of the bulk metal electrode since these are finite in the actual calculation. (The bulk Fermi level can only be obtained in the thermodynamic limit). Instead, Ep is set by imposing overall charge neutrality in the cluster. The above expression of the Green's function in non-orthogonal basis is formally correct, but it is not the standard one as commonly appears in the literature:

G (±) (£) = [tfS - F - E (±) (£)] ~ •

(8)

The disappearance of two S matrices in the definition of the Green function is compensated by a redefinition of the density matrix which now becomes the "stantard" one:

P =- - f

F

Im [ G ( + ) ( £ ) 1 &E.

(9)

The integral in Eq. (9) or Eq. (6) can be efficiently calculated along a contour in the complex plane[19,18, 27]. We now force GAUSSIAN98/03 to evaluate F using the density matrix defined in Eq. (6) instead of the standard one obtained from filling up the molecular orbitals obtained from diagonalizing the Fock matrix[33]. The procedure is repeated until self-consistency is achieved in the density matrix to the desired accuracy. This is, in fact, the hardest part, computationally speaking and our main achievement. It is also interesting to note that the applicability of this approach can be easily implemented in any quantum-chemistry or ab initio code as long as it is based on a localized orbital expansion of the self-consistent, single-electron wavefunctions. Finally, as regards the DFT calculations for the core clusters we will currently use the B3LYP functional[21] and the basis sets and core pseudotentials described in Refs. [34, 35, 36]. We will check the robustness of our results against different functional and basis sets when necessary. Another advantage of using Green's functions is that we can now obtain the transmission probability using a more general expression than that provided in Eq. (1)[9]: T{E) = Tr[rh(E)G^(E)rR(E)G^(E)}.

(10)

In this expression Tr denotes the trace over all the orbitals of the cluster are given by i[S+(L>(£!) - £~, L) (.E)]. Finally, in the matrices TR^(E) order to single out the contribution of individual channels to the current, one can diagonalize the transmission matrix. While the size of the matrix FiG^riiG^ can be as large as desired, the number of eigenvalues with

Molecular Electronics with Gaussian98/03

11

a significant contribution will be typically much smaller and it will be determined by the narrowest part of the nanocontact or the molecule. Notice that the expression for the transmission probability in Eq. 10 is directly applicable in non-orthogonal basis sets as long as one uses Eq. 8 to define the Green's function instead of the mathematically more appropriate definition of Eq. 5. The transmission can also be found in a slightly different form. We first rewrite Eq. 10: T(E) = Tr [TlL/2(E)GW(E)TR(E)1/2TR(E)^2G^(E)r^2(E)\

, (11)

where we have simply made use of the cyclic property of the trace after splitting the coupling matrices into two. By identifying

t(E) = i f 2 (£)G«(£)rf (£),

(12)

T(E) = Tr[t(E)tf(E)].

(13)

we can write The matrix t and its adjoint tf axe known as the transmission matrices in scattering theory and their product is a hermitian matrix with real eigenvalues. This confirms that the transmission probability obtained from Eq. (10) is a real quantity. Finally, we should stress that in any self-consistent field approximation only G(E-F) has a strict meaning. In order to obtain the actual dependence on energy of the zero-bias conductance, Q(E), which would correspond to the conductance for different values of an external gate potential that can charge or discharge the system (and change the Fermi energy), one must perform a new self-consistent calculation for each different value of the Fermi energy. We have analyzed the extent of this problem on some occasions and found that the quantity Q(E) evaluated for a charge neutral system can serve as a good approximation to the real conductance as a function of Ep. This partially justifies plotting Q(E) for a range of energy values around the Fermi energy as routinely done in the literature. This problem deserves, as will be shown below, further consideration when bound or quasibound degenerate states are present. 3.2.1

Example: Aluminum nanocontacts

Before addressing the subtleties of non-equilibrium transport and moving onto the detailed analysis of various basic experiments in molecular electronics, we would like first to present a simple case to illustrate the previous discussion. An interesting and relatively simple system where our method can be tested is an Al metallic nanocontact. From all the metallic elements studied experimentally[6] Al is the least understood despite of the fact that is one of the most accesible from a computational point of view. A complete theoretical study of electrical transport in metallic atomic contacts

12 J. J. Palacios et al.

Figure 4: (a) Conductance around the Fermi energy (set to zero) for the nanocontact shown in the inset, (b) Same as in (a) for the nanocontact shown in the inset, (c) Transmission of the individual channels as a function of energy for the nanocontact in (a). The highest trasmitting ones are doubly degenerate, (d) Same as in (c), but for the nanocontact in (b). requires a realistic modelization of the formation process of these nanocontacts. Some structural studies for Al using molecular dynamics[15] and 06 initio relaxations[37] have been reported. However, given the illustrative character of what follows, we will simply consider two archetypal atomic structures that resemble those appearing in the last stages of the breaking process of Al atomic contacts and may be responsible for the last conductance plateau: Single-atom contacts and short atomic wires. The first structure we consider consists of two opposite pyramids grown in the (001) direction and "glued" by a single atom [see inset in Fig. 4(a)]. Bulk distances are taken for the whole structure and it is understood that the appropriate Bethe lattice has been added to the outer planes of the core cluster. Single-atom contacts have been studied in the past with modified tightbinding models[14] which impose local charge neutrality. As shown in Fig. 4(a), we find that, even in this simple case, the ab initio results differ qualitatively from those obtained with these models. The value of Q around the Fermi level is « 3<7o which is remarkably different from the value of ** Go reported in Ref. [14]. This discrepancy is due to a combination of facts. First, the hopping parameters that reproduce bulk properties in tightbinding models are not adequate for atoms with low coordination numbers (like those at the tips), being these typically smaller than the 06 initio ones. Second, there are non-negligible contributions from next-to-near-neighbor

Molecular Electronics with Gaussian98/03

13

hoppings in Al, which, are usually neglected in parametrized tight-binding models[14]. This translates into the fact that the number of conducting channels at the Fermi energy is larger than 3 as obtained in Ref. [14] [see Fig. 4(c)]. The second structure we have considered is almost identical to the previous one, except for having two atoms instead of a single one forming a short chain (a dimer) at the neck [see inset in Fig. 4(b)]. This structure has been chosen on the basis of extensive molecular dynamics and ab initio simulations we have performed which often show the formation of the Al dimer before the final breaking (a similar conclusion has been recently reached as reported in Ref. [37]). The distance between tip atoms has been chosen to sustain a slight stretch. As can be seen in Fig. 4(b), this apparently harmless structural modification get us closer to the typical conductance of the last plateau in the breaking process of an Al nanocontact (G ~ So)- The number of conducting channels is now three as seen in Fig. 4(d). These results are fairly robust against changes in the number of planes forming the pyramids (i.e., the cluster size) and the crystallographic orientation of them. We should mention once again, however, that talking about an exact value for the conductance has little meaning since this will always be slightly dependent on the detailed atomic structure close to the contact which changes from experimental realization to experimental realization. Nevertheless, within this inherent statistical uncertainty, our results are fairly consistent with the average experimental results. With the hope that this example has already convinced the reader about the predictive and quantitative power of this ab initio methodology we move on to more complex problems. 3.3

Methodology: Finite bias, finite temperature

When one is interested in the response to a finite bias, Q(V), as obtained from an equilibrium calculation, can still be a good approximation to the real differential conductance dl/dV. Strictly speaking, however, what one needs to do is to calculate the current / by integrating the transmission probability in a energy window eV: 9P

I=~

f°°

T(E, V) [fL(E - eV/2) - fK(E + eV/2)]dE,

(14)

h J-oo where / L and / R are the Fermi distribution functions of the left and right electrodes, respectively. Notice that this innocent-looking expression is not a trivial generalization of Eq. (1) extended to finite bias voltages and finite temperature. The transmission coefficient that appears in the integrand T(E, V) can depend on temperature, but, most importantly, depends on V. To calculate this transmission coefficient, knowledge of the Fock matrix in the presence of a bias voltage is required which, in turn, depends on

14 J. J. Palacios et al.

the density matrix out of equilibrium. Based on standard non-equilibrium Green function techniques, it has been repeatedly shown in the literature how to do this [9]

P = - - f°° [S-1G<(E)S"1] dE,

(15)

where G<(E) = tG<">(£) [fL(E)Th(E)

+ fR(E)TR(E)}

G<+>(£).

(16)

There are several technical and conceptual issues regarding the computation of the above equations that need some discussion. We refer the reader to the literature for a detailed account of them[19, 25, 18]. Here we will simply mention two issues. Maybe the most important conceptual issue is that related to the fact that using DFT for non-equilibrium situations is not fully justified: DFT is a ground-state theory. We will make use of it simply because, at present, there is no better way to deal with out-of-equilibrium problems in an operative way. The second issue is a technical one that, in our view, has not received due attention in the literature and that we find it worth a more detailed discussion[25]: The out-of-equilibrium electrostatics. In most self-consistent approaches[16, 19, 26, 18] the Poisson equation is solved with boundary conditions appropriate for the electrode geometries (as mentioned, typically infinite planes) and the bias applied. In other words, an external and uniform electric field is superimposed to the self-consistent field created by the molecule or supermolecule. In the Keldysh formalism, where one imposes an electrochemical potential difference /J,^C — fj^c = eV,

the electric field does not need to be added if a significant part of the metallic electrodes has already been included in the central cluster. Thus, one can consider realistic electrode geometries if necessary. To be more specific, let us separate \iec into two contributions: Mec = Mchemical + ^electrostatic >

(17)

where we define Mchemical = ^electrostatic

=

lim

6N—>0

SEcole/SN

Vui^SEee/SN.

ECoie is the standard energy contribution to the total energy from the cores of the atoms and See is the contribution from the electron-electron interaction. Note that this definition is not a standard one in the sense that one usually thinks of external fields when referring to the electrostatic potential contribution to the electrochemical potential. Instead we will make use of this term when referring to any contribution coming from electron-electron

Molecular Electronics with Gaussian98/03

Figure 5: Schematic picture of the development of a potential drop between electrodes separated by a barrier on imposing an electrochemical potential difference between them, (a) Before self-consistency begins the chemical potential in electrodes is different by an amount eV and the bottom of the conduction bands is the same for both electrodes. This situation corresponds to a diffusion problem, (b) After self-consistency has been achieved the chemical potential difference practically vanishes in favor of an electrostatic potential difference between eletrodes. interactions. Strictly speaking, however, one should not use the term electrostatic potential since there are exchange and correlation contributions included in DFT. Specifically, SEcore = tr[(JP • hcorj SEn = tr[£P-hee], where hcore and hee are the core and interaction matrices composing the Fock matrix. In a self-consistent procedure with overall charge neutrality one electrode gets charged and the other discharged by the same amount. Once self-consistency has been achieved, an electrostatic potential difference V between atoms one or two layers inside opposite electrodes develops while their chemical potentials are similar, i.e., they remain neutral. On the other hand, the electrostatic potential drop will be smaller than V between atoms at the surface of opposite electrodes since they carry the charges. We show this process schematically in Fig. 5.

3.3.1

Example: Gold nanocontacts

In order to illustrate the previous discussion, we carry out calculations for Au nanocontacts. It is well known that Au can form atomic chains of up to 7-8 atoms[6]. Two types of fee core cluster models are considered now: (a)

15

16

J. J. Palacios et cd.

Figure 6: Average on-site energies of the atoms in the Au cluster shown in the inset for zero bias (white dots) and 5 V (dark dots). two (111) planes containing 19 atoms each plus a four atom chain labeled Aul9-Au4-Aul9 (a nanocapacitor can be modelled by taking out the fouratom chain leaving a wide gap between the planes); (b) two (lOO)-oriented pyramidal clusters plus a four-atom chain labeled Aul4-Au4-Aul4 [(100)oriented double-pyramid nanocontacts like those in previous section, Au5Au5 and Aul4-Aul4, have also been considered]. As for the case of Al in previous section, interatomic bulk distances have been taken for the cluster (2.88 A). All results correspond to zero temperature and an external bias voltage in the range 0-5 V. Figure 6 shows the average on-site energies (5d6s6p) on all atoms of the nanocontact Aul9-Au4-Aul9 for 0 and 5 V. This magnitude reflects only the potential energy on each atom, not telling us anything about the chemical potential (charge) on them. As can be seen, at 5 V bias the major drop in the potential occurs at the chain/plane contacts. While the total potential drop is 4.44 eV the drop between the first and the last chain atoms is 1.67. In the case of zero bias the results are symmetric with respect to the geometric center of symmetry, as expected. For finite bias the potential drop between the left electrode and the first atom in the chain is 1.92 eV while it is only of 0.85 eV between the chain end and the right electrode. This feature seems common to all previously reported results and reflects not only bulk band structure features, but also details of the nanocontact geometry [18]. The voltage-dependent transmission T(E, V) for V = 0 and 5 V is shown in Fig. 7 and changes significantly. For instance, the gap below -0.5 eV

Molecular Electronics with Gaussian98/03

12

" Au

0.9 H

0.6 •

0.3

tjx/y?^ f^'

i l l [W^ j

-2.5 -1.5 -0.5 0.5 1.5 Energy (eV)

2.5

Figure 7: Tranmission versus energy for V=0 (continuous line) and 5 V (dashed line) for the Aul9-Au4-Aul9 nanocontact shown in Fig. 6.

Table 1: Fittings of the numerical results for the variation with applied voltage V of the difference between the average on-site energies of the atomic orbitals at the outer planes in clusters similar to those of Figures 1 and 2 (see text) and of current through them, with functions AE = aVb and I/QQ = cVd, respectively (AE, I/QQ and V given in eV). cluster Au5-Au5 Aul4-Aul4 Aul4-Au4-Aul4 Aul9-Au4-Aul9 Aul9-vacumm-Aul9

0.803 0.840 0.882 0.882 0.928

a b 0.935 0.993 0.971 1.003 1.001

e 0.521 0.668 0.805 0.886

d 1.068 1.067 0.950 0.91

17

18 J. J. Palacios et al.

is partially filled. In both cases the conductance never goes above QQ as corresponds to an s-like element such as Au. It is worth noting that although the total current calculated by integrating the zero-bias transmission T(E, V = 0) in an energy window {-eV/2, eV/2] does not differ much from that obtained with the full non-equilibrium approach (not shown), the differential conductance is significantly different. In Table I we report the fittings to the numerical results for the difference between the average energies of all atoms in the left and right planes in the previous nanocontact and between the two outer planes in the pyramidal nanocontacts versus the applied bias V. The fittings were done with aVb, instead of assuming a linear relationship from the start. All regression coefficients were higher than 0.999. Although a and b are always rather close to 1, some significant features are worth of comment. There is a steady increase of a and b as the size of the cluster increases. The values closest to one are found for the capacitors (the nanocontacts with an « 14 A gap). In the latter case the coefficient a is only 7% smaller than 1, while the exponent cannot in practice be differentiated from one. As explained above, the small deviations with respect to the "ideal" behavior are due to the finite charge that those planes carry. As expected, as the size of the cluster increases, the results steadily approach the macroscopic behavior where the applied electrochemical potential difference eV and the potential drop coincide. These results prove our previous statement: No external field needs to be added for sufficiently large electrodes. The results for the current versus voltage were also fitted to I/Go = bVd (see Table I). In this case the lowest regression coefficient is 0.99 (in some cases the results are better fitted with second order polynomia). The significant deviations from a linear behavior (Ohm's law) are not surprising in systems as small as those investigated here and have been reported previously [18].

4

Towards a quantitative description of single-molecule junctions

The following cases illustrate how the technique we have just presented in previous section can be of great help in the interpretation of various experiments related to the field of molecular electronics. We have chosen to focus on the transport properties of three paradigmatic molecules: H2, Ceo, and a (5,5) metallic carbon nanotube. These molecules have been selected on the basis of either their practical and basic interest. At the same time they will take us on a trip through the three main experimental techniques used nowadays to create single-molecule junctions: MCBJ's, STM's and conventional lithography.

Molecular Electronics with Gaussian98/03

19

Figure 8: (a) Conductance as a function of energy (referred to the Fermi energy) for the (lll)-oriented 6-3-1-3-6 clusters with (dashed line) and without (dotted line) mirror symmetry and for the (lOO)-oriented 9-4-1-4-9 cluster (solid line); (b) same as in (a) but for the (lll)-oriented 6-3-1-1-3-6 clusters with (dashed line) and without (dotted line) mirror symmetry and for the (lOO)-oriented 9-4-1-1-4-9 cluster (solid line); (c) transmission eigenvalues for the 6-3-1-3-6 cluster with mirror symmetry [see inset in (a)]; (d) same as in (c) but for the 6-3-1-1-3-6 cluster [see inset in (b)] with mirror symmetry.

4.1

Platinum nanocontacts with H2 molecules

We start our set of examples by showing how the methodology just described can shed light on a recent experiment [38, 39] of electronic transport in Pt nanocontacts where, apparently, H2 molecules anchor themselves to the metal and modify the standard conductance of pure Pt. The H2 molecule is the simplest molecule and is ideal to carry out a detailed structural and transport analysis. The main conclusion in this section is the following: Contrary to the authors' explanation for the conductance histograms where the main peak around Q sw Go (for Go = 2e2/h) is attributed to a configuration where a H2 molecule bridges the gap between Pt electrodes (see also Ref. [40]), we attribute this peak to a situation where the H2 molecule dissociates at the nanocontact and forms a stable complex Pt2H2 where the two H atoms chemisorb between electrode tip atoms [see left inset in Fig. 10]. We speculate that the bridge with H in molecular form could account for the peak around Q « O.lSo reported in Ref. [38], but not commented on there. As a starting point we consider the conductance of a clean Pt nanocon-

20

J. J. Palacios et al.

tact without H2. As mentioned on several occassions in this review, it would be desirable to simulate the whole breaking (or formation) process of the nanocontact on stretching (or indentation). This is, unfortunately, computationally not feasible from first principles for Pt. As we did for Al, we select a few representative atomic arrangements for the lowest conductance plateau out of the many possible. There is "visual" evidence for Au[41] that confirms the formation of pyramids on both (111) and (001) stretching lattice directions in mechanical break junction experiments. Assuming the same for Pt given the lattice similarities, we have chosen to study the conductance for the following nanocontact atomic configurations: 9-4-1-4-9 (single-atom contact) and 9-4-1-1-4-9 (chain contact) for the (001) direction and 6-3-1-3-6 (single-atom contact) and 6-3-1-1-3-6 (chain contact) for the (111) direction (the numbers indicate the number of atoms per plane). We have considered two possible plane orderings for the (111) direction: 6B-3A-.. .-3A-6B and 6C-3A-. ..-3C-6A (remember that the bulk one is ... A-B-C-A-B-C . . . ) . These nanocontacts correspond to cases with and without mirror symmetry with respect to the middle plane perpendicular to the stretching axis z [the (001) nanocontact has also been chosen with mirror symmetry]. Specifically, we have taken bulk atomic distances for the atoms in the outer planes, but we have performed full relaxation in all the other coordinates, including the distance between the two outer planes. The resulting structures are thus expected to be representative of the ones responsible for the last(first) conductance plateau on stretching(indentation) cycles. The conductance reaches « 2ft> at the Fermi energy (set to zero) for the two short-chain cases with mirror symmetry and drops slightly for the other one [see Fig. 8(b)]. Two conduction channels mainly contribute to the conductance in the three clusters considered with the two-atom chain [see Fig. 8(d)]. Given the polycrystalline nature of the electrodes, one should not expect mirror symmetry to be the dominant situation in the experiment so we can expect an average conductance around Q « 1.75ft)- This value nicely coincides with the position of the lowest conductance histogram peak reported in Ref. [38]. The conductance of the single-atom contacts presents similar values for the (111) direction [although, as Fig. 8(c) shows, three channels now contribute to transport], but is appreciable higher in the (001) direction [see Fig. 8(a)]. Although we cannot afford a statistical analysis, the fact that the position of the lowest conductance histogram peak shifts to higher values when single-atom contacts are favoured (on only considering indentation cycles or higher bias voltages as reported in Ref. [42]) is consistent with our results. The conductance is also here noticeably dependent on the detailed atomic arrangement of the electrodes (note that the conductance increases appreciably on the scale of 1 eV below and above the Fermi energy). This is due to the fact that the Fermi level lies on the edge of the d band for bulk and the contact atom has almost bulk coordination. This is also fairly consistent with the fact that Pt histogram peaks are broad in

Molecular Electronics with Gaussian98/03 _274.07 p

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Figure 9: Total energy as a function of the distance of the H2 molecule to the axis of the nanocontact for the cluster shown in the insets. The relative orientation of the molecule has been forced to that shown in the insets (along the axis), but the rest of coordinates are allowed to relax. The upper inset corresponds the minimum of the curve. This minimum indicates that a certain amount of strain is necessary in order for the molecule to move in between the two Pt atoms of the tips. comparison to, e.g., those of Au[6]. When the same conductance measurements are performed in a H2 atmosphere the histograms change noticeably. Inelastic vibrational spectroscopy along with theoretical support lead the authors in Ref. [38] to conclude that a H2 molecule can position itself along the stretching axis of the Pt nanocontact right before it breaks apart. The authors claim that this configuration accounts for the main histogram peak for conductance where Q » (?o- With this claim in mind we consider nanocontacts like that shown in the inset of Fig. 8(b), but with a H2 molecule anchored in between electrode tip atoms. The results presented below are for the smallest cluster in the (001) direction, 4-1-H2-1-4, but we have performed similar calculations for (111) nanocontacts for reassurement. We maintain bulk interatomic distances for the four atoms in the two outer planes whose variable respective distance d mimics the effect of the piezo voltage. For every d we let the positioning of the H atoms and that of the two contacting Pt atoms relax, starting from a configuration where the H2 molecule sits along the stretching axis. Full circles in upper curve in Fig. 10 represent the total energy vs. d for all the stable configurations we have found in which the molecule stays on the zaxis (see right inset in Fig. 10). Beyond d « 9.5A the molecule sticks to one of the electrodes or dissociates. Below d « 8.25A the molecule tends to loose

21

22 J. J. Palacios el al. -274.001

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9

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-274.12 -

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6.0

6J

7.0

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d (A)

8J

9.0

9.5

10.0

10.5

Figure 10: Total energy of a H2 molecule in a Pt nanocontact versus distance between electrodes (outer planes). Upper curve circles correspond to the H atoms positioned along the stretching axis (2) in molecular form (see right inset) and lower curve circles correspond to the H atoms positioned on the plane perpendicular to the stretching axis (see left inset). its alignment with the 2-axis. (As a guide to the eye we also present with empty circles the energy for configurations where the molecule is forced to stay along the z-axis). In all these points the H2 retains its molecular form. We also observe that a certain amount of stress is necessary in order for the molecule to position itself in between Pt atoms. This can be more easily seen in Fig. 9 where we see a minimum in the bonding energy at a position w 1 A away from the nanocontact. At an intermediate point (d « 9 Ain Fig. 10) the H-H bond distance and the H-Pt distance turn out to be similar to those proposed in Ref. [38]. Figure 11 (a) shows Q{E) for d = 9.25A (we chose the distance that more closely reproduced the vibrational mode frequency reported in Ref. [38]. The conductance at the Fermi energy turns out to be remarkably smaller than Qo- A similar calculation for the whole range of stable configurations shown in Fig. 10 gave a conductance in the range 0.2-0.5C/O- To our ease this result agrees with the intuitive picture: Closed-shell molecules cannot conduct if the molecular character is maintained after contact with the electrodes and the charge transfer is much smaller than one. Both conditions seem to hold here. The charge transfer from the H2 molecule to the electrodes is typically « 0.15 or smaller and the bonding and antibonding molecular levels maintain their character since they are clearly visible in the density of states (not shown here) around -6 eV and 17 eV, respectively. A value of conductance close to Qo is still possible through hybridization of the Pt states with those of the molecule[40]. In our calculations, however, this does not occur at the Fermi energy, but at E « —1.5 eV. We believe that

Molecular Electronics with Gaussian98/03 3i

1

-r

I

°-2

- 1 0

Energy (eV)

p

(a) A

1

-

2

23

^

-

1

!

0

Energy (eV)

(b) •

1

2

Figure 11: (a) Conductance as a function of energy (referred to the Fermi energy) for the cluster shown in the inset where a H2 molecule sits along the stretching axis of the Pt nanocontact (d = 9.25A). Dotted lines in (c) represent the transmission of the individual eigenchannels for the cluster in (a), (b) Same as in (a) but for the structure shown in the inset where the two H atoms form a double atomic bridge (d = 6.5A). Dotted lines in (d) represent the transmission of the individual eigenchannels for the cluster in (b). this H2 bridge actually accounts for a secondary peak around Q =» 0.1ft) n °t discussed, but clearly present in the reported data[38]. So, what accounts for the main conductance peak at « Go?. There is little doubt that this peak is H-related, but here we propose an alternative to the H2 molecular bridge discussed above. From the previous calculations we observe a "natural" tendency for the molecule to stand up-right in between Pt atoms for small distances between outer planes. We now perform calculations similar to the previous ones, but initially placing the molecule axis perpendicular to the stretching 2-axis. We let the Pt tip atoms and the H atoms relax towards equilibrium for several values of d without assuming the H2 to be in molecular form. The total energy of the stable configurations we have found are represented by full circles in the lower curve in Fig. 10. (Again, for completeness, we also present results when the H atoms are not allowed to leave the plane perpendicular to the 2-axis). The two H atoms are only in molecular state at the upper limit of this curve (d « 8 A). For smaller values of d the H2 molecule breaks apart and the H atoms position themselves up to 2.0 A apart from each other forming a double atomic H bridge as the one shown in the left inset of Fig. 10. The results of conductance for d = 6.5A (a distance for which vibrational modes are closer to the experimental ones) are depicted in Fig. ll(b). The conductance at the Fermi energy approaches closely QQ. The eigenchannel transmission results

24 J. J. Palacios et al. _274,04 |

-274.08

8

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.

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Figure 12: Total energy as a function of the distance of the H2 molecule to the axis of the nanocontact for the cluster shown in the insets. The relative orientation of the molecule has been forced to that shown in the insets, but the rest of coordinates are allowed to relax, including the H-H distance. The lower inset corresponds to the molecule far from and perpendicular to the axis, The only energy minimum is found at zero distance when the H atoms separate (see upper inset). are also consistent with the noise analysis in Ref. [38] since, the fact that fluctuations do not strictly vanish at QQ may indicate that more than one channel is contributing to the conductance. At this point the distance between tip Pt atoms is approximately that of bulk Pt and that between tip Pt atoms in Fig. 8(b). When comparing with the results presented in Fig. 8(b) we see that the H seems to block out one of the channels. It is important to finally stress here that the exact value of the conductance is not very dependent on the basis set and the functional. However, we note once again that the atomic configuration of the electrodes in the cluster and the model used to represent the bulk of these electrodes introduces a source of uncertainty in the results. We believe that this is the source of the discrepancy between our results and those presented in Refs. [38] and [40]. After considering various other possibilities in terms of cluster sizes and orientations, we can safely conclude that the stark difference between the conductance of the two basic bridge structures studied in this work is conclusive. Dissociation of the H2 molecule at low temperature is not that surprising. Although it is well known that the molecule is not dissociated upon adsorption on flat, perfect surfaces (and our procedures were checked to reproduce this result), it may well be possible in the presence of low-coordinated Pt (such as in the present case or in strongly stepped surfaces). We have studied the formation probability of these structures letting the H atoms fully relax starting from H2 in molecular state far from the nanocontact. Although a

Molecular Electronics with Gaussian98/03 25 full dynamical adsorption analysis is beyond the scope of this work, we have found that these novel configurations can be reached starting from molecular H fax away from the nanocontact without surpassing activation barriers (see Fig. 12). We have also found higher-energy stable configurations or metastable local minima; for instance, on some relaxations we have found that a single atom locates between the Pt electrodes while the other moves onto the electrode surface. Although these metastable configurations can play an active role in the formation of the actual nanocontact and in the histograms, we defer the taxonomy of this problem for future work. 4.2

Scanning tunneling spectroscopy of fullerenes adsorbed on Au surfaces

Since its discovery in 1982 by Binnig et a/. [43] the Scanning Tunneling Microscope (STM) has proved to be an indispensable tool to study the mechanical, electrical, and geometrical properties of surfaces, adsorbates and, as mentioned in the introduction, to build metallic nanocontacts and nanodevices. The tremendous power of STM comes both from its highly local character, which allows close inspection at the atomic level, and its dependence on the electronic structure of the substrate being scanned. This is reflected by Tersoff and Hamann's formula[44, 45] who, simplifying Bardeen's first-order formalism [46] by approximating the electronic structure of the tip, arrive at the following equation: fEF+eV

Joe /

pt(E-eV)Ps(E,rt)dE.

(18)

JEF

As deduced from this formula, the current I increases with the density of states (per unit volume and energy) in the tip, pt(E), and the density of states in the sample localized around the radius of curvature of the tip, ps(E, rt), both lying in the energy window of width eV between the electrochemical potentials of the electrodes. Bardeen's approximation consists in assuming that the states on each side of the energy barrier (built up between the electrodes by their different work functions) decay in the vacuum, being their mutual overlap the origin of the exponential decay of the tunneling current with tip-sample distance. Actually, following Lang[47], the barrier may be simplified by a square potential well of width s, and the tunneling current attains the following form: pEF+eV

Joe /

JEF

.

pt(E - eV)ps(E)e-2s^('t'+EF-E)+eVdE

(19)

where cj> represents the work function of the sample. Unfortunately, the above formulae deduced within Bardeen's formalism are not valid when tip and sample come in closer contact. This occurs, for

26 J. J. Palacios et al.

Figure 13: Left panel: Normalized conductance derived from CITS measurements by Rogero et al. [48] for the two types of C ^ adsorption sites on Au(lll). Right panel: CITS image by Rogero et aZ.[48] at a bias of +0.6V, where the difference in brightness between on-top and bridge sites is clearly seen. The circle and triangle indicate the points where the curves shown in the left panel were obtained. The Fermi level is set to zero. [Reprinted with permission from Ref. 48 (Figs. 4d, 4c)© 2002 American Institute of Physics]. instance, when the STM tip is used to fabricate nanocontacts (see, e.g., Ref. [6]). Therefore, a computational method incorporating Landauer's formalism is much more useful if one is interested in studying STM experiments beyond the tunneling regime. An efficient implementation of such a theoretical framework has been explained in Sec. 3 of this work, which has also the appealing feature of being an ab-initio method, without the need to introduce ad-hoc parameters to describe the tip-sample interaction[47, 49, 50]. In the following we will compare the results achieved with the GECM with those from a STM study by Rogero et al.[4S] of Ceo adsorbed on Au(lll). To be more specific, what these authors perform is a spectroscopic study named Current Imaging Tunneling Spectroscopy (CITS). This technique measures the tunneling current at each point during a topographic scan for a range of bias voltages applied to the sample-tip system. The numerical differentiation of each curve provides the corresponding conductance profile. From these curves it is possible to construct an image of the conductance at a given bias for each point of the topographic scan. This type of measurements provide a tremendous amount of information about the electronic structure of the sample around the Fermi energy. However, taking full profit of these type of experiments requires a theoretical method able to corroborate the interpretation of the measurements. We give next the details of

Molecular Electronics xuith Gaussian98/03

27

Figure 14: Conductance spectra calculated with our method for the on-top and bridge adsorption sites in the region around the Fermi level (here set to zero). The upper left inset shows the same data for the on-top geometry on a wider energy range, where the fullerene gap can be clearly appreciated. The other insets show the atomic models we have used in the calculations. Rogero et al.'s work, starting with a brief overview of the adsorption of Cgo on metallic surfaces. Later on we will discuss their main achievements as compared with our own findings using the the GECM[51]. Nucleatkm and growth of Ceo monolayers and thin films on Au(lll) surfaces has been widely studied[52, 53, 54, 55, 56, 57]. Although binding between gold and Ceo is weaker than with other metals [55, 58, 57], it is far from negligible; the adsorption energy is estimated around 40-60 kcal/mol[53, 57]. In fact, adsorption of Ceo is able to lift the well-known 23 x y/3 Au(lll) reconstruction, and photoemission studies of Ceo on polycrystalline Au[59, 60] and Au(lll) surfaces{57] revealed energy shifts indicative of LUMO hybridization and charge transfer from Au to the adsorbed fullerene molecule. At high coverages, closed-packed layers grow with the thermodynamically most stable adsorbate phase being a (2\/3 x 2\/3) R30° structure with a nearly perfect lattice matching in which all the molecules are in equivalent surface sites [53]. Apart from this, another superstructure forms with crystallographic directions matching those of the substrate, resulting in fullerene molecules sitting on different adsorption sites[53, 55]. The proposed 11 x 11 Cgo coverage[53, 55] for this superstructure has been recently discovered to be composed of a smaller 2x2 grid by Rogero et al. [48]. The main features that characterize the adsorption of the above-mentioned Ceo adlayers can be drawn from Fig. 13, where we show the CITS data obtained by Rogero et oZ.[48]. The image appearing in the right panel of Fig.

28 J. J. Palacios et al.

13 was obtained at a bias of +0.6V[48]. The curves plotted in the left panel of Fig. 13 correspond to the conductance profiles measured at the points indicated with the circle and triangle[48]. From the different position and width of the peaks appearing in the conductance profile Rogero et al. deduce the following facts: 1. The relatively weak C6o-Au(lll) binding reflects on the sharp form of the peaks, with the Ceo LUMO resonance shifted towards the Fermi level by charge transfer from the substrate, yielding a HOMO-LUMO gap of 2.3 eV. 2. The two curves appearing in the left panel of Fig. 13 are interpreted as corresponding to a 2x2 superstructure of the adsorbed layer that places the Cgo molecules on two different adsorption sites (bridge and on-top) of the underlying substrate surface: the interaction of the fullerene with the adsorption site being responsible for the two type of spectra. We now proceed to check if the interpretation of the experimental facts by Rogero and coworkers is confirmed by our calculations. With this aim we calculate the conductance profile corresponding to the two geometries that characterize the 2x2 Ceo superstructure (see insets in Fig. 14). The results are plotted in Fig. 14. From this figure it is immediately apparent one of the main facts found by Rogero et al. and previous authors; namely, that the interaction between the molecule and the surface is not very strong. This is reflected in the sharp peaks that reveal the underlying positioning of the Ceo orbitals, from which we estimate the HOMO-LUMO gap to be 2.9 eV, in good accordance with the experimental CITS results [48]. On the other hand, the relative positioning and height of the conductance maxima between the on-top and bridge geometries is also reproduced. The form of the peaks reflects, as suggested by Rogero et al.,[48] the different interaction of the molecules with the adsorption sites as deduced from the Potential Energy Scan (PES) shown in Fig. 15. For each adsorption site we plot DFT calculations when a Ceo molecule approaches the Au(lll) surface towards the on-top and bridge sites, respectively, with either a sixmember ring or a five-member ring facing the surface. The same equilibrium distance (2.75A) is obtained for all the geometries, which coincides with that found in Ceo-gold nanobridges[61] and is consistent with Altman and Colton suggestion of no height differences between adsorbed Ceo molecules on Au(lll)[53, 54]. As seen from Fig. 15 the adsorption energy is smaller when the molecule sits on top of a gold atom (16 kcal/mol) than when the fullerene binds to a bridge site (35 kcal/mol), which explains the narrower and higher form of the conductance maximum corresponding to the on-top geometry with respect to the bridge site. Once the aforementioned values are corrected by considering the lateral interaction of the Ceo monolayer [57],

Molecular Electronics with Gaussian98/03 300

2

r—•—i—•—i—•—i—•—,—•—i—•—i—•—i—•—i

5

0

-

1 o\ \l oil *\\

• t

E/kcalmol ' -JQO

• • o a

• Hexagon facing on-top • Hexagon facing bridge o Pentagon facing on-top o Pentagon facing bridge

\

50 -

-

%^

-50 _100

I—•

1

:

:

'

'

1.5

'

2

•

'

•

'

•

'

'

2.5 3 3.5 Distance C,0-surface / A

'

4

•

'

•—'

4.5 5

>

Figure 15: DFT Potential Energy Scan of a Ceo molecule approaching the two types of Au(lll) surface sites (on-top and bridge). Results with an hexagon and a pentagon of the fullerene facing the surface are included for completeness. The energy of the separated Ceo + surface is set to zero. estimated from the Lennard-Jones potential to be of about 25 kcal/mol[57], they lie within the experimental margins discussed above. The stronger Au-C6o interaction when the molecule sits on the bridge site is consistent with a larger amount of charge transferred from the gold surface to the molecule: 0.8 electron, as compared to the value of 0.5 electron obtained for the on-top site. This gap accounts for the different alignment of the Fermi level, which for the bridge geometry lies closer to the LUMO-derived orbitals, in complete agreement with Rogero et al.'s findings. The amount of transferred charge calculated by us is in accordance with recent results derived from photoemission spectra[57], which give 0.8±0.2 electrons per fullerene molecule adsorbed on Au(lll). Finally, we point out the fact that there exist minor differences in binding energy with respect to the symmetry axis that points towards the surface. Actually, this coincides with the fact that no predominant molecular orientation had been found [48] and is also indicative of the relative weak C6o-Au(lll) interaction, which allows a large degree of rotational freedom on the adsorbed fullerene. As mentioned before, our method is not only applicable to the tunneling regime but also at tip-sample contact distances. This is reflected in Fig. 16 where we plot the conductance of the system for the two types of adsorption sites as the tip moves towards the adsorbed Ceo- The two maxima correspond to conductance channels coming from the first three C6o LUMO-derived orbitals, whose degeneracy has been partially removed due to the interaction with the gold surface[48, 57] in two sets of two (broader

29

30 J. J. Palacios et al. 1.5 |

.

1

.

.

.

1.5 I—.—i—.—i—.—i—.—I

1 B.«

O

- 3 - 2 - 1 0 ft"»'«v

1

D=1E.DA D.15.5A D = is DA

:', : ;ji

Ail -0.5

0

0.5 *

Energy / eV 1.5

|

.

1.5 i—.

1

1

i

i

Q

/.I n

o

•

. . \[« '

14

D.140A

[AL, . , . JtA-J

m\

D.I«A

-» -! -1 ° '

0.5 -

-0.5

1

i——T—.—i

- 3 J/

o

.

1

i

="-S:S{

| U

0

0.5 Exergi'UV

1 •

Figure 16: Top panel: Conductance profile from tunneling to contact regime of a Ceo molecule adsorbed on top of a gold atom in a Au(lll) surface. Bottom panel: Same as above for the Ceo molecule adsorbed on a bridge site in a Au(lll) surface. Distance (in Amgstron) between gold surface and tip displayed in the legend. The Fermi level has been set to zero in both sets of curves. The insets in both panels plots the same data on a wider energy range at tip-surface distances of 13.5 and 14.5 A.

Molecular Electronics with Gaussian98/03 1.5 I

.

1

X

S O

.

\

A \

_4 [—, 1i

\ \-,

\

O—e On-top 4—A Bridge 11

1

.

1 12

, 1 13 Distance A

31

,

,—Mt— 14 •

\

\ ^

*J\

12 13 Tip-surface distance/ A

14 r

Figure 17: Maximum conductance from the second peak of Fig. 16 for both geometries. The tunneling regime can be distinguished by the exponential decay in the conductance beyond 13A. The inset shows the same data with a logarithmic scale for the conductance. peak) and one (sharper peak) resonances, respectively. These two peaks also appear at tunnel tip-surface distances in the experimental CITS data of Rogero et al. (see Fig. 13). We can check that, as the tip comes in closer contact with the fullerene, their interaction alters the size, width, and positioning of the peaks. The net result is an increase in the conductance to non-negligible values and a larger shift of the LUMO-derived C6o orbitals towards the Fermi level due to the larger amount of charge transferred from the tip atoms to the fullerene as they approach each other. This fact explains the slight closure of the HOMO-LUMO gap as the tip approaches the molecule, which is apparent after inspecting the insets of Fig. 16. On the other hand, the relative shape of the peaks corresponding to the on-top and bridge geometries found at tunneling distances is maintained as the tip approaches the sample. This still reflects the differences in binding strength commented above. Figure 17 shows the value of the conductance maxima of the second peak vs. tip-surface distance. We can see the change of slope due to the different conductance regimes, with the exponential decay typical of tunneling appearing beyond 13A, which agrees well with the experimental value reported by Joachim et al.[50] of 13.2A. The departure from a linear trend in the logarithmic representation of the conductance is related to the abovementioned closure of the Ceo HOMO-LUMO gap. Structural deformation of the fullerene cage has not been considered in our calculations since we

32 J. J. Palacios et al.

Figure 18: Two atomistic models for the CNT-electrode contact: (a) an open (5,5) carbon nanotube end-contacted to (111) surfaces and (b) a similar nanotube side-contacted. have focused on the change from tunneling to contact regimes, where the deformation of Ceo by the tip is negligible [50]. In summary, we have shown that the so-called GECM method is able to accurately reproduce CITS spectra. The example of Ceo adsorbed on Au(lll) represents just a starting point, but the method looks promising as a valuable tool in the interpretation of STM and CITS spectra up to the contact regime. The main results of this section have been published in Ref. [51]

4.3

Carbon nanotubes with realistic contacts

The controversy on the observed electrical transport properties of carbon nanotubes (CNT's) is due to our lack of control and understanding of their contact to the metallic electrodes. It is clear now that the CNT-electrode contact influences critically the overall performance of the CNT and that it is crucial to lower the inherent contact resistance to achieve the definite understanding of the intrinsic electrical properties of CNT's[62]. In order to determine the relevant factors behind the contact resistance so that this can be pushed down to the quantum limit per channel RQ = I/Go, a big experimental effort has been made both in CNT growth and lithographic techniques[63, 64]. While considerable progress has already been made, it is fair to admit that theoretical support has not been determinant in this regard. A limited number of works have addressed this issue previously [65, 66, 19, 67, 68], but a full analysis of this problem requires the use of the state-of-the-art ab initio techniques to calculate electrical transport like the one we are describing in this review and these are still under developement. As we have already mentioned in previous sections, there are multiple difficulties beyond the usual computational ones in first-principles calculations. The main one is that the actual atomic structure of the electrode (and probably that of the CNT) at the contact are unknown and, most likely, change from sample to sample even when fabricated under the same

Molecular Electronics with Gaussian98/03

Figure 19: (a) Conductance as a function of energy for an N = 10 (5,5) open metallic nanotube end-contacted to Al(lll) surfaces [see Fig. 18(a)]. The Fermi energy is set to zero. Inset: Schematic band structure of the metallic nanotube showing the four states responsible for the resonances, (b) Transmission as a function of energy for the highest conducting eigenchannels.

conditions. It is our believe that atomic-scale modeling, however, can still be of guidance in the interpretation of the experiments and to the future design of operational devices with CNT's. There are two ingredients in this puzzle: The atomic-scale geometry and the chemical nature of the electrode. In order to address this issue we study open single-walled metallic (5,5) CNT's contacted in two representative forms (see Fig. 18) to Al, Ti, and Pd electrodes which are among the most commonly used metals in the experiments. We find that CNT's contacted to Al in end-contact geometry [see Fig. 18(a)] the two CNT bands couple weakly to the electrodes. Moreover, we find that the two bands couple very differently to the electrodes (one of them is almost shut down for transport) and do not mix. For the side-contact geometry [see Fig. 18(b)] the coupling is the same for both bands, but equally weak as for to the end-contact geometry. Figure 19(a) shows Q{E) for a (5,5) metallic CNT composed of N = 10 carbon layers (diameter « 6.8 A and length «s 11.2 A) that has been endcontacted [Fig. 18(a)] to Al(lll) surfaces. The end-carbon-layer-surface

33

34

J. J. Palacios et al.

distance has been optimized to a value of 1.8A. It is important to obtain a correct estimate of this distance since the charge transfer and the contact coupling are fairly sensitive to it and cannot be chosen whimsically. Small variations of a few per cent away from the optimum contact distance increase the energy of the contact by a large amount. In particular, a decrease of the contact distance by a few tenths of an A, which might improve the contact transmission, is energetically prohibited. Four resonances can be seen in the conductance around the Fermi energy (set to zero) in Fig. 19. These resonances can be easily traced back to four extended states of the isolated finite CNT[69]. Two of them (A;i, fo) originate in the bonding (TT) band of the CNT and the other two (fc*,^) in the antibonding (TT*) band (see inset in Fig. 19). The resonances have different widths for different bands indicating that they couple very differently to the electrodes. Moreover, the two bands do not mix with each other. This is more clearly seen in Fig. 19(b) where we show the highest transmission eigenvalues of the transmission matrix. Two independent channels exhibit resonances in the energy window (« 3.5eV) around Ep where only the TT and TT* bands can contribute to transport. This result is consistent with the fact that TT* states, of large angular momentum, do not couple to the low-angular momentum states of the electrode, while 7T states, of low angular momentum, couple more easily[65, 66]. Notice that there is a charge transfer from the metal to the CNT, but this mainly localizes at the end carbon layer (fa 0.2 electrons per carbon atom) and it does not affect the overall band positioning in the center of the CNT. The specific band origin of the resonances is nicely confirmed by their evolution on the length of the CNT presented in Fig. 20. We present the conductance for TV = 8,9,10,11,12, and 13 carbon-layer CNT's. The opposite signs of the group velocity for the TT and n* bands make the quasi-bound states belonging to the TT* band shift down in energies while those belonging to the 7T band shift up as N increases. As expected from a simple particlein-a-box argument applied to finite CNT's[69], for N — 3Z, where I is an integer, we should expect two states with the same wave vector kn but in different bands to coincide at the Fermi energy. Naively one should thus expect Q = 4e2/h[70}. Our results for the contacted N = 9 and N = 12 CNT's show otherwise: Two resonances never coincide at the Fermi level. The reason is that Coloumb blockade prevents two (band and/or spin) degenerate quasibound states to be filled up at the same time and degeneracies are removed. From Figs. 20(b) and (e) we estimate the charging energy to be w 0.3 eV in these CNT's which is smaller than the single-particle level spacing as confirmed by experiments [63]. We have analyzed this Coulomb blockade effect in detail for the N = 9 CNT. For a partially discharged CNT the two resonances labeled fci and kl coincide in energy above the Fermi energy and the conductance reaches there Ae2/h (see Fig. 21). For the neutral system this degeneracy is partially removed and the conductance drops. The spin degeneracy removal due to Coulomb blockade requires technically

Molecular Electronics with Gaussian98/03

Figure 20: Conductance as a function of energy for an N = 8(a), N = 9(b), N = 10(c), N = ll(d), N = 12(e), and AT = 13(f) (5,5) open metallic nanotube end-contacted to Al(lll) surfaces [see Fig. 18(a)]. The Fermi energy 3,

°-2

,

,

-

,

1

1

0

,

Energy (eV)

,

,

1

Figure 21: Conductance as a function of energy for an N = 9 (5,5) open metallic nanotube end-contacted to Al(lll) surfaces [see Fig. 18(a)] for different values of the Fermi energy or charge of the system Q=-3,-2,-1,0,1,2,3. The Fermi energy has been always set to zero so that curves shift downward as Q increases.

35

36 J. J. Palacios et al.

Figure 22: Same as in Fig. 19, but for an N = 15 (5,5) open metallic nanotube side-contacted to Al(lll) surfaces [see Fig. 18(b)]. challenging open shell calculations and is currently under study. If the interpretation of the different coupling strengths of the CNT bound states with the Al electrodes is correct and angular momentum considerations are relevant, similar couplings should be expected for both bands if no axial symmetry is present. This is the case for the other contact geometry considered in this work [see Fig. 18(b)]. Figure 22 shows results for an AT = 15 CNT side-contacted to Al(lll) surfaces (the CNT-surface distance has been optimized to 2.3A). Conductance resonances come in pairs in the relevant energy window which is what is expected for an N = 15 CNT. More importantly, all of them present similar widths, confirming our expectations. Contrary to the previous geometry, localized end states[69] influence the coupling around leV for this contact geometry where mixing with the CNT extended states takes place. Our results for the coupling strength with Al contacts are consistent with previous studies where jellium models were considered as contacts[66], and with those in Ref. [19], but we do not subscribe previous ab-initio results presented in Ref. [71] based on what it seems to be more realistic contact models similar to ours. We now complete our study for end-contacted CNT's considering Ti, and Pd electrodes (see Figs. 24 and 23). Pd behaves very much like Al, except that the coupling to the CNT is clearly stronger and the overall conduc-

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Figure 23: Same as in Fig. 19, but for an N = 11 (5,5) open metallic nanotube end-contacted to Pd(lll) surfaces (see inset).

Figure 24: Same as in Fig. 19, but for an N = 10 (5,5) open metallic nanotube end-contacted to Ti(lll) surfaces (see inset).

37

38 J. J. Palacios et al. tance higher. Apart from this difference in the coupling strength, the bands do not mix and couple, as well, very differently to the electrode [see Fig. 23(b)]. In contrast to Al and Pd electrodes, where Q exhibits resonances, Q presents an oscillatory behavior for Ti around E-p [see Fig. 24(a)], with the two bands exhibiting similar coupling to the electrodes. This is accompanied by band mixing as the anticrossings in the transmission eigenvalues reveal in Fig. 24(b). This is, to our knowledge, the first numerical evidence of what has been hinted at on the basis of indirect first-principles calculations[68] and what has recently been observed in experiments[64]: Early 3-d elements as Ti are probably the best choice for making high-transparency contacts to CNT's compared to more traditional metals such as Al and Pd (apparently due to the presence of d-states at the Fermi energy). Although perfect transparency at the contact is never achieved, our calculations indicate that properly engineered Ti contacts are a good bet for future perfect contacts to CNT's. At this point, however, we can only speculate on the possibility of perfect transparency for other Ti electrode geometries. The optimal contact distance is similar to that of Al but the charge transfer is « 0.4 electrons per C atom at the end layers. There seems to be a correlation between charge transfer and contact transparency if we compare Al and Ti, but this is no longer the case when comparing Al and Pd and we cannot draw a clear correlation between these two quantities at the present time.

5

Open issues

We would not like to finish this review without enumerating some of the issues in nanoscale and molecular tranport that we have not addressed here and that are still open (maybe we should say even more open than those discussed in these notes): • Electron phonon coupling. A number of effects derived from the interaction between the electronic and nuclear degrees of freedom has been observed in a number of cases[61] and it is nowadays the subject of experimental and theoretical scrutinity[72]. This is closely related to how inelastic effects modify elastic transport and how they can be included in Landauer's formalism. • Structural relaxation in a non-equilibrium situation. The way a current can modify the atomic structure of the nanoscopic region that sustains it has been only recently addressed[73, 74] and, although, it seems to have a small effect, it is no clear whether or not it can be completely neglected from the overall picture. • Beyond a single-particle description. Present theoretical work relies heavily on the single-particle description of the electronic structure.

Molecular Electronics with Gaussian98/03 39

Many-body physics like the Rondo-effect recently observed in single molecules [75] cannot be addressed within these simplified methodologies.

6

Acknowledgements

Financial support from the Spanish MCYT (grants BQU2001-0883, PB960085, and MAT2002-04429-C03), FEDER european funds, and the Universidad de Alicante is gratefully acknowledged.

A

Bethe lattices

In this appendix we discuss how selfenergies for Bethe Lattices (BL) used to describe the leads are calculated. A BL is generated by connecting a site with N nearest-neighbors in directions that could be those of a particular crystalline lattice. The new N sites are each one connected to N — 1 different sites and so on and so forth. The generated lattice has the actual local topology (number of neighbors and crystal directions) but has no rings and, thus, does not describe the long range order characteristic of real crystals. Let n be a generic site connected to one preceding neighbor n — 1 and N — 1 neighbors of the following shell (n + i with i — l,..,N — 1). Dyson's equation for an arbitrary non-diagonal Green function is written as,

(EI-Eo)G n , f c =V B , B _iG n _ u +

Yl

V «,* G a

(20)

i=l,...,JV-l

where A; is an arbitrary site, E the energy, and Vjj is a matrix that incorporates interactions between orbitals at sites i and j (bold capital characters are used to denote matrices). Eo is a diagonal matrix containing the orbital levels and I is the identity matrix. Then, we define a transfer matrix as Ti_i,iGj_ij = Gij

(21)

Multiplying Eq. (1) by the inverse of G n _i i n we obtain,

(El - E 0 )T n _ l i n = V n , n _! + I

Yl Vni J T B _i, n

\»=1,...,JV-1

(22)

/

Due to the absence of rings the above equation is valid for any set of lattice sites, and, thus, solving the Bethe lattice is reduced to a calculation of a few transfer matrices. Note that a transfer matrix such as that of Eq. (2) could also be defined in a crystalline lattice but, in that case it would be useless.

40

J. J. Palacios et al.

Eq. (3) can be solved iteratively, -i

T B _i, n = El-Eo-

Yl

- l

Vn.n-1

^n,iTn,i

(23)

i=l,...,JV-l

If the orbital basis set and the lattice have full symmetry (including inversion symmetry) the different transfer matrices can be obtained from just a single one through appropriate rotations. However this is not always the case (see below). Before proceeding any further we define selfenergies that can be (and commonly are) used in place of transfer matrices, Sij = V y T y

(24)

Eq. (4) is then rewritten as, -i

£«-!,« = VB_i,B EI-Eo

Y,

S ».'

-i

V»-i,»

W

i=l,...,N-l

where we have made use of the general property V n ) n _i = Vn_i „• As discussed hereafter, in a general case of no symmetry this would be a set of N coupled equations (2iV if there is no inversion symmetry). Symmetry can be broken due to either the spatial atomic arrangement, the orbitals on the atoms that occupy each lattice site, or both. When no symmetry exists, the following procedure has to be followed to obtain the selfenergy in an arbitrary direction. The method is valid for any basis set or lattice. Let T\ be the N nearest-neighbor directions of the lattice we are interested in and Vn the interatomic interaction matrix in these directions. To make connection with the notation used above note that the vector that joins site n - 1 to site n, namely, r n - r n _ ! would necessarily be one of the lattice directions of the set 71. The selfenergies associated to each direction have to be obtained from the following set of 2N coupled selfconsistent equations, S Ti = V n [El - E o - (ET - E ^ ) ] - 1 V*, (26) S n = V n [El - E 0 - ( E T - S T i )]" : V t ,

(27)

where i = 1,..., N and fj = — TJ. VTi is the interatomic interaction in the Tj direction, and S x and S ^ are the sums of the selfenergy matrices entering through all the Cayley tree branches attached to an atom and their inverses, respectively, i.e., N

Sr = £ S T i

(28)

Molecular Electronics with Gaussian98/03 41 N

Sf = E s ^ -

(29)

»=1

This set of 2N matricial equations has to be solved iteratively. It is straightforward to check that, in cases of full symmetry, it reduces to the single equation. The local density of states can be obtained from the diagonal Green matrix, j

P

Gn,n= £ I - E 0 - Y, S n

(30)

t=l,..,JV

Finally, as regards the tight-binding parameters, which should include only nearest-neighbors interactions, there is no unique way of choosing them. The ones we generically use were obtained through fittings to the electronic bulk band structures[23], but the reader is free to choose any other criterion that fits better the problem at hand.

B

Green functions and non-orthogonal basis sets

In this appendix we briefly discuss how Eqs. 6 and 7 for the density matrix and the total number of particles in a nonorthogonal basis are derived. Working with a nonorthogonal basis requires the use of the metrics

£ > Q > S^1 < %| =/• a/3

(31)

Changing to an orthogonal basis \<j>a > can be done through the equation |^>=^5^1/2|^> 0

(32)

Then an expression which allows to obtain the matrix elements in an orthogonal basis of a general matrix A from those in the non-orthogonal basis, can be easily derived, Ao = S~1/2ANoS-1/2 (33) Using the above metrics and the definition of the Green's function operator, which includes the operator selfenergies,

[(£ ± iS)I -F- tf} - E^}] (?W = /,

(34)

Eq. (5) for the matrix elements of the Green's function in a nonorthogonal basis inmediately follows.

42

J. J. Palacios et al.

The local charge density is written in terms of the Green's function as [29]

p(r) = - - f

F

K J-oo

Im < r|G (+) (£)|r >

= - I fEFlmJ2Mr)S^G{+\E)S^;(r), J-°°

afrS

(35)

while the total number of particles is in its turn given by [33] N = f drp(r) = Y, Sa0Pap = Tr [P • S]. J

(36)

0a

References [1] A. Nitzan and M. A. Ratner. Electron transport in molecular wire junctions. Science, 300(5624): 1384-1389, 2003. [2] A. Aviram and M. A. Ratner. Chem. Phys. Lett, 29:277, 1974. [3] M. A. Reed, C. Zhou, C. J. Muller, T. P. Burgin, and J. M. Tour. Science, 278:252, 1997. [4] B. Mann and H. Kuhn. J. Appl. Phys., 3:4398, 1971. [5] C. Dekker. Physics Today, 52:22, 1999. [6] N. Agrait, A. Levy Yeyati, and Jan M. van Ruitenbeek. Physics Reports, 377:81-279, 2003. [7] N. Agrait, J. C. Rodrigo, and S. Vieira. Phys. Rev. B, 47:12345, 1993. [8] N. Agrait, G. Rubio, and S. Vieira. Phys. Rev. Lett, 74:3995, 1995. [9] S. Datta. Electronic transport in mesoscopic systems. Cambridge University Press, Cambridge, 1995. [10] Mukunda P. Das and Frederick Green. In J. da Providencia, editor, Proceedings of XXVI International Workshop on Condensed-Matter Theories. Nova Science, New York, 2003. [11] J. A. Torres and Saenz J. J. Phys. Rev. Lett, 77:2245-2248, 1996. [12] A. Garcia-Martin, J. A. Torres, and J. J. Saenz. Phys. Rev. B, 54:1344813451, 1996. [13] Y. Meir and N.S. Wingreen. Phys. Rev. Lett, 68:2512, 1992.

Molecular Electronics with Gaussian98/03

[14] J. C. Cuevas, A. Levy-Yeyati, and A. Martm-Rodero. Phys. Rev. Lett., 80:1066, 1998. [15] A. Hasmy, E. Medina, and P. A. Serena. Phys. Rev. Lett, 86:5574, 2001. [16] N. D. Lang. Phys. Rev. B, 52:5335, 1995. [17] K. Hirose and M. Tsukada. Phys. Rev. B, 51:5278, 1995. [18] M. Brandbyge, J. L. Mozos, P. Ordejon, J. Taylor, and K. Stokbro. Phys. Rev. B, 65:165401, 2002. [19] J. Taylor, H. Guo, and J. Wang. Phys. Rev. B, 63:245407, 2001. [20] M. Di Ventra, S. G. Kim, S. T. Pantelides, and N. D. Lang. Phys. Rev. Lett, 86:288, 2001. [21] M. J. Frisch, G. W. Trucks, H. B. Schlegel, M. A. Robb G. E. Scuseria, J. R. Cheeseman, V. G. Zakrzewski, J. A. Montgomery, Jr., R. E. Stratmann, J. C. Burant, S. Dapprich, J. M. Millam, A. D. Daniels, K. N. Kudin, M. C. Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B. Mennucci, C. Pomelli, C. Adamo, S. Clifford, J. Ochterski, G. A. Petersson, P. Y. Ayala, Q. Cui, K. Morokuma, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. Cioslowski, J. V. Ortiz, A. G. Baboul, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R. L. Martin, D. J. Fox, T. Keith, M. A. AlLaham, C. Y. Peng, A. Nanayakkara, C. Gonzalez, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, J. L. Andres, C. Gonzalez, M. Head-Gordon, E. S. Replogle, and J. A. Pople. GAUSSIAN98, Revision A.7, Gaussian, Inc., Pittsburgh PA, 1998. [22] J. J. Palacios, A. J. Perez-Jimenez, E. Louis, and J. A. Verges. Phys. Rev. B, 64:115411, 2001. [23] J. J. Palacios, A. J. Perez-Jimenez, E. Louis, E. SanFabian, and J. A. Verges. Phys. Rev. B, 66:035322, 2002. [24] J. J. Palacios, A. J. Perez-Jimenez, E. Louis, and J. A. Verges. Nanotechnology, 12:160, 2001. [25] E. Louis, J. A. Verges, J. J. Palacios, A. J. Perez-Jimenez, and E. SanFabian. Phys. Rev. B, 67:155321, 2003. [26] P. S. Damle, A. W. Ghosh, and S. Datta. Phys. Rev. B, 64:201403, 2001. [27] Y. Xue, S. Datta, and M. A. Ratner. J. Chem. Phys., 115:4292, 1991.

43

44 J. J. Palacios et al. [28] J. Heurich, J. C. Cuevas, W. Wenzel, and G. Schon. Phys. Rev. Lett., 88:256803, 2002. [29] E. N. Economou. Green's functions in Quantum Physics. Number 7 in Springer Series in Solid State Physics. Springer, 1970. [30] J. Taylor, H. Guo, and J. Wang. Phys. Rev. B, 63:121104, 2001. [31] Y. Fujimoto and K. Hirose. Phys. Rev. B, 67:195315, 2003. [32] N. D. Lang and M. DiVentra. arXiv:cond-mat/0307442. [33] A. Szabo and N. S. Ostlund. Modern Quantum Chemistry. Hill, New York, 1989.

McGraw-

[34] L. F. Pacios and P. A. Christiansen. J. Chem. Phys., 82:2664, 1985. [35] M. M. Hurley, L. F. Pacios, P. A. Christiansen, R. B. Ross, and W. C. Ermler. J. Chem. Phys, 84:6840, 1986. [36] R. B. Ross, J. M. Powers, T. Atashroo, W. C. Ermler, L. A. LaJohn, and P. A. Christiansen. J. Chem. Phys., 93:6654, 1990. [37] P. Jelinek, Ruben Perez, Jose Ortega, and Fernando Flores. arXiv:condmat/0211488. [38] R. H. M. Smit, Y. Noat, C. Untiedt, N. D. Lang, M. C. van Hemert, and J. M. van Ruitenbeek. Nature (London), 419:906, 2002. [39] R. H. M. Smit. PhD thesis, Universiteit Leiden, 2003. [40] F. Pauly J. Heurich, J. C. Cuevas, W. Wenzel, and G. Schon. arXiv:cond-mat/0211635. [41] L. G. C. Rego, A. R. Rocha, V. Rodrigues, and D. Ugarte. arXivxondmat/0206412. [42] Y. Noat S. K. Nielsen, M. Brandbyge, R. H. M. Smit, K. Hansen, L. Y. Chen, A. I. Yanson, F. Besenbacher, and J. M. van Ruitenbeek. arXiv:cond-mat/0212062. [43] G. Binnig, H. Rohrer, Ch. Gerber, and E. Weibel. Appl. Phys. 40:178, 1982. [44] J. Tersoff and D. R. Hamann. Phys. Rev. Lett, 50:1998, 1983. [45] J. Tersoff and D. R. Hamann. Phys. Rev. B, 31:805-813, 1985. [46] J. Bardeen. Phys. Rev. Lett, 6:2654, 1961. [47] N. D. Lang. Phys. Rev. B, 34:5947-5950, 1986.

Lett,

Molecular Electronics with Gaussian98/03 45

[48] C. Rogero, J. I. Pascual, J. Gomez-Herrero, and A. M. Baro. J. Chem. Phys., 116:832-836, 2002. [49] V. Meunier and Ph. Lambin. Phys. Rev. Lett, 81:5588-5591, 1998. [50] C. Joachim, J. K. Gimzevski, R. R. Schlittler, and C. Chavy. Phys. Rev. Lett, 74:2102-2105, 1995. [51] A. J. Perez-Jimenez, J. J. Palacios, E. Louis, E. SanFabian, and J. A. Verges. CHEMPHYSCHEM, 4, 2003. [52] R. J. Wilson, G. Meijer, D. S. Bethune, R. D. Johnson, D. D. Chambliss, M. S. de Vries, H. E. Hunziker, and H. R. Wendt. Nature, 348:621-622, 1990. [53] E. I. Altman and R. J. Colton. Surf. ScL, 279:49-67, 1992. [54] E. I. Altman and R. J. Colton. Surf. ScL, 295:13-33, 1993. [55] E. I. Altman and R. J. Colton. Phys. Rev. B, 48:18244-18249, 1993. [56] D. Fujita, T. Yakabe, H. Nejoh, T. Sato, and M. Iwatsuki. Surf. Sci., 366:93-98, 1996. [57] C.-T. Tzeng, W.-S Lo, J.-Y Yuh, R.-Y. Chu, and K.-D. Tsuei. Phys. Rev. B, 61:2263-2272, 2000. [58] T. Hashizume, K. Motai, X. D. Wang, H. Shinohara, Y. Saito, Y. Maruyama, K. Ohno, Y. Kawazoe, Y. Nishina, H. W. Pickering, Y. Kuk, and T. Sakurai. Phys. Rev. Lett., 71:2959-2962, 1993. [59] T. R. Ohno, Y. Chen, S. E. Harvey, G. H. Kroll, J. H. Weaver, R. E. Haufler, and R. E. Smalley. Phys. Rev. B, 44:13747-13755, 1991. [60] S. J. Chase, W. S. Bacsa, M. G. Mitch, L. J. Pilione, and J. S. Lannin. Phys. Rev. B, 46:7873-7877, 1992. [61] H. Park, J. Park, A. K. L. Lim, E. H. Anderson, A. P. Alivisatos, and P. L. McEuen. Nature (London), 407:57-60, 2000. [62] S. Frank, P. Poncharal, Z. L. Wang, and W. A. de Heer. Science, 280:1744, 1998. [63] Hyongsok T. Soh, Calvin F. Quate, Alberto F. Morpurgo, Charles M. Marcus, Jing Kong, and Hongjie Dai. Appl. Phys. Lett, 75:627, 1999. [64] J. Kong, E. Yenilmez, T. W. Tombler, W. Kim, H. Dai, R. B. Laughlin, L. Liu, C. S. Jayanthi, and S. Y. Wu. Phys. Rev. Lett, 87:106801, 2001.

46

J. J. Palacios et al.

[65] H. J. Choi, J. Ihm, Y.-G. Yoon, and S. G. Louie. Phys. Rev. B, 60:R14009, 1999. [66] M. P. Anantram. Appl. Phys. Lett, 78:2055, 2001. [67] A. N. Andriotis, M. Menon, and G. E. Proudakis. Appl. Phys. Lett., 76:3890, 2000. [68] C. K. Yang, J. Zhao, and J. P. Lu. Phys. Rev. B, 66:041403, 2002. [69] A. Rubio, D. Sanchez-Portal, E. Artacho, P. Ordejon, and J. M. Soler. Phys. Rev. Lett, 82:3520, 1999. [70] D. Orlikowski, H. Mehrez, J. Taylor, H. Guo, J. Wang, and C. Roland. Phys. Rev. B, 63:155412, 2001. [71] M. B. Nardelli, J. L. Fattebert, and J. Bemholc. 64:245423, 2001.

Phys. Rev. B,

[72] Yu-Chang Chen, Michael Zwolak, and Missimiliano Di Ventra. arXiv:cond-mat/0302425. [73] Z. Yang and M. Di Ventra. Phys. Rev. B, 67:161311, 2003. [74] M. Brandbyge, K. Stokbro, J. Taylor, J. L. Mozos, and P. Ordejon. Phys. Rev. B, 67:193104, 2003. [75] J. Park, A.N. Pasupathy, J. I. Goldsmith, C. Chang, Y. Yaish, J. R. Petta, M. Rinkoski, J. P. Sethna, H. Abruna, P. L. McEuen, and D. C. Ralph. Nature (London), 417:722-725, 2002.

Chapter 2: Molecular Dynamics Simulations of Single Molecule Atomic Force Microscope Experiments

Wiestew Nowakab and Piotr E. Marszatekb "Institute of Physics, Nicholaus Copernicus University Grudziadzka 5, 87-100 Torun, Poland hDepartment

of Mechanical Engineering and Materials Science CBIMMS, Duke University, Durham, 27708 NC, USA

Abstract The ultimate goal of nanosciences is to develop tools for spatial and chemical control of single molecules. A number of experimental techniques have emerged in recent years which allow for mechanical manipulations of single molecules including single molecule fluorescence, optical and magnetic tweezers, atomic force spectroscopy. Proper understanding of experimental results requires adequate theory. In this paper we present an overview of recent advances and trends in applications of computer simulations aimed at understanding of atomic force microscope (AFM) experiments. Special attention is devoted to those variants of classical molecular dynamics (MD) simulations, such as steered MD or biased MD, which are particularly helpful in interpretation of experimental data obtained for single biopolymer molecules. A short meta-review (review of recent reviews) of advances in the MD technology is also presented.

47

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W. Nowak and P. E. Marszalek

Glossary of Terms AFM ATP ADP BMD BPTI CHARMM DFT DNA FF FN FPS GTP IMD Ig LCD MD MscL NAMD2 PC PMF RC REMD QM/SMD QSAR RNA SCC-DFTB SMD TMD UV WHAM

Atomic Force Microscope Adenosine TriPhosphate Adenosine DiPhosphate Biased Molecular Dynamics Bovine Pancreatic Trypsin Inhibitor Chemistry at Harvard Macromolecular Modeling Density Functional Theory Deoxyribonucleic Acid Force Field Fibronectin Force Probe Microscopy Guanosine TriPhosphate Interactive Molecular Dynamics Immunoglobulin Liquid Crystal Device Molecular Dynamics Mechanoselective Ion Channel Name of Molecular Dynamics Code Personal Computer Potential of Mean Force Reaction Coordinate Random Expulsion Molecular Dynamics Quantum Mechanical Steered Molecular Dynamics Qualitative Structure Activity Relationship Ribonucleic Acid Self Consistent Charge Density Functional Theory based Tight Binding Method Steered Molecular Dynamics Targeted Molecular Dynamics Ultra Violet Weighted Histogram Analysis Method

1. Introduction Single molecule experiments are somehow at variance with the idea of "standard" chemistry that typically deals rather with myriads of molecules than a single one (Avogadro number is ~1023). However, one should remember, that the ultimate limit of miniaturization is perhaps at a single

Molecular Dynamics Simulations

49

molecule level.1 The quest for better and better computers, digital memories, transducers, sensors, etc. is the driving force of the nanotechnological revolution. Clearly, single molecule manipulations and chemistry performed on a couple of molecules are in focus of many research labs' activities. There are numerous techniques used to observe and manipulate single molecules. Description of those experimental approaches and new results were subject of many reviews and will not be repeated here. Optical methods of fluorescence2"4 based on scanning confocal microscopy, near field scanning optical microscopy and fluorescence correlation spectroscopy5"7 are very popular in biologically oriented studies.8"10 Optical tweezers,11"13 where light is used to manipulate molecules attached to dielectric beads,10 came a long path from the studies of single myosin molecules binding to a single actin filament14 to the observation of DNA packing inside lambda phage.15'16 Sometimes magnetic tweezers, known for longer time,17 are better suited for performing advanced research on single molecules.18'19 The first single-molecule AFM experiments, initiating the field of force probe spectroscopy (FPS), were performed in 1994 by E.-F. Florin, V. Moy and H. Gaub.5 In their experiments the unbinding force of the individual ligand - receptor pair in the important biotin-streptavidin complex was measured using Atomic Force Microscope (AFM). The principle of FPS is rather simple: a sharp tip of the AFM cantilever is attached to a single molecule while the other end of the molecule is firmly attached to a moving support. The motion of this support is controlled by the application of voltage to the piezo device. The elongation of the molecule requires a force and the cantilever bends under this force. Its minute deformation is usually detected by measuring changes in the intensity of the laser beam reflected from the tip part of the cantilever and projected onto a split photodiode. Since a proper calibration procedure allows for a precise determination of the cantilever force constant, and the position of the piezo stage is strictly controlled, a dependence of the molecule size with respect to the applied force may be determined with high accuracy. The basic principles of AFM experiments are presented in Fig. 1

50

W. Nowak and P. E. Marszatek

Principles of tinglt^molerule fartv
Fig. 1. The principle of AFM spectroscopy. The AFM was used, for example, in studies of unfolding single molecules20 composed of titin21'22 or fibronectin23 domains and in the measurement of the elasticity of various polysaccharides. Since piconewton forces are registered, the technique is sensitive enough to study such basic chemical processes as chair-boat transitions in glucose24"25 and other mechanically induced conformational changes26"28 in biopolymers29 or on their surfaces.30 The AFM applications in nanotechnology are enormous and range from DNA based ink31 to MILIPEDE - an innovative idea of a terabyte storage device based on the array of AFM nanotips32.

Molecular Dynamics Simulations

51

A <£•••"'•'•

'-

v

4

B

_^JL&~^~** —«*-«

•

W~;" 100 Ml

Carrion-Vazquez, Oberhauser, Fisher, Marszalek. Li & Fernandez. (2000} Prog. Bicphys. MoL Bid 74, 63-91

Fig. 2. An example of a force spectrogram of a protein composed of many identical repeats of the 127 domain of titin (A). Consecutive peaks on force - extension plot correspond to unfolding of individual Ig domains (B). The detailed analysis of peak shapes and SMD simulations reveal that main unfolding events are related to concerted rupture of several hydrogen bonds between A'-G |3 stands of Ig domains of titin (C). (Fig. 2 is reprinted from ref. [104], © 2000, with permission from Elsevier.)

The reader interested in experimental details33 is referred to the overview of AFM studies of titin and polysaccharides and application of this technique for chemical analysis purposes, presented by Fisher et al.26'34' An excellent and comprehensive review of experimental biophysical techniques used for stretching single molecules of DNA35, RNA and proteins has been published quite recently36. Force spectroscopy of single-biomolecules was also described by Gaub et al. in two short reviews.37'38 Very recent results on single molecule studies of protein folding which combine fluorescence and force spectroscopy were summarized by Rief et al.39'40 A comprehensive review of the measurements of protein-protein interaction forces was published by Leckband41 and instrumentation hints were given by Reiner et al.42 Methodological issues on using AFM in protein folding research were

52

W. Nowak and P. E. Marszalek

discussed by Best et al.,43 in biochemistry by Ikai et al.44 and in single bond detection by Williams.45

2. Meta-review on molecular dynamics Huge progress in computer technology brought MD simulations of biomolecules to nanosecond and even microsecond time regime. Many excellent books and papers cover the progress in methodology of molecular simulations and recent progress in such studies. Thus, here we present only a brief listing of available readings which indicate good sources of information on particular topics. The classical book by Haile,46 explains in detail the technical background of molecular dynamics methods. The book is rather old, so more recent books by Frankel47 or Field48 may be recommended, as well as the comprehensive, interdisciplinary oriented textbook by Schlick,49 written primarily for graduate students. For those who prefer a physicists' point of view on MD, a detailed description of algorithms by Kholmurudov et al.50 might be of great interest. An important subject of enzymatic reactivity and its theoretical description in the framework of stochastic methods and nonlinear dynamics was recently covered by authors contributing to the book edited by Werner Ebeling et al.51 In addtion to monographs or textbooks, there are also numerous journal review articles describing the applications of computer modelling to chemistry and biology. Perhaps all researchers active in the field would appreciate the historic overview of MD development presented by Karplus.52 Van Gunsteren gave a personal perspective on the recent progress (2002) in molecular dynamics simulations.53 The emerging and fast growing field of "ab initio" molecular dynamics calculations was described in detail by Hutter.54 Parinello et al.55 focused their description of the "ab initio" MD field on applications of the popular Carr-Parrinello method. Karplus and McCammon published a brief account on the role of the modeling of flexbility in uderstanding functions of biomolecules.5 Goodfellow at al.57 presented the MD methodology in simple terms and put forward some representative examples of applications. In particular, a review of studies of medically important proteins, such as p53 and HIV related, is included in this paper, along with research on protein foldingunfolding phenomena, ligand-target interactions, and a brief description of the methods of analysis of the MD results. Protein folding has been studied quite extensively in recent years, since it is perhaps one of the greatest unsolved general problems of structural molecular biology. Among numerous reviews of this subject, the most recent onces were published by Karplus et al.58'59 Vendruscolo and Paci discussed the

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53

interplay between theoretical approaches and folding experiments.60 Fersht and Daggett gave a detailed, atomistic picture of folding (and unfolding) of selected proteins from ns to microsecond.61'62 The search for a unifying mechanism of protein folding was at least partially succesful - it seems that the preformed secondary structure elements of protein colapse to the transition state which eventually make the final fold.63'64 Another large area of applications of MD methods are studies of biochemical reactions.65'66 The important field of enzymatic catalysis was critically examined by Warshel.67 Karplus tried to put forward arguments that proteins behave differently than simple chemicals.68 Dynamical aspects of chemical catalytical reactions may be modeled computationally,69 but the rigorous description of the processes of creating and breaking of the chemical bonds requires full quantum-mechanical treatment.70 Reaction kinetics is related to the changes in free energy (AF or AG) of reactants and products. Computer methods are mature enough to give realistic numbers of AF for many systems,71 though indeed the energy landscapes of proteins or biopolymers are particularly complex.72 The AFM type of experiments, in which an additional external force is added to the molecule, make these hypersurfaces even more complex and difficult to calculate. This is because typical parametrizations of forcefields are tuned to equilibrium structures of the standard benchmark molecules, and these structures are strongly disturbed by the external force. To get reasonable convergence in studies of biologically relevant conformations or realistic values of free energy estimates, one has to rely on the ergodic hypothesis, i.e. to assume that a simulation is long enough to cover all of the important regions of the phase space of the dynamical system under study. The progress in simulation time was spectacular: from a mere 60 ps simulations of BPTI by Levitt, done in 1983, through 5 ns simulations of chymotrypsin inhibitor by Dagett (1995) and the famous 1 microsecond simulation of villin fragment by Duan and Kollman in 1998, to the current distributed computing projects, which use thousands of processors to collect information for a single protein folding on a milisecond timescale.73 However, long time MD simulations are still rather expensive and rare.74'75 There is a need for approximate, but effective methods able to grasp the basic features of conformational transition on such long times. One approach, reviewed here, is to use an external, controlled force to brake a computer simulation time barrier. The stochastic approach proposed by Elber is also a promising alternative with respect to this goal.76 The scope of the applications of computational MD techniques is huge, and encompasses different topics such as high preassure simulations of biomolecules,77 drug design,78 QSAR application of DFT based methods in

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W. Nowak and P. E. Marszalek

medicinal chemistry,79 studies of metalloproteins,80 ion channels,81 membrane transporter proteins,82 just to name a few. In this review we focus on the possible applications of modeling in interpretation of AFM single molecule experiments. The most recent literature (2000 - 2003) related to forced conformational transitions will be reviewed in greater detail.

3. Variants of MD methods used in force spectroscopy of single molecules Molecular dynamics simulation, due to its very nature, is particulary suitable for modeling single molecule experiments. Perhaps the first application of the MD method in interpretation of AFM experiments on streptavidin-biotin complex rupture forces was done by Grubmiiller et al.83 in 1996. However, the strategy of realizing "in silico" a transition from one predefined (and known) conformation to the other one is not new and may be traced back, at least, to the classical reaction path studies by Elber and Karplus84 published in 1987. The review of such computational approaches was published by Elber85 and now this research has evolved into advanced long time stochastic studies of biological processes.86 The original idea of using an externally applied force to induce a "controlled" conformational change was put forward by Schlitter et al. in 1993.87 They proposed a method called targeted molecular dynamics (TMD) to enforce a transition between the initial state and a target state defined as points in multidimensional space. The first application was to the T -> R transition in hemoglobin. The TMD paved the road to more popular techniques - steered MD (SMD)88'89 and biased MD90 (BMD).

3.1 Steered MD The initial development of SMD was triggered by the needs of Duke Univeristy (Durham, NC, USA) biochemist Jane Richardson, who wanted to study "interactively" cysteine connecting loops in a model protein SScorin. The group of J. Hermans (UNC, Chapell Hill, USA) modified their computer codes in 1996 to make such forced manipulations on protein structure possible.88 Next year the first paper on SMD came from the K. Schulten group (UIUC, Illinois, USA).89 In this work the authors investigated the biotinavidin complex using a wide range of forces and various pulling speeds, much more related to real AFM experiments than those used by Grubmiiller et al.83 in their seminal study of the same molecular sytem. The conclusion of this

Molecular Dynamics Simulations

55

paper89 was rather pesimistic: since simulation and experiment timescales vary in some six orders of magnitudes, calculated forces are always too high in comparison to experimental ones. Later Galligan et al.91 studied the same system using Brownian dynamics and successfully reproduced experimental logaritmic dependence of a rupture force on the loading rate. The technique of umbrella sampling92 is commonly used to drive the system studied by MD simulations along the selected reaction coordinate. Adding an external potential keeps the desired value of the coordinate as long as necessary to collect conformations required for the calculation of free energy change (potential of mean force, PMF). The effect of external potential should be as small as possible in order to take away its bias from the Hamiltonian. In SMD the external force, usually time-dependent, pulls the system along some choosen degrees of freedom. The situation now is somewhat different: we do seek strong perturbations and large conformational changes, with external force changing rapidly in time. In contrast to classical free energy computations, termodynamic nonequilibrium is involved in this type of simulations. In SMD there are two typical protocols: SMD with constant pulling velocity and constant force. The standard way of applying external forces is to restrain some selected part of the system studied (for example one atom, a ligand or a group of atoms) by a harmonic (or other) potential. The restrain potential is localised in space, in the sense that the point in x0 e R3 is "a source" of this potential U, and there is a certain conformation of the molecule which gives the lowest energy for the constrained system. The harmonic constrain may be defined as: U = k(x-x o -d) 2 /2,

(1)

where k is the stiffnes of the external spring (restraint), d = vt, and v is a constant velocity of the motion of the restraint point. One can easily see that the corresponding external force exerted on the system is: F = k(xo+d-x).

(2)

The meaning of this force is simple - it corresponds to a molecule being pulled by a harmonic spring with its end moving with velocity v. This is exactly the situation one has in AFM experiments, where the molecule is being pulled by the tip of the cantilever. Sometimes another variant of SMD is used. The restraint point is selected far from the atoms of the molecule, the point remains fixed in space but the spring constant changes linearly with time:

56

W. Nowak and P. E. Marszalek

F = at(xo-x).

(3)

There are several reviews of applications of SMD and technical details related to the best simulation strategy published by scientists collaborating with the Schulten group.93'94 Special attention should be devoted to a very recent review by Tjakhorshid et al.95 The authors describe the results of the simulations of nearly 20 molecular systems - as different as a bilayer composed of 200 lipids, membrane-water interface, binding of estrogen receptor to DNA, apolipoprotein, calmodulin, Rieske subunit motion in cytochrome bcl complex,96 bacteriorhodopsin and purple membrane, aquaporin-1, photosynthetic light harvesting system, fibronectin type IIIi0 or CD2-CD58 complex. New results of very advanced calculations on F0-ATPase and FpATPase complexes presented in this review are of particular interest, since SMD type calculations were used to study these molecular engine molecules by adding an external torque to the carefully selected parts of the ATPase assambly. It is interesting to note, that the hardware requirements have changed immensly in the last 5 years: from one 4-processor HP computer used in early SMD studies to two 512-processor clusters applied recently to 327 K atoms of the F,-ATPase complex.95 With an inexpensive (below $20 K ) home built 16processor linux cluster and free academic software, one can run SMD simulations for 30-40 K atoms systems at a reasonable rate of 1-2 ns/day. With the advent of LCD displays having stereo capabilities, perhaps new tools of 3D modeling, similar to virtual labs proposed by Prins et al.,97 will allow for collective work in virtual reality using, for example, haptic devices. Such a device, within the framework of interactive molecular dynamics98 (IMD) was successfully applied in studies of the GlpF membrane channel and glycerol kinase." However, as far as we know, no SMD studies based on a virtual reality lab and related to AFM experiments were reported so far. Problems with matching SMD and AFM timescales may be aleviated by calculating the potential of mean force from SMD trajectories. Gilllingsrud et al.' 00 presented three time series analysis techniques: based on the weighted histogram analysis method (WHAM), van Kampen's Q-expansion and minimization of the Onsager-Machlup action. As an example, the PMF of extraction of lipids from a membrane was calculated.l00'10' 3.1.1 Titin It seems that a huge (1 urn long) muscle protein titin is the favorite object of both AFM and SMD single molecule studies. Titin is responsible for passive elasticity of muscle. It consists of hundreds of immunoglobulin-like domains

Molecular Dynamics Simulations 57

(Ig, seven-stranded P-sandwich) and fibronectin (FN) domains and less structured sequences, such as PEVK. The Gaub group102 studied single titin molecules using AFM, and they have obtained a picture of the unfolding of individual domains. The Fernandez group, investigated the dependence of unfolding forces in titin models upon stretching velocity1 The mechanism of Ig domains unfolding involves rupture of two sets of hydrogen bonds.104 Mutagenesis work proved this hypotesis, by demonstrating that the "hump" which is clearly visible in the first peak of the force spectrogram of the 127 module is absent in its K6P mutant. The mutant is not able to make the first set of hydrogen bonds that link A and B strands of I2721'105 (see Fig. 3).

1279-K6P mutant polyprotein 1000 I

g-

i

800

I

600

/

-200 \ f 0

20 40 60 80 100 120 140 160 180 200 Extension (nm)

Mamzalek, Lu, Li, Carrion-Vazquez, Oberhauser, Schulten, Fernandez (1999). Nature 402, 100-103.

Fig. 3. Experimental force-extension curve of the engineered polyprotein composed of 9 repeats of the K6P mutant of the 127 domain of cardiac titin. Reproduced with permission of ©Nature Publishing Group. The interpretation of AFM experiments as sequential unfolding of consecutive domains was supported by SMD simulation by Lu et al., which was performed for a single 127 domain of titin in water.106 Two sets of hydrogen bonds were indicated in the domain: while A-B ani-parallel strands could be easily separated (force <50 pN), the parallel A'-G stands could be ruptured in a concerted way only under the action of a large force.107 Later on, it has been found that this key event of 127 unfolding must be accompanied by extensive rearrangements of water molecules. These events impose conditions on the time length of SMD simulations with explicit solvent. m > m - ]10

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W. Nowak and P. E. Marszalek

Further studies (2001) performed by the Schulten group were devoted to the mechanism of refolding of the single 127 domain. During 2 ns SMD simulations two refolding patterns have been observed: seperated A'-G strands reformed the majority of their native hydrogen bonds, while for A and B strand 2 ns time was too short to recover their hydrogen bonding.''' Since not all Ig units of titin are identical, SMD was used to study the differences in the unfolding mechanism between some domains, namely II and I27.112 SMD suggests that, in contrast to 127, II does not have unfolding intermediates. Details of folding-refolding dynamics of the II domain were described by Gao et al. in a separate paper.113 Interestingly, in this work the important stabilizing effect of the disulfide bridge was easily discovered by modeling oxidized and reduced forms of the protein. The S-S bridge limits the maximum extension of the II titin unit. However, very recent AFM results published by Li and Fernandez114 put in doubt the special role of S-S bridges in II modules of titin. More analytical models of protein elasticity probed by AFM and related to Ig domains were presented by Zhang and Evans.115 Best et al.116 used engineered titin modules to test their method of searching for the unfolding mechanism. Some summary information on titin SMD studies may be found in the paper by Isralevitz et al.' 17 Very recently Best et al. have claimed,118 based on MD simulations and analysis of protein mutagenesis data, that a stable intermediate is indeed involved in 127 unfolding,119 but it is related rather to side chain interactions than to hydrogen bond ruptures, as was previously assumed. The unfolding intermediate has also been found in AFM experiments by Schweiger et al.120 done for the Ig rod domain of filamin from Dictyostelium discoideum. The report on AFM experiments performed on the isolated, huge titin molecule was published by Kellermayer et al.121 3.1.2 Fibronectin and other elastic proteins Fibronectin (FN) is another interesting protein extensively studied both by the AFM and MD methods. It is found in the extracellular matrix in all vertebrates. Elastic FN fibrils connect cells, they may stretch four times as long as their relaxed length, and play a role in signalling. Two identical subunits form a dimeric molecule consisting of more than 20 modules on each part. There are three types of monomers (I-III), a module FN-III10 was most often studied computationally.122 The first AMF measurements done by Rief et al.123 have shown that FNIII modules of titin exhibit 20% lower unfolding forces than Ig domains. At the same time Oberhauser et al., who studied FNIII modules from tenascin124, have found that these forces are 50% smaller, and that tenascin FN modules refold twice as fast as those from fibronectin. Since

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the overall fold of both types of FNIII modules is very similar, observed differences in rupture forces (75-200 pN) are hard to explain without atomistic simulations of dynamics. This hierarchy in elasticity of individual FN modules may have a physiological role.23 Early simulations of FN suffered from limited simulation times and a rather crude treatment of the solvent. Kramer et al.125 used the SMD method to study the behavor of the FNIIIio module under an external force. A small water sphere was used in the simulations. Since timescales were much shorter than in the experiments, a few hundred pN forces had to be applied to unfold this P-structure and to observe that unfolding starts from the rupture of the G strand.125 Simulations have shown that residues in the loop 78-80 (RGD) are "strategically" located close to a membrane protein integrin, and act as a mechanoswitch during unfolding. Further studies by the same group gave a better picture of the mechanism of FNIII unfolding for different modules.126 Paci and Karplus90 prefered to use an implicit solvent model together with the biased MD method (yi). A few dozen of the unfolding computer experiments (0.2-1 ns) were carefully analyzed. Differences between FNIII9 and FNIII 10 folding pathways were highlighted. For the first time an unfolding path for the FIII9 - FNIII I0 dimer was presented. Perhaps the most advanced simulations of FNIII modules were performed by Gao et al.122 They used a huge (370 A long) water box with 130 K atoms to avoid artifacts caused by a spherical solvent shell which undergoes a large deformation during the stretching of the protein. This study suggests that there are at least three distinct unfolding pathways, but their discrimination, based solely on fast SMD simulations, is not possible. A good discussion of FN dynamics may be found in the recent review by Tajkhorshid et al.95 In 2003 Li and Makarov suggested127 that the current time scale gap between AFM experiments and MD simulations (simulations are typically 106 times too fast) may be eliminated by describing the unfolding dynamics in terms of diffusive motion along a predefined unfolding coordinate R in the presence of external driving potential and the potential of mean force G(R). The other degrees of freedom are hidden in a viscous force. The PMF G(R) is computed from a series of equilibrium MD runs with the fixed values of R, and friction coefficients are extrated from a series of SMD simulations. It is notable, that the authors were able to calculate the unfolding free energy barrier of titin at zero force, which was almost identical to that obtained experimentally. Two distinct unfoding pathways of titin are predicted by this model. Mechanical properties of elastin, yet another protein found in the extracellular matrix, especially in relation to the temperature factor, were also computationally studied,128"132 but more traditional techniques of MD

60

W. Nowak and P. E. Marszatek

simulations were used in this case. Spectrin repeats and its two mutants were recently studied by AFM and SMD techniques as well.133 3.1.3 SMD simulations indirectly related to AFM experiments The SMD method was used in several important studies of ligand-protein interactions134 for systems which have not yet been subject to any AFM type experiments. Since the number of such applications will grow, we present a brief account of the recent papers. In 1999 Kosztin et al. presented unbinding pathways of the retinoic acid hormone from its nuclear receptor h-RAR. Three hypothetical paths were selected. The SMD calculations indicated that one of those is not probable and that alternative pathways may be used by the hormone for binding and unbinding. A detailed analysis of possibile close contacts, salt bridges and hydrogen bonds along the path has improved our understanding of this medically important receptor and may help to develop new drugs. In 2001 Heymann and Grubmiiller136 performed a detailed study of forced unbinding of spin-labeled dinitrophenyl hapten from the monoclonal antibody AN02 using a variant of the SMD method called force probe molecular dynamics (FPMD). Large flexibility of the binding pocket region has been found. A large (6 kcal/mol) entropic contribution was calculated to the ligandprotein bond. Interestingly, no differences in unbinding forces were found in two Y33F and I96K mutants of the antibody. Mechanosensitive channel with large conductance (MscL) can be gated by tension in the cellular membrane. Gullingsrud et al, following their initial, classical 3 ns MD study of the gating mechanism,137 used SMD in a recent 10 ns study138 to add radial external forces (70 pN) in the critical part of the channel to induce its opening. In contrast to the previous calculations by Ma et. al. in which TMD was used139, the authors have found that the iris-like motion is probably involved in the process of the channel opening. Both TMD and SMD confirm tandem-like mechanism of MscL sensitivity to the membrane tension. GTP hydrolizing G proteins are the part of the signal transduction system that control proliferation, differentiation and metabolism. Among them Ras p21 attracted particular attention, since it works like a molecular switch, changing states between the GDP-bound inactive form and the GTP-bound active form. The conformational changes related to this process were studied by Ma and Karplus140, Diaz et al.141 using TMD. Recently Kosztin et al.142 added an "artificial" force attached to two carefully selected atoms, in order to estimate the work related to the transition between tense (T) and relaxed (R) states of the so called switch II region in h-RAS protein. It has been found that the force generation proceedes in two steps via a short lived (< Ins)

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metastable T state formed just after the GTP hydrolysis, and the amount of work related to this process is around 1 kcal/mol. Cell-cell adhesion is mediated by proteins. The interaction between the Tlymphocyte adhesion receptor CD2 and its ligand CD58 was studied by Bayas et al.143 using the SMD method. Two regimes - constant force and constant velocity - were used to enforce detachment of these two proteins. The mechanism of the forced detachment depends strongly on the loading rate. At faster loading rates (70 pN/ps) the proteins unfold before detachment, at slower rates (7 pN/ps) they just separate without significant initial distortions. These calculations clearly show that multiple conditions in the SMD protocol should be tried during studies of protein complexes, before conclusions on interactions mechanisms are reached. Elastic properties of a P-selectin construct consisting of a lectin domain and epithelial growth factor-like domain was studied using SMD by Lu and Long144. Groups of hydrogen bonds were identify to rupture during forced dissociation of this complex, which is important in inflammatory proceses. Best et al.145 posed an interesting question whether proteins, which unlike titin, are not designed for mechanical function can withstand large forces before unfolding. An artificial complex of barnase with the 127 domain of titin was studied by AFM and computationally (3-6 ns SMD trajectories). The results indicate that barnase unfolds at much lower forces than the titin fragment. Since unfolding rates of both proteins in solution are comparable, this indicates that the mechanism of mechanically forced unfolding may be different than the chemically induced one. 3.2 Biased JMD The first application of an interesting variant of enforcing conformational transitions was proposed by Paci and Karplus.90 As the authors say, their method is based on the unpublished work by P. Ballone and S. Rubini, who used biased forces to study the crystallization process. The idea of applying very small perturbations for inducing desirable conformational changes, also related to the earlier work by Harvey and Gabb,146 is the following. The reaction coordinate (RC) has to be selected first. Sometimes it is very complex and difficult to determine or even guess. Usually we do not know how the studied process proceedes. Let us assume that the unfolding of a protein induced by a force attached to C-terminus of the protein backbone is to be studied, and RNc is the distance between the N-terminus and the C-terminus. For example, Paci and Karplus, in their study of the relation of the unfolding process to the topology of the protein,147 used as the reaction coordinate p(t) = R2NC • It is worth to mention,

62

W. Nowak and P. E. Marszalek

that these authors used the implicit Gaussian model for the solvent. They claim that the adiabatic relaxation of the solvent is necessary for fast unfolding studies, and that such a simplified approach is closer to the real AFM experiments. Time dependent bias force is: Fb = 7(P -Pmax),

(4)

but only when p(t) < pn,ax(t), otherwise the force is set to zero. The parameter pmax (t) is the maximum value of p reached by the reaction coordinate at times less or equal to t. Thus, the force is nonzero only when thermal fluctuations tend to decrease the selected distance (RC). One can guess that if an energy barrier appears on the selected structural transformation, then the bias force is large, in the other regions its value is not high. The values of y parametres are adjusted by a trial and error procedure, depending on the process studied and assumed timescale, suitable to the forced structural transition under study. In this paper147 values of y around 0.0012 kcal- mol'1 A"4 were used in biasing forces applied in order to unfold a range of P-sandwich and a-helical proteins. It is worth to note that 1 kcal- mol 1 A"1 = 69.4786 pN. BMD has been used in several iluminating studies. Recently, Paci et al.148 have applied BMD (together with the constant force MD simulation) to reveal atomistic details of unbinding of fluorescein from hepten-antibody complex. The same system was simulated by Heynman and Grubmixller,'49 using the SMD method, so the two approaches may be compared. The biased MD technique is useful in generating molecular states, which are not easy amenable to experiment. For example, Paci et al.150 used this method to obtain nearly 400 clusters of partially unfolded states of human <xlactalbumin. A clever selection of the radius of gyration as a reaction coordinate, based on the idea of Marchi and Ballone,151 and using an implicit solvent made the effective generation and analysis of so many semi-stable protein conformations possible.

4. Free energy profiles in pulling experiments Relating single molecule measurements to thermodynamics is not an easy task, since AFM experiments are not performed on ensambles of molecules and the processes studied are not in equilibrium. Free energy profiles along selected reaction coordinate or potential of mean force (PMF) have a special importance at a single molecule level.152'153 In 1997 Balsera et al.154 presented a one-dimensional stochastic model for the reconstruction of PMF along the path of force induced conformational

Molecular Dynamics Simulations 63

transitions in a molecular system. The model, based on suggestions by Evans and Ritchie,155 is useful in the analysis of ligand unbinding. Dissipation is related to a friction coefficient. The accuracy of the potential energy function evaluated is determined from fluctuations which are proportional to the amount of irreversible work deposited into the system by external forces. In this paper uncertainties 8F in the calculated forces are estimated to be about 50 pN for a 10 ns simulation. Let us assume that the initial state / is in equilibrium with the heat bath at temperature T (unextended molecule). The parameter X, related to a controllable degree of freedom, describes a transition of the system from state / to state / (extended molecule). Changes in X result in expenditure of some amount of work W, changes of free energy of the system AG and the exchange of heat with the bath. In 1997 C. Jarzynski156 proved an equality that relates the irreversible work W, performed during /->/transition of the system, to the equilibrium free energy difference between states / a n d / , AG: <e-W/kT> =

e-AG/kT

( 5 )

This equation, further analysed by Jarzynski,157'158 is very important and became the basis of estimates of thermodynamic quantities in single molecule experiments. In 2001 Hummer and Szabo159 presented a simple recipe on how to reconstruct free energy surfaces from multiple nonequillibrium pulling experiments. Perhaps one of the first applications of this formalism to SMD computer experiments was the work done by Jensen et al.160, who studied conduction of glycerol through the membrane aquaglycerolporin channel. PMF calculated from Jarzynski's equality gives a reasonable understanding of molecular events in this highly selective system. The calculated highest energy barrier of 7.3 kcal/mol compares well with the experimental Arrhenius activation energy 9.6 ±1.5 kcal/mol.160 A similar methodology has been used by Amaro et al.161 in their research of ammonia transfer in the HisF enzymatic protein, which is important in histidyne synthesis. A total of 40 ns of the simulation time (many trajectories of 1.2 ns each) was used in order to collect reasonable statistics. Park et al.' 62 very recently applied Jarzynski's equality to free energy calculations from SMD of helix-coil transitions of deca-alanine. The limited number of 10 trajectories was generated and the second order cumulant expansion of rigorous formulas yielded the most accuarate estimates of free energy changes. The authors notice, that the average of the exponential work (Eq. 5) is dominated by trajectories corresponding to small W values, that only arise rarely. Practical applications are limited to slow processes, for which the fluctuations of W are comparable to temperature. A related paper by

64

W. Nowak and P. E. Marszalek

Zuckerman and Woolf163 gives a new theory of errors inherent to nonlinear averages, such as those used in free energy calculations. It is worth to note that Jarzynski's equality has not been proven experimentally until recently.164 The work by Liphardt et al.164 that verifed Jarzynski's equality was named one of the top 10 scientific breakthroughts in 2003 by "Science" magazine. The authors tested Jarzynski's equality by performing irreversibly and reversibly mechanical stretching of a single RNA molecule between two conformations. Application of this equality to the irreversible work trajectories recovers the AG profile of the stretching process to within kT/2 of its best independent estimate, i.e. the mean work of reversible stretching. Other approaches relating thermodynamics to SMD were suggested in the literature. As we mentioned before, in 1999 Gullingsrud et al.100 proposed three time series analysis methods, aimed at the reconstruction of PMF from SMD data on forces and positions. One technique, based on WHAM, views stretching as a quasi-equilibrium process. It has been found not to be very promising in the interpretation of AFM experiments. The other two, one based on van Kampen's Q-expansion and the second on the least squares minimization of the Onsager—Machlup action with respect to the choice of PMF, assume a Langevin description of the dynamics in order to account for the non-equilibrium character of SMD data. A MD trajectory is equivalent to a particle moving in one-dimensional potential PEM W(z) which has to be found in the course of analysis from the equation: 7z

= F(z,t)-^^-

+ cr{(t).

(6)

oz Here y is a friction coefficient, calculated from the velocity autocorrelation function, F(z,t) is the applied external force, and o£(t) is the Gaussian white noise; the fluctuation-dissipation theorem gives a relation o2 =2ykT, where T is temperature and k is Boltzmann's constant. The methods were tested on artificial potentials and a phospholipid membrane monolayer system.100 In AFM experiments a cantilever tip picks up a protein molecule attached to the surface in a random fashion. Therefore some force spectrograms are difficult to interpret, since the mode of coupling is not known "a priori". The same difficulty is related to the problem of selecting RC in SMD or BMD simulations. In principle one may try all possible (and reasonable) models of performing the desired conformational transition, but this is not feasible. In certain studies, such as ligand dissociation from the site buried in an enzyme, the random expulsion molecular dynamics (REMD) scheme proposed by Wade et al. may help.165 The random force is acting on the center of mass of

Molecular Dynamics Simulations

65

the ligand. The direction of the force is kept constant for a chosen time interval, unless the ligand encounters rigid parts of the cavity. Then a new direction of the force is chosen. The substrate probes different regions of the protein until it finds the exit pathway. Ludemann et al.166 used the REMD method to determine the most probably exit path of camphor from the cytochrom P450cam active site. The points determined from REMD, located outside the protein, were then used to attach a force, which extracts the ligand from the protein. The magnitude of the SMD unbinding forces is reasonable (500-1000 pN) and compares well with the forces obtained in earlier studies of similar systems.83'135

5. The most recent results The progress in MD simulations of single molecule experiments and forced conformational transitions is very fast. In 2003 Brockwell et al.167 have shown that pulling geometry to a large extent determines forces observed during unfolding. In all-P protein, E21ip3, unolding forces differ by an order of magnitude, depending on the force direction chosen. Such an anisotropy may have physiological role. Lorenzo et al.168 calculated molecule-membrane interaction forces. To illustrate the principle of the method a small aminoacid alanine was modeled on a us timescale in an explicit solvent environment. Non-specific interactions at water-membrane interface were determined. Hummer and Szabo169 developed a simple procedure for extracting kinetic data from pulling experiments. This approach was tested on computer simulation data for titin. Special attention should be devoted to simplified mechanical models of proteins. For example, Cieplak et al.170171 used Go-like models to perform comparative forced unfolding of various secondary structures. Thermal and mechanical unfolding may be studied in that way. The sequence of events during the unfolding of titin is similar to those observed in fully atomistic simulations.171 However, availability of large thousand-processors clusters, make record breaking, huge SMD simulations possible. Brockman et al.172'173 have performed ns simulations (280 000 h CPU time) of dynamics of an extremely interesting system - a dual-domain molecular motor F^o-ATPase. This enzyme may work in two modes. The off-membrane Fi domain utilizes the transmembrane proton gradient to synthetize ATP from ADP and P;. Alternatively, it may hydrolize ATP and release energy. In both modes torsional motion of the central part of the enzyme is involved.172 It is known

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W. Nowak and P. E. Marszalek

that the isolated ATPase may work as a molecular nanomachine. Brockman et al. studied a sequence of conformational transitions in Y\ subunit induced by an external torque applied to the cental "y-stalk". It has been found that some parts of the enzyme ((3E subunit) behave as a prestretched "springs" during the enzymatic cycyle.173 Schulten at al.95 reported recently SMD simulations of both model Fo (112 K atoms) membrane and Fj (327 K atoms) off-membrane subunits of this ATPase. The principal goal of the work was to understand events that couple proton translocation across the membrane with the mechanical rotation of the Cio subunit oligomer, which is in the contact with the central y-stalk. This simulation was not able to predict all steps in the activity cycle of Fo, but helped to check numerous hypotheses on the mechanism of the ATPase activity. The TMD method was used by Ma et al.174'175 to generate trajectories for the bovine mitochondrial Fl-ATPase, related to elementary 120 degrees rotations of the central y subunit. This part has a special "ionic track" of Arg and Lys residues that plays a role in the guiding motions of |3 subunits. Excited state dynamics opens new challenges and opportunities for computer simulations.176 The first application of ab inito MD to isomerisation phenomena in bacteriorodopsin was done by Hayasi et al.177 According to the simulations, the protein plays a crutial role in the kinetics and selectivity of the photoisomerisation of the retinal. It discriminates from many photoisomerization paths the one which initiaties the process of light signal transduction. We stress the future role of the modeling of excited states, since optomechanical experiments on single molecules have been already performed in H.Gaub's laboratory.178'179 A peptide with multiple photoactive azobenzene groups incorporated in the backbone was synthetized. Cis to trans-azo switching was induced by UV irradiation and the changes in single molecule length were registered by the AFM technique. The molecule was shortened by light and at the same time it was able to perform some mechanical work. Thus, optomechanical single molecule energy conversion was demonstrated.179 It seems that performing SMD simulations of this very interesting system is only a matter of time.

6. MD results for stretching of polysaccharides Polysaccharides are not so popular in biophysical or chemical studies as proteins or nucleic acid, albeit their physiological role is enormous.180

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67

Mechanical properties of simple polysaccharides were studied by AFM single molecule spectroscopy. In 1997 Rief et al,181 studied dextran linked to a gold surface and tried to explain the force-extension curves using MD simulations for 5 ring model systems and CHARMM force field. Li et al.182 measured by AFM the difference in a-(l,4) and P-(l,4) linked polysaccharides (amylose vs. cellulose). Heymann and Grubmuller183 performed MD simulations of stretching 16-mer models of these two polysaccharides, using EGO code and constant velocity pulling regime both in vacuo and in TIP3P water. In order to correctly reproduce the experimental plateau on the forceextention curves of amylose, these authors had to introduce artificial conformational constraints to set up the anti-parallel orientation of individual a-(l,4)-glucose units. In 1998 Marszaiek et al.24 demonstrated that polysaccharide elasticity is governed by chair-boat transitions of gluocopyranose rings. AFM experiments have shown that oxidation of the ring nearly eliminates the enthalpic component of the elasticity of dextran and amylose. The same group put forward a hypothesis that stretching of the pyranose ring may serve as a part of molecular signaling: it is pleausible that the fragment of pectin, namely a-(l-4)-D-galactouronan, acts as a two state elastic switch.25 Quantum chemical optimization of geometries of standard 4 Ci, boat and 'C 4 conformations of galatouronic acid, performed on the B3LYP/6-31* level, supported this hypothesis. The AFM force spectroscopy technique may be used for analytical purposes. In 2001 Marszaiek et al.34 have shown that individual molecules of floridean starch may be effectively identified as contaminats in the solutions of agarose or ^.-carrageenan due to distinct molecular fingerprints of these polysaccharide species. Chair-boat transitions in a single polysaccharide molecule may be observed in force-ramp AFM experiments.184 In this regime a pulling force is linearly increasing and sudden geometrical changes of the pulled molecule may be easily identified. An analytical two-state model developed by Marszaiek et al.184 allows for the quantitative characterization of fundamental ring transitions. Extensive quantum mechanical studies of barriers to forced transitions in sugar rings were performed by O'Donoghue and Luthey Schulten.185 The HF calculations show that the twist-boat conformation of aD-glucose is the most probable stable minimum in a stretched polysaccharide and the barrier for the chair to boat transition in a-D-glucose ranges from 14 to 15 kcal/mol. Recently, results of single molecule AFM measurements of elasticity of heparin have been published.186 Elastic properties of polysaccharides with 1-6 links are more complex than those with 1-4 links. Recenly we have found by stretching single pusrulan molecules in the AFM instrument that P-l—>6 linkages produce an unusual hookean-like elasticity.187 Pusrulan [a P~(l—>6)-linked glucan, (see Fig. 4)] is

68

W. Nowak and P. E. Marszalek

similar to dextran [an oc-(l—>6)-linked glucan] but equatorial position of glycosidic bonds (CpOi) in pustulan, and the axial one in dextran, lead to clear differences in mechanical properties of a and P-(l-6)-glicosidic linkages. In contrast to cellulose whose 1-4 links allow for fairly free rotations of the sugar rings in the polymer chain pustulan experiences rotational restrictions during the stretch, and therefore its force-extension relationships strongly deviate from the simple freely rotating chain-like relationship for cellulose. Additional work is used to rotate the aglycone bond (O6-C6) about the C6-C5 bond, because this rotation contributes to the extension of pustulan but is absent in cellulose. The elasticity of dextran is different from both cellulose and pustulan and suggests that its oc-(l—^-linkages induce not only the rotation about the C6-C5 bond but also trigger various ring instabilities. We applied SMD simulations to investigate the deformations of the pyranose ring induced by forces attached to the ring at carbon atom d and through a rotatable C6-C5 bond. Using NAMD2 code188 running on a linux cluster in parallel mode we stretched pustulan and dextran decamers immersed in water boxes of such a size that at least a 6 A layer of TIP3P water molecules was present at each step of the simulations. After a 1-5 ns equilibration at 300K (dielectric constant e = 1) the structures were stretched by applying force (spring constant of 68 pN/nm) to the Oi atom of the last sugar and moving this spring at a constant velocity of 10"4 A /fs. To check the effect of the pulling speed on the results a control 200 ns simulation in vacuum with e = 80 were also performed Selected results187 are presented in Fig. 4. A comparison of AFM data and the detailed atomistic picture emerging from computer modeling, indicates that one can control the orientation of an atomic "crank": the O6-C6 bond and reorient it mechanically from the gauche-trans (gt) or gauche-gauche (gg) position to the less favourable energetically but more extended trans-gauche (tg) position. These rotations produce a hookean-like elasticity of pustulan but do not affect the chair conformation of the pyranose ring. However, deforming the pyranose ring by oc-1—>6 linkages, using the O6-C6 cranks and axial C r O | levers, such as those in dextran, not only reorients the O6-C6 bond to the tg position but also produces ring instabilities, which flip the pyranose structure to a boat-like conformation. Studies of polysaccharide elasticity and mechanical properites of organic

rings expand single molecule mechanochemistry and are important for future nanotechnological applications.

Molecular Dynamics Simulations 69

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70

W. Nowak and P. E. Marszalek

1. Quantum molecular dynamics of amylose stretching Quality data may be obtained from forced MD simulations of single molecules only when the force field (FF) is not flawed. The results of modeling are force field dependent. The development of FFs was centered around equilibrium benchmark structures so the quality of the potentials applied to molecules stretched far from energy minima is not necessarely the best. Thus, there is high demand for a fully quantum-mechanical, forcefield free, calculation for streteched biopolymers. In a recent paper189, which is a result of a collaborative effort among Yang, Marszalek, Nowak and their students, Lu et al189 have presented a practical solution to this long lasting problem. We used for the first time a quantum mechanics based methodology to investigate how a polysaccharide immersed in water, responds to a stretching force. The amylose fragment composed of ten sugar rings was placed in a TIP3P water box (30 A x 30 A x 80 A; 8004 atoms) and was treated by the self consistent charge density functional theory-based tight-binding method (SCC-DFTB).190 Explicit long range van der Waals forces were used. The total energy of the system including the interactions between amylose and water was calculated with the divide-and-conquer linear-scaling QM/MM model developed by the Yang group191 Our tests performed for amylose dimers indicate a very high similarity of potential energy surfaces calculated by the SCC-DFTB method and the DFT B3LYP method (6-31 lg++** basis set was used). The results of the amylose stretching are presented in Fig. 5.

Molecular Dynamics Simulations OH

1 0,70

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,

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,

, 0.95

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Normalized extension (x/IJ Fig. 5 Results of SMD simulations of the amylose stretching. Black trace: the experimental force-extension relationship obtained by AFM. Dark gray trace: the force-extension relationship obtained by QM/SMD simulations. Light gray trace: the O1-C1-C2-O2 dihedral angle averaged over the glucose units. The extension, x, was normalized by the molecule length, lc, determined at the force of 1620 pN.

The QM/SMD simulations represent the mechanical properties of amylose in water remarkably well. The force-induced chair to boat transitions (represented by the dihedral angle shown in Fig. 5) coincide with the plateau region of the experimental force-extension curve, as detailed analysis of trajectories shows. The increasing tension forces the neighboring glucose rings to rotate around the O1-C4' bonds. The rings orient themselves in a parallel fashion, even though they were initially in an anti-parallel orientation. The elasticity of amylose to a large extent depends on a complex pattern of the breakage/creation of hydrogen bonds. Without a proper solvatation it is not possible to reproduce the value of the plateau force. These calculations,189 though quite time-consuming, bring a promise for doing realistic simulations of nanomechanical interactions in any biologically important glycoproteins on a routine basis, without the need of tedious optimization of force field parameters.

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W. Nowak and P. E. Marszalek

Concluding remarks The new methods and protocols of performing MD simulations - TMD, SMD, BMD - greatly help to interpret single molecule experiments. The progress is very fast and trends are clear: forced transitions may bridge mechanics of molecules with chemistry. Better simulation protocols and better hardware will reduce the gap in timescales between SMD or BMD simulations (ns) and AFM experiments (ms). The findings of computational efforts, sometimes unexpected, lead to new ideas, sugest new mutagenesis work and new experimental tests. Piconewton forces observed in AFM or optical tweezer setups are ultimate mechanical signals sent up from the nanoworld to keen observers who are now well prepared to receive and interpret them.

Acknowledgements This work was supported by the NSF (WN and PEM), KBN and GRUMK (WN).

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

Robertson, N. & McGowan, C.A. A comparison of potential molecular wires as components for molecular electronics. Chem. Soc. Rev. 32, 96-103 (2003). Plakhotnik, T., Donley, E.A. & Wild, U.P. Single-molecule spectroscopy. Ann. Rev. Phys. Chem. 48, 181-212 (1997). Kuhnemuth, R. & Seidel, C.A.M. Principles of single molecule multiparameter fluorescence spectroscopy. Single Molecules 2, 251-254 (2001). Orrit, M. Single-molecule spectroscopy: The road ahead. J. Chem. Phys. 117, 10938-10946(2002). Florin, E.L., Moy, V.T. & Gaub, H.E. Adhesion Forces between Individual Ligand-Receptor Pairs. Science 264, 415-417 (1994). Medina, M.A. & Schwille, P. Fluorescence correlation spectroscopy for the detection and study of single molecules in Biology. Bioessays 24, 758-764 (2002). Bohmer, M. & Enderlein, J. Fluorescence spectroscopy of single molecules under ambient conditions: Methodology and technology. Chemphyschem 4, 793-808 (2003). Haran, G. Single-molecule fluorescence spectroscopy of biomolecular folding. J. Phys.-Cond. Mat. 15, R1291-R1317 (2003). Forkey, J.N., Quinlan, M.E., Shaw, M.A., Come, J.E.T. & Goldman, Y.E. Threedimensional structural dynamics of myosin V by single-molecule fluorescence polarization. Nature 422, 399-404 (2003).

Molecular Dynamics Simulations

73

10. Mehta, A.D., Rief, M., Spudich, J.A., Smith, D.A. & Simmons, R.M. Singlemolecule biomechanics with optical methods. Science 283, 1689-1695 (1999). 11. Molloy, J.E. & Padgett, M.J. Lights, action: optical tweezers. Contemporary Physics 43, 241-258 (2002). 12. Coirault, C , Pourny, J.C., Lambert, F. & Lecarpentier, Y. Optical tweezers in biology and medicine. M S-Medecine Sciences 19, 364-367 (2003). 13. Schwille, P. & Kettling, U. Analyzing single protein molecules using optical methods. Curr. Opin. Biotech. 12, 382-386 (2001). 14. Finer, J.T., Simmons, R.M. & Spudich, J.A. Single Myosin Molecule Mechanics Piconewton Forces and Nanometer Steps. Nature 368, 113-119 (1994). 15. Purohit, P.K., Kondev, J. & Phillips, R. Mechanics of DNA packaging in viruses. Proc. Natl. Acad. Sci. 100, 3173-3178 (2003). 16. Bustamante, C , Smith, S.B., Liphardt, J. & Smith, D. Single-molecule studies of DNA mechanics. Curr. Opin. Struct. Biol. 10, 279-285 (2000). 17. Smith, S.B., Finzi, L. & Bustamante, C. Direct Mechanical Measurements of the Elasticity of Single DNA-Molecules by Using Magnetic Beads. Science 258, 1122-1126(1992). 18. Zlatanova, J. & Leuba, S.H. Magnetic tweezers: a sensitive tool to study DNA and chromatin at the single-molecule level. Biochemistry and Cell Biology-Biochimie Et Biologie Cellulaire 81,151-159 (2003). 19. Romano, G., Sacconi, L., Capitanio, M. & Pavone, F.S. Force and torque measurements using magnetic micro beads for single molecule biophysics. Optics Comm. 215, 323-331 (2003). 20. Basche, T., Nie, S. & Fernandez, J.M. Single molecules. Proc. Natl. Acad. Sci. USA 98, 10527-10528 (2001). 21. Marszalek, P.E., Lu, H., Li, H.B., Carrion-Vazquez, M., Oberhauser, A.F., Schulten, K. & Fernandez, J.M. Mechanical unfolding intermediates in titin modules. Nature 402, 100-103 (1999). 22. Li, H.B., Linke, W.A., Oberhauser, A.F., Carrion-Vazquez, M., Kerkviliet, J.G., Lu, H., Marszalek, P.E. & Fernandez, J.M. Reverse engineering of the giant muscle protein titin. Nature 418, 998-1002 (2002). 23. Oberhauser, A.F., Badilla-Fernandez, C , Carrion-Vazquez, M. & Fernandez, J.M. The mechanical hierarchies of fibronectin observed with single-molecule AFM. J. Mol. Biol. 319, 433-447 (2002). 24. Marszalek, P.E., Oberhauser, A.F., Pang, Y.P. & Fernandez, J.M. Polysaccharide elasticity governed by chair-boat transitions of the glucopyranose ring. Nature 396,661-664(1998). 25. Marszalek, P.E., Pang, Y.P., Li, H.B., El Yazal, J., Oberhauser, A.F. & Fernandez, J.M. Atomic levers control pyranose ring conformations. Proc. Natl. Acad. Sci. (USA) 96, 7894-7898 (1999). 26. Fisher, T.E., Marszalek, P.E. & Fernandez, J.M. Stretching single molecules into novel conformations using the atomic force microscope. Nat. Struct. Biol. 7, 719724 (2000). 27. Zhang, L , Wang, C , Cui, S.X., Wang, Z.Q. & Zhang, X. Single-molecule force spectroscopy on curdlan: Unwinding helical structures and random coils. Nano Letters 3, 1119-1124(2003).

74

W. Nowak and P. E. Marszalek

28. Onoa, B., Dumont, S., Liphardt, J., Smith, S.B., Tinoco, I. & Bustamante, C. Identifying kinetic barriers to mechanical unfolding of the T-thermophila ribozyme. Science 299, 1892-1895 (2003). 29. Hansma, H.G., Pietrasanta, L.I., Auerbach, I.D., Sorenson, C , Golan, R. & Holden, P.A. Probing biopolymers with the atomic force microscope: A review. J. Biomaterials Sci.-Polymer Ed. 11, 675-683(2000). 30. Jandt, K.D. Atomic force microscopy of biomaterials surfaces and interfaces. Surf. Sci. 491, 303-332 (2001). 31. Demers, L.M., D. S. Ginger, S.-J. Park, Z. Li, S.-W. Chung & Mirkin, C.A. Direct Patterning of Modified Oligonucleotides on Metals and Insulators by Dip-Pen Nanolithography. Science 296, 1836-1838 (2002). 32. Durig, U., Cross, G., Despont, M., Drechsler, U., Haberle, W., Lutwyche, M.I., Rothuizen, H., Stutz, R., Widmer, R., Vettiger, P. "Millipede" - an AFM data storage system at the frontier of nanotribology. Tribology Letters 9, 25-32 (2000). 33. Ludwig, M., Rief, M., Schmidt, L., Li, H., Oesterhelt, F., Gautel, M. & Gaub, H.E. AFM, a tool for single-molecule experiments. Appl. Phys. - Mat. Sci. & Proc. 68, 173-176 (1999). 34. Marszalek, P.E., Li, H.B. & Fernandez, J.M. Fingerprinting polysaccharides with single-molecule atomic force microscopy. Nat. Biotech. 19, 258-262 (2001). 35. Bustamante, C , Bryant, Z. & Smith, S.B. Ten years of tension: single-molecule DNA mechanics. Nature 421, 423-427 (2003). 36. Strick, T.R. et al. Stretching of macromolecules and proteins. Rep. Prog. Phys. 66, 1-45 (2003). 37. Clausen-Schaumann, H., Seitz, M., Krautbauer, R. & Gaub, H.E. Force spectroscopy with single bio-molecules. Curr. Opin. Chem. Biol. 4, 524-530 (2000). 38. Engel, A., Gaub, H.E. & Muller, D.J. Atomic force microscopy: A forceful way with single molecules. Curr. Biol. 9, R133-R136 (1999). 39. Zhuang, X.W. & Rief, M. Single-molecule folding. Curr. Opin. Struct. Biol. 13, 88-97 (2003). 40. Rief, M. & Grubmuller, H. Force spectroscopy of single biomolecules. Chemphyschem 3, 255-261 (2002). 41. Leckband, D. Measuring the forces that control protein interactions. Ann. Rev. Biophys. Biomol. Struct. 29, 1-26(2000). 42. Riener, C.K., Stroh, CM., Ebner, A., Klampfl, C , Gall, A.A., Romanin, C , Lyubchenko, Y.L., Hinterdorfer, P. & Gruber, H.J. Simple test system for single molecule recognition force microscopy. Analytica Chimica Ada 479, 59-75 (2003). 43. Best, R.B., Brockwell, D.J., Toca-Herrera, J.L., Blake, A.W., Smith, D.A., Radford, S.E. & Clarke, J. Force mode atomic force microscopy as a tool for protein folding studies. Anal. Chim. Ada 479, 87-105 (2003). 44. Ikai, A., Afrin, R., Sekiguchi, H., Okajima, T., Alam, M.T. & Nishida, S. Nanomechanical methods in biochemistry using atomic force microscopy. Curr. Prot. & Pept. Sci. 4, 181-193 (2003). 45. Williams, P.M. Analytical descriptions of dynamic force spectroscopy: behaviour of multiple connections. Analyt. Chim. Ada 479, 107-115(2003).

Molecular Dynamics Simulations

75

46. Haile, M. Molecular Dynamics Simulation: Elementary Methods, (John Wiley, 1992). 47. Frankel, D. & Smit, B. Understanding Molecular Simulation, (Academic Press, 2001). 48. Field, M.J. A Practical Introduction to the Simulation of Molecular Systems, (Cambridge University Press, 1999). 49. Schlick, T. Molecular Modeling and Simulation - An Interdisciplinary Guide., (Springer, New York, 2002). 50. Kholmurodov, K.T., Altaisky, M.V., Puzynin, I.V., Darden, T. & Filatov, F.P. Methods of molecular dynamics for simulation of physical and biological processes. Physics of Particles and Nuclei 34, 244-263 (2003). 51. Werner Ebeling, L.S.-G., Yuri M. Romanovsky. Stochastic dynamics of reacting biomolecules, (World Scientific, Singapore, 2003). 52. Karplus, M. Molecular dynamics of biological macromolecules: A brief history and perspective. Biopolymers 68, 350-358 (2003). 53. Hansson T, O.C., van Gunsteren W. Molecular dynamics simulations. Curr. Opin. Struct. Biol. 12, 190-196(2002). 54. Marx, D. & Hutter, J. Ab initio molecular dynamics: Theory and Implementation, in Modern Methods and Algorithms of Quantum Chemistry, Vol. 1 (ed. Grotendorst, J.) 301-449 (John von Neumann Institute for Computing, Julich, 2000). 55. Carloni, P., Rothlisberger, U. & Parrinello, M. The role and perspective of a initio molecular dynamics in the study of biological systems. Ace. Chem. Res. 35, 455464 (2002). 56. Karplus, M. & McCammon, J.A. Molecular dynamics simulations of biomolecules. Nat. Struct. Biol. 9, 646-652 (2002). 57. Moraitakis, G., Purkiss A. G, Goodfellow J. M. Simulated dynamics and biological molecules. Rep. Prog. Phys. 66, 483-406 (2003). 58. Zhou, Y.Q. & Karplus, M. Interpreting the folding kinetics of helical proteins. Nature 401, 400-403 (1999). 59. Lazaridis, T. & Karplus, M. Thermodynamics of protein folding: a microscopic view. Biophys. Chem. 100, 367-395 (2003). 60. Vendruscolo, M. & Paci, E. Protein folding: bringing theory and experiment closer together. Curr. Opin. Struct. Biol. 13, 82-87 (2003). 61. Mayor U., Guydosh, N.R., Johnson, CM., Grossmann, J.G, Sato S., Jas G.S., Freund S.M., Alonso D.O., Daggett V., Fersht A.R. The complete folding pathway of a protein from nanoseconds to microseconds. Nature 421, 863-867 (2003). 62. Fersht, A.R. & Daggett, V. Protein folding and unfolding at atomic resolution. Cell 108, 573-582(2002). 63. Daggett, V. & Fersht, A.R. Is there a unifying mechanism for protein folding? Trends inBiochem. Sci. 28, 18-25 (2003). 64. Daggett, V. Molecular dynamics simulations of the protein unfolding/folding reaction. Ace. Chem. Res. 35, 422-429 (2002). 65. Warshel, A. & Parson, W.W. Dynamics of biochemical and biophysical reactions: insight from computer simulations. Quart. Rev. Biophys. 34, 563-679 (2001). 66. Warshel, A. Molecular dynamics simulations of biological reactions. Ace. Chem. Res. 35, 385-395 (2002).

76

W. Nowak and P. E. Marszalek

67. Warshel, A. Computer simulations of enzyme catalysis: Methods, progress, and insights. Ann. Rev. Biophys. Biomol. Struct. 32, 425-443 (2003). 68. Karplus, M. Aspects of protein reaction dynamics: Deviations from simple behavior. J. Phys. Chem. B 104, 11-27 (2000). 69. van Speybroeck, V. & Meier, R.J. A recent development in computational chemistry: chemical reactions from first principles molecular dynamics simulations. Chem. Soc. Rev. 32,151-7 (2003). 70. Bala, P., Grochowski, P., Nowinski, K., Lesyng, B. & McCammon, J.A. Quantum-dynamical picture of a multistep enzymatic process: Reaction catalyzed by phospholipase A(2). Biophys. J. 79, 1253-1262 (2000). 71. Simonson, T., Archontis, G. & Karplus, M. Free energy simulations come of age: Protein-ligand recognition. Ace. Chem. Res. 35,430-437 (2002). 72. McCammon, A.J. & Wolynes, P.G. Theory and simulation - Enlarging the landscape - Editorial overview. Curr. Opin. Struct. Biol. 12, 143-145 (2002). 73. Pande V.S., Baker, I., Chapman, J., Elmer, S.P., Khaliq, S., Larson, S.M., Rhee, Y.M., Shirts, M.R., Snow, C D . , Sorin, E,J., Zagrovic, B. Atomistic protein folding simulations on the submillisecond time scale using worldwide distributed computing. Biopolymers 68, 91-109 (2003). 74. Daggett, V. Long timescale simulations. Curr. Opin. Struct. Biol. 10, 160-164 (2000). 75. Straub, J.E., Guevara, J., Huo, S.H. & Lee, J.P. Long time dynamic simulations: Exploring the folding pathways of an Alzheimer's amyloid A beta-peptide. Ace. Chem. Res. 35, 473-481 (2002). 76. Elber, R. Novel methods for molecular dynamics simulations. Curr. Opin. Struct. Biol. 6,232-235(1996). 77. Paci, E. High pressure simulations of biomolecules. BBA-Protein Struct, and Mol. Enz. 1595, 185-200(2002). 78. Wong, C.F. & McCammon, J.A. Protein flexibility and computer-aided drug design. Ann. Rev. Pharm. Toxicol. 43, 31-45 (2003). 79. Sulpizi, M., Folkers, G., Rothlisberger, U., Carloni, P. & Scapozza, L. Applications of density functional theory-based methods in medicinal chemistry. Quant. Struct.-Act. Relat. 21, 173-181 (2002). 80. Banci, L. Molecular dynamics simulations of metalloproteins. Curr. Opin. Chem. Biol. 7, 524 (2003). 81. Giorgetti, A. & Carloni, P. Molecular modeling of ion channels: structural predictions. Curr. Opin. Chem. Biol. 7, 150-156(2003). 82. Dwyer, D.S. Molecular modeling and molecular dynamics simulations of membrane transporter proteins. Methods Mol. Biol. 227, 335-50 (2003). 83. Grubmuller, H., Heymann, B. & Tavan, P. Ligand binding: Molecular mechanics calculation of the streptavidin biotin rupture force. Science 271, 997-999 (1996). 84. Elber, R., Karplus, M. A method for determining reaction paths in large molecules: Application to myoglobin. Chem. Phys. Letters 139, 375-381 (1987). 85. Elber, R. Reaction Path Studies of Biological Molecules, in Recent developments in theoretical studies of proteins., Vol. 7 (ed. Elber, R.) (World Scientific, Singapore, 1996). 86. Elber, R., Ghosh, A. & Cardenas, A. Long time dynamics of complex systems. Ace. Chem. Res. 35, 396-403 (2002).

Molecular Dynamics Simulations

77

87. Schlitter, J., M. Engels, P. Kruger, E. Jacoby & A. Wollmer. Targeted molecular dynamics simulation of conformational change - application to the T to R transition in insulin. Mol. Sim. 10, 291-308 (1993). 88. Leech, J., Prins, J.F. & Hermans, J. SMD: Visual Steering of Molecular Dynamics for Protein Design. IEEE Comp. Sci. Eng. 3, 38-45 (1996). 89. Izrailev, S., Stepaniants, S., Balsera, M., Oono, Y. & Schulten, K. Molecular dynamics study of unbinding of the avidin-biotin complex. Biophys. J. 72, 15681581 (1997). 90. Paci, E. & Karplus, M. Forced unfolding of fibronectin type 3 modules: An analysis by biased molecular dynamics simulations. J. Mol. Biol. 288, 441-459 (1999). 91. Galligan, E., Roberts, C.J., Davies, M.C., Tendler, S.J.B. & Williams, P.M. Simulating the dynamic strength of molecular interactions. J. Chem. Phys. 114, 3208-3214 (2001). 92. Torrie, G.M. & Valleau, J.P. Nonphysical sampling distribution in Monte Carlo free energy estimation: umbrella sampling. J. Comput. Phys. 23, 187-199 (1977). 93. Izrailev, S., Stepaniants, S., Isralewitz, B., Kosztin, D., Lu, H., Molnar, F., Wriggers, W. & Schulten, K... Steered molecular dynamics, in Algorithms for Macromolecular Modelling (eds. Deuflhard, P. et al.) (Springer Verlag, New York, 1998). 94. Isralewitz, B., Baudry, J., Gullingsrud, J., Kosztin, D. & Schulten, K. Steered molecular dynamics investigations of protein function. J. Molec. Graph. Model. 19, 13-25(2001). 95. Tajkhorshid, E., A. Aksimentiev, I. Balabin, M. Gao, B. Isralewitz, J. C. Phillips, F. Zhu & Schulten, K. Advances in Protein Chemistry, in Protein Simulations, Vol. 66 (eds. Eisenberg, D. & Kim, P.) (Elsevier, 2003). 96. Izrailev, S., Crofts, A.R., Berry, E.A. & Schulten, K. Steered molecular dynamics simulation of the rieske subunit motion in the cytochrome bc(l) complex. Biophys. J. 77, 1753-1768 (1999). 97. Prins, J.F., Hermans, J., Mann, G., Nyland, L.S. & Simons, M. A virtual environment for steered molecular dynamics. Future Generation Computer Systems 15, 485-495 (1999). 98. Stone, J., Gullingsrud, J., Grayson, P. & Schulten, K. The 2001 ACM Symposium on Interactive 3D Graphics: A system for interactive molecular dynamics simulation, pp 191-194 (ACM SIGGRAPH, New York, 2001). 99. Grayson, P., Tajkhorshid, E. & Schulten, K. Mechanisms of selectivity in channels and enzymes studied with interactive molecular dynamics. Biophys. J. 85, 36-48 (2003). 100.Gullingsrud, J.R., Braun, R. & Schulten, K. Reconstructing potentials of mean force through time series analysis of steered molecular dynamics simulations. J. Comp. Phys. 151, 190-211 (1999). 101.Stepaniants, S., Izrailev, S. & Schulten, K. Extraction of lipids from phospholipid membranes by steered molecular dynamics. J. Mol. Model. 3, 473-475 (1997). 102.Rief, M., Gautel, M., Oesterhelt, F., Fernandez, J.M. & Gaub, H.E. Reversible unfolding of individual titin immunoglobulin domains by AFM. Science 276, 1109-1112(1997).

78

W. Nowak and P. E. Marszalek

103.Card on-Vazquez, M., Oberhauser, A.F., Fowler, S.B., Marszalek, P.E., Broedel, S.E., Clarke, J. & Fernandez, J.M. Mechanical and chemical unfolding of a single protein: A comparison. Proc. Natl. Acad. Sci. (USA) 96, 3694-3699 (1999). 104. Carrion-Vazquez, M., Oberhauser, A.F., Fisher, T.E., Marszalek, P.E., Li, H.B. & Fernandez, J.M. Mechanical design of proteins-studied by single-molecule force spectroscopy and protein engineering. Prog. Biophys. & Mol. Biol. 74, 63-91 (2000). 105.Li, H.B., Carrion-Vazquez, M., Oberhauser, A.F., Marszalek, P.E. & Fernandez, J.M. Point mutations alter the mechanical stability of immunoglobulin modules. Nat. Struct. Biol. 7, 1117-1120 (2000). 106.Lu, H., Isralewitz, B., Krammer, A., Vogel, V. & Schulten, K. Unfolding of titin immunoglobulin domains by steered molecular dynamics simulation. Biophys. J. 75,662-671(1998). 107.Lu, H. & Schulten, K.. Steered molecular dynamics simulation of conformational changes of immunoglobulin domain 127 interpret atomic force microscopy observations. Chem. Phys. 247, 141-153 (1999). 108.Lu, H. & Schulten, K. The key event in force-induced unfolding of titin's immunoglobulin domains. Biophys. J. 79, 51-65 (2000). 109.Lu, H., Krammer, A., Isralewitz, B., Vogel, V. & Schulten, K. Computer modeling of force-induced titin domain unfolding, in Elastic Filaments of the Cell, Vol.481 143-162(2000). 110.Isralewitz, B., Gao, M., Lu, H. & Schulten, K. Forced unfolding or mutant titin immunoglobulin domains with steered molecular dynamics. Biophys. J. 78, 28A28A (2000). 111.Gao, M., Lu, H. & Schulten, K. Simulated refolding of stretched titin immunoglobulin domains. Biophys. J. 81, 2268-2277 (2001). 112.Gao, M., Lu, H. & Schulten, K. Unfolding of titin domains studied by molecular dynamics simulations. J. Muscle Res. Cell. Motil. 23, 513-21 (2002). 113.Gao, M., Wilmanns, M. & Schulten, K. Steered molecular dynamics studies of titin II domain unfolding. Biophys. J. 83, 3435-3445 (2002). 114.Li, H. & Fernandez, J.M. Mechanical Design of the First Proximal Ig Domain of Human Cardiac Titin Revealed by Single Molecule Force Spectroscopy. J. Mol. Biol. 334, 75-86 (2003). 115.Zhang, B. & Evans, J.S. Modeling AFM-Induced PEVK Extension and the Reversible Unfolding of Ig/FNIII Domains in Single and Multiple Titin Molecules. Biophys. J. 80, 597-605 (2001). 116.Best, R.B., Fowler, S.B., Toca-Herrera, J.L. & Clarke, J. A simple method for probing the mechanical unfolding pathway of proteins in detail. Proc. Natl. Acad. Sci. (USA) 99, 12143-12148 (2002). 117.Isralewitz, B., Gao, M. & Schulten, K. Steered molecular dynamics and mechanical functions of proteins. Curr. Opin. Struct. Biol. 11, 224-230 (2001). 118.Best, R.B., Fowler, S.B., Herrera, J.L., Steward, A., Paci, E. & Clarke, J. Mechanical unfolding of a titin Ig domain: structure of transition state revealed by combining atomic force microscopy, protein engineering and molecular dynamics simulations. J. Mol. Biol. 330, 867-77 (2003). 119.Fowler SB, Best RB, Toca Herrera JL, Rutherford TJ, Steward A, Paci E, Karplus M, Clarke J. Mechanical unfolding of a titin Ig domain: Structure of unfolding

Molecular Dynamics Simulations

79

intermediate revealed by combining AFM, molecular dynamics simulations, NMR and protein engineering. J. Mol. Biol. 322, 841-849 (2002). 12O.Schwaiger, I., Kardinal, A., Schleicher, M., Noegel, A.A. & Rief, M. A mechanical unfolding intermediate in an actin-crosslinking protein. Nat. Struct. Mol. Biol. 11, 81-85 (2004). 121.Kellermayer, M.S.Z., Bustamante, C. & Granzier, H.L. Mechanics and structure of titin oligomers explored with atomic force microscopy. BBA-Bioenergetics 1604,105-114(2003). 122.Gao, M., Craig, D., Vogel, V. & Schulten, K. Identifying unfolding intermediates of FN-III10 by steered molecular dynamics. J. Mol. Biol. 323, 939-950 (2002). 123.Rief, M., Gautel, M , Schemmel, A. & Gaub, H.E. The mechanical stability of immunoglobulin and fibronectin III domains in the muscle protein titin measured by atomic force microscopy. Biophys. J. 75, 3008-3014 (1998). 124.0berhauser, A.F., Marszalek, P.E., Erickson, H.P. & Fernandez, J.M. The molecular elasticity of the extracellular matrix protein tenascin. Nature 393, 181185(1998). 125.Krammer, A., Lu, H., Isralewitz, B., Schulten, K. & Vogel, V. Forced unfolding of the fibronectin type III module reveals a tensile molecular recognition switch. Proc. Natl. Acad. Sci. (USA) 96, 1351-1356 (1999). 126.Craig, D., Krammer, A., Schulten, K. & Vogel, V. Comparison of the early stages of forced unfolding for fibronectin type III modules. Proc. Natl. Acad Sci. (USA) 98, 5590-5595 (2001). 127.Li, P.C. & Makarov, D.E. Theoretical studies of the mechanical unfolding of the muscle protein titin: Bridging the time-scale gap between simulation and experiment. J. Chem. Phys. 119, 9260-9268 (2003). 128.Li, B., Alonso, D.O.V., Bennion, B.J. & Daggett, V. Hydrophobic hydration is an important source of elasticity in elastin-based biopolymers. J. Am. Chem. Soc. 123, 11991-11998(2001). 129.Li, B., Alonso, D.O.V. & Daggett, V. The molecular basis for the inverse temperature transition of elastin. J. Mol. Biol. 305, 581-592 (2001). 130.Li, B., Alonso, D.O.V. & Daggett, V. Stabilization of globular proteins via introduction of temperature-activated elastin-based switches. Structure 10, 989998 (2002). 131.Li, B. & Daggett, V. Molecular basis for the extensibility of elastin. J. Muscle Res. CellMotil. 23, 561-573 (2002). 132.Li, B. & Daggett, V. The molecular basis of the temperature- and pH-induced conformational transitions in elastin-based peptides. Biopolymers 68, 121-129 (2003). 133.Altmann, S.M., Grunberg, R.G., Lenne, P.F., Ylanne, J., Raae, A., Herbert, K., Saraste, M., Nilges, M. & Horber, J.K.H Pathways and intermediates in forced unfolding of spectrin repeats. Structure 10, 1085-1096 (2002). 134.Verkhivker, G.M., Bouzida, D., Gehlhaar, D.K., Rejto, P.A., Freer, S.T. & Rose, P.W. Complexity and simplicity of ligand-macromolecule interactions: the energy landscape perspective. Curr. Opin. Struct. Biol. 12, 197-203 (2002). 135.Kosztin, D., Izrailev, S. & Schulten, K. Unbinding of retinoic acid from its receptor studied by steered molecular dynamics. Biophys. J. 76, 188-197 (1999).

80

W. Nowak and P. E. Marszalek

136.Heymann, B. & Grubmuller, H. Molecular dynamics force probe simulations of antibody/antigen unbinding: entropic control and nonadditivity of unbinding forces. BiophysJ. 81, 1295-1313 (2001). 137.Gullingsrud, J., Kosztin, D. & Schulten, K. Structural determinants of MscL gating studied by molecular dynamics simulations. Biophys. J. 80, 2074-2081 (2001). 138.Gullingsrud, J. & Schulten, K. Gating of MscL studied by steered molecular dynamics. Biophys. J. 85, 2087-2099 (2003). 139.Kong, Y., Y. Shen, T. E. Warth & Ma, J. Conformational pathways in the gating of Escherichia coli mechanosensitive channel. Proc. Natl. Acad. Sci. (USA) 99, 5999-6004 (2002). 14O.Ma, J. & Kaplus, M. Molecular switch in signal transduction: Reaction paths of the conformational changes in ras p21. Proc. Natl. Acad. Sci. (USA) 94, 1190511910(1997). 141.Diaz, J.F., Wroblowski, B., Schlitter, J. & Engelborghs, Y. Calculation of pathways for the conformational transition between the GDP and GTP bound states of the Ha-ras-p21 protein. Calculations with explicit solvent simulations and comparison with calculations in vacuum. Proteins Structure Function and Genetics 28, 434-51 (1997). 142.Kosztin, I., Bruinsma, R., O'Lague, P. & Schulten, K. Mechanical force generation by G proteins. Proc. Natl. Acad. Sci. (USA) 99, 3575-3580 (2002). 143. Bayas, M.V., Schulten, K. & Leckband, D. Forced detachment of the CD2CD58 complex. Biophys. J. 84, 2223-2233 (2003). 144.Lu, S. & Long, M. Forced extension of P-selectin construct using steered molecular dynamics. Chinese Sci. Bull. 49, 10-17 (2004). 145.Best, R.B., Li, B., Steward, A., Daggett, V. & Clarke, J. Can non-mechanical proteins withstand force? Stretching barnase by atomic force microscopy and molecular dynamics simulation. Biophys. J. 81, 2344-2356 (2001). 146.Harvey, S.C. & Gabb, H.A. Conformational transition using molecular dynamics with minimum biasing. Biopolymers 33, 1167-1172 (1993). 147.Paci, E. & Karplus, M. Unfolding proteins by external forces and temperature: The importance of topology and energetics. Proc. Nat. Acad. Sci. (USA) 97, 65216526 (2000). 148.Paci, E., Caflisch, A., Pluckthun, A. & Karplus, M. Forces and energetics of hapten-antibody dissociation: A biased molecular dynamics simulation study. J. Mol. Biol. 314, 589-605 (2001). 149.Heymann, B. & Grubmueller, H. AN02/DNP-hapten unbinding forces studied by molecular dynamics atomic force microscopy simulations. Chem. Phys. Letters 303,1-9(1999). 150.Paci, E., Smith, L.J., Dobson, C M . & Karplus, M. Exploration of partially unfolded states of human alpha-lactalbumin by molecular dynamics simulation. J. Mol. Biol. 306, 329-347 (2001). 151.Marchi, M. & Ballone, P. Adiabatic bias molecular dynamics: a method to navigate the conformational space of complex molecular systems. J. Chem. Phys. 110,3697-3702(1999). 152.Keller, D., Swigon, D. & Bustamante, C. Relating single-molecule measurements to thermodynamics. Biophys. J. 84, 733-738 (2003).

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153.Fox, R.F. Using nonequilibrium measurements to determine macromolecule freeenergy differences. Proc. Natl. Acad. Sci. (USA) 100, 12537-12538 (2003). 154.Balsera, M., Stepaniants, S., Izrailev, S., Oono, Y. & Schulten, K. Reconstructing Potential Energy Functions from Simulated Force-Induced Unbinding Processes. Biophysical! 73, 1281-1287(1997). 155.Evans, E. & Ritchie, K. Dynamic strength of molecular adhesion bonds. Biophys. J. 72, 1541-55 (1997). 156.Jarzynski, C. Equilibrium free energy differences from nonequilibrium measurements: a master equation approach. Phys. Rev. E 56(1997). 157.Jarzynski, C. Equilibrium free energies from nonequilibrium processes. Ada Phys. Pol. B 29, 1609(1998). 158.Jarzynski, C. How does a system respond when driven away from thermal equilibrium? Proc. Nat. Acad. Sci. (USA) 98, 3636 (2001). 159.Hummer, G. & Szabo, A. Free energy reconstruction from nonequilibrium singlemolecule pulling experiments. Proc. Natl. Acad. Sci. (USA) 98, 3658-3661 (2001). 16O.Jensen, M.O., Park, S., Tajkhorshid, E. & Schulten, K. Energetics of glycerol conduction through aquaglyceroporin GlpF. Proc. Natl. Acad. Sci. (USA) 99, 6731-6736(2002). 161.Amaro, R., Tajkhorshid, E. & Luthey-Schulten, Z. Developing an energy landscape for the novel function of a (beta/alpha)8 barrel: ammonia conduction through HisF. Proc. Natl. Acad. Sci. (USA) 100, 7599-604 (2003). 162.Park, S., Khalili-Araghi, F., Tajkhorshid, E. & Schulten, K. Free energy calculation from steered molecular dynamics simulations using Jarzynski's equality. J. Chem. Phys. 119, 3559-3566 (2003). 163.Zuckerman, D.M. & Woolf, T.B. Theory of a systematic computational error in free energy differences. Phys. Rev. Lett. 89, 180602 (2002). 164.Liphardt, J., Dumont, S., Smith, S.B., Tinoco, I. & Bustamante, C. Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski's equality. Science 296, 1832-1835 (2002). 165.Ludemann, S.K., Lounnas, V. & Wade, R.C. How do substrates enter and products exit the buried active site of cytochrome P450cam? 1. Random expulsion molecular dynamics investigation of ligand access channels and mechanisms. J. Mol. Biol. 303, 797-811 (2000). 166.Ludemann, S.K., Lounnas, V. & Wade, R.C. How do substrates enter and products exit the buried active site of cytochrome P450cam? 2. Steered molecular dynamics and adiabatic mapping of substrate pathways. J. Mol. Biol. 303, 813-830 (2000). 167.Brockwell, D.J., Paci, E., Zinober, R.C, Beddard, G.S., Olmsted, P.D., Smith, D.A., Perham, R.N. & Radford, S.E. Pulling geometry defines the mechanical resistance of a beta-sheet protein. Nat. Struct. Biol. 10, 731-7 (2003). 168.Lorenzo, A.C., Pascutti, P.G. & Bisch, P.M. Nonspecific interaction forces at water-membrane interface by forced molecular dynamics simulations. J. Comp. Chem. 1A, 328-339 (2003). 169.Hummer, G. & Szabo, A. Kinetics from nonequilibrium single-molecule pulling experiments. Biophys J 85, 5-15 (2003).

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17O.Cieplak, M., Hoang, T.X. & Robbins, M.O. Thermal folding and mechanical unfolding pathways of protein secondary structures. Proteins-Structure Function and Genetics 49, 104-113 (2002). 171.Cieplak, M., Hoang, T.X. & Robbins, M.O. Folding and stretching in a go-like model of titin. Proteins-Structure Function and Genetics 49, 114-124 (2002). 172.Bockmann, R.A. & Grubmuller, H. Nanoseconds molecular dynamics simulation of primary mechanical energy transfer steps in Fl-ATP synthase. Nat. Struct. Biol. 9, 198-202 (2002). 173.Bockmann, R.A. & Grubmuller, H. Conformational Dynamics of the F(l)-ATPase beta-Subunit: A Molecular Dynamics Study. Biophys. J. 85, 1482-91 (2003). 174.Ma, J.P., Flynn, T.C., Cui, Q., Leslie, A.G.W, Walker, J.E. & Karplus, M. A dynamic analysis of the rotation mechanism for conformational change in F-lATPase. Structure 10, 921-931 (2002). 175.Ma, J.P, Flynn, T.C., Cui, C , Leslie, A.G.W., Walker, J.E. & Karplus, M. A dynamic analysis of the rotation mechanism for conformational change in F-lATPase (vol 10, pg 921, 2002). Structure 10, 1285-1285 (2002). 176.Doltsinis, N.L. & Marx, D. First principles molecular dynamics involving excited states and nonadiabatic transtitions. J. Theoret. Comput. Chem. 1, 319-349 (2002). 177.Hayashi, S., Tajkhorshid, E. & Schulten, K. Molecular dynamics simulation of bacteriorhodopsin's photoisomerization using ab initio forces for the excited chromophore. Biophys. J. 85, 1440-1449 (2003). 178.Hugel, T., Holland, N.B., Cattani, A., Moroder, L., Seitz, M. & Gaub, H.E.Singlemolecule optomechanical cycle. Science 296, 1103-1106 (2002). 179.Holland, N.B., Hugel, T., Neuert, G., Cattani-Scholz, A., Renner, C , Oesterhelt, D., Moroder, L., Seitz, M. & Gaub, H.E. Single molecule force spectroscopy of azobenzene polymers: Switching elasticity of single photochromic macromolecules. Macromolec. 36, 2015-2023 (2003). 180.Ramesh, H.P. & Tharanathan, R.N. Carbohydrates - the renewable raw materials of high biotechnological value. Crit. Rev. Biotechnol. 23, 149-73 (2003). 181.Rief, M., Oesterhelt, F., Heymann, B. & Gaub, H.E. Single molecule force spectroscopy on polysaccharides by atomic force microscopy. Science 275, 12951297(1997). 182.Li, H., Rief, M., Oesterhelt, F., Gaub, H.E., Zhang, X. & Shen, J. Single-molecule force spectroscopy on polysaccharides by AFM - nanomechanical fingerprint of alpha-(l,4)-linked polysaccharides. Chem. Phys. Letters 305, 197-201 (1999). 183.Heymann, B. & Grubmueller, H. Chair-boat transitions and side groups affect the stiffness of polysaccharides. Chem. Phys. Letters 305, 202-208 (1999). 184.Marszalek, P.E., Li, H.B., Oberhauser, A.F. & Fernandez, J.M. Chair-boat transitions in single polysaccharide molecules observed with force-ramp AFM. Proc. Natl. Acad. Sci. (USA) 99, 4278-4283 (2002). 185.O'Donoghue, P. & Luthey-Schulten, Z.A. Barriers to forced transitions in polysaccharides. J. Phys. Chem. B 104, 10398-10405 (2000). 186.Marszalek, P.E., Oberhauser, A.F., Li, H.B. & Fernandez, J.M. The force-driven conformations of heparin studied with single molecule force microscopy. Biophys. J. 85, 2696-2704 (2003). 187.Lee, G., Nowak, W., Jaroniec, J., Zhang, Q. & Marszalek, P.E. Nanomechanical control of glucopyranose rotamers. J. Am. Chem. Soc, 126, 6218-6219 (2004).

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188.Kale, L., Skeel, R., Bhandarkar, M., Brunner, R., Gursoy, A., Krawetz, N., Phillips, J., Shinozaki, A., Varadarajan, K., Schulten, K. NAMD2: Greater scalability for parallel molecular dynamics. J. Comp. Phys. 151, 283-312 (1999). 189.Lu, Z., Nowak, W., Lee, G., Marszalek, P.E. & Yang, W. The elastic properties of single amylose chains in water: a quantum mechanical and AFM study. J. Am. Chem. Soc, (2003) in press. 19O.Cui, Q., Elstner, M., Kaxiras, E., Frauenheim, T. & Karplus, M. A QM/MM implementation of the self-consistent charge density functional tight binding (SCC-DFTB) method. J. Phys. Chem. B 105, 569-585 (2001). 191.Liu, H., M. Elstner, E. Kaxiras, T. Frauenheim, J. Hermans & Yang, W. Quantum mechanics simulation of protein dynamics on long time scale made possible. Proteins: Structure, Function and Genetics 44, 484-489 (2001).

Chapter 3: Molecular Dynamics Simulations of a Molecular Electronics Device: The NanoCell

Jorge Seminario, Pedro Derosa\ Luis Cordova6 and Brian Bozardc Chemical Engineering Department, Texas A&M University, College Station, Texas 77843, USA "Institute for Micromanufacturing/Department of Physics, Louisiana Tech. University, Ruston, LA71272, USA ^Department of Electrical Engineering, University of South Carolina, Columbia, SC 29208, USA (Former affiliation for all authors) cSpace

and Naval Warfare (SPAWAR) Systems Center, Charleston, SC 29419-9022, USA

Abstract Molecular dynamics simulations using ab initio force fields and signal processing techniques are combined to analyze the dynamic properties of a minimum unit of a field programmable random molecular array also called nanocell. We analyze stretching, angular, and torsional internal modes at several temperatures of operation and their correlations as well as the possibility of signal modulation and processing at terahertz frequencies. The proposed scenario is another alternative for molecular electronics, which is being developed at very low frequencies of operation using the electronic states of molecules, clusters and other nanoscopic systems.

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1. Introduction The use of small molecules to complement silicon-based electronic devices is under intensive development [1-7]; each molecule occupies an area between 105 to 106 [2, 5] times smaller than conventional electronic devices and therefore provides hopes to go beyond the limits set by the use of bulk silicon devices. On the other hand, molecules would allow us to develop several other applications directly related or reminiscent of biological systems. All this impetus has created the new field of molecular electronics (moletronics). Moletronics focuses on the development of electronic devices using small molecules with feature sizes on the order of one or a few nanometers [2, 5]. These very small feature sizes are well below the minimum feature size of present electronic devices (-45 nm in the year 2004). Notice that a high doping of 1018 cm"3 corresponds to an average separation between nearest impurities of 10 nm thus a radical change of technology would be needed when approaching such a size. The ultimate goal of miniaturization is to use the minimum amount of atoms per electronic function. However, this goal poses a strong challenge since molecules with dimensions smaller than 10 nm cannot be assembled as standard semiconductor devices using modern lithographic techniques; the alternative is the use of other techniques such as molecular self-assembly and vapor deposition, which bring a strong random component into the field. Deposition techniques yield patterns resulting in minimum units that can be thoroughly analyzed but it will be required that they can be reconfigured (programmed) in order to compensate for their lack of addressability [8, 9]. Thus, programmability of devices at the molecular level compensates for their lack of addressing. Addressing is one of the most important properties of standard microelectronics. Fortunately, programmability of non addressable molecular devices can be achieved using molecules featuring highly nonlinear current-voltage characteristics like negative differential resistance (NDR) [1]. Experiments showing NDR and their use as memories have been already reported [10, 11], and a strong effort is presently underway in order to demonstrate programmable molecular devices [1,6, 12]. Addressability during the construction and operation of molecular devices is a challenge. Both self-assembling [5, 13] and direct assembling [14] have been proposed for the construction and they depend on the chemical affinity of terminal groups attached to the "electrically active" part of the molecule. Due to the non-deterministic characteristics of these processes, molecules are expected to be arranged in quasi-random configurations, which together with their small size, make them no directly addressable. The conversion, reconfiguration, or reprogramming during operation of a randomly constructed

Molecular Dynamics Simulations of a Molecular Electronics Device 87

circuit relies on the use of molecules with highly nonlinear features that allow the system to have alternative states able to be set externally after construction to perform a specific function [1, 8,9, 12]. Molecular circuits will be created by chemical means inside externally addressable units; however the inside of these units cannot be addressed except by the external programming; these small units are called nanoCells. The nanoCell is one of the most important and viable concepts for the development of electronics beyond the deterministic or lithographic approach of present C-MOS integrated circuits. Very few scenarios have been proposed to the use of molecular-based devices in order to bypass problems undermining miniaturization or scaling-down processes for integrated circuits through the introduction of a revolutionary approach in which molecules or nanoclusters of atoms are used to perform logical operations complementing or extending the capabilities of deterministic ones. The nanoCell concept takes advantage of the great ability of chemistry to synthesize molecules with specific characteristics with tolerances of fractions of one nanometer; such a tolerance is far from the reach of any IC fabrication technique, which is limited to practically two dimensions. However molecules are created in a single flask in quantities in the order of Avogadro's number ~6xlO23 and they are not individually addressable or interconnectable thus it is impossible to read or write any information in a deterministic manner. A way out of this problem is to address a molecular circuit constructed by chemical means, of a size such that is addressable by deterministic techniques (like present lithography), and indirectly program the individual molecules for performing specific and more complex functions than their microelectronics counterparts. This almost unthinkable approach is possible if the molecules (assembled almost randomly) can be chosen with programmable features. It has been already shown in the literature that even two single molecules with strong non linear characteristics can perform programmable functions thus compensating for the lack of addressability of single molecules [1, 8, 9, 12]. The applications of these nanoCells can be easily extended interconnectiong long biomolecules like DNA or one dimensional wires. This macroscopic systems in one dimension (or backbone) and nanoscopic in the other two allow the transmission of vibrations along the molecule backbone due to the presence of actual bands where vibrational modes become part of a one-dimensional continuum and thus signals entered in one place of a long molecule can be read and processed in other places far away in the molecule as it was already suggested ealry in 1941 [15]. In this work, we study one of the possible minimum programmable structures, a square consisting of four active molecules interconnected by four Au clusters. This arrangement is one of the two most likely allowed to take

88 J. Seminario et al.

place when self-assembling on a substrate (the other one being a triangle) [12]. We focus this study on the thermal stability of the system and the correlation among vibrations in the molecules. Ab initio calculations, molecular dynamics simulations, and signal processing techniques are combined to achieve this aim. MD simulations provide time dependant information of the system, and signal processing techniques are used for the analysis of the time-domain signal in order to show the strong correlations among the dynamics of internal degrees of freedom and to suggest a new scenario for the use of molecular arrangements as signal processing devices. A detailed analysis of the frequency spectra yields correlations between several local vibrations that can be used to include vibrational signals externally introduced through the contacts of the molecular device. Although these correlations between internal modes are not surprising, the question is if using the natural amplitude oscillations of the vibrational modes, which can be experimentally detected, will allow the detection of mixed signals in localized sites. If this were possible, several single operations could be implemented, enabling molecular processors dealing with signals in the THz range of frequencies. We propose that signals can be encoded as the movement of atoms that can be detected observing the changes in the electrical characteristics of the molecule. For instance, when the displacement, ? f of atoms associated to a vibrational mode changes the dipole moment of the molecule ^, a signal proportional to the so called infra red (IR) intensity is obtained from IR Intensity <* — \dr )

and when atoms displacements affect the polarizability,

«, a signal

proportional to the so called Raman intensity is obtained, Raman Intensity

dr )

These two intensities which correspond to the most common experimental spectroscopic techniques, can be unambiguously determined theoretically, and are potential candidates to be used as means for encoding information because, on the other hand, induced external changes in the dipole and polarizability by external fields yield changes in the displacements of the atoms in the molecule. Thus, signals are internally represented by atom displacements and transduced to IR and Raman intensities.

Molecular Dynamics Simulations of a Molecular Electronics Device 89

2. Procedure and Methods This study focuses on a molecular unit consisting of four A11405 clusters interconnecting four active molecular devices (Figure 1). The Au clusters are roughly spherical with diameter o f - 2 . 3 nm. The molecule, 2,5-dinitro-l,4diethynylphenylthiolate-benzene, dinitro hereafter, is a phenyl-based oligomer known to perform negative differential resistance NDR, a very important feature for molecular electronics applications [10, 16, 17]. We combine molecular dynamics simulations, able to simulate time evolution of the system at different temperatures, with digital signal processing techniques, able to analyze the time-domain signals obtained by the MD simulations. The initial geometry of the system is obtained minimizing the energy of the whole structure using the fast minimizer tool [18] from the Cerius2 [19] suit of programs. The MD runs solve Newton's equations simultaneously for all atoms in the system using the leap frog algorithm [20] as implemented in Cerius2 [19]. We use the universal force field (UFF) [21], which is parameterized for all the elements in the periodic table and validated using experimental data for organic molecules [22], main-group compounds [23], and metal complexes [24]. UFF uses combination rules based on elements, hybridization, and connectivity to determine interaction parameters from a set of parameters characterizing each atom type. In this work, modifications are performed for Au-Au and Au-S interactions with respect to those in the UFF [21, 25]. We chose a 12-10 potential

with XAU-AU - 2.884 A and Xs.Au - 2.350 A. For a more general description of the UFF, the reader is referred to the original work of Rappe [21], and to the compilation and implementation done by Senn [25]. The Au-Au interaction with a 12-10 potential, which is an approximation from a more realistic Sutton-Chen many-body potential [26, 27]. The equilibrium distance Au-Au is set to 2.884, which corresponds to the Au-Au distance between nearest neighbors in an fee Au lattice,[28] and it is within the equilibrium geometries of 2.6 and 3.0 A, depending of the size and shape of the cluster obtained from ab initio calculations [29]. The interaction energy DQ of 14.7 kcal/mol is obtained from the experimental cohesive energy of 87.5 kcal/mol [28], which accounts for 12 bonds per atom, but each bond involves two atoms; thus, the interaction energy is six times smaller than the cohesive energy per atom.

90 J. Seminario et al.

Figure 1. (Caption given on the next page)

Figure 2. (Caption given on the next page)

Molecular Dynamics Simulations of a Molecular Electronics Device 91

Figure 1. Minimum programmable unit in square conformation consisting of four Au clusters interconnected with four "dinitro" molecules consisting of four A11405 clusters interconnecting four active molecular devices. The Au clusters are roughly spherical with diameter of ~2.3 nm. The molecule, 2,5-dinitro-l,4-diethynylphenylthiolatebenzene, also called dinitro is a phenyl-based oligomer known to perform negative differential resistance NDR. Figure 2. Time and frequency domain plots of the C-H bond vibration at 10 K for a) the C-H in the top ring closer to the central ring and b) the C-H in the central ring. Time domain signals are shown on the left and frequency spectra are on the right. The analyzed bonds are boxed on the molecule. The S-Au is also chosen to be a 12-10 potential in order to allow the S alligator clip to attach to or to dissociate from the Au clusters. The equilibrium bond length is set to 2.35 A, which is based on several density functional theory calculations on similar systems, some of them already reported[30], however, Do is set to 14.7 kcal/mol in order to make this bonded interaction simulated by nonbonded fields and compatible with calculated geometrical structures. Therefore, the Au-S interaction is chosen based on a precise reproduction of geometrical features, rather than energetic ones. Nevertheless, the bond energy of a S-Au bond is 38 kcal/mol, which can be directly associated with roughly three Au atoms, thus corresponding to an average of 13.0 kcal/mol per interaction, which is consistent with the 14.7 kcal/mol chosen for geometrical purposes. This is simply an average to have an idea of the magnitude of the energies involved; it does not mean that the S atom bonds equally to three Au atoms as already shown theoretically [31-34] [16] and experimentally [35-37]. The vibrational analysis is performed using a series of simple filters applied to the time dependant atomic displacement signals. Briefly, a Hamming window is first applied to reduce spectral leakage. Lowpass filters, bandpass filters, and signal rectifiers are then used to isolate selected vibrations of interest. A very detailed description of the methods used in this paper can be found elsewhere [38].

3. Results Table 1 shows average bond length and vibrational frequencies obtained for all the bonds analyzed in this study, when available, comparison to experimental values of the corresponding bond stretching modes are provided. We focus our study on the top and central rings of the three-ring molecule (Figure 1). It is expected that the bottom ring behave similarly to the top ring. Two CH bonds,

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one located on the top ring and the other in the central ring, are chosen for this study since they represent two different environments and can provide complementary information about other vibrations in the molecule. The C-H on the top ring (C-Hcc) is located nearby the ethynyl group separating the two rings; the C-H bond in the central ring (C-HNo2) can provide information about the NO 2 substituents. The other vibrations considered in this work are the C-C triple bond in the ethynyl group joining the top ring to the central ring, two resonant bonds on the top ring, the C-S bonds and C-S-Au angle, the scissoring and rotation of the NO 2 , and the relative rotation between the top and the central ring. Table 2 summarizes the results obtained for NO 2 modes (scissoring, rotation). Table 1. MU4 molecular dynamics results for bond stretchings. We show more than one value of frequency when more than one peak of high intensity is observed in the frequency spectrum. Bond T(K) Average Vibrational Vibrational Experimental bond Frequency Frequency Frequency length (A) (THz) (cm'1) (cm'1) 10 1.084 92.4 3082 C H 100 1.084 92.4 3082 CC 300 1.085 92.5 3085 3O51i 1000 1.089 92.6 3089 3002 [39] C R NO2

rr-r-i

10 100 300 1000 10

1.084 1.084 1.085 1.090 1.206

92.3 92.3 92.5 92.4 54.9

3079 3079 3085 3082 1831 2038 2038 2038

J^m 3000[4°]

"CTnple

10 ° 300 1000

L206 1.206 1.208

6L1 61.1 61.1

C-C Resonant 1,6

10 100 300 1000

1.400 1.400 1.401 1.404

34.3/47.6 47.6 47.6 27.7/47.4

1144/1588 1588 1588 924/1581

10 100 300 1000

1.401 1.402 1.402 1.406

49.2/55.9 48.9 55.7 55.7/48.9

1431/1865 1631 1858 1858/1631

1500[40]

10

1.789

—2

1434

794 [41]

C

C-C Resonant 4,5 C-S

, 2 1 2I00[4

°]

1600.

Molecular Dynamics Simulations of a Molecular Electronics Device 93 Bond

T(K)

Average Vibrational Vibrational Experimental bond Frequency Frequency Frequency (cm' 1 ) (cm' 1 ) length (A) (THz) 1.789 m2 1434 624 [42] 100 1.791 —2 1441/1144 582 [43] 300 1000 1.796 —2 1194/530/1438 ' Average value for the C-H vibration is 3035 cm"1, 3044 cm"1, 3059 cm"', and 3067 cm"1, corresponding to the Wilson vibrational modes 2, 7, 13, and 20, which belong to the C-H stretching region in a benzene molecule [44]. 2' There are too many peaks, several of them around the experimental frequencies; however, it is not possible an unambiguous determination. Table 2. MU4 molecular dynamics results: NO2 scissoring and rotational modes. We show more than one value of frequency when more than one peak of high intensity is observed in the frequency spectrum. Mode

O-N-O scissoring

10 100 300 1000

Average Value (degree) 118.6 118.4 118.3 118.0

NO 2 rotational

10 100 300 1000

23.6 24.1 26.7 77.5

T (K)

Vibrational Frequency (THz) 27.4/26.3 48.5/27.4 26.3/27.4/48.5 27.4/26.3

Vibrational Frequency (cm"') 914/877 1618/914 877/914/1618 914/877

1.6 1.3/1.6 1.3/1.6

53.4 43.4/53.4 43.4/53.4

Experimental Values (degree) 124.4 (angle ONO)

4. C-H Bonds These bonds only differ on their van der Waals (vdW) interactions showing only different magnitudes around the same frequency in the spectra. The CHCc and C-HN02 time and frequency domains plots at 10 K are shown in Figure 2a and b, respectively. Their equilibrium bond length and a vibration frequency are very similar and not affected by their environment; however, the maximum displacement, which is 0.016 A for C-HCc is only 0.010 A for CHNO2 because the NO2 restricts the H movement. In fact, the frequency spectrum for C-HN02 (Figure 2c) shows a strong line located at about 27 THz, which is the characteristic frequency for the scissoring modes of the NO 2 (Table 2). Two other strong lines at 49 and 55 THz are also common to both

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the C-H vibrations and the scissoring mode; however, these lines are associated with the C-C resonant bond vibration (vide infra).

Figure 3. Time and frequency domain plots of the C-H bond vibration at 100 K. for a) the C-H in the top ring closer to the central ring and b) the C-H in the central ring. Time domain signals are shown on the left and frequency spectra are on the right. The analyzed bonds are boxed in the molecule.

The characteristic spectra at 100 K are shown in Figure 3 for the two C-H bonds. Frequency and bond length do not change with respect to those at 10 K; however, the oscillation amplitude increases almost three times for C-H cc (0.044 A), and more than three times for C-HN02 (0.036 A) with respect to the observed amplitudes at 10 K.

Molecular Dynamics Simulations of a Molecular Electronics Device 95

Figure 4. Time and frequency domain plots of the C-H Bond vibration at T = 300 K for a) the C-H in the top ring closer to the central ring and b) the C-H in the central ring. Time domain signals are shown on the left and frequency spectra are on the right. The analyzed bonds are boxed in the molecule.

At 300 K (Figure 4), both the equilibrium bond length and the oscillation frequency slightly increase (0.001 A and 0.1 THz respectively) and the maximum displacement increases three times with respect to that at 100 K (0.095 for C-H cc and 0.087 A for C-H N02 ).

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V

Figure 5. Time and frequency domain plots of the C-H bond vibration at 1000 K for a) the C-H in the top ring closer to the central ring and b) the C-H in the central ring. Time domain signals are shown on the left and frequency spectra are on the right. The analyzed bonds are boxed on the molecule. At 1000 K (figure 5), the peak at the characteristic frequency of the C-H vibration spreads out and increases in intensity. All other peaks in the spectra are very small; as a consequence, modulation is small. The equilibrium bond length shows the largest increase but still limited due to the harmonic force field used to simulate this bond, which does not allow bond dissociations only allowed for Au-Au and Au-S bonds [38]. The vibrational frequency did not change with the temperature. The maximum displacement duplicates with respect to the case at 300 K (0.186, and 0.205 A, respectively, for C-HCc and C-HNo2)- 1000 K is likely beyond the operating temperature for this system.

Molecular Dynamics Simulations of a Molecular Electronics Device 97

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Figure 6. Time and frequency plots of the upper C-C bond in the top benzene ring at a) 10, b) 100, c) 300, and d) 1000 K. Time domain signals are shown on the left and frequency spectra are on the right. The analyzed bond is boxed on the molecule. The dominant frequency of each of the C-H vibrations we analyzed for the 10, 100, 300, and 1000 K simulations ranges from 92.3 THz (3079 cm-1) to 92.6 THz (3089 cm"1) (Table 1). For these cases, the stretching vibrational frequencies lie very close to the experimental values 3035 cm"1, 3044 cm"1, 3059 cm"1, 3067 cm"1 corresponding to the Wilson vibrational modes no. 2, 7, 13, and 20, of the C-H stretching region in a benzene molecule [44].

5. C-C Resonant Bonds The time and frequency domain spectra for two C-C resonance bonds in the upper benzene ring are shown in Figure 6 and 7. Each figure shows the vibration spectra of the 10, 100, 300, and 1000 K simulations for each of these bonds. These vibrations are much less symmetric across their mean value than the C-H bond vibrations. Figure 6 shows the results for the upper C-C bond in the benzene ring. The average length of this bond ranges from 1.400 A to 1.404 A (Table 1) as the temperature increases from 10 K to 1000 K. The bond length is almost the same up to 300 K (1.400 A for 10 and 100 K and 1.401 A for 300 K) and goes to 1.404 at 1000 K. The frequency spectra for these vibrations have many well-defined peaks that are present at all temperatures, but with varying magnitudes. The peak that is consistently dominant relative to other peaks over the entire range of temperatures is located at 47.6 THz (1586 cm"1) and could be assigned to the characteristic frequency for this vibration; however, this is not the highest peak for all the cases. A second peak at ~49 THz is also present and could be due to the bottom C-C resonant bond as explained in the following paragraph. The presence of several dominant peaks at all temperatures is an indication of a strong correlation that the C-C resonant bond has with other vibrations in the system. There is an isolated peak located above 90 THz, which comes from the C-H vibration. Figure 7 shows the results for the lower C-C bond in the top benzene ring. The average length of this bond ranges from 1.401 A to 1.406 A as the temperature increases from 10 K to 1000 K. The frequency content of this bond vibration spectrum is similar to that of the upper C-C bond shown in Figure 6. A well-defined peak at 49 THz (1633 cm"1) dominates the spectra for the lower C-C resonant bond. This frequency was also observed in the top C-C resonant bond with less intensity than the one at ~47 THz. Conversely, a

Molecular Dynamics Simulations of a Molecular Electronics Device 99

small peak at ~47 THz is also observed in the spectra of the bottom C-C resonant bond. We can conclude, based on the relative intensity of these two peaks that the top C-C bond vibrates with a characteristic frequency of 47.6 THz while the bottom C-C bond characteristic frequency is slightly shifted to 49 THz. The C-C resonance bond vibration spectra vanishes after 60 THz, and a well-defined peak takes place in the neighborhood of the C-H vibration, which is around 92.5 THz. For the most part, the C-C vibration spectra are characterized by several well-defined peaks in the range of about 0 to 70 THz. These characteristics peaks can also be observed in the C-H vibration spectra. This confirms the correlation between the C-H bond vibrations and the vibrations of their neighboring C-C bonds. We can conclude that within the C-C vibrations, we also find the contribution of the C-H bond vibrations, and vice versa.

100 J. Seminario et al.

Figure 7. Time and frequency plots of the lower C-C bond in the top benzene ring at a) 10, b) 100, c) 300, and d) 1000 K. Time domain signals are shown on the left and frequency spectra are on the right. The analyzed bond is boxed on the molecule.

Molecular Dynamics Simulations of a Molecular Electronics Device 101

Figure 8. Time and frequency domain plots of the C-C triple bond vibration at a) 10, b) 100, c) 300, and d) 1000 K. Time domain signals are shown on the left and frequency spectra are on the right. The analyzed bond is boxed on the molecule.

6. C-C Triple Bond The C-C triple bond located between the upper and middle rings in the molecular device is analyzed in Figure 8. The average length is 1.206 A up to 300 K, and 1.208 A for 1000 K. These values are really close to the experimental bond length for ethyne (1.203 A [45]). At 10 K (Figure 8a), the dominant peak in the frequency spectrum is located at 54.9 THz (1831 cm"1), which is intermediate between the expected vibrational frequency of C-C resonant bond (-48 THz) and C-C triple bond of about 61 THz (Table 1)

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showing the strong effect that the ring vibrations have on this bond. For the 100, 300, and 1000 K spectra shown in Figure 8b to d, the dominant peak is located at ~61 THz (2033 cm"1), which is in perfect agreement with the experimental C-C vibrational frequency of 2011 cm"' [46] for ethyne HCCH. A line at ~61 THz is also present at 10 K, but is not dominant.

Molecular Dynamics Simulations of a Molecular Electronics Device 103

Figure 9. Time and frequency domain plots of the C-S bond vibration results at a) 10, b) 100, c) 300, and d) 1000 K. Time domain signals are shown on the left and frequency spectra are on the right. The analyzed bond is boxed on the molecule.

7. C-S Bond The vibrational spectra for the C-S bond located at the top of the molecular device are shown in Figure 9. The dominant frequency for 10, 100, and 300 K is around 43 THz (1434 cm"1), which is certainly not due to the intrinsic C-S expected to be -20 THz [41-43] (Table 1), but rather to contributions from other vibrations that have been mixed into the C-S bond vibration. The resonant C-C bond on the top of the ring has a vibration around this value, for instance. A line at around 17 THz is present in the frequency spectrum at all temperatures and it corresponds to the characteristic frequency for this vibration. The spectrum in this range of frequencies shows several peaks from neighbor atoms vibrations that makes difficult to characterize it.

104 J. Seminario et al.

i

3

Molecular Dynamics Simulations of a Molecular Electronics Device 105

Figure 10. Time and frequency domain plots of the Au-S-C angle bending at a) 10, b) 100, c) 300, and d) 1000 K . Time domain signals are shown on the left and frequency spectra are on the right. The analyzed bond is boxed on the molecule.

8. Angle Au-S-C The results of the Au-S-C angle bending mode are shown in Figure 10. The average value for this angle increases from 98° to 120° as the temperature increases from 10 to 1000 K. Ab initio calculations of the C-S-Au predict for this angle a value in that range (-105°) [16, 33], and a temperature increase is expected to yield a wider distribution of angles due to thermal agitation. Notice that we have not calculated ab initio values for this angle within the force field; S is not ligand to Au in our simulation thus a Lennard-Jones (LJ) type of interaction is responsible for this grouping. LJ force field potential is a two-body interaction, which is a particular case of the many-body Sutton-Chen many-body potentials [26, 27]. Then it follows that the Au-S-C equilibrium angle is a result mainly of the one to one interactions with surrounding atoms. It can be inferred, when comparing to ab initio values, that the model used for the S to Au atom interactions reproduces similar physical parameters of the SAu contact. The vibration of the S to Au distance yields a spectra showing low frequencies contributions. These contributions have several very close peaks of similar amplitude indicating that the angle along Au-S-C is affected by low frequency vibrations of nearby atoms, like the Au-Au vibration in the contact. As for the C-S bond, the abundant low frequency contributions from the contacts makes difficult to characterize this vibration.

106 J. Seminario et al.

Figure 11. Time and frequency domain plots of the O-N-O angle bending at a) 10, b) 100, c) 300, and d) 1000 K. Time domain signals are shown on the left and frequency spectra are on the right. The analyzed bond is boxed on the molecule.

Molecular Dynamics Simulations of a Molecular Electronics Device 107

9. Angle O-N-O The results for the O-N-0 angle bending are shown in Figure 11. The average value for this angle decreases from 118.6° to 118.1° as the temperature increases from 10 to 1000 K. The spectra for this angle-bending mode have several well-defined peaks. At all temperatures there are three peaks located around 25, 26, and 27.5 THz from which the peak at 27.5 THz dominates at all temperatures (except at 300 K where the one at 26 THz is slightly more intense). The strong peak located at 48.5 THz is due to the stretching mode of the C-C bonds in the aromatic ring and modulates the scissoring vibrational mode of the NO 2 .

(

108 J. Seminario et al.

Molecular Dynamics Simulations of a Molecular Electronics Device 109

Figure 12. Time and frequency domain plots of the C-C-C-C dihedral at a) 10, b) 100, c) 300, and 1000 K. Time domain signals are shown on the left and frequency spectra are on the right. A zoom to the 0-5 THz region is shown below the corresponding time and frequency spectra. The analyzed bond is boxed on the molecule.

10. Relative Rotation Between Rings (Dihedral) In order to study the relative rotation between two consecutive benzene rings, we define an "improper" dihedral angle between the planes of the top and central rings, which represents the relative orientation of the planes of each ring. The fact that the two C atoms, one in each ring, defining this dihedral are not formally bonded does not affect this definition since they are practically co-linear with the two C atoms connecting them. We study the cosine of the dihedral angle instead of the angle itself, in order to avoid discontinuities inherent to the +180 to -180 jump. The results obtained for the C-C-C-C dihedral angle rotation at 10, 100, 300, and 1000 K are shown in Figure 12. For the 10 and 100 K dihedral rotations, shown in Figure 12a and b, the cosine of the angle stays close to 1 indicating that the dihedral in both cases oscillates around its initial value of

110 J. Seminario et al.

approximately 0°. The amplitude of oscillation increases when going from 10 to 100 K; however, it is unlikely that the rings will complete a rotation with respect to one another at these temperatures. For the 300 and 1000 K dihedral rotations shown in Figure 12c and d, several complete ring rotations are observed. The frequency spectrum of the relative rotation of the rings, is characterized by several low frequency (less than 1 THz) peaks located very close to each other as shown in the high resolution plots shown below the corresponding time and frequency spectra in Figure 12 (notice the different vertical scale between the low and high frequency detail plot). The magnitude of the low frequency peaks is clearly much larger than the high frequency peaks. Peaks between 0.5 and 1 THz reduce their intensity with T; whereas for 10 K, the region in the spectra with important peaks goes from 0 to 1 THz, for 1000 K this region goes from 0 to 0.5 THz. The high frequency detail is very noisy and it is very difficult to obtain information from it.

Molecular Dynamics Simulations of a Molecular Electronics Device 111

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(d) Figure 13. Time and frequency domain plots of the O-N-C-C dihedral at a) 10, b) 100, c) 300, and d) 1000 K, a zoon to the 0-5 THz regime is provided for this case. Time domain signals are shown on the left and frequency spectra are on the right. The analyzed bond is boxed on the molecule.

11. NO2 Rotation To analyze the rotation of the NO 2 , we defined a dihedral angle between the planes defined by O-N-C and N-C-C, respectively. We perform the analysis in the same way as for the C-C-C-C dihedral. The results for the O-N-C-C dihedral rotation are shown in Figure 13. At 10, 100, and 300 K the NO 2 does not perform a complete rotation as it can be seen in Figure 13a, b, and c, it oscillates around its initial position of approximately 0°, though the amplitude of oscillation increases with temperature. Only at 1000 K does the NO2 make complete rotations Figure 13d. The spectrum for the NO2 rotation is less noisy than the one for the rings relative rotation; however, the main frequency components spread and shift to lower values when T increases, as it was observed for the C-C-C-C rotation. For 10 K, the higher intensity components are around 1.6 THz; for 100 and 300 K they are below 1.6 THz, whereas for 1000 K, they are below 0.5 THz.

12. Correlation Between Vibrations Several correlations between vibrations can be observed; for instance, the C-H bond affects neighboring vibrations, C-C resonant bonds, sharing one C atom with a C-H bond on the top ring, show a lone peak above 90 THz at all temperatures (Figure 6 and 7). The influence of the CH vibration is instead

Molecular Dynamics Simulations of a Molecular Electronics Device 113

very small on the C-S bond, where a very small peak appear about 90 THz (Figure 9), and it is inexistent on the C-C triple bond (see Figure 8) The effect of the scissoring mode of the NO 2 in the central ring seems to be more delocalized along the molecule than other vibrations; it affects the dynamics of other bonds, and angles even if they are not close by. A peak around -27 THz appears in several other vibrations located even on the top ring. Both C-H resonant bonds, one on the top ring and the other on the central ring, at all temperature, show a distinguishable peak at 27 THz (Figures 2 to 5). A very small peak at 27 THz can be observed on the C-C triple bond (Figure 8) and the C-S vibration (Figure 9), although the peak in those vibrations is just too small to conclude beyond doubt that NO2 is influencing these vibrations. The C-S-Au angle seems not to be affected by the NO 2 scissoring mode (Figure 10). The two C-C resonant bonds we treat in this work, both located on the top ring, show a well defined, and of a relatively high intensity, peak located at ~28 THz (Figure 6 and 7), a small isolated peak at 28 THz is also observed in the spectra of the NO 2 rotation (Figure 13), It is expected that the NO2 rotation have some correlation to the scissoring mode, we can conclude then, that the peak at 28 THz is due to the NO2 scissoring mode, slightly shifted to higher frequencies. Several peaks around 27 THz exist in the relative ring rotation spectrum that could correspond to the three peaks observed in the NO 2 scissoring mode (Figure 12), although the spectra in that region is very noisy and it is difficult to assure such a correlation. Two main frequencies, close to each other, are found for the C-C resonant bonds on the top ring; however, the relative amplitude depends on the position of the bond in the ring, the bond on the top has a main frequency of -47 THz whereas for the C-C at the bottom, the frequency at 49 THz dominates. The NO 2 scissoring mode also shows a peak at 48 THz corresponding to a nearby C-C resonant bond. It is difficult to follow the relative ring rotation and the NO 2 rotation in other vibrations due to the low frequency the former two rotations have and the very noisy nature of the spectra at low frequency.

13. Discussion and Conclusions Frequency analysis applied to individual bonds, angles, and dihedrals, is proposed as an alternative to the usual vibrational mode analysis. The analysis we carried out in this work has several advantages, it is simpler than identifying normal modes that in a system as big as the one in this work is a nearly impossible task. Normal modes are perpendicular to each other, thus

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correlations between different vibrations in the molecule cannot be obtained from that analysis. Normal modes usually involve several vibrations along the molecule, all vibrating at the same frequency; conversely, each vibration can be part of several normal modes. The vibration of a bond, angle, or dihedral is projected onto several normal modes thus analyzing normal modes only partial aspects of the dynamics of each vibration can be seen. The analysis we perform here, provides the entire frequency spectra involved in each vibration; then correlating vibrations by identifying similar frequency components, the effect that a particular vibration has on one part of the molecule can be inferred. Of particular interest is the correlation between a vibration in the molecule that can be controlled externally and other vibrations. For instance the relative rotations between the central ring and the other two, can be controlled by an external field if, for instance a dipole moment is created in the central ring by adding adequate substituents. We found that some vibrations, such us the C-H vibration, affects strongly neighboring vibrations, however, the rotation of the NO 2 group also affects vibrations relatively further from it, C-C resonant bond, C-S vibration and C-H vibration are affected by NO 2 rotation. In addition to the vibrational properties, MD is also used to obtain operational information, we determine that 1000K is most probably beyond the operational T for this device. The procedure used in this work is very promising although simple. By analyzing spectral information on one bond, we can infer how to excite another bond; thus, the dynamics of a different bond can be modified and therefore applications in molecular electronics, and signal processing, among others, are possible. A potential scenario for the implementation of molecular electronic circuits is introduced by applying digital signal processing techniques to results from classical molecular dynamics simulations of a molecular system interconnected by nanosize gold clusters. Under this new scenario, signals can be introduced, processed, and read through interactions with the internal vibrational modes of the small molecular unit. At least modulation operations are intrinsically inherent to any molecular system. We demonstrated how molecular dynamics techniques can be used for the analysis of molecular electronic circuits; this is important since it will provide ideas as how the molecular circuits can be implemented to perform signal processing. A modified force-field is implemented for Au-Au and S-Au reproducing very well geometrical features of these interactions, and yielding calculated frequencies in very good agreement with available experimental information. On the other hand, analyzing the vibrational spectra of each bond, angle or

Molecular Dynamics Simulations of a Molecular Electronics Device 115

dihedral of interest provides an alternative to normal mode analysis, which can be complicated for a system as big as the one treated in this work. Since normal modes are orthogonal to each other, an internal vibrational mode, such as a bond stretching, may be part of more than one normal mode; thus the analysis of normal modes only yields partial information for each internal vibration. The frequency analysis focuses on oscillations to obtain all the frequencies involved, rather than focusing on normal modes where several oscillations of the same frequency are combined. Depending on a proper selection of molecules, temperatures of ~300 K are acceptable, and the possibilities for analog signal processing using molecules are wide open. Low frequency oscillations can be externally controlled by electric fields, thus modulating higher frequency oscillations. Thus, we demonstrate numerically how molecular dynamics techniques can be used for the analysis of molecular electronic circuits, and how a molecular circuit can be implemented to perform signal processing. It is also shown that the level of coupling between modes is enough to allow its decoupling using standard techniques. Analyzing the vibrational spectra of bonds using digital signal processing techniques provides an alternative to normal mode analysis. A particular internal vibrational mode, such as a bond stretching, may have contributions from more than one normal vibrational mode. The analysis of normal modes only yields partial information for each particular internal vibration; however, the analysis reported in this brief finds correlations between different vibrational modes, allowing signal modulation and processing. The application of an input signal in one region of the molecular circuit yields the modulation of bonds in other regions. Depending on the range of the vibrations wavelengths, we can find out if they can be excited (optically, bias potential, or in other ways), thus we find that modulation can be performed using standard IR and Raman spectroscopies. A near-future application is the development and design of molecular-size filters and the implementation of cellular automata techniques. Thus chemical derivatives of particular molecules can be proposed in order to use a characteristic of a particular bond to perform signal processing; therefore, a carefully study of the effect of several types of bonds when more than one signal is applied needs to be determined. This detailed analysis of the frequency spectra yields correlations between several local vibrations that can be used to include vibrational signals externally introduced through the contacts of the molecular device. Although these correlations between internal modes are not surprising, the question is if using the natural amplitude oscillations of the vibrational modes, which can be experimentally detected, will allow the detection of mixed signals in localized sites. If this were possible, several single operations

116 J. Seminario et al.

could be implemented, enabling molecular processors dealing with signals in the THz range of frequencies. We propose that signals can be encoded as the movement of atoms that can be detected observing the changes in the electrical characteristics of the molecule. For instance, when the displacement, f, of atoms associated to a vibrational mode changes the dipole moment of the molecule <", a signal proportional to the so called infra red (IR) intensity is obtained from IR Intensity «= —

KorJ and when atoms displacements affect the polarizability, <*, a signal proportional to the so called Raman intensity is obtained, Raman Intensity«

V& )

These two intensities correspond to the most common experimental spectroscopic techniques, and can be unambiguously determined theoretically, and are potential candidates to be used as means for encoding information because, on the other hand, induced external changes in the dipole and polarizability by external fields yield changes in the displacements of the atoms in the molecule. Thus, signals are internally represented by atom displacements and transduced to IR and Raman intensities. Depending on a proper selection of molecules, temperatures of ~300 K are acceptable, and the possibilities for analog signal processing using molecules are wide open. Low frequency oscillations can be externally controlled by electric fields, thus modulating higher frequency oscillations. For example, inducing a motion pattern on the Au clusters (input signal), can modulate the C-H oscillation amplitude (output signal).

Acknowledgments We highly appreciate the support of ARO under grants DAAD19-00-1-(0154, 0592, 0634), DARPA under grant N00014-01-1-0657. We also thank David McClard, Nagendra Vadlamani, Jimena Bastos, and Sridhar Bingi for their assistance with the details of the manuscript.

Molecular Dynamics Simulations of a Molecular Electronics Device 117

References (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)

(14) (15)

Seminario, J. M.; Cordova, L. E.; Derosa, P. A., "An ab initio approach to the calculation of current-voltage characteristics of programmable molecular devices," Proc. IEEE, vol. 91, pp. 1958-1975, 2003. Tour, J. M.; Kosaki, M.; Seminario, J. M, "Molecular Scale Electronics: A Synthetic/Computational Approach to Digital Computing," J. Am. Chem. Soc, vol. 120, pp. 8486-8493, 1998. Ellenbogen, J. C; Love, J. C, "Architectures for Molecular Electronic Computers: 1. Logic Structures and an Adder Designed from Molecular Electronic Diodes," Proc. IEEE, vol. 88, pp. 386-426, 2000. Joachin, C; Gimzewski, J. K.; Aviram, A., "Electronics Using HybridMolecular and Mono-Molecular Devices," Nature, vol. 408, pp. 541-548, 2000. Kwok, K. S.; Ellenbogen, J. C, "Moletronics: Future Electronics," Mater. Today, vol. 5, pp. 28-37, 2002. Tour, J. M.; Reed, M. A.; Seminario, J. M.; Allara, D. A.; Weiss, P. A., "Molecular Computer," in US Patent 6,430,511, 2002. Joachim, C, "Bonding more atoms together for a single molecule computer," Nanotechnology, vol. 13, pp. R1-R7, 2002. Seminario, J. M.; Cordova, L. E., "Toward Multiple-Valued Configurable Random Molecular Logic Units," Proc. IEEE Nanotechnology Conf, vol. 1, pp. 146-150,2001. Seminario, J. M.; Cordova, L. E.; Derosa, P. A., "Search for Minimum Molecular Programmable Units," Proc. IEEE Nanotechnology Conf., vol. 2, pp. 421-424,2002. Chen, J.; Reed, M. A.; Rawlett, A. M.; Tour, J. M., "Large On-Off Ratio and Negative Differential Resistance in a Molecular Electronic Device," Science, vol. 286, pp. 1550-1552, 1999. Reed, M. A.; Chen, J.; Rawlett, A. M.; Price, D. W.; Tour, J. M., "Molecular Random Access Memories," Appl. Phys. Lett., vol. 78, pp. 3735-3737, 2001. Tour, J. M.; VanZandt, W. L.; Husband, C. P.; Husband, S. M.; Wilson, L. S.; Franzon, P. D.; Nackashi, D. P., "NanoCell Logic Gates for Molecular Computing," IEEE Trans. Nanoteck, vol. 1, pp. 100-109, 2002. Tour, J. M.; Jones, L., II; Pearson, D. L.; Lamba, J. S.; Burgin, T.; Whitesides, G. W.; Allara, D. L.; Parikh, A. N.; Atre, S., "Self-Assembled Monolayers and Multilayers of Conjugated Thiols, a,w-Dithiols, and Thioacetyl-Containing Adsorbates. Understanding Attachments Between Potential Molecular Wires and Gold Surfaces," J. Am. Chem. Soc, vol. 117, pp. 9529-9534., 1995. Harriott, L. R., "Limits of Lithography," Proc. IEEE, vol. 89, pp. 366-374, 2001. Szent-Gyorgyi, A., "Towards a New Biochemistry," Science, vol. 93, pp. 609611,1941.

118 J. Seminario et al.

(16) Seminario, J. M.; Zacarias, A. G.; Derosa, P. A., "Analysis of a dinitro-based molecular device," J. Chem. Phys., vol. 116, pp. 1671-1683, 2002. (17) Chen, J.; Wang, W.; Reed, M. A.; Rawlett, A. M.; Price, D. W.; Tour, J. M, "Room-Temperature Negative Differential Resistance in Nanoscale Molecular Junctions," Appl. Phys. Lett, vol. 77, pp. 1224-1226, 2000. (18) Molecular Simulations Inc., User Guide. San Diego: Molecular Simulations, Inc, 1997. (19) Molecular Simulations Inc., "Cerius2." San Diego, 1999. (20) Allen, M. P.; Tildesley, D. J., Computer Simulation of Liquids. Oxford: Clarendon Press, 1990. (21) Rappe, A. K.; Casewit, C. J.; Colwell, K. S.; Goddard III, W. A.; Skiff, W. M., "UFF, a Full periodic Table Force Field for Molecular Mechanics and Molecular Dynamics Simulations," J. Am. Chem. Soc, vol. 114, pp. 1002410035, 1992. (22) Casewit, C. J.; Colwell, K. S.; Rappe, A. K., "Application of a Universal ForceField to Organic Molecules," J. Am. Chem. Soc, vol. 114, pp. 10035-10046, 1992. (23) Casewit, C. J.; Colwell, K. S.; Rappe, A. K., "Application of a Universal ForceField to Main Group Compounds," J. Am. Chem. Soc, vol. 114, pp. 1004610053, 1992. (24) Casewit, C. J.; Colwell, K. S.; Rappe, A. K., "Application of a Universal ForceField to Metal-Complexes," Inorg. Chem, vol. 32, pp. 3438-3450,1993. (25) Senn, H. M., "Ph.D. thesis, Diss. ETH No. 13972, ETH, URL: http://ecollection.ethbib.ethz.ch/show?type=diss&nr= 13972." Zurich, 2001. (26) Sutton, A. P.; Chen, J., "Long-Range Finnis Sinclair Potentials," Phil. Mag. Lett, vol. 61, pp. 139-146, 1990. (27) Liem, S. Y.; Chan, K., "Simulation study of platinum adsorption on graphite using the Sutton-Chen potential," Surface Science, vol. 328, pp. 119-128, 1995. (28) Lide, D. R., "CRC Handbook of Chemistry and Physics," vol. 79th Edition 1998-1999. Boca Raton: CRC Press, 1998. (29) Seminario, J. M.; Tour, J. M., "Systematic Study of the Lowest Energy States of Aun (n=l-4) Using DFT," Int. J. Quantum Chem., vol. 65, pp. 749-758, 1997. (30) Derosa, P. A.; Zacarias, A. C; Seminario, J. M., "Application of Density Functional Theory to the Study and Design of Molecular Electronic Devices: The Metal-Molecule Interface," in Reviews in modern quantum chemistry, Sen, K. D., Ed. Singapore: World Scientific, 2002, pp. 1537-1567. (31) Seminario, J. M.; Zacarias, A. G.; Derosa, P. A., "Analysis of a Dinitro-Based Molecular Device," J. Chem. Phys., vol. 116, pp. 1671-1683, 2002. (32) Seminario, J. M.; De La Cruz, C. E.; Derosa, P. A., "A Theoretical Analysis of Metal-Molecule Contacts,"./. Am. Chem. Soc, vol. 123, pp. 5616-5617, 2001. (33) Derosa, P. A.; Seminario, J. M., "Electron transport through single molecules: Scattering treatment using density functional and green function theories," Journal ofPhysical Chemistry B, vol. 105, pp. 471-481, 2001.

Molecular Dynamics Simulations of a Molecular Electronics Device 119

(34) (35) (36)

(37)

(38) (39) (40) (41)

(42) (43) (44) (45) (46)

Seminario, J. M.; De la Cruz, C. E.; Derosa, P. A., "A theoretical analysis of metal-molecule contacts," J. Am. Chem. Soc, vol. 123, pp. 5616-5617, 2001. Ulman, A., "Formation and Structure of Self-Assembled Monolayers," Chem. Rev., vol. 96, pp. 1533-1554, 1996. Heister, K.; Rong, H.-T.; Buck, M.; Zharnikov, M.; Granze, M., "Odd-Even Effects at the S-Metal Interface and in the Aromatic Matrix of BiphenylSubstituted Alkanethiol Self-Assembled Monolayers," J. Phys. Chem. B, vol. 105, pp. 6888-6894,2001. Haag, R.; Rampi, M. A.; Holmlin, R. E.; Whitesides, G. M., ".Electrical Breakdown of Aliphatic and Aromatic Self-Assembled Monolayers Used as Nanometer-Thick Organic Dielectrics," J. Am. Chem. Soc, vol. 121, pp. 78957906, 1999. Seminario, J. M.; Derosa, P. A.; Bozard, B. H.; Chagarlamudi, K., "Vibrational Study of a Molecular Device using Molecular Dynamics Simulations," J. Nanoscience Nanotech, vol. 5, pp. 1-11, 2005. Li, J.; Li, B.; Zhang, X. C; Su, R. Z., "The study of flame retardants on thermal degradation and charring process of manchurian ash iignin in the condensed phase," Polymer Degradation and Stability, vol. 72, pp. 493-498, 2001. "http://wwwchem.csustan.edu/Tutorials/INFRARED.HTM," 1998. Lu, G. W.; Xia, H. R.; Wang, X. Q.; Xu, D.; Chen, Y.; Zhou, Y. Q., "Raman scattering investigation of the zinc cadmium tetrathiocyanate single crystals," Materials Science and Engineering B-Solid State Materials for Advanced Technology, vol. 87, pp. 117-121, 2001. Kang, Y. U.; Hwang, T. S.; Song, H. Y.; Son, W. K.; Park, J. K, "The synthesis and characteristics of soluble PAES/PEES copolymers for heat resistant ion exchange membrane," Polymer-Korea, vol. 23, pp. 1-7, 1999. Tada, H.; Ito, S., "Conformational change restricted selectivity in the surface sulfonation of polypropylene with sulfuric acid," Langmuir, vol. 13, pp. 39823989, 1997. Alcolea, M., "Scaling Factors for the Prediction of the Frequencies of the Ring Modes in Benzene Derivatives," J. Phys. Chem. A, vol. 103, pp. 11366-11377, 1999. Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A., Ab Initio Molecular Orbital Theory. New York: Wiley, 1986. Strey, G.; Mils, I. M., J. Mol. Spectrose, vol. 59, pp. 103, 1976.

Chapter 4: Computation of Excited State Potential Energy Surfaces via Linear Response Theories Based on State Specific Multi-Reference Coupled Electron-Pair Approximation Like Methods

Sudip Chattopadhyay, Dola Pahari, Uttam Mahapatraa and Debashis Mukherjee6 Department of Physical Chemistry Indian Association for the Cultivation of Science Calcutta 700 032, India * Department of Physics, Darjeeling Government College Darjeeling 734 101, India hAlso

at Jawaharlal Nehru Centre for Advanced Scientific Research Bangalore 560 064, India Formulation and pilot numerical implementation of linear response theories (LRT) for excited state potential energy surfaces (PES) are presented, which are based on a class of state specific (SS) multi-reference coupled electron-pair approximation (CEPA)-like methods (SS-MRCEPA). These theories,termed as MRCEPA-LRT, are suited for the excited states whose ground state either have intruders at some region of the PES, or displays real or weakly avoided crossings. Genesis of the SS-MRCEPA approximations is traced from the parent full-blown state specific multi-reference coupled cluster theory (SS-MRCC), and its efficacy to provide accurate PES for the ground state in an intruder-free manner is emphasized. Two CEPA versions are discussed, and the MRCEPA-LRT emerge as the LRT based on the two SS-MRCEPA. They provide size-intensive energies in the sense that they yield excitation energies for the fragments in the limit of noninteracting fragments, just like in the single-reference CCLRT. The set of MRCEPA-LRT developed in this article bears close structural resemblance to our LRT based on the SS-MRCC method (MR-CCLRT), just as the SS-MRCEPA does to the parent SS-MRCC method. Pilot applications to the PES of some low-lying excited states of the P4 model and Li2 molecule, whose ground states possess quasi-degeneracy or face intruders indicate very satisfactory performance, comparing quite favorably with the results of the full-blown MR-CCLRT, despite the neglect of a host of complicated terms.

121

122 S. Chattopadhyay et al.

Glossary of Terms ACPF AQCC BW CAS CAS-CI CC CCSD(T) CEPA CEPA(D) CI EPV FCI HF LRT MBPT MR MR-CISD MRCC MR-ACPF MR-AQCC MRCEPA MR-MBPT MR-CCLRT MRCEPA-LRT mH PES RS SCF SD SR SRCC SR-CCLRT SR-CISD SS SS-MRCC SS-MRCEPA

Averaged Coupled Pair Functionals Averaged Quadratic Coupled Cluster Brillouin-Wigner Complete Active Space Complete Active Space based Configuration Interaction Coupled Cluster A specific scheme incorporating non-iterative inclusion of triples Coupled Electron-pair Approximation CEPA with diagonal dressing Configuration Interaction Exclusion Principle Violating Full CI Hartree-Fock Linear Response Theory Many-body Perturbation Theory Multi-reference Multi-reference Configuration Interaction Single and Double Multi-reference Coupled Cluster Multi-reference Averaged Coupled Pair Functionals Multi-reference Averaged Quadratic Coupled Cluster Multi-reference Coupled Electron-pair Approximation Multi-reference Many-body Perturbation Theory State Specific Multi-reference Coupled Cluster based LRT State Specific Multi-reference Coupled Electron-pair Approximation based LRT milli Hartree Potential Energy Surface Rayleigh-Schrodinger Self Consistent Field Single and Double Single Reference Single Reference Coupled Cluster Single Reference Coupled Cluster based LRT Single Reference Configuration Interaction Single and Double State Specific State Specific Multi-reference Coupled Cluster State Specific Multi-reference Coupled Electron-pair Approximation

Computation of Excited State Potential Energy Surfaces 123

1

Introduction

There are now well-established theories of electron correlation for closedshell systems such as configuration interaction (CI) method [l], many-body perturbation theory at low orders [2], coupled electron-pair approximation (CEPA) methods [3], and their different variants [4; 5; 6], and singlereference coupled cluster (SRCC) theory at various levels of truncations [7; 8]. While the truncated CI's, like the single and double(SD) excitations on a single reference function (SR-CISD), do not respect the important property of size-extensivity [9], the other methods mentioned above explicitly satisfy this criterion and have thus become increasingly popular; in particular the SRCC method with SD [7; 8] and approximate noniterative inclusion of triples, SR-CCSD(T) is being widely used nowadays [10]. The CEPA-like approximations can be viewed both as Cl-like methods corrected for inextensivity by deleting the disconnected terms in the CISD equations and as suitable approximation of the SRCC theory where only the EPV (Exclusion Principle Violating) terms stemming from powers of the cluster operators are retained. On the other hand, development of correlation theories for excited or ionized states, for biradicals or for studying potential energy surfaces (PES) involving quasi-degeneracy of varying degrees-involving real or weakly avoided crossings-has been a real challenge. Because of its structural simplicity, the variational approach was the most straightforward to generalize, leading to the multi-reference CI method with the inclusion of linearly independent single and double excitations out of the reference functions (MR-CISD) [ll]. This has been used quite extensively. The MR-CISD method suffers from lack of size-extensivity, and some multireference generalizations of Davidson corrections have been proposed [12; 13; 14] to correct for this empirically. Several CEPA-like modifications have also been formulated, where all the disconnected terms are eliminated by inspection, and only the EPV terms are retained to various degrees [6; 15]. These modifications usually start with a reference function ip0, denned in a complete active space (CAS), and include single and double excitations from the CAS function. In almost all the methods, the combining coefficients Cf, accompanying the model functions <^ in the final function are kept frozen in their values as computed while generating tpQ via a CAS-SCF or CAS-CI function. For real or weakly avoided crossings in a PES, there would generally be a strong influence of the crossing state on the target state of interest, which would lead to substantial revision of the coefficients c^ in

124 S. Chattopadhyay et al.

any theory based on an MR function generated from a set of
Computation of Excited State Potential Energy Surfaces 125

a MRCEPA approach couched in a dressed CI form, which makes systematic inclusion of other terms stemming from the powers of cluster operators somewhat difficult to discern. The method of Mahapatra et al. [27; 28] is a full-blown state specific MRCC (SS-MRCC) , displaying manifest size-extensivity and size-consistency. The formalism of Masik and Hubac [26] leads to simpler equations than in the formulation of Mahapatra et al. [27; 28], though it is not fully size-extensive. We have suggested recently appropriate size-extensive MRCEPA like approximants [16; 17] to the parent SS-MRCC formalism of Mahapatra et al. [27; 28], and preliminary numerical applications indicate effectiveness of these methods. They avoid intruders naturally whenever the target state energy remains rather away from the virtual function energies. While this is always achievable for the ground states of a system, even those with pronounced MR character and showing real or avoided crossings, a straightforward application of the SR-MRCC or the associated SS-MRCEPA approximants derived therefrom, would lead in general to intruder problem. To get around the difficulty, we have recently developed a linear response version [17; 29] of the parent SS-MRCC theory, where the excited states are generated by solving an eigenproblem generated by an excitation operator on the ground state. The linear response theory (LRT), termed by us as MRCCLRT, led to a very good performance to generate several PES for excited states of different symmetries for difficult systems such as BeH2. Since the SS-MRCEPA methods proposed by us provide rather accurate energies as compared to the parent SS-MRCC method, despite the neglect of a plethora of nonlinear terms, it seems worthwhile to develop linear response theories based on our recently formulated SS-MRCEPA. We present in this article formulation of such an approach, alongwith some pilot numerical applications to the excited PES for systems showing quasi-degeneracy in their ground states to indicate the efficacy of this approach. The success of the linear response approach based on a given form of the ground state is predicated by the quality of the ground state function. Although the LRT based on SRCC theories are in vogue for quite some time [30; 31; 32; 33; 34], not much has been attempted to generalize this to the MR case. Pal et al. [35] has developed MR-based LRT based on effective hamiltonian approach in the frame work of the valence-universal formalism. Response to a time dependent field and its static limit yielding time-independent equations has been derived by Ajitha and Pal [36]. Another MRCC based LRT has been proposed by Tenno et al. [37] in the

126 S. Chattopadhyay et al.

state-universal setting. Szalay [38] has developed a constrained variational approach for the MRCC method, which is relevant in the context of the linear response method, where the first derivative amplitudes are naturally not required in the expression for the first derivative energy. A linear response extension of the state specific MRCEPA methods such as MR-ACPF (multi-reference averaged coupled pair functionals) and MRAQCC (multi-reference averaged quadratic coupled cluster) was formulated by Szalay and co-workers [39]. Gadanitz and Ahlrichs [4] introduced the MR-ACPF based on the CEPA-type arguments. Latter on, Szalay and Bartlett developed a new method called MR-AQCC [5; 6] which in sprit and level of approximations is closely related to MR-ACPF. The size-consistent self-consistent complete active space singles and doubles CI [(SC)2CASCISD] has been applied very recently by Martin et al. [40] to computation of vertical excitation as well as the ionization energies. As mentioned earlier, Chattopadhyay et al. [29] have recently proposed a LRT based on the SSMRCC theory (MR-CCLRT) using the same strategy as that of the SRCC based LRT (SR-CCLRT) [30; 31; 32; 33; 34] for computing the excited state energies. It has been shown that the MR-CCLRT formalism [29] is sizeintensive in nature [34; 41], in the sense of providing excitation energies of fragments in the limit of non-interacting fragments. The organization of the paper is as follows. In Sec.2, after a brief discussion of the structure of our SS-MRCEPA and our SS-MRCC based linear response theory, MR-CCLRT [29], we discuss two physically motivated approximations of graded complexity, based on our parent MR-CCLRT generated from the SS-MRCC guided by the same type of approximations we made in arriving at the SS-MRCEPA schemes. Issues such as sizeintensivity and orbital invariance will also be addressed to. In Sec. 3, we present the numerical examples to test the potentiality of our MRCEPALRT as compared to the FCI values. Sec. 4 summarizes the main thrusts of the article, alongwith the concluding remarks.

2 2.1

Theoretical Developments The SS-MRCC Theory and Its CEPA-like

Approximants

Before presenting the formulation of the linear response method (MRCEPALRT), based on our SS-MRCEPA, we would first introduce and discuss the

Computation of Excited State Potential Energy Surfaces 127

structure of the parent SS-MRCC equations. This would serve both to fix the notations for the key ingredients entering our formalism, and also to motivate towards the approximations warranted in the CEPA-like methods. We would discuss then emergence of two CEPA-like approximants to the SS-MRCC method. We would next introduce the essentials of the fullblown MR-CCLRT derived as the appropriate linear response version, and then proceed to approximate them to generate MRCEPA-LRT in a manner entirely analogous to the steps taken to reach the SS-MRCEPA schemes from the SS-MRCC method. We start out with a CAS spanned by a set of model functions <£M, which also specifies the set of doubly occupied core orbitals, the set of active orbitals and also the set of virtual orbitals. We write the ground state function t/j0 in a cluster expansion representation of the component of a SS wave operator defined for each (^ it acts on; ^ = ^exp(r")^cM

(1)

Our theory is quite flexible regarding the choice of the coefficients cM. We may, if we so desire, keep them frozen at their initial values obtained by the CAS-SCF or CAS-CI procedure, or we may determine them selfconsistently alongwith the cluster amplitudes of TM. Since each virtual function \i c a n be reached from every model function <^>M, there is a redundancy in the number of cluster amplitudes. In SS-MRCC theory, this redundancy is exploited by positing sufficiency conditions to attain the twin desirable goals: (a) maintenance of manifest size-extensivity and (b) avoidance of intruders in a natural manner. The working equations for determining the cluster amplitudes leading to excitations to \i from
(Xi\HM»K

+ £<X<|exp(-T
(2)

V

where H^ — i7exp(J>) and Htlv=((j>ll\Hv\<j)v). Xi's are all the excited states reached by the action of TM on 4>\i • From now on, we assume that we solve for the coefficients c^ selfconsistently by diagonalizing the effective matrix H^:

128 5. Chattopadhyay et al.

^HpVcv

= Ecll

(3)

V

We note that the set cM and T** are coupled through Eq.(2) and Eq.(3). Solving these coupled set of equations, we obtain both the cluster amplitudes and the converged coefficients from the diagonalization. We now discuss the development of coupled electron-pair like approximations of our SS-MRCC theory leading to the SS-MRCEPA methods. There could be several routes to the CEPA-like approximations starting from the SS-MRCC theory which, while capturing the physical essence of the CEPA-like schemes for the single-reference situation, emphasize different aspects of technical details. The fundamental idea in any CEPA-like method is to eliminate the size-inextensive terms coming from the energy containing term of a MR-CISD. They are not cancelled automatically in a MR-CISD if multiple excitations beyond the CISD space are not included. One may imagine that the simplest way to eliminate them is to identify the disconnected terms and subtract them explicitly in the MR-CISD equations. As we would emphasize below, in contrast to the single-reference situation, this analysis does not exactly carry through in the multi-reference case. It turns out that the disconnected looking terms in the cluster expansion representation of the function ipo have both connected and disconnected components, and some counter terms which automatically arise in the cluster expansion eliminate just the disconnected portion. The difference of the term with the disconnected component and the counterterm is connected, but these two entities contain different cluster operators TM and Tv. This kind of cancellation with counterterms would have been very hard to guess in a MR-CISD setting. What should obviously be a common characteristic is the inclusion of the singles and doubles space interacting via the hamiltonian. In the multiple cluster expansion of the wave-function, what constitutes the SD-space is not unique, and it is here that the SS-MRCEPA methods have a freedom of choice. One may, for example, emphasize that one should include only the single and double excitations {xf} reached by the action of TM on <^>M in Eq.(2) for every \f used in the projection. Another possibility could be to include all the virtual functions which are obtained from single and double excitations from the entire model space. In this latter situation, a virtual function which is singly or doubly excited with respect to a given cp^ may be more than doubly excited with respect

Computation of Excited State Potential Energy Surfaces 129

to another model function Vy and more than quadratic powers of cluster operator are generally needed for inducing excitation to all functions in the CISD space. There are structural differences in the CEPA-like schemes obtained with the above two choices. In the former (scheme I), upto quadratic powers of one-body operators Tf can be included, and the quadratic powers of T% should include only the EPV terms. In the latter scheme (scheme II), upto quartic powers of TM leading to the singles-doubles excitation space with respect to the model space can appear. In the most general equation of the Scheme II, as in Eq.(4), the projector QSD corresponds to the singles and doubles projector of the entire model space figures. However, in our numerical implementation, we would approximate our equations by neglecting more than quadratic powers of every TM, and retain only the EPV terms coming from the powers of T2M. In this case, the projector reduces to QM, the projector to the virtuals which are singles and doubles with respect to the model function (p^, and the CEPA approximation reduces to what we called scheme I above. The most general CEPA-like equations for both the schemes can generically be written as

(xt\H^(P + QsD)exp(T^\^) + £< X f | exp(-T") exp(T")|^) <^|ffM (P + QsD)exp{T")\tf>u)cv = 0 V/,/i

(4)

As we emphasized above, we would rather use the less general QM in our present article in the numerical implementation. To see the explicit form of the CEPA-like approximations, we write the above equation in long hand as follows: \{XiW\^) I

+

to|[#,m^> II

+ T,AXi\exp(-T>l)exp(Tl/)\(j>tt)H^Cv IV

+

!<Xj|[[ff,TV]|^>]c M III +

other t e r m s

=0

(,s K

'

There are several features of the equations which should be emphasized here: (a) the combining coefficients cM are not frozen at some preassigned values, but are iteratively updated to their relaxed values; (b) the first three terms have explicitly connected algebraic structure in case T^ is connected; (c) among the quadratic terms appearing in III, the terms with one-body excitations are still doubles while the product of doubles or of singles and doubles lead to virtual functions \i which are more than doubly excited with re-

130 S. Chattopadhyay et al.

spect to (j)^; (d) the fourth term contains two pieces, one containing T" and the other containing TM, which may be individually disconnected but the entire term containing the matrix element {xi\ exp(—TM)exp(T")|0M)H'A1^ is a connected entity [27; 28]. The aspect (d) is what we emphasized earlier as generating a connected entity from two individually disconnected pieces, and which could not have been derived from the MR-CISD equations. For a proof of this assertion, we refer to our earlier publications [27; 28]. This feature of the SS-MRCC equation, retained in generating the SS-MRCEPA equations is very important for our purpose, since in any approximation or truncation scheme, each term of the two pieces must be treated on the same footing to ensure connectivity. Thus the term IV has to be computed in a manner which uses the same approximation for the matrix elements H^. Using the same kind of considerations as were used to generate the various CEPA-like approximations from the SRCC-SD equations, we confine the rank of cluster operators TM to at most two-body excitations. Thus for each <j>n, the x;' s used in the projection in Eq.(5) above must be at most doubly excited with respect to <^M. Moreover, the term III should include product of single excitations leading to the doubles, and should exclude all other product terms which are more than doubly excited with respect to M. However, depending on the sophistication of the CEPA-like approximation, we will include the EPV terms of III coming from the product of doubles to various degrees. For every xi reached by a T/* from 0M, these EPV terms of III must include at least one Tf operator while the other TM can be any one of the possible excitations which have at least one orbital in common with those involved in T/\ To show this, we express the leading terms of Eq.(5) explicitly in the following form

l(Xi\H\») I

+ Em(Hlm - H^Slm)t^

HE'mnd'm^t^ III(b)

II

-

Arf

III(a)

+ Ev{Xl\Tv\ll)HllvCl/ IV(a) +other terms

= Stfcu IV(b)

,cs ^

where we have shown the genesis of all the terms by the same labels by which they were indicated in Eq.(5). All CEPA schemes neglect the terms not explicitly shown in Eq.(6). The leading terms of the two pieces of the term IV containing T" and Tu are now separately indicated as IV(a) and

Computation of Excited State Potential Energy Surfaces 131

IV(b) respectively. The 'energy' £ is taken as a parameter, whose actual value depends on the approximation used in H^. AM is the EPV correction in III(a) coming from product of doubles or of singles and doubles, whose actual form again depends on the approximation used in our SS-MRCEPA. The negative sign of this term in Eq.(6) is to keep conformity with the analogous correction term used in the SR case. The term III(b) denotes the doubles contribution confined to the product of two singles (which is indicated by the prime in the sum). We note at this stage that the terms I-III are entirely similar to those in the SRCEPA method. In fact if we neglect the terms IV(a) and IV(b), then we get precisely SRCEPA-like methods for each cp^. But this approximation is too drastic in the sense that it entails intruders (because of the implicit presence of (Hu - H^) in Eq.(6)), and thus belies one of the very basic goal which motivated our SS developments. We recall at this stage the feature which we highlighted earlier: while two pieces of IV, i.e. IV(a) and IV(b) may be disconnected, the entire term IV taken together is a connected entity. The two pieces IV(a) and IV(b) serve two specific and distinct purposes in this regard. A portion of the term in IV(a) for v = /i cancels both H^^im and A^tf terms of II and III(a), while the rest of the expressions for v ^ [i of IV(a) correct for the lack of extensivity coming from the terms appearing in Et^c^. We now discuss two specific types of CEPA-like approximations. Entirely in line with the CEPA(O) scheme for the SR case, if we ignore AM entirely in Eq.(6), then we should start generating an analogous SSMRCEPA(O) scheme. We still have the freedom of approximating H^v independently, but in the spirit of quasi-linearization, we now approximate Hfj,v as just H^v Then £ becomes just EQ, the CAS energy itself. This set of approximations leads to what we call the SS-MRCEPA(O). Written in long hand, it takes the form

{(Xi\H\^) + ]T(ff,m - E08lm)C + ^ Xy r o > n «]c,. + J2(xi\Tl'\MH^cv=0 with Eo satisfying the equation

V

(7)

132 S. Chattopadhyay et al.

We now discuss the most sophisticated approximation of AM, by retaining all the EPV terms. This is analogous to what was suggested by Malrieu and co-workers [15] in the SR case, which we denote as CEPA(D). The qualifier D indicates that all such terms can be incorporated in the exact CEPA by a diagonal dressing of the SDCI equations [15]. We approximate A^ by all the terms containing all TMs with at least one orbital in common with those appearing in t*. A^ in each equation for t% is thus /-dependent, and we indicate this by A^. As emphasized earlier, we have to treat the two pieces of IV on the same footing in their handling of H^,. We approximate this by its quadratic expansion involving double commutators. The diagonal term H^ appearing in IV(a) of Eq.(6) for a given I can be written as

Hnv — Hw + ^ ^ + ^ N E P V

(8)

where A^ is the correction for TM with at least one orbital in common with those labeling tf. The rest of the term has no orbital in common with those appearing in tf. With this observation, it is clear that the term (H^^ + AjJ<$jm in Eq.(6) is cancelled by the first two terms of Eq.(8). Denoting the energy £ as £QEPA(D)> Eq.(6) for this scheme, referred to as SS-MRCEPA(D), takes the form:

KXl\H\^) + Y,Wlm ~ (£CEPA(D) + £|,NEPv)Mtf + m

(9) where £ satisfies the equation

V

To see the avoidance of the intruders, we rewrite the SS-MRCEPA equations in the following form _ (Xl\H\,*} + I Em,n ff*m.n« + E ^ W I ^ M ^ / C M )

*' ~

S-Hu-D^

(10)

/1n

,

Computation of Excited State Potential Energy Surfaces

133

H^v — H^ for SS-MRCEPA(O), and has a multi-commutator structure for SS-MRCEPA(D). £ = Eo for SS-MRCEPA(O), and is £CEPA(D) for SS-MRCEPA(D). D^ = 0 for SS-MRCEPA(O) and has the value as given by Eq.(8) for SS-MRCEPA(D). Since D^ is small, unlike in the effective hamiltonian based theories, (£ — Hu — Djj will always be robust if £ is well-removed from any Hu, even when H^ is close to this Hu. Intruders are thus avoided naturally. Structurally, the SS-MRCEPA(O) and the SS-MRCEPA(D) bear close resemblance to the recently developed size-extensive SSMR perturbation theory (SS-MRPT) [42; 43] of the Rayleigh-Schrodinger (SS-MRPT(RS)) and the Brillouin-Wigner (SS-MRPT (BW)) types, respectively, generated from the SS-MRCC theory, but are generally more accurate than the perturbative schemes. There is a real advantage of approximating the SS-MRCC theory in the SS-MRCEPA form if we can retain the bulk of the correlation energy captured by the full-blown SS-MRCC theory, while we save a lot of computer time since we neglect a host of nonlinear terms. Our results, discussed in Sec.3, will amply demonstrate the accuracy of the SS-MRCEPA schemes. 2.2

Linear Response Theory Based on SS-MRCC (MRCCLRT) and Its Approximants Based on SS-MRCEPA

As discussed earlier, the major thrust in this paper is the development and pilot numerical implementation of the linear response theories based on the SS-MRCEPA methods, discussed in the preceding subsection. For the excited states whose ground states are multi-reference in nature, this is a very convenient and compact approach. Before embarking on the detailed considerations leading to the MRCEPA-LRT based on the response function derived from SS-MRCEPA, it is pertinent to discuss at this stage the essential issues of the linear response theory based on the parent SS-MRCC (MR-CCLRT) itself. In the CC based linear response theory, the excited functions are written in terms of the ground state function via the action of a suitable excitation operator. In the spirit of the SR-CCLRT [30; 32; 33; 34; 31] we posit on our excited state, \tpk)-, the following Ansatz: h/>*> = $ > x p ( T " ) 5 £ | < ^

(11)

134 S. Chattopadhyay et al.

\tl>k) satisfies the Schrodinger equation (12)

H\il>k) = Ek\il>k)

In close analogy with the SR-CCLRT, we include in each excitation operator S% all the virtual excitations from |0M), and retain the cluster components TM of the multi-reference ground state (eq(l)) in our Ansatz for \tpk}- In addition, we also want to change the relative weights of the various |^M)'s in the function 1^*)- With this choice, we can represent S% as

SZ = *fl + SZ

with S» = Y,*?kY?' m

(13)

where Y™^ creates a virtual function \Xm) from \4>n)- It is evident that there are several S£'s which excite to the same \xm) from several | ^ ) ' s . Thus, we have to invoke a set of sufficiency conditions to determine them. We have

£[exp(T^)|^><^|^S£|^> + exp(r")5f |^>(^|ff M |^)

+exP(r'X°|<M<<Mtf^> + E e x P( r A I )^^^l^> + exp(r")S^QH M |^) + exp(T")a^gH M |^>]c M = Efc^exp(T^^|^)^

(14)

where Q is the full virtual space projector. Following the same manipulations leading to the SS-MRCC equations [27; 28], we interchange the labels H and v in the above equation in the first three terms and equate terms with same fi. For each virtual space projection onto (x;| exp(-T^) we have

(Xl\%S%\^ + (x^QH^K + {xilHMsfcp + ^{Xi\ exp(-T") e x p C n S j ^ X ^ l i ^ l ^ V

+ (xi\ exp(-T") eMTv)\^)(^\HvSvk\4>v)cv +{Xi\ exp(-T*) exP(Tn\^)(^\Hu\
(15)

Computation of Excited State Potential Energy Surfaces 135

The sufficiency conditions imply that each virtual function (x;| can be reached from several <£M's and the projection for each (xi\ has to be done for each |(/>A). Also, each ket \xm) appears with a different coefficient, depending on how it is generated via various 5£'s acting on the associated \^)'s. To keep this information as a mnemonic, we would from now on denote (x;| as (xjj if it is used for projection from Eq.(15) for a certain \(j),j) and function \xm) as |x™) if it is obtained by Y™^ acting on |^M) From now on, we shall use the coefficients d^'s and d™fc's defined by

<*„* = 4 %

(16)

<^k = *HW

(17)

Using the above equations, we can rewrite Eq. (15) with the notation |x™) for \xm) reached from | ^ ) : m m

+ Y,(xU exp(-r") eXp(T'')Yvmi\li)(r\Hv\v)d%t vm

+ J^ixU exp(-r^) ex P (T-)|^)(^|F^t|^>C fc vm

+ E<^l

ex P(- r ")

exp(T")|^)(^|^|^)^ fc = Ekd\k

(18)

The third term in Eq. (18) involves QH^ acting on \^), which implies that QH^\
Y,{\\HvS^\v)cv + J2(4>x\HAv)sfcv = Ekcxsf

(19)

which on simplification generates,

£<0A|j?,*;mti6,><e* + ^ ( ^ I ^ I ^ ) ^ = EkdXk vm

v

(20)

136 5. Chattopadhyay et al.

The above MR-CCLRT equations can be written compactly as a matrix eigenproblem. Once the cluster amplitudes of TM for all /i's have been found out by solving the SS-MRCC equations, we can get the associated excited state energies relative to the state \ip) by solving the MR-CCLRT equations. An attractive feature of the MR-CCLRT is that the computed excitation energies are size-intensive [29] in the sense that they become asymptotically equal to the sum of the fragment excitation energies in the limit of noninteracting fragments. We will now develop the corresponding MRCEPA-LRT using exactly the same approximations on MR-CCLRT as has been used in the SS-MRCC theory to arrive at the SS-MRCEPA theories. To derive the working equations for the SS-MRCEPA based response theories, we start with the parent MR-CCLRT equations (18) and (20) for the excited states and make approximations akin to what have been used to arrive at the SS-MRCEPA(O) and SS-MRCEPA(D) theories from the SS-MRCC theory for the ground state. Both the MRCEPA-LRT schemes can be subsumed in the same type of working equations, with appropriate definitions of certain dressed operators. In our SS-MRCEPA-like methods, we ignore more than double excitations out of any given ^ leading to a xjj a n d quasi-linearize the terms in [exp(—TM)exp(T")] accordingly. The working equations of the virtual space projections of the CEPA-based MR-CCLRT Eq. (18) for both the CEPA-based schemes can be simplified as follows

m m vm

+ Y.(xlMTl/-T»)\>1)H
+E<^i(r"-ni^><^|SM^t|^>ccfc = ^ 4 * vm

(21)

Computation of Excited State Potential Energy Surfaces 137

The corresponding model space equation is

X ^ I H ^ I ^ C * + Y,{^\HAMdvk = EkdXk vm

(22)

v

In the above working equations of the MRCEPA-LRT, some appropriate approximant to the operator H^ figures whose actual form depends on whether we are considering SS-MRCEPA(O) or SS-MRCEPA(D). We give below the explicit forms of H^, and the dressed operators H^ and H"^ for different SS-MRCEPA methods: For SS-MRCEPA(O) based response equations

\

m,n

(23)

/

H"v = Hv

(24)

However, for the SS-MRCEPA(D) based response equations %v

(25)

= {^\Hv\4>u)

(26)

H'l = Hv

H^L

\

(27)

+ HTt + Hfi+lY, oljWin + \YS ^JUL) m,n

m,n

/

It should be mentioned here that there is at least one orbital in common between t2m and t£n in the term £ m , n < 4 , « O L Just as in the parent MR-CCLRT [29], we can cast the working equations of MRCEPA-LRT also in the following matrix form: / Hvu 'Hlim\(dvk\_

( d»k \

{ nlv nlm ){dfk )-Ek\d^k

)

The elements of the operator H for the two MRCEPA-LRT are given by

(^Wfa) = (^[Hvfo)

(28)

{^\n\xT) = {^iWrFlM

(29)

138 S. Chattopadhyay et al.

(xlM») = <x^|<M<W + (ti\(Tv - T»)\^)H> »v

(30)

(timix?) = ixUiT" - ^ i r ' i ^ i g y . +{xlii\Yv^Qli[Hl,}\v)8liV + ( x U f ^ l - M V +(X1MT" - T^)\^)(^\WVY^\4>V)

(31)

Once the cluster amplitudes of TM for all /z's have been found out by solving the SS-MRCEPA equations, we can get the associated excited state energies relative to the state \ip) by solving the MRCEPA-LRT equations by constructing the matrix of % via Eqs.(21-31). The 5 amplitudes are confined to single and double excitations only. It is evident that if one drops the excitation operator, Y™^ from the working equations of MRCEPA-LRT they naturally reduce to the corresponding ground state CEPA theories, SS-MRCEPA. This implies that SS-MRCEPA are the completely unperturbed versions of MRCEPA-LRT, which is why we coin the term, MRCEPA-LRT, for the linear response theory based on the SS-MRCEPA method. We not only get the excited state energies but also the ground state energy by solving the MRCEPA-LRT equations. It is important to mention that, just as in the full-blown MR-CCLRT [29], one can prove in the same manner that the underlying operators in the eigenvalue problem of MRCEPA-LRT are connected and the possible excitation energies also satisfy the size-intensivity criteria [29; 34; 41]. We conclude this section with the following comments. In general, the CEPA-like methods do not possess the invariance property with respect to separate unitary transformation of orbitals within the manifold of doubly occupied, active and the virtual orbitals. A rigorous formulation of MRCEPA theories with this property is rather hard. Interestingly, the SSMRCEPA(O) discussed in this paper does possess this property. The more elaborate MRCEPA(D) unfortunately does not. We are recently exploring a somewhat less elaborate SS-MRCEPA method, where this invariance is restored, but at a price of having to include in the method a small set of more than doubly excited CI space functions to maintain the invariance [44]. These and the use of CAS-SCF orbitals to study their performance in the MRCEPA context are some of the aspects of study for us in the near future.

Computation of Excited State Potential Energy Surfaces 139

3

Results and Discussions

In our two applications, we employ a two-electron/two orbital, designated as (2 x 2) model space, consisting of two active orbitals, denoted as a and b and they belong to two different symmetries. This ensures the complete model space is effectively two-dimensional, spanned by fa = [core]a2 and 2 = [core]b2. The spin-adaptation of the working equations pertaining to both SS-MRCEPA and MRCEPA-LRT is a rather simple exercise, since the reference determinants are closed shell singlets. In order to examine the efficacy and performance of our methodology, we have applied our MRCEPA-LRT to study the PES of the excited states of two molecules whose ground states display some typical difficulties of treating states with quasi-degeneracy: (i)the ground state possesses a pronounced MR character at some geometry in the PES, (ii) for these illustrative molecular systems, the configurational and orbital quasi-degeneracies vary continuously with geometry, (iii) there are intruders to some reference determinant at some regions of the PES of the ground state. The performance of our MRCEPA-LRT is strongly dependent on the ability of the parent SS-MRCEPA methods to tackle both quasi-degeneracy and the intruder problem. In our numerical applications, we have studied PES of some low-lying excited states of the P4 model and Li2 molecule. Our formalism treats both these determinants (fa and fa>) o n the same footing, and we have two distinct options to choose our orbitals: (a) we may choose either of the two determinants to generate the HF orbitals, or (b) we may use the orbitals obtained from a CAS-SCF calculation. It turns out that for the molecules studied by us in this paper, there is the following rough general trend: the PES of the ground state obtained either by the SS-MRCEPA or as the lowest root from the MRCEPA-LRT is improved generally, by going over the CAS-SCF orbitals. For P4 model, the results of the ground state obtained by CEPA(O) method show improvement by around 0.3-0.9 mH and for CEPA(D) this varies from around 0.03-0.45 mH. At the square geometry (highly degenerate case), this improvement is maximum. The performance of the MRCEPA-LRT for the excited PES with the CAS orbitals is not uniformly better however, over the entire PES. For the P4 model, the results obtained by either of the two variants of MRCEPA-LRT are altered by 1-3 mH relative to the actual choice of the orbitals, as described above. Similarly, for L12 molecule, the ground state energies obtained using CAS orbitals are improved by around 0.4-1.5 mH

140 S. Chattopadhyay et al.

and 0.42-0.8 raH in case of CEPA(O) and CEPA(D) methods respectively. In the dissociation region, the improvement in either method using CAS orbitals over the HF one is negligibly small (around 0.006 mH). However, for Li2 molecule, the performance of excited PES exhibited by either of the two methods of MRCEPA-LRT are changed between 0.87-3.4 mH relative to the actual choice of the orbitals. Thus, there is no significant loss in accuracy even if we use the HF orbitals, but it saves the time for optimization. Also, the CAS-SCF orbitals for the ground state tend to over-emphasize the importance of the nondynamical correlation for the ground state, which can get altered substantially for the excited states. This is why, we will present here the results obtained with HF orbitals using one of the two model functions. The actual model function used by us is described below for each system. To assess the performance of our methods for studying the low-lying PES, we compare our results obtained with those from full CI (FCI) calculations in the same basis. The required molecular integrals and the different CI results were generated by the electronic structure package, GAMESS. The two sets of unknowns, the cluster amplitudes and the combining coefficients, are coupled in the SS-MRCEPA equations. They are solved in a nested iterative loop. In the micro-iteration the cluster amplitudes are solved for fixed coefficients till a local convergence is reached. The coefficients are then updated by diagonalizing the matrix H^ in the macroiteration. Once these unknowns in the MRCEPA are determined, we construct the matrix of Ji, and get the required eigenvalues and vectors from the MRCEPA-LRT eigenproblem. It should be noted that, unlike in the case of SR-CCLRT, our MRCEPALRT equations are defined in a space of overcomplete basis: each virtual function \xm) appears several times-generated from various | ^ ) ' s , with attendant coefficients d™k. If the number of linearly independent functions in the Hilbert space is Nh, then only Nh roots are meaningful. The rest of the roots are extraneous. These are easily diagnosed from the very small value of the norm of these functions. A. P4 Model System Our first test case is the simultaneous stretching of both H — H bonds in P4 (rectangular i/ 4 model). The P4 model introduced by Jankowski and

Computation of Excited State Potential Energy Surfaces 141 -2.000 -j

1

-2.025 -2.050 •

-2.075 ^

-2.100-

3 ^

• -2.125-

> g

' -2150"

, \

1

s^ A

i

\

/ \

/ \ \

/ /

\ \

g -2,75:

/ /

\

-2.200 -

.

-2.225 -

^

/

MRCEPA(0)-LRT MRCEPA(D)-LRT

/ —

F

C

|

\

/

\

,

-2.250 -

-2.275 -I

/

^ ^ ^ ^ - ^ ^ ^ ^ ^ ^ ^ ^

, 2

,

1 3

,

, 4

,

R (a.u.)

, 5

,

, 6

Fig. 1 : PES of some low-lying states of the P4 model systerr (Energy of the 11A2 state shifted down by 0.25 a.u.)

Paldus [45; 46] and later also studied extensively by several workers [28; 42; 47; 48; 49] is a rather simple test system and thus well suited to explore the validity and efficacy of our formalism. The geometries considered here are the same as in Ref.[45]. The geometry of P4 model is specified by the distance of approach of the two H2 molecules, R, which is varied from the pure square geometry and then gradually elongated to the rectangular geometry. The configurational quasi-degeneracy involving
142 S. Chattopadhyay et al.

2 : la\2a\ becomes more pronounced with lowering of the values of R. For R=2.0 a.u. we reach the situation where the two configurations become exactly degenerate. With increasing R, the degeneracy is lifted very rapidly. We have used the DZP basis taken from Ref.[46] and the HF orbitals of the function fa in our calculations. In this model system, all molecular calculations are carried out using C ^ symmetry.

-1-55-1 1Ba -1.60-

1

.^p.

1B\

•

\ \

-1.65-

\

\

^v

-1.70=

\

-1.75-

^ * * » « - ^ ^ ^ _ _ _ ^ ^

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>'/^

•

//

"185"

if

MRCEPA(0)-LRT

/

190

MRCEPA(D)-LRT

2'A,/ -1.95-

FCI

/

-2.00 2

3

4

5

6

R (a.u.) Fig. 2 : PES of some low-lying states of the P4 model systerr

The results for SS-MRCEPA(O) and SS-MRCEPA(D) based linear re-

Computation of Excited State Potential Energy Surfaces 143

sponse methods for the PES of states of various symmetries are plotted in Figs. 1 and 2. The corresponding FCI results are also shown. In order to show PES of several states in the same graph, we have sometimes shifted down the energies of some higher-lying states by a fixed amount, as explained in the captions for these graphs. From this we can see that: (i) The FCI values and the results obtained by the MRCEPA-LRTs for the ground state, l 1 ^ ! as well as excited state, 11A2 are very close and uniform for all points on the PES. For the I1.02 state, the scenario is slightly different. For this state, the performance of our methods is also quite good. However, for very small values of R, our results are found to be slightly away from the corresponding FCI values. (ii)At small values of R, the performance of our MRCEPA-LRT for I1.©! state is quite uniform and the deviation is fairly small with respect to the FCI results. But the deviation of MRCEPA-LRT results from the FCI values is not that small for this state at larger values of R. (iii) Although the performance of MRCEPA-LRT for 21Ai state is generally good at small and intermediate values of R, it becomes somewhat worse with increase in the values of R. Both Figs. 1 and 2 show that at larger values of R the deviation of MRCEPA(O)-LRT from the FCI for the 2xAx and l 1 ^ states is smaller than the MRCEPA(D)-LRT, the reason for which is not clear to us at this point. B. Li2 Molecule The Li2 molecule is a well studied system [28; 42; 50; 51] to test the efficacy of the various SR and MR methods. It is now well known that the ground state of the Li2 molecule requires a two-determinantal description in the zeroth order. In this calculation we have used a two-dimensional CAS defined by the two determinants (pi = lojlo^2o| and fa — \a'1g\a2u2u'2u. This is a good example to test the performance of our methods within a small reference space, since there is a quasi-degeneracy of the reference determinants at large internuclear distances in the PES of the ground state and there are intruders to <j>2 at the intermediate distances. We have used a small basis (DZ) [52] for the study of the Li2 PES of some low-lying electronic states, using D2h point group at each step of calculation. The configuration <j>\ has been chosen as the HF function in our calculations. Using these, we can compare the potentiality of our MRCEPA-LRT with the FCI values for all the states studied.

144 5. Chattopadhyay et al. -14.845 -i

1

-14.850 -

^ * ^ *

-14.855-

/

-14.860^

/«!/r:rrrTr^

-14.865-

/

>, -14.870-

\ " \ \ 1

35

5 •j - 1 4 - 8 7 5 "

I / /

1 -14.880-

I /

MRCEPA(0)-LRT MRCEPA(D)-LRT

/ /

\

-14.885-

/ /

/

I

\

•v'v

FCI

/

\l/

-14.890-

'

i

4

'

i

6

•

i

8

•

i

10

•

R (a.u.)

i

12

'

I

14

'

I

16

'

Fig. 3 : PES of some low-lying £ states of Li2 molecule (Energy of the £u* state shifted down by 0.06 a.u.)

We plot the results in Figures 3-5 to check the uniformity and accuracy of the PESs of various low-lying excited states obtained using our newly developed MRCEPA-LRT alongwith the corresponding FCI values. Again, as in the case of P4, we have sometimes shifted down the energies of the PES for some higher-lying various states by a fixed amount, as explained in the captions of the corresponding figures. We have tagged the symmetries of the different energy states according to the I?ooft point group, though we have performed the calculation within the £>2/i framework. It is evident

Computation of Excited State Potential Energy Surfaces 145

that the MRCEPA-LRT results are in close proximity to the FCI values and quite uniform throughout the PES, including the points of intruders and degeneracies. The performance of our SS-MRCEPA based linear response methods for Li2 molecule is quite good for low-lying £ states as is evident from the Figs. 3 and 5. From Fig. 4, it is clear that the performance of our newly developed MRCEPA-LRT to compute the PESs of Ug and n u states is not so satisfactory as that of the performance for the previously mentioned low-lying £ states. To underline the performance of the MRCEPA-LRT vis-a-vis the parent MR-CCLRT in predicting the excitation energies, we present in Table-I the vertical excitation energies of the Li2 molecule at the equilibrium ground state geometry obtained from both MRCEPA(O)-LRT and MRCEPA(D)-LRT as well as the full-blown MR-CCLRT methods alongwith with the FCI results.

Table-I FCI, MRCEPA(O)-LRT, MRCEPA(D)-LRT and MRCCLRT vertical excitation energies (in mH) for a few low-lying singlet excited states of Li2 at the equilibrium distance (R=5.0512 a.u.) of the ground state. Values within the parentheses indicate the deviation from FCI. ~

Methods

£+

II M

ng

S~

S~

FCI MR-CCLRT

68.203 68.23 (-0.027)

109.723 109.56 (0.163)

112.709 112.524 (0.185)

173.4 173.598 (-0.198)

206.726 206.896 (-0.17)

MRCEPA(0)-LRT MRCEPA(D)-LRT

69.397 110.633 113.451 174.455 207.755 (-1.194) (-0.91) (-0.742) (-1.055) (-1.029) 68.221 109.543 112.502 173.713 207.144 (-0.018) (0.183) (0.207) (-0.313) (-0.418)

From the table, it is evident that the numerical accuracy achieved by the MRCEPA-LRT is quite close to that of the parent MR-CCLRT approach. This indicates that SS-MRCEPA based LRT (MRCEPA-LRT) is a very useful and good approximant of the full-blown SS-MRCC based LRT (MRCCLRT), although SS-MRCEPA neglects most of the complicated nonlinear terms of the parent SS-MRCC method using a series of physically motivated graded approximation schemes. The method works very well to compute the PES of various low-lying exited states for ground states with

146 S. Chattopadhyay et al. -14.760 -i

1

.14765.

I i I

-14.770-

3.

> ^

u

-14.775-

MRCEPA(O)-LRT MRCEPA(D)-LRT FCI

!l

\l

' 1

-14.780-

\

-14.785-

\

-14.790-

-14.795 -I

"'^:^?'

\.

1 4

,

, 6

,

s^^"***^

1 8

,

, 10

,

, 12

,

, 14

,

, 16

R (a.u) Fig. 4 : PES of some low-lying n states of Li2 Molecule

varying degrees of MR character and displaying real or avoided crossings. The general trend of the results for both P4 and Li2 systems is very encouraging indeed. We do not want to draw definite conclusions regarding the relative performances of MRCEPA(0)-LRT and MRCEPA(D)-LRT from these pilot calculations. Both perform quite well. As Table-I indicates, the excitation energies computed by MRCEPA(D)-LRT are of better quality at the ground state equilibrium geometry of Li2, though MRCEPA(D)LRT sometimes show greater variations at some other geometries.

Computation of Excited State Potential Energy Surfaces 147 -14.66 -i

.1467.

1

-14.683

MRCEPA(O)-LRT MRCEPA(D)-LRT FCI

-\ " \ \

'

\

>£- -14.69-

\ \ \

5K

O>

<5

'

\

UJ -14.70-

-14.71 -

-14.72 A

\

j ;' i

, 4

>v

,

, 6

,

, 8

,

, 10

,

, 12

,

, 14

,

, 16

(a.u.) 2 ' states of Li2 Molecule Fig. 5 : PES of someRlow-lying

4

Summary

In this paper, we proposed two variants of linear response methods developed for excited states from the state specific multi-reference CEPA theories for the associated ground state. These theories are designed to generate excited state energies of a system whose ground state is multi-reference in character and also show intruders. Around the quasi-degenerate configurations, the effective hamiltonian based methods are expected to work well,

148 S. Chattopadhyay et al.

but they will fail in the regions of the PES where intruders are present. But our SS-MRCEPA based LRT performs with equal efficacy in the entire range of the PES for the various excited states. Numerical test calculations on P4 model and Li2 molecular systems support the above facts. A comparison of our results with those obtained from the FCI in the same basis indicates that the computed excited state energies are very accurate and uniform in nature throughout the entire range of the PES, and this is generally valid for states with various spatial symmetries. Thus our MRCEPA-LRT serves as a good model to compute accurate and uniform PES of the excited states. This fortifies our belief that the MRCEPA-LRT methods are useful and good approximants of the parent MR-CCLRT, despite the neglect of a host of terms present in the parent theory. The results are encouraging for the systems we have studied using our MRCEPA-LRT, which leads us to express cautious optimism about its general potentiality. 5

Acknowledgments

DM wishes to thank the CSIR, INDIA for financial support [Proj. No. 01 (1624)/EMR-II)]. DP thanks the CSIR, INDIA for the research fellowship. This article is dedicated to P Ca rsky, I Hubac and M Urban on the happy occasion of their reaching sixty.

Bibliography 1. I. Shavitt (1975) in Methods of electronic structure theory, Vol. 3, H. F. Schaefer III (ed.) Plenum Press, NY. 2. R. J. Bartlett and W. D. Silver (1975) Int. J. Quantum Chem. S9, 183; J. A. Pople, J. S. Binkley and R. Seeger (1976) Int. J. Quantum Chem. S1O, 1. 3. W. Meyer (1973) J. Chem. Phys. 58, 1017; W. Meyer (1989) Theor.Chim. Ada 35, 277; A. C. Hurley (1976) in Electron Correlation in Small Molecules, Academic Press, London; R. Ahlrichs, H. Lischka, V. Staemmler and W. Kutzelnigg (1975) J. Chem. Phys. 62, 1225; W. Kutzelnigg (1977) in Methods of Electronic Structure Theory, H. F. Schaefer III (ed.) Plenum Press, NY. 4. R. J. Gadanitz and R. Ahlrichs (1988) Chem. Phys. Lett. 143, 413. 5. P. G. Szalay and R. J. Bartlett (1995) Chem. Phys. Lett. 214, 481; L. FustiMolnar and P. G. Szalay (1996) J. Phys. Chem. 100, 6288.

Computation of Excited State Potential Energy Surfaces 149 6. P. G. Szalay (1997) in Recent Advances in Coupled-Cluster Methods, Ed. R. J. Bartlett, World Scientific, Singapore and references therein. 7. J. Cizek (1966) J. Chem. Phys. 45, 4256; J. Cizek (1969) Adv. Chem. Phys. 14, 35; J. Paldus, J. Cizek and I. Shavitt (1972) Phys. Rev. A 5, 50. 8. R. J. Bartlett (1989) J. Phys. Chem. 93, 1697; R. J. Bartlett (1995) in Modern Electronic Structure Theory, D. R. Yarkony (ed.) World Scientific, Singapore. 9. R. J. Bartlett (1981) Ann. Rev. Phys. Chem. 32, 359. 10. J. Noga and R. J. Bartlett (1987) J. Chem. Phys. 86, 7041; E. E. Scuseria and H. F. Schaefer III (1988) J. Chem. Phys. 152, 382; K. Raghavachari, J. A. Pople, E. S. Replogle and M. S. Head-Gordan (1990) J. Chem. Phys. 94, 5579; P. Piecuch, S. A. Kucharski and R. J. Bartlett (1999) J. Chem. Phys. 110, 6103. 11. R. J. Buenker and S. D. Peyerimhoff (1974) Theor. Chim. Acta 35, 33; P. E. M. Siegbahn (1980) J. Chem. Phys. 72, 1674; R. J. Buenker and S. Krebs (1999) in Recent Advances in Multireference Methods K. Hirao (ed.) World Scientific, Singapore and references therein. 12. S. R. Langhoff and E. R. Davidson (1974) Int. J. Quantum Chem. 8, 61; E. R. Davidson (1974) in The World of Quantum Chemistry, R. Dauel and B. Pullman (eds.) Reidel, Dodrecht; J. P. Francois and R. Gybels (1990) Chem. Phys. Lett. 172, 346. 13. J. P. Malrieu (1982) Theor. Chim. Acta 61, 163. 14. M. R. A. Blomberg and P. E. M. Seigbahn (1983) J. Chem. Phys. 78, 5682; W. D. Laidig and R. J. Bartlett (1984) Chem. Phys. Lett. 104, 424. 15. J. P. Daudey, J. L. Heully and J. P. Malrieu (1993) J. Chem. Phys. 99 1240; I. Nebot-Gil, J. Sanchez-Marin, J. P. Malrieu, J-L. Heully and D. J. Maynau (1995) J. Chem. Phys. 103, 2576. J. P. Malrieu, I. Nebot-Gil and J. Sanchez-Marin (1994) J. Chem. Phys. 100, 1440. 16. S. Chattopadhyay, U. S. Mahapatra, B. Datta and D. Mukherjee (2002) Chem. Phys. Lett. 357, 426. 17. S. Chattopadhyay, U. S. Mahapatra, P. Ghosh and D. Mukherjee (2002) in Low-Lying Potential Energy Surfaces, Eds. M. R. Hoffmann and K. G. Dyall ACS Symposium Series No. 828, Washington, DC. 18. B. Brandow (1967) Rev. Mod. Phys. 39, 771; I. Lindgren (1974) J. Phys. B, 7, 2441. 19. D. Mukherjee, R. K. Moitra, and A. Mukhopadhyay (1977) Mol. Phys. 33, 955; I. Lindgren (1978) Int. J. Quantum Chem. S12 33,; I. Lindgren, and D. Mukherjee (1987) Phys. Rep. 151, 93; D. Mukherjee, and S. Pal (1989) Adv. Quantum Chem. 20, 561. 20. B. Jeziorski, and H. J. Monkhorst (1981) Phys. Rev. A 24, 1668; B. Jeziorski, and J. Paldus (1988) J. Chem. Phys. 88, 5673. 21. T. H. Schucan and H. A. Weidenmiiller (1972) Ann. Phys. 73, 108; G. Hose and U. Kaldor (1979) J. Phys. B 12, 3827. 22. J. P. Malrieu, Ph. Durand and J. P. Daudey (1985) J. Phys. A: Math. Gen.

150 5. Chattopadhyay et al.

18, 809; J. L. Heully and J. P. Daudey (1988) J. Chem. Phys. 88, 1046. 23. D. Mukhopadhyay, B. Datta and D. Mukherjee (1992) Chem. Phys. Lett. 197, 236; B. Datta, R. Chaudhuri and D. Mukherjee (1996) J. Mol. Struct: THEOCHEM 361, 21. 24. A. Landau, E. Eliav, Y. Ishikawa and U. Kaldor (2000) J. Chem. Phys. 113, 9905. 25. J. P. Malrieu, J. P. Daudey and R. Caballol (1994) J. Chem. Phys. 101, 8908; J. Meller, J. P. Malrieu and J. L. Heully (1995) Chem. Phys. Lett. 244, 440; J. Meller, J. P. Malrieu and R. Caballol (1996) J. Chem. Phys. 104, 4068. 26. J. Masik, and I. Hubac (1997) in Quantum Systems in Chemistry and Physics: Trends in Methods and Applications, R. McWeeny et al. (eds.) KA, Dordrecht; I. Hubac, J. Prittner and P.Carsky (2000) J. Chem. Phys. 112, 8779. 27. U. S. Mahapatra, B. Datta, B. Bandyopadhyay and D. Mukherjee (1998)Adv. Quantum Chem. Vol. 30 D. Hanstrop and H. Persson (eds.) Academic Press Inc. San Diego. 28. U. S. Mahapatra, B. Datta and D. Mukherjee (1998) Mol. Phys. 94, 157; U. S. Mahapatra, B. Datta and D. Mukherjee (1999) J. Chem. Phys. 110, 6171. 29. S. Chattopadhyay, U. S. Mahapatra and D. Mukherjee (1999) in Recent Advances in Multireference Methods K. Hirao (ed.) World Scientific, Singapore; S. Chattopadhyay, U. S. Mahapatra and D. Mukherjee (2000) J. Chem. Phys. 112, 7939. 30. H. J. Monkhorst (1977) Int. J. Quantum Chem. S l l , 421; E. Dalgaard and H. J. Monkhorst (1983) Phys. Rev. A28, 1217. 31. H. Nakatsuji and K. Hirao (1978) Chem. Phys. Lett. 59, 362; H. Nakatsuji and K. Hirao (1988) J. Chem. Phys. 68, 2053; H. Nakatsuji and K. Hirao (1979) Chem. Phys. Lett. 67, 329, 334; K. Hirao (1993) J. Chem. Phys. 79, 5000. 32. D. Mukherjee and P. K. Mukherjee (1979) Chem. Phys. 39, 325; B. Datta , P. Sen and D. Mukherjee (1995) J. Phys. Chem. 99, 6441. 33. H. Sekino and R. J. Bartlett (1984) Int. J. Quantum Chem. S18, 255; P. Piecuch and R. J. Bartlett (1999) Adv. Quantum Chem. Vol. 34, 295 and references therein. 34. H. Koch and P. J0rgensen (1990) J. Chem. Phys. 93, 3333; H. Koch, R. Kobayashi, A. Sanchex de Mears and P. J0rgensen (1994) ibit 100, 4993. 35. S. Pal (1989) Phys. Rev. A 39, 39; S. Pal (1992) Int. J. Quantum Chem. 41, 443. 36. D. Ajitha and S. Pal (1997) Phys. Rev. A 56, 2658. D. Ajitha, N. Vaval and S. Pal (1999) J. Chem. Phys. 110, 2316. 37. S. Tenno, S. Iwata, S. Pal and D. Mukherjee (1999) Theor. Chem. Ace. 102, 252. 38. P. G. Szalay (1995) Int. J. Quantum Chem. 55 , 15. 39. P. G. Szalay, T. Muller and H. Lischka (2000) Phys. Chem. Chem. Phys. 2,

Computation of Excited State Potential Energy Surfaces 151 2067. 40. I. Martin, C. Lavin, Y. Perez-Delgado, J. Pitarch-Ruiz and J. Sanchez-Marin (2001) J. Phys. Chem. A 105, 9637. 41. D. Mukhopadhyay, S. Mukhopadhyay, R. Chaudhuri andD. Mukherjee (1991) Theor. Chim. Ada 80, 441. 42. U. S. Mahapatra, B. Datta and D. Mukherjee (1999) Chem. Phys. Lett. 299, 42; U. S. Mahapatra, B. Datta and D. Mukherjee (1999) J. Phys. Chem. A 103, 1822. 43. P. Ghosh, S. Chattopadhyay, D. Jana and D. Mukherjee (2002) Int. J. Mol. Set. 3, 733. 44. A. Deb, S. Chattopadhyay, D. Pahari and D. Mukherjee (2003) (to be communicated). 45. K. Jankowski and J. Paldus (1983) Int. J. Quantum Chem. 18, 1243. 46. J. Paldus, P. E. S. Wormer and M. Benard (1988) Coll. Czech. Commum. 53 1919. 47. S. Zarrabian and J. Paldus (1990) Int. J. Quantum Chem. 38, 761; K. Jankowski, J. Paldus and J. Wasilewski (1991) J. Chem. Phys. 95 , 3549. 48. J. P. Finley, R. Chaudhuri and K. F. Freed (1995) J. Chem. Phys. 103, 4990. 49. K. Jankowski and I. Grabowski (1995) Int. J. Quantum Chem. 55, 205. 50. U. Kaldor (1991) Theo. Chim. Ada 80, 427; U. Kaldor (1990) Chem. Phys. Lett. 140, 1. 51. I. Garcia-Cuesta, J. Sanchez-Marin, A. Sanchez de Mears and N. Ben Amor (1997) J. Chem. Phys. 107, 6306. 52. T. H. Dunning Jr. (1970) J. Chem. Phys. 53, 2823.

Chapter 5: Modelling of Anisotropic Exchange Coupling in Rare-Earth -Transition-Metal Pairs: Applications to Yb3+-Mn2+ and Yb^-Cr34 Halide Clusters and Implications to the Light Up-Conversion

M. Atanasova, C. DauP, H.U. GiideP a Institute of

General and Inorganic Chemistry Bulgarian Academy of Sciences Acad. G. Bontchev Str. Bill, 1113 Sofia, Bulgaria b Departement

de Chimie, Universite de Fribourg, Ckdu Musee 9, CH-1700 Fribourg, Switzerland

c Departement fur

Chemie und Biochemie, Universitdt Bern, Freiestrasse 3, CH-3000 Bern 9, Switzerland

Abstract A procedure for calculating magnetic exchange coupling constants in rare earth (RE)-transition metal (TM) dimers is presented. In a first step RE-TM transfer (hopping) integrals between orbitals carrying unpaired (magnetic) electrons on RE and TM are determined from a MO-calculation utilising the concept of orbital exchange pathway and effective Hamiltonian theory. In a second step, many-electron wave-functions for ground or excited electronic states on RE and TM are constructed in which spin-orbit coupling (which dominates for the RE) is fully (variationally) taken into account. Finally, the data from steps 1 and 2 are combined to obtain numerical values for kinetic exchange integrals (within Anderson's superexchange theory) utilising a nonHeisenberg (orbital dependent) exchange operator. The MO's corresponding

153

154 M. Atanasov, C. Daul and H. U. Giidel

to the d(Mn) and f(Yb) magnetic orbitals have been used to also calculate ferromagnetic contributions to the exchange coupling. The model is applied to Yb3+-Mn2+ dimers with corner, edge and face shared octahedra and Cl" and Br" ligands. For that purpose we utilise a spectroscopically adjusted (to spectra of single nucleous octahedral Mn2+, Cr3+ and Yb3+ complexes) parameterisation of the Extended Huckel theory. The expediency of the approach is discussed based on a comparison between calculated and experimental (available from inelastic neutron scattering experiments) anisotropic exchange parameters in Br3Cr3+-Ji(Br3) Yb3+Br3 dimers. Ferromagnetic exchange parameters for the ground state and the lowest emitting excited state increase from corner to edge to face Mn-Cl-Yb sharing thus varying in the same way as the efficiency of the up-conversion found for such species. This lends support for an exchange mechanism for this phenomenon postulated previously.

1. Introduction Interactions between transition metals (TM) and rare earths (RE) in solids have been of great interest because of their ability to combine photons of low energy to photons of higher energy (up-conversion, UC) '"2. It was discovered that pairs of Mn2+ and Yb3+ bridged by three or one Cl" ions are able to convert IR radiation at about 10300 cm"1 ( corresponding to the 2F7/2->2F5/2 excitation at Yb3+ ) into red (14700 cm"1, CsMnBr3:Yb - face sharing 3) or orange (16200 cm"1, Rb2MnCl4:Yb - corner sharing 4) luminescense, both corresponding to the "^Tig—>6Aig transition within the Mn2+ ion. Based on spectra in high resolution an exchange UC mechanism has been postulated. Surprisingly, no theoretical work in that direction have been done yet, in spite of the fact, that theories of excitations in pairs of exchange coupled TM are well documented5"10 and tested by experiment n"18. In this study we describe a procedure allowing to treat exchange interaction within a cluster of one TM and one RE, bridged by common ligands. Utilising MO wavefunctions and energies we base our model on the concept of localized magnetic orbitals allowing to treat both cluster ground and excited states and to account for both antiferromagnetic (kinetic) and ferromagnetic (potential) exchange. After an introduction into the modelling of exchange interactions using Anderson's exchange theory including ions in non-degenerate (Section 2.1) and degenerate (Section 2.2) electronic states, we focus on one example known from literature 19, that of CoCl2(H2O)2:Mn2+ with Mn2+ and Co2+ ions in the pair possessing ground states without and

Modelling of Anisotropic Exchange Coupling 155

with orbital degeneracy, respectively. For didactical purposes and including mathematical details as well, we derive expressions for the anisotropic exchange coupling tensor of a Mn2+-Co2+ pair for a general (without assuming any symmetry) bridging geometry. In Section 3 we further introduce a model for treating exchange coupling in TM(Mn2+)-RE(Yb3+) pairs. To approximate the model parameters we make use of a spectroscopically adjusted extended Hueckel theory which we introduce in Section 3.3. As a first application an elaborated example - that of MnC^4" and YbCl63" octahedra sharing one Cl" ligand will be presented in detail (Section 3.6.). In Section 4 we present additional applications to edge and face shared octrahedra of the same ions both in their ground state and in the lowest emitting excited state. In Section 4.2 the relevant aspects of our results regarding the mechanism of up-conversion will be discussed.

2. Modelling Exchange Interactions: Anderson's Exchange Theory 2.1. Exchange interactions for orbitally non-degenerate states Electrons of d (TM) and f(RE) orbitals are quite localized. Therefore, a good starting point in describing exchange coupling between electrons on different centers is to consider unperturbed d and f orbitals and to correct their wavefunctions affected by the surrounding ligands. Taking two TM centers with unpaired electrons occupying d^ orbitals on each (Fig.l) we consider magnetic orbitals dli and dl2 composed of d^1 and d^2 orbitals on centers 1 and 2 and (smaller) tails of the ligand (pz) (pL1 and
(1)

OCl-XdlLl+SdiLl 0C2 ~ Xd2L2 + Sd2U

p~s dd

Overlap integrals d-L (SdL ) and d-d (Sdd) are small with a covalency parameter %<jL, typically in the range 0.1 - 0.5. Within the basis of the four Slater determinants |dli+dl2+|, |dli+dl2"|, |dlfdl2+|, |dlfdl2"| we can construct one

156 M. Atanasov, C. Daul and H. U. Giidel

triplet and one singlet functions (eqs. 2). They are eigenfunctions of the Heisenberg exchange operator (eq.3) with the eigenvalues given in eq.4. The exchange coupling constant JJ2 describes the exchange splitting of the two multiplets Es-Er=J12. It is a phenomenological parameter, to be deduced from experiment (magnetic susceptibility, spectroscopy, inelastic neutron scattering) or calculated theoretically. T T : Idlx'd^l,(l/V2)[|dl,+dl2|+|dlrdl2+|], |dlfdl2"l ¥ s = (l/V2)[|dl1+dl2|-|dl1dl2+j] Hexc=-Jl2SlS2

(2.1) (2.2) (3)

ET=-(1/4)J 1 2

(4.1)

Es= (3/4)J12

(4.2)

Fig. 1 Localized magnetic orbitals in Anderson's super exchange model. Within the exchange theory of Anderson20 and using perturbation theory, Ji2 is expressed as a sum of two terms (eq.5). Ji 2 ^2Ki 2 , the Heisenberg exchange integral (eq.6) is a positive quantity, tending to stabilize the triplet over the singlet state. It reflects the gain of potential energy due to the Fermi hole, which excludes an electron of a given spin in the vicinity of a reference electron with the same spin (potential exchange). The Ji 2tf = -4T]22/U term describes the gain of kinetic energy due to the delocalization (described by the tails in eqs 1), tending to stabilize a singlet ground state (kinetic exchange).

Modelling of Anisotropic Exchange Coupling 157

As is seen, it is counteracted by the energy U, i.e. the increase of Coulomb repulsion when moving from singly (cuVcuV ) to doubly occupied (dli2 or dl22 ) orbitals dli and dl2. U is a large quantity, typically in the range 5-8 eV. The crucial quantities K12 and the hopping integral Ti2 can be expressed using the explicit forms of dli and dl2 (eq.7 and 8). As expected, T !2 is intrinsically connected with the covalency of the metal-ligand bond. Eq.(8) describes an useful approximation for Ki2, expressing this quantity in terms of the intraatomic exchange integral of the bridging ligand, weighted by orbital reductions factors (Xi2 and OC22 The expression of the exchange integral (Eq.5) reflects the configurational mixing of the singlet function (Eq.2.2) with the charge transfer excited states |dli+dlf| and |dl2+dl2"| which stabilize *FS against T T . In this form the equation for Jn can be easily generalized for magnetic centers (like high-spin Mnn (d5)) with more than one magnetic electrons. In this case a sum of terms over different exchange pathways, Jf2 (Eq.9) have to be included in the expression for Ji2, with nj and n2 accounting for the number of unpaired electrons on center 1 and 2, respectively. Ji2 3Ji2f+Ji2at2K12-4T122/U K12=JJdlI(r1)*dl2(r1)(l/rI2)dl1(r2)dl2rr2)*dT:dT2=[dl1dl2|dl,dl2] T 12 ~ N , N 2 [CC2 (d 1 1 h I 9 L 2 ) + a,(d 2 I h I
J12=~J- Z ^ nln2

iel,je2

(5) (6) (7) (8)

(9)

2.2. Exchange interactions in the case of orbital degeneracy The description of exchange coupling in terms of a single (scalar) parameter Ji2 (isotropic Heisenberg-Dirac-van Vleck Hamiltonian, Eq.3) is only valid if electronic states on ions 1 and 2 are orbitally non-degenerate. This is generally the case for half filled shells of TM (octahedral d3, d8, d5 (highspin)) and if spin-orbit coupling is negligible. For orbitally degenerate states and/or cases of strong spin-orbit coupling the isotropic model is generally not valid; orbital degrees of freedom intermix with spin ones to lead in general to an anisotropic exchange tensor, j = |j.. j (i j=x,y,z). The Hamiltonian for the spin coupling now takes the form:

158 M. Atanasov, C. Daul and H. U. Gildel

Uxx J*y Hexc=-SlJS2=-[six,Sly,Slz_Jyx

JASA

J^ Jyz S2y =

(10)

~^xx^lx^2x ~Jyy^\y^2y ~^zz^\z^2z ~ ~ Jxy^lx^2y ~^yx^ly^2x ~^xz^U^2z ~^zx^\z^2x ~^yz^ly^2z ~^zy^lz^2y

2.2.1. A prototype example: CoCl2(H2O)2:Mn2+ 19. In CoCl2.2H2O the cobaltous ions are surrounded by a nearly perfect square arrangement of four Cl ions and two water molecules along the b-axis. The crystalline field symmetry around Co2+ is nearly tetragonal with a principle axis (z) parallel to the b-axis (Fig.2). Co2+ possesses a high-spin *Ti ground state with orbital components which can be classified according to the three eigenstates of a fictitious orbital angular momentum of (1=1, lz= ±1,0). The spin-orbit interaction and the tetragonal crystal field in the 4T1 basis can be calculated making use of angular momentum operators 1 and Iz: H=A/I.S0-8(Iz2-2/3) (11) Here So is the spin operator acting on the s=3/2, ms=±3/2, ±1/2 spinfunctions of Co2+ and A,'=-(3/2)kX, - the spin orbit constant (X, Co2+ free ion=180 cm'1, k - a reduction factor). In CoCl2(H2O)2:Mn2+ Mn2+ substitutes for Co2+ and the 6Ai ground state of Mn2+ is exchange coupled with the ground state of a neighbour Co2+ ion (Fig.3). The latter is orbitally degenerate and the spin coupling becomes orbital dependent, since different orbitals on Co overlap in a different way with d-orbitals on Mn2+. In this sence the present example closely resembles that of a Yb3+-Mn2+ pair, however the orbital degeneracy here is only 3 (^T{) instead of 7 (one hole in the 4f-sub-shell of Yb3+). For didactic purposes it is useful to consider this example in some detail, before going to the more complicated cases of Mn2+-Yb3+ pairs. The exchange interaction between Co2+ and Mn2+ can be written in the general form of Eq.12; here J(lz,lz') is the exchange integral between Mn2+ and the lzand lz' are the 1=1 orbital components of Co2+ (Eq.13).

Hex=-OSo.S = - E J C U / ^ S r S

(12)

lz,lz'

Piziz- is an operator connecting the lz and lz' d-functions of Co2+ and So and S are spin operators for Co2+(So=3/2) and Mn2+(S=5/2). In the expression for J(lzJz') (eq 13) singly occupied d^-orbitals of Mn2+ are subsummed and

Modelling of Anisotropic Exchange Coupling 159

integrals (Uhld^) and (djh|l z ') of electronic hopping between Mn2+ and Co2+ describe various exchange pathways, allowing for off-diagonal (lz*lz') terms (non-diagonal exchange). Such terms may arise in ligand fields of low symmetry. J ( W z ' ) = - £ (UhldpXdJhllzVCSU)

(13)

M

i—^

JD \

. T. /

r—'•

T

j

I

cW

T Z*y)

b(z)

1 A

° O "•^ Co

° Mn

Co-Cl 2.45,2.48A Co-0 2.04 A

Fig. 2 Coordination geometries and magnetic ordering structure in the lowtemperature modification of CoCl2.2H2O. A

As is derived in Appendix 1, the orbital operator O can be expressed in terms of angular momentum (Ij, i=x,y,z) and the identity 1 operators as follows: 6 = [J(l,l)-J(0,0)]lz2 + V2 J'(l,0) ( U Z + U , ) - V2J"(l,0) (Iylz+lzly) + j'(i,-i)(ix2-iy2) - J"(i.-i) (Uy+M0+J(0,0) i

(14)

160 M. Atanasov, C. Daul and H. U. Giidel

ci^pci H 2 O©«Mn II ^OH2

C1QJPC1 H2OCHCon8S0OH2 Fig. 3 The Mn2+-Co2+ dimer units in Mn2+ doped CoCl2.2H2O. From the 9 components of J(lz,lz') (lz,l2'=±l,0) tensor only 6 are independent; the following relations hold (see Appendix A.I.): J*(lz,lz')=J(lzMz) J(1,O)=-J(O,-1) J(1,1)=J(-1,-1)

(15)

The J(12,1Z) parameters (l^V) are generally complex numbers. In eq 14 we denote their real and imaginary parts by J'(lz»lz) and J"(l z ,l z ), respectively. Since crystal field and spin-orbit coupling energies (Eq.ll) are much larger than J(lz,lz') we take the eigenstates of this Hamiltonian and focus on the ground state Kramers doublet of Co2+ which can be written in the following form:

H)-HH*H'4

(16)

Here, mj and nis in |m,,m s ) stay for Co2+ orbital and spin quantum numbers. Values of 8 (Eq.ll) and a,b,c (Eq.16) have been determined by Oguchi in such a way, that theoretical g-values match the experimental ones: 8 =5kX , a=895, b=-0.288, c=0.346.22 The separation between the ground and the first excited state (260 cm"1) is much larger than the exchange energy. Therefore excited states can be neglected and exchange tensor components can be obtained applying the operator (Eq. 12 and 14) on the right hand side of A

eq.16. Let us calculate as an example J^ and Jxx. Applying the operator O on

Modelling of Anisotropic Exchange Coupling 161

the function +_L\ of Co2+ we notice that the effect of the operators lz2 and 2/ lx2-ly2 of relevance for J^ and JU is just to produce the following orbital replacements: lz2: P( 1 -> 1), P(-1 ->-1) and multiplying by 1, P( 0-»0) and multiplying byO (17) lx2-ly2: P(l->-l) and P(-1->1) and multiplying by 1 Keeping this in mind expressions for J^ and JM are obtained for example taking:

-JzzSzSz I \

2/

5l\=_6Sz(Co)Szl\

2 2/

5\\

2/

2 2/

(18)

-JxxsA I\ i,l\ =-6 S,(Co)Sx i\ i l \ 2/

2'2/

2'2/

2/

In eq.18 s ^ and S^Co) are spin-operators acting on the left and right sides of eq.16. We notice that the operators S^ for Mn2+ lead to the same result on both sides of eq 18 and can thus be dropped out. Keeping in mind that sx leads to changes of mz by ±1 and integrating we obtain:

J

--

/I \2

Sz

1\ 2/

/=2

(20S*(C0)l)

(Jjos^i) J--

, i a \"2Sx2/

(19) -2\-2OSx(CO)|2/

Here we make use of the relations s z i \ = i i \ and sx J_\_! _ i \ to give 2/ 2|2/

2/

2

2/

(1/2) for the denominators in Eq.19. Now J^ and J^ are derived as follows; we apply first the Sz and Sx operators on the right hand side of eq.16 and make the substitutions21 :

162 M. Atanasov, C. Daul and H. U. Giidel

Sz 2,2\ = 3 3 3\ 2 2/

Sz

3 A= I 3 1 U 2

2 2 2/

2 2/

2 2 2/

l\

1 3 A.

2 2/

2 2 2/

sx 2 , 2 \ = ^ 2 , I \ ; s x 2 , I \ = ^ 2 , 2 \ + 2,_I\ ; 2 2/

2 2 2/

2

2 2/

2 2/

s,2,_I\ = 2,I\ + ^2,_2\; 2/

2 2

2 2 2/

(2(u)

(20.2)

2 2/

2/

We obtain:

6sz(Co)|^=6[|a|-l,mb|o)^-Ic|l,-^j=[J(l,l)-J(0,0)]. Tla _ lf 2\_I c i,_I\]+J(O,o)[2a _!,2\ + I b Of I\-Ic i,-l\" [2

2/

2

2/J

|_2

2/

2

2/2

2/_

Jzz=2/- OSz(Co) I \ = (3a2-c2)J(l,l)+b2J(0,0)

(21a)

6 sx(co) I\= 6 [ ^ fl _ 1 ,I\ + ^o,2\ +6 o i _l\ + ^J lf _2\ + J lf I\ ]= 2/

=[J(I,D-J(O,O)]

2

2 / 2

2/

r^ a |_ 1 ) i^ c | u _|^ + c | 1 ; ^i+

2 / 2 ( 2 / 1

^ = 2 / - ! OS x (CoJi\= 2V3acJ(l,l)+2b2J(0,0)+2c2J'(l,-l)

2/

(21b)

Expressions for the Jy (ij=x,y,x) parameters are listed in Table 1. The J(l,l) and J(0,0) terms lead to an axially symmetric exchange tensor (Jzz^Jxx=Jyy)5 while the off- diagonal term J'(l>-1) introduces additionally orthorombicity (J^Jyy). The J"(l,-1) , J'(l,0) and J"(l,0) parameters lead to

Modelling of Anisotropic Exchange Coupling 163

the off-diagonal exchange [ Jy (i*j=x,y,z)] . The non-diagonal part of the exchange interaction (Eq. 10) can be written using expressions in Table 1 as: 2(V3ab-3bc)J'(l,0)(s x S z + s z S x ) + 2(V3ab-bc)J"(l,0)(s y S z + s z S y ) + 2(bc-V3ab)J'(l,0)(s x S z - s 2 S x ) + 2(3bc-V3ab)J"(l,0)(s y S z - s z S y )

(22)

Table 1. Expressions for the exchange-tensor components for a Co2+-Mn2+ exchange pair in a general coordination geometry around Co2+ and Mn2+ of having no symmetry. Jxx=2V3acJ(l,l)+2b2J(0,0)+2c2J'(l,-l) Jyy=2V3acJ(l,l)+2b2J(0,0)-2c2J'(l,-l) J2Z=(3a2-c2)J(l,l)+b2J(0,0) J xy =-2c 2 J"(l,-l)

J^c'r'Cl,-!) J^bcJ'O.O) Jzx=2(2bc-V3ab)J'(l,0) Jyz=2bcJ"(l,0) J^VSab^bcygO)

As seen from (Eq.22) the non-diagonal exchange interaction can be decomposed into a symmetric part and an antisymmetric part (DzyaloshinskiMoriya exchange). We note that in eq.22 quantization axes have been adopted to make the g-tensor diagonal (main axes of the g-tensor). These anisotropic exchange interactions have important consequences for the EPR spectra of Mn2+, when doped into the CoCl2.2H2O lattice. Below T=17.5K COCI2.2H2O is an antiferromagnet with sub-lattice magnetization directed along the b-axis and longitudinal components (along the b-axis , Fig.2) of the exchange integrals much larger than the transversal ones. A succession of antiferro- ferri- and ferromagnetic transitions appear upon increasing magnetic field directed along the the b-axis. Below the ordering temperature internal fields due to the anisotropic J-tensor can lead to splittings of the S=5/2 levels of Mn2+. It has been shown 19, based on symmetry arguments, that exchange coupling between Mn2+-Co2+ pairs along the c(x) axis and within the ab(zy) plane are described by the Hamiltonians: Hexc(c)= - - C sxSx - j ; sySy- J°B s2Sz (23)

164 M. Atanasov, C. Daul and H. U. Giidel

H eX c(ab)=-/i c s x S x - / ^ sySy-Jlzz s 2 S z - / ^ s x S y -J^ SyS,-/^ sxSz-J'y, sySz

(24)

Below the ordering temperature the spins of Co2+ are ordered, with an easy direction along the z-axis (Fig.2) i.e. they all are in a — —\ state. 2*2/ Applying the sx and sy operators on the —,—)

state leads to the spin-flip

state 1 _ l \ at a higher energy. Let us denote its energy with respect to 2'

2/

1 _L\ by Eo (HeXc(c)) and Ei (HeXC(ab)). The following matrix elements 2'2/ mixing the ground magnetic state with the spin-flip state are derived:

(HKHH) = -{ y » 5 '4 J - 5 ' (i4h-Hi-i) = -i J1 - s «-i J ^ We focus now on H<>xc (c) and notice that E o » J^,J°

(25) (26) (see below).

Applying perturbation theory, and rearranging terms we obtain the following spin Hamiltonian for Mn2+: Heff(c)= -^(J°J

+/;2)[5(5

+l)-S/]

(27)

The contribution to the D-parameter from this term is obtained by summing the contributions from the two Co2+ ions (above and below the Mn2+ ion), i.e. multiplying Eq(27) by 4 (=22). We get D = - ^ (JaJ+Jl2)

(28)

Infrared resonance absorption of clustered spins and critical magnetic fields for antiferro-, ferri- and ferromagnetic transitions 23 allow to obtain

Modelling of Anisotropic Exchange Coupling 165

EO=22.1 cm'1, 7^=0.59 cm"1 and ^ = ^ eq 28 we obtain:

= a J ° , a < l . Substituting into

D=0.032 a 2 cm"1

(29)

This value of D is much smaller than the experimental one and of opposite sign (D=-0.13 cm"1) . Let us turn now to Heff within the ab-plane. Contributions to D from the diagonal part of HeXC(ab) are again negligibly small and we focus here at the off-diagonal part (the last two term of the matrix element eq 25). We obtain: Heff(ab)= _ J ^

5 ( 5

+ 1) + (

^

+ r

(30)

; -j^)Sl\

The contribution to the D-parameter from this term is obtained by summing the contributions from the four Co2+ ions in the plane surrounding each Mn2+, i.e. multiplying Eq.30 by 16(=42). Substituting expressions of J\zijXyz

Jly

md

from Table 1 we get

D=

-^r(JlJ Ex

+Jlyz2-Jl2)

This

is negative

0

0

0

= -^r{4b2c2[J'm2 El

and

matches

the

+/"(l,0) 2 ]-4cV"(l -I) 2 } sign

from

(31) experiment if

J\z + J\z > J\y • Concluding this section, we can state, that anisotropic exchange coupling can produce measurable effects in the EPR spectra of the Mn2+ ion, when entering as impurity in CoCl2.H2O. However, the large number of model parameters and lack of numerical estimates for these parameters did not allow to make more conclusive statements. In Section 3 we describe a procedure allowing to make this possible.

166 M. Atanasov, C. Daul and H. U. Gudel

3.

Magnetic Exchange in Mn 2+ (Cr 34 )-Yb 3+ Pairs

3.1. Electronic structure and energy levels ofYb3+ in cubic and lower symmetric ligand fields With one hole in the Yb3+ f-shell (f13-configuration) we can deal with electronic states of Yb3+ in a one electron description, keeping in mind that electron-hole equivalence requires to turn signs of all one-electron operators (ligand field and spin-orbit coupling). When treating energy levels of felements one usually starts from real f-functions - fLy^x^y2)], f[xyz], ffyz2], ftz3], flxz2], f M x V ) ] and ffxCx^y2)] which for a well oriented Yb-Cl(Br) bond direction ( along the z-axis) and a cylindrical symmetry transform as the irreducible representations
(32)

Estimations of the AOM parameters of a and n -type - eCT and eK (see Section 3.3) yield values for ea(eK) of 338(64) and 208(37 cm"1) for Yb-Cl and Yb-Br respectively, leading to the order of increasing energy from a2U to t2u to t]u. Symmetry lowering from Oh to C4v, C2v and C3v in solid networks composed of octahedra sharing one, two or three ligands, respectively, leads to corresponding symmetry splittings: Oh

ti u t2u a2u

C4V

C2v

C3V

ai+e ai+bi+b 2 ai+e bi+e a 2 +b!+b 2 a 2 +e b2 ai ai

We note that AOM parameters for RE are one order of magnitude smaller compared to their typical values in 3d metals complexes, and also, in the case of Yb 3+ , compared to the spin-orbit coupling constant (£(Yb 3+ )=-2900 cm"1). It follows that spin-orbit coupling leads to a strong deviation from the regime of a strong ligand field with f-orbitals optimally aligned for metal-ligand

Modelling of Anisotropic Exchange Coupling 167

overlap; it is thus reasonable to start from the free-ion levels of Yb3+ - 2F7/2 < separated in energy by (7/2)£ =10000 cm"1 and to consider ligand field as small perturbation. However for practical purposes we consider all effects on the same footing starting from real f-orbitals but diagonalising the full 14x14 matrix of spin-orbit coupling (Appendix 2) and ligand field (see Sections 3.2,4). In cubic symmetry the 2F7/2 ground state of Yb3+ splits into twb Kramers doublets F 6 and F 7 and a quartet F8, while the 2F5/2 excited state of Yb3+ gives rise to Eg and F 7 . From the spectrum of Cs2NaYbCl6 containing the octahedral YbCU3" chromophore 25, the positions of these multiplets have been located at (in a n 1 ) 362 (r 8 ), 701(T7) 10243 (r 8 ) and 10708 (r 7 ) with F6 - the ground state. These become additionally split at sites with a lower symmetry as found in Cs3Yb2Br9 (C3v site ) and MnCl2:Yb3+ (C2v site), see Section 3.2. 2Fs/2

3.2. Exchange Hamiltonian and configuration state basis functions for Mn2+(Cr3+)-Yb3+pairs Kramers doublets and quartets of Yb3+ in octahedral ligand field can be regarded as fictitious spins of 1/2 (F6, F 7 ) and 3/2 (F8) respectively. Let us denote fictitious spin-operators acting on these functions by sx, sy and sz and the corresponding spin-operators for the S=5/2(3/2) spin functions of Mn2+(Cr3+) by Sx,Sy,Sz. In the magnetic dimers considered in this work, we are concerned with site symmetries C2v, C3v and C4v (for Yb3+ and the TM octahedra sharing common edge, face and corner, respectively). In these symmetries the exchange Hamiltonian takes a diagonal form (Eq.33). The exchange parameters J^, J ^ , Jyy can be calculated by letting H e x ^ -JzzSzSz-JXX SxSj-Jyy SySy

(33)

Hexc act on product functions

, = l \ c = l\ (Yb3+-Mn2+) or „ = i\\ s = 2\ 2/ 2/ 2/| 2/ (Yb3+-Cr3+) and equating the result with the effect of the spin Hamiltonian Hexc' (Eq.34) acting on linear combinations (due to spinA

orbit mixing) of orbital (via the orbital operator O) times spin functions (via the spin operators s z ', s x ' and s y ' of the genuine spin 1/2) representing the Kramers state of Yb3+. We utilize wave functions for Kramers doublets and quartets with proper symmetry with respect to the time-reversal operator to ensure correct phases. It is practical to

168 M. Atanasov, C. Daul and H. U. Giidel A

express O in terms of angular momentum operators (see Section 2.2.1). In octahedral field the orbital momenta of the f-orbitals (1=3) are only partly quenched; t2, ti orbital functions behave as angular momenta for 1=1 (eq.35) and a2 as a pseudo-scalar. In order to utilize this equivalence we make use of A

f-functions described by Griffith26 (Eq.36) to allow to set up the O operator in the form given by (Eq.37); we denote by ± l " , 0 " and ±l',0' the components of X2 and ti which obey relations (35). The splitting of the f-basis into two types of orbital momentum functions leads to corresponding sets of orbital exchange parameters, which we group in Eq.37 into an isotropic J(a2,a2) term and anisotropy parameters for axial - J ( l " , l " ) , J(0",0") (t2type); J(1\1'X J(0',0') (t r type) and, additionally (in two-fold symmetries) orthorhombic J(l",-1") (t2-type) and J(l',-1') (ti-type) exchange coupling terms. HeXc = - 6 (sz'Sz+s,'SI+sy'Sy)

(34)

t2 <=> (1/2)1 t, »-(3/2)l

(35)

^-° = -i| f( - 3)+ i| f(1) t 2 ( 0 > 7? f { - 2 ) + 7? f ( 2 ) t2(O =

-^|f(3)+3|f(-1)

^ - - ^ - " - ^ t1(0') = f(0) = fz3

tl(1') =

-3| f ( 1 ) -i| f ( - 3 )

a2= "i f( - 2)+ i f(2) A

O= axial term: (l*2(t2)[J(l ",1 ")-J(0",0")]+0«2(ti)[J(l M >J(0',0')]+ IJ(0",0")+I. J(0\0')+

(36)

Modelling of Anisotropic Exchange Coupling 169

orthorhombic term: ( l ^ H y 2 ^ ) ) J(l",-l")+(lx2(tl)-ly2(tl)) J(l ',-1')+

(37)

scalar term: +J(a2,a2) 5.5. Parameterization of the ligand field in TM and RE complexes: a spectroscopically adjusted EH-theory Exchange integrals in TM-RE pairs depend on the electronic states of the TM and RE involved and on a large number of orbital exchange parameters (Eq.37) which need to be calculated in advance. Parameterizations of the LF in TM complexes using density functional theory (DFT) have been quite successful in recent time allowing to calculate almost parameter-free ground and excited state multiplets of TM complexes . In contrast, DFT applications on RE complexes turn out to present severe problems, presumably due to basis sets and/or functionals for 4f-orbitals which are not well established yet. Thus, a DFT calculation 29 on the octahedral YbCl63" unit using triple zeta basis of 4f and a LDA 30 plus PW96 31 functional and average occupancy (13/7 electrons on each f-orbital) yields a a2u
170 M. Atanasov, C. Daul and H. U. Giidel

when trying to calculate exchange coupling in Yb3+-Mn2+ pairs; the attempt to let such calculations converge failed already at the SCF step; scrambling of 4f(Yb), 3d(Mn) and 3p(Cl) orbitals leads to oscillations in the charge density, between spatially separated but otherwise energetically close ranges. In contrast, less sophisticate Extended Hiickel (EH) calculations have lead to reasonable result as is seen by the results included in Table 2. In the EHmethod 32, diagonal elements of the one-electron operator for a given orbital (HH) are approximated by the valence state ionization potential from that orbital (VSIP);; (Eq.38), while off-diagonal elements Hy are taken to be proportional to diatomic overlap integrals Sy (Eq.39). The parameter k can be adjusted such, as to fit spectroscopic lODq values; we take k=1.25(1.75) for Mn2+(Cr3+) leading to lODq parameters 6700(15000) cm"1 close to experiment 7500(12400)cm"1 . Adopted values of the VSIP for 4f (Yb) and 4p(Br) (12.36 eV and -14.20 eV) are altered compared to tabulated ones (-13.86 and 13.10 eV 33) to ensure a larger 4f(Yb)-3p(Cl) and 4p(Br) energy separation. As for the spin-orbit coupling constant of Yb3+ we take the value of-0.353 eV from a relativistic ZORA calculation of the free Yb3+ ion. Using this spectroscopically adjusted EH-theory a reasonable agreement between calculated and spectroscopic ground and excited energy levels for YbCl63" and YbBr63" chromophors has been achieved (Table 3). (38) (39)

Hfi=KVSIP)fi Hij = k.Sij(Hii+Hij)/2

Table 3. Spectroscopic data on the ground and excited state energy levels (in cm"1) of YbCl/" and YbBr63" chromophores and their values calculated using EHT. Energy level I MnCl2:Ybj+ C2v 1J I Cs3Yb2Br9 C 3 / ; I CszNaYbC^CV exp. calc.EHT exp. calc.EHT exp. calc.EHT 2F7/2 gr.state 0

0

1 2 3

268 371 608

0

362 435 733

0

114 140 441

0

218 276 439

0

0

T6(2)

362 385 T8(4) 701 703 T7(2)

F5/2 exc.state 0* 10179 10193 10119 10102 10243 10203 Tg(4) 1* ...? 10242 10146 10135 10708 10657 T7(2) 2* I 10764 10670 \ 10590 10402 | Experimental values taken from: 1) P.Gerner, H.U.Guedel, personal communication; 2) M.P. Hehlen and H.U.Gudel, J.Chem.Phys. 98(3) (1993) 1768; 3) R. Schwartz, Inorg. Chem. 16 (1977) 1694.

Modelling of Anisotropic Exchange Coupling 171

3.4. Integrals of electronic 4f(Yb3+)-3d(Mn2+ or Cr3*) hopping EH calculations yield d- and f-orbitals which can be easily identified by their dominating percentage of d- and f-character, respectively. For the purpose of evaluating effective integrals of electron d<=>f hopping we cut the ligand part from these MO's and take the 12x12 matrix containing MO coefficients (in columns) for d and f-orbitals only. Let us denote this matrix by U and the diagonal matrix containing the respective MO energies by A. From these two matrices we can generate an effective one-electron Hamiltonian which contains on the block diagonals 5x5 and 7x7 ligand field matrices for d (Mn(Cr)) and f (Yb) electrons and the 5x7 matrix of integrals of d<=>f hopping on the off-diagonal. This is accomplished as follows; we introduce the ovelap matrix S(Eq.4O). One can easily find S~1/2 using the following recipe; get eigenvalues A,(diagonal matrix) and eigenvectors (in columns) U,. S"1/2 is then given by eq.41. As has been shown by DesCloizeaux 34 the effective matrix h is calculated following Eq.42. S=U.UT S- 1 / 2 =U S A; 1 / 2 U S T h = S"1/2UAUTS-1/2

(40) (4i) (42)

Calculations on RE-TM dimers have been done (see Section 3.3) using a basis of real d(TM ), f(RE) and s and p(ligand) functions. For the sake of applying the operator 6 we get the matrix h' (Eq.43), by using the matrix T (see Appendix A.2) which transforms the 1=3, mi=0,±l, ±2, ±3 functions into t 2 (0", ±1"), t!(0',±r) and a2 (eq.36), while leaving the basis of real dfunctions (d^, d^yz and d^.^xy) unchanged: h'= T"'hT

(43)

5.5. Two electron exchange integrals involving 4f(Yb3+) and 3d(Mn2+ or Cr3*) In calculating two-electron exchange integrals we made use of approximation (Eq.8). In other words, from the MO's dominated by the d- and f-orbitals we take only that part residing on the bridging ligand (1). Let d^ and fv be representative for two such functions; we have:

i

172 M. Atanasov, C. Daul and H. U. Oiidel

fv= Y^PJ

< 44 >

Then the exchange integral is approximated by a sum of intra atomic exchange integrals on the bridging ligands (1) times squares of the coefficients c and c y , the sum extending over

(45) p-orbitals belonging to the same bridging ligand and summation being taken over all bridging ligands:

[d,fv|d^fv]=XX44fi:;

(46)

3.6. An elaborated example: exchange coupling in ClsMnClYbCls6' (corner sharing) As an example of application of our method we consider the Cl5MnClYbCl56" dimer with a linear Mn2+-Cl-Yb3+ bridging geometry. We consider here the exchange coupling between the ground states of Mn2+ (6Ai) and Yb3+ (T6) (C4v symmetry). Diagonalizing the Hamiltonian of the spin-orbit coupling plus the ligand field we get the following expressions for the components of the Kramers doublet F6:

KH-^ffH) |-iH'iH«4)-H>

(47)

With a=0.020 b=-0.563 c=0.826; we note the dominance of the ti(±l',0') orbitals involved in stronger a-interactions and the negligible participation of the weaker ^-interacting t2(±l",0") orbitals. In Table 4 we list Mn2+ -Yb3+ hopping integrals ( d j h | f v ) and two-electron exchange integrals [dM fv| dM fv] (in parenthesis), which we calculate following the recipes in Sections 3.4 and 3.5, respectively. From these data we calculate orbital exchange parameters, (J(v,v)), where we differentiate between

Modelling of Anisotropic Exchange Coupling 173

contributions from kinetic (Jk(v,v), Eq.(48)) and potential (Jp(v,v),Eq(49)) exchange; values of these parameters are listed in Table 5.

h(y^=

^l(*Mf>u\hM

(48)

(49)

W,v)=}0j^\dM Derivation of expressions for J^ and Jxx=Jyy is straightforward: Jzz:

11/2>= a(-l ",-l/2)+b (O',l/2) +c (1 ',-1/2) sz' 11/2>= -(l/2)a(-l",-l/2)+(l/2)b (O',l/2) -(l/2)c (1 ',-1/2) Osz' | l/2>=-(l/2)aJ(l",l")(-l",-l/2)+(l/2)b J(0',0') (O',l/2) -(l/2)c J(l',l')(l',-l/2) Ja= 2 = -a2 J(l ",1 ")+b2 J(0',0')-c2 J(1 ',1')

(50)

sx' I l/2>=(l/2)a(-l",l/2)+(l/2)b(O',-l/2) +(l/2)c (l',l/2) Os x '|l/2>=(l/2)aJ(l",l")(-l",l/2)+(l/2)b J(l',l')(l',l/2) inT 2 <-l/2 | Osx' | l/2>= b2 J(0',0')

J(0',0') (O',-l/2) +(l/2)c (51)

Using these expressions, we obtain after substitution of a,b,c and Jk(v,v') and Jp(v,v) values of J^ and Jxx=Jyy listed in Table 6 (last column). Following the same procedure exchange coupling constants for Mn2+-Yb3+ clusters with face (Cl, Br), edge and comer (Cl) sharing geometries (Figure 4) have been calculated. They are included in Table 6.

,

-11.483 -11.483 -10.571 -10.571 -10.617 -10.617

-11.486 -11.486

-11.486 | -11.486 |

yz(t2g) yz(t2g)

xz(t 2 g ) xz(t 2g )

I

xy(t2g) _xy(t ^2g) _x2^-y2(eg) x2-y2(eg) ^(eg) z^eg)

,

t2-i" t2-l" -12.337 -12.337 0.0 0.0 (oo) (O0) 0.0 0.0 (0.0008) (0.0008) 0.0 0.0 (0^0) (0^0) 0.0089i 0.0089i (0.0013) (0.0013) -0.0089 I -0.0089 (0.0009) I (0.0009)

I

.

,

t21" t 2 -0" I t21" t 2 -0" -12.337 -12.337 -12.337 -12.337 0.0 0.0 0.0 0.0 (oo) £ao) (O0) (O0) 0.0 0.0 0.0 0.0 (0.0002) (0.0008) (0.0002) (0.0008) 0.0 0.0 0.0 0.0 (£0) (O0) (£0) (O0) 0.0 0.0089i 0.0 0.0089i (0.0023) (0.0013) (0.0023) (0.0013) 0.0 -0.0089 0.0 1 (0.0007) [ -0.0089 (0.0009) 1 (0.0007) [ (0.0009)

I

.

ti-o tpOJ -12.26 -12.269 0.0 0.0 (o.o; (0.0) 0.0 0.0 (0.0; (0^0) -0.066 -0.0666 (O0 (O0) 0.0 0.0 (OXV (OX)] 0.0 0.0 (0.0 | | (0.0)

— ,I

trr tpT -12.267 -12.267 0.0 0.0 (o.o) (0.0) 0.0 0.0 (0.0005) (0.0005) 0.0 0.0 (OO) (O0) -0.0098i -0.0098i (0.0012) (0.0012) -0.0098 [ -0.0098 (0.0007) | (0.0007)

1

,

.

|

,

UV_ -12.267 0.0 (0.0) 0.0 (0.0005) 0.0 (O0) -0.0098i (0.0012) -0.0098 | (0.0007)

a2 -12.360 0.0 (0.0) 0.0 (0.0002) 0.0 (0.0) 0.0 (0.0024) 0.0 (0.0008)

Table 4. Orbital exchange constants (hopping integrals) (d^ | h | fv) and two-electron exch exchange integrals [d^ fv| d^ fv] (in parenthesis) of a Cl5MnClYbCl56" dimer with linear Mn2+-Cl-Yb3+ sharing (corner shared share MnCl64" and YbCl^3") octahedra. Energies of the ligand field splitted d- and f-orbitals are also listed (all energies in eV).

°^ ^ S &

. i"

|_

b

^

O

176 M. Atanasov, C. Daul and H. U. Giidel

Modelling of Anisotropic Exchange Coupling 175

f z

fz

MnO 2 :Yb 3+

Rb2MnCl4:Yb3+

CsMnBr3:Yb3+ Fig.4 Mn-ligand-Yb sharing geometries of Yb-Mn dimers in Yb3+ doped Mn2+ halogenide solids; for RbMnCl3:Yb3+ and CsMnCl3:Yb3+ both comer and face sharing is encountered.

176 M. Atanasov, C. Daul and H. U. Giidel

Table 5. Kinetic and potential (in parenthesis) exchange parameters (in units 10"4 eV2/3U[5U] and \QAI5 eV, respectively) for Mn2+-Yb3+ and Cr^-Yb3* halide bioctahedral complexes with face, edge and corner sharing bridging geometries Octahedra

I CrBr3Yb

Sharing: t2: J(l",l") J(0",0") J(l",-1") ti:

J(l',l') J(0',0') J(l',-1') a2: J(a2,a2)

I

I MnBr3Yb

I MnCl3Yb

I MnCl2Yb

I MnClYb

face

face

face

edge

corner

-4.31 (6.49) -0.31 (1.83) 0 (O0)

-5.26 (3.22) -5.68 (1.58) 0 (O0)

-12.90 (5.21) -6.48 (4.02) 0 (O0)

-3.64 (6.70) -16.20 (1.00) 3.35 (0.0)

-1.60 (30.00) 0.00 (32.00) 0.00 (0.0)

-7.14 (7.10) -74.65 (3.81) 0 (O0)

-4.35 (3.04) -47.33 (1.24) 0 (O0)

-9.58 (8.80) -59.02 (4.30) 0 £00)

-8.60 (6.70) -7.34 (1.80) 7.19 (0.0)

-1.88 (24.00) -44.36 (0.00) 0.00 (0.0)

-2.28 (5.40)

0.00 | (34.00)

-16.72 (16.93)

I

-5.68 (6.77)

-5.55 I (18.90)

|

4. Applications, Results and Discussion 4.1 Exchange integrals in Mn +-Yb

and Cr +-Yb

+

pairs

4.1.1 Exchange coupling in Br3CrBr3YbCl33" with Cr3"1" and Yb3+ in their groimd states With three unpaired electrons in the t2g shell, CrBr63' possesses a 4A2 ground state, while YbCl63" is in a T4 ground state (notations for the C3v site symmetries). Exchange constants (Jzz= -0.0064 eV and Jxx=Jyy=0.0052 eV) deduced from neutron scattering experiments 35 indicate exchange coupling to be highly anisotropic - antiferromagnetic along the Yb-Cr chain and ferromagnetic in perpendicular to this direction. In a qualitative agreement with this interpretation of the neutron scattering data, our calculations show

Jyy

Jxx

Jm

-11.20 -0.64 (0.30) -11.39 0.52 (1.45) -11.39 0.52 I (1.45)

I Br3CrBr3YbBr33" calc. exp.(meV) exc.st. 1.31 (0.60) -2.25 (1.40) -2.25 (1.40)

gr.st. -6.97 (0.10) -9.95 (1.75) -9.95 | (1.75)

gr.st. -5.34 (0.20) -7.28 (0.60) -7.28 | (0.60) exc.st. -0.55 (0.40) -2.00 (0.60) -2.00 (0.60)

I Cl3MnCl3YbCl33'

I Br3MnBr3YbBr33~ gr.st. -1.19 (4.43) -1.98 (0.23) -6.94 | (0.83)

e: exc.st. -I-0.80 i(3.94) (0.40 ((1.74) -4.74 (2.74) (

5" I CUMnC^YbCL, CUMnC^

gr.st. exc.st. -12.77 -1.35 (16.00) (10.00) -14.06 0.00 (0.00) (8.00) -14.06 0.00 (0.00) (8.00)

I Cl5MnClYbCl55"

and lO"4^ eV, respectively) of Table 6. Kinetic and potential (in parenthesis) exchange contributions (in units 10"4eV2/3U[5U] /3U[' the anisotropic exchange Hamiltonian Hexc=~Jxx s^-Jyy sy Sy - J^ sz Sz for CrI"[Mnn]-Ybin 1halide face, edge and corner shared bi-octahedra; Sj, Sj (i=x,y,z) - spin -operators acting on the (fictious) (1/2) and (3/2[5/2]) spin-functions on Yb and the TM Cr[Mn], respectively. I

I

•i

CO

?r

| «; 3' ^ a* I | | ts

Modelling of Anisotropic Exchange Coupling 177 179

178 M. Atanasov, C. Daul and H. U. Giidel

(Table 6) J^ to be mainly antiferromagnetic; ferromagnetic coupling in the z direction is calculated about four times smaller than that in the x-y plane, the latter compensating perhaps with excess the negative J^Jyy. With one unknown parameter U and with different approximations inherent in calculating kinetic and potential contributions to J, we are not able to calculate total values of J ( a sum of two times the ferromagnetic term plus four times the antiferromagnetic term). One can speculate and get U=2.35 eV - not quite unreasonable but still smaller than expected based on experience and take a scaling of the ferromagnetic terms by a factor of 5 in order to bring numerical data quantitatively agree with experiment. A good agreement between calculated and experimental exchange integrals using an EH-approximation for the integrals of Cr-Yb hopping, but accounting for electronic correlation within a CI treatment has been reported by Mironov et al 36 . 4.1.2. Exchange constants for Mn2+ ground state - Yb3+ ground and excited states of Mn2+-Yb3+ dimers with corner, edge and face sharing. Comparing data for the Cl- clusters we conclude that antiferromagnetic (kinetic) exchange parameters with Yb3+ in the ground state are strongest in the Mn-Yb dimers with corner sharing, about twice weaker for face sharing and the smallest for edge charing, with a disctinct anisotropy in the latter case. Calculated values for the antiferromagnetic coupling with Yb3+ in the emitting excited Kramers doublet are much smaller than those for dimers with Yb3+ in the ground state. Going to the potential (ferromagnetic) contributions for Yb3+ in its ground state, we notice the large anisotropy with J^Jyj^Jzz for face sharing , J ^ J y ^ O for corner sharing and Jzz^JxxJyy for edge sharing. Focussing on the ferromagnetic term Jzz, we notice the tremendous increase of this parameter from face to edge to corner sharing topology. Again concerning J^ it is also worth to note that exchange coupling between 6Aj(Mn2+) and the lowest emitting Kramers doublet of Yb3+ is not as much smaller compared to the coupling in the 6A](Mn2+)-Yb3+ ground state, as is the case for antiferromagnetic coupling.

4.2. Exchange coupling and its implications for the mechanism of the up-conversion of light in Mn +-Yb + dimers Electronic transitions accompanied by a spin-flip within a TM are spinforbidden, however they can gain intensity by exchange coupling to other ions. Focusing on a Mn2+-Yb3+ pair, we take Mn2+ in its ground state 6Ai and Yb in a fictitious spin Vz state to represent its ground state or some excited

Modelling of Anisotropic Exchange Coupling 179

doublet state. These states can couple with spins parallel or antiparallel to each other to give pair states with total spins S=3 and 2, respectively. The flip of one spin within the d5 configuration of Mn2+ site leads to *Ti and *T2 excited states. A parallel coupling between the S=3/2 spins of Mn2+ and that of s=l/2 of Yb3+ leads now to the same total spins S=2 as the one for the antiferromagnetically coupled S=5/2(6Ai) - s=l/2 pair; the transition between these two states corresponds formally to a 6Ai—»4Ti, ^ 2 excitation within a single nuclei Mn2+ complex; it is not spin-forbidden with respect to the cluster total spin (Fig.5). As has been shown by experiment 6 ' n and theory 5 ' 6 ' 7 , such transition can gain intensity by a mechanism which is not identical, but closely related to that of the exchange coupling between magnetic ions.

T l

l o r 2

S=3/2 / —5 C

Mn

,

S=l

^

\ S=2 \

Yb

Fig.5 Exchange mechanism of intensity gain for intra-configurational spin-forbidden transition at octahedral Mn2+ site in Mn2+-Yb3+ halide bridged dimers. In general terms, the interaction between the radiation field and a pair of paramagnetic ions a and b is described by the operator H' (Eq.52). E is the electric field, n^y are vector parameters describing the dipole moment of the transition between the singly occupied orbitals i and j on centers a and b, respectively; Saj and Sy are vector operators which act on the genuine spins of electrons on orbitals i ang j . We should note that the spin part of H' is identical with the

180 M. Atanasov, C. Daul and H. U. Giidel

H^SS^-EXs^)

(52)

iea jEb

one of the exchange operator Eq.3; indeed the latter equation can be rewritten from a form involving total spins Si and S2 for ions a and b into one involving contributions from separate orbitals:

H-'-SIW'-^)

(53)

tea jeb

It has been shown7 that the parameters n^bj equal the derivative of Jaibj with respect to E, i.e. naibj

~ (~^~)W

K

'

In other words, to consider the mechanism of pair-transitions implies to reconsider the mechanism of superexchange in the presence of an electric field. This leads to the following expression for n ^ 6 : 4(ai|p|bj)h(bj->ai) ^(uJplaiXaibjIJbju)^-^) | K**~ AE(bj->ai) & AE(ai->u) 2(ai|p|v)(vbj[|bjai)nv ti AE(v-^ai) [expressions obtained by interchanging ai and bj in the above] (55) Here p is the dipole moment vector, n^, the number of electrons in the fi,v-th orbital, AE(bj—»ai) the energy required to transfer one electronfrombj to ai, h(bj—»ai) - the transfer integral between bj and ai and {aibj\bj/j) the two-electron integral: (aibj\\bJu)=j\ai(l)bj(2)^-bj(l)/l(2)dTldT2

(56)

All summations over (j. and v are to be carried out over all the empty, singly and filled orbitals except for ai and bj. The operator H' and the n^i parameters defined by Eqs.52 and 55 are quite general and allow to obtain the intensity of any transition involving local excitations either of a or b or local excitations in which both electronic states on a and b change (see below). Appications of this formalism has been precluded due to the large number of model parameters and the lack of any practicable computational scheme to allow calculation of these quantities from fist principles. That is why, applications of the model has been confined only to qualitative interpretations of spectroscopic data6'13"18.

Modelling of Anisotropic Exchange Coupling 181

In what follows, we study in some detail the correlation between J]2 and the intensity of a pair transition. We take as a simple model example now a pair TM(d2)-RE(f1) (Fig.6). The S=l (triplet, T) and ^(doublet, D) states of TM and RE couple to yield pair states of total spin 3/2 (quartet, Q) and V2, with the wavefunction of the latter (Ms=l/2 component) given by:

¥ £ =^T(MS =l)D(ms = - l / 2 ) - ^ 7 W , =0)D(ms =1/2) =

(57)

We consider the transition from this state into the excited doublet state in which one spin of TM has flipped to give a singlet state (S) TM(S=0)RE(s=l/2); it is described by(Ms=l/2 component): V&=S(M, =0)D(ms=V2) = ^\d;d-r\-^\d;d;r\

tfl S=0

S=l/2

S=3/2 (^Q MS=l/2 S^ll S=l/2 / D T D

Fig.6 A simple model example of intensity enhancement of the spin-forbidden S=l—>S=0 (spin-flip) transition of a transition metal with two unpaired electrons(such as Ni2+ in octahedral coordination) and S=l/2 ion (such as Cu2+ or Yb3+(pseudo-spin >/,)).

(58)

182 M. Atanasov, C. Daul and H. U. Gildel

The two states interact with S=l/2 charge transfer states in which one electron has jumped from TM to RE, L/ 2+ / + /~|,L/, + / + /i,or from RE to TM, d*d-d*\, d*dXd' • This mixing is the cause both for the exchange 1

1

2|

1 2

2

stabilization of »p£ over the »?£, = d*d^f*

(described by the kinetic

exchange integral jkn ) and the intensity of the *F^, - » ¥ ^ , pair transition. As shown in Appendix A.3, using both perturbation theory and applying explicitly the operator H'(eq.52), the ^F^, - ^ ^ transition probability, is (neglecting higher order two-electron terms) proportional to the square of the matrix element (yfD \H\^) •

W*->-^^W^ + W^>]~

(59)

Let us compare this result with Eq.(60) for the exchange integral Jn:

Jk =-h

2[

!

,

*

]

h 2f

1

|

1

We see that in addition to the sum over squares of the hd2fand common to \ fofi, H^SD) '2 (hd2f.hd}f)

an( ^

*^i2 > t ' i e r e ^s

an

1 (60) hdlf,

interference term

which appears in the intensity expression. This can add or

subtract from the first one, to show that, in general, | (w°D //j*F^) 12 and J\2 will not be interrelated. However, as it frequently appears, one of the two terms, that for hd2f or hdxfy is small or zero and both | Nf^ M Y ^ , ) I2 and Jj2 are dominated just by one leading term. It then follows that for such cases the intensity of pair transitions will be proportional to J\2. Our results show, that this is the case for corner (see Table 5) and face sharing; in both cases, the exchange Mn2+-Yb3+ is governed by one single dominant contribution - the one between the d^ (Mn2+) - ti(O') f^ (Yb3+) orbitals: in the case of edge sharing several competing terms occur. In Fig.7 we have plot the Jkzz values for the complexes studied; we see that Mn2+-Yb3+ antiferromagnetic coupling is largest for corner sharing; it is tempting to relate

Modelling of Anisotropic Exchange Coupling 183

this result with the experimental finding, that up-conversion is most efficient in this case; the underlying electronic excitation involves a change of the electronic state created after the first absorption 6Ai(Mn2+)-2F7/2(Yb3+) - » 6Ai(Mn2+)-2F5/2(Yb3+) to a state where simultaneously Yb 3+ turns back into the ground state 2F5/2, while Mn 2+ (6Ai) becomes excited into the ''Tig state.

• Jzz A

y . y ^

\

MnCl7Yb

/\

-4 -

/ MhBrj^Ns/

-8 '

-12 -

MnCl.Yb

exc. state

\ \

MnClYb \

\

gr. state

\*

Fig.7 Antiferromagnetic J^ (kinetic) exchange coupling constants for Mn +-Yb3+ dimers with octahedrally coordinate Mn2+ and Yb3+ and three, two and one sharing halide atoms (face-, edge- and corner shared octahedra). Turning to the bilinear dependence of the intensity of pair transitions on terms of the form hlf,h\fwn& hlf.h2f we notice the following; integrals of

184 M. Atanasov, C. Daul and H. U. Giidel

TM-RE hopping, hxf and hlf, are expressed in terms of hdl and hfl metalligand hopping integrals as follows:

(61)

hdf~_htL==_±£==

J I.

\Q

t gr. state

- ^ZZ

12 -

/ /

g .

/ /MnClYb

4" -

7 exc.state

Jl x* o vu MnBr3Yb

y S

MnCl2Yb

MnCl3Yb Fig.8 Ferromagnetic J^ (potential) exchange coupling constants for Mn2+-Yb3+ dimers with octahedrally coordinate Mn2+ and Yb3+ and three, two and one sharing halide atoms (face-, edge- and corner shared octahedra).

Since coefficents cM and cfl (Eq.46) depend linearly on hdi and hji as well, one can expect that integrals of potentials exchange Jf2 and the transition probability | (w^ H^¥^ \2 will show-up again in a symbathic behaviour,

Modelling of Anisotropic Exchange Coupling 185

they both increase with h. Here cross terms, hlf .h2f with contributions from different ligand orbitals are automatically included (which is the missing terms in the comparison of | (*F£ I t f 1 ^ ) ^

with

^u)-

R is

striking, that

values of J?z plotted in Fig. 8 follow the same trend as the one found experimentally for the efficiency of light conversion; they increase from face to edge and corner sharing (cf with the ./,* plot, Fig 7) . We attribute this difference to mixed terms (h d l f . hdlf) which do not appear in J\2, but which are important in the case of edge sharing. The correlation does not change qualitatively when comparing exchange integrals for the ground and for the lowest emitting excited state of Yb3+ (see potential exchange terms in Table 6).

5. Conclusions 1) Exchange coupling between TM(Mn2+) and RE(Yb3+) has been studied experimentally with the aid of understanding the mechanism of up-conversion of light [3,4]. The analysis of the experimental data up to date has lead to the proposition that the phenomenon is governed by exchange coupling between the TM and RE. The modelling of exchange coupling between Mn2+ and Yb3+ lends support to this hypothesis; we find that both kinetic and potential contribution to the exchange coupling constant are largest for a corner sharing Mn2+-Yb3+ dimer. We further get the interesting result, that ferromagnetric exchange integrals exactly mirror trends born out by experiment: efficiency of up-conversion increases from face to edge to corner shared Mn-Yb ions. 2) We propose a procedure of calculating exchange coupling constants based on the following steps: (i) In a first step RE-TM transfer (hopping) integrals between orbitals carrying unpaired (magnetic) electrons on RE and TM are determined from a MO-calculation utilising the concept of orbital exchange pathway and effective Hamiltonian theory, (ii) In a second step, manyelectron wave-functions for ground or excited electronic states on RE and TM are constructed in which spin-orbit coupling (which dominates for the RE) is fully (variationally) taken into account, (iii) Finally, the data from steps 1 and 2 are combined to obtain numerical values for kinetic exchange integrals (within Anderson's super exchange theory) utilising a non-Heisenberg (orbital

186 M. Atanasov, C. Daul and H. U. Giidel

dependent) exchange operator. The MO's corresponding to the d(Mn) and f(Yb) magnetic orbitals have been used to also calculate ferromagnetic exchange integrals as well. Our method has been implemented at the level of an extended Huckel model (EHT), where a restricted number of parameters has been adjusted to fit spectra in high resolution of separate MnCl64" and YbC^3" chromophores. However, the procedure is quite general and can be used in combination with any method (DFT or ab-initio) which is able to yield reasonable account for the magnetic orbitals involved. In this respect we have to state, that available basis functions and/or functionals within the DFT methodology are not sufficient yet to do this. We motivate further work for improving DFT for rare earths.

Appendices Appendix 1 A

Derivation of the orbital operator (O) for exchange coupling in the case of orbital degeneracy. The orbital operator O, representing the exchange coupling tensor J within the 1=1 basis "J(U) J(l,0) J(l-l)" J = J(0,l) J(0,0) J(O,-l) (A.1.1) j ( - u ) j(-i,o) j ( - i - i ) can be expanded into a series of orbital operators l»ly,lz acting on angular momentum eigen functions for 1=1, mi=0, ±1 and the identity operator 1 as follows: 6 = alz2 +b(UE+Ux)+c(IyIz+lzly) + d(lx2-ly2) +e (Uy+M*) + f 1

(A. 1.2)

The coefficients a to f can be determined in the following way; we substitute the matrices of the operators 1*, ly and lz and 1 2 1 :

Modelling of Anisotropic Exchange Coupling 187

l

o -4= o V2

x

= —

0

4i

o ^ L o 4i

—

1= —

J i

y

o -^ o

V 2

0

r

—

l

n r

[i o ol

z

4 i

o 4= o

= 0 0 0

-, p oo

1 = 0 1 0

o o - i o

-^(b-ic) V2 f

i (A.1.3)

into eq (A. 1.2) to obtain, after collecting terms: a+f

o

d-ie

6 = 4=0> + ic) ^ ( - b + ic) < A - L4 ) V2 V2 d + ie -pr(-b-ic) a+f V2 The comparison with eq (A. 1.1) yields the following connection between the parameters a to f and the components of the exchange tensor: J(l,l)=J(-l,-l) = a+f; J(l,O)=J'(l,O)+iJ"(l,O)= 1 ( b _ i c ) =J( 0 ,l)*; J(l,-l)=J'(l,-l)+iJ"(l,-l)= (d-ie)=J(-l,l)*; J(O,-I)=J'(O,-l)+iJ"(O,-l)= - L (_b+ic)=J(-l,0)'; V2 We thus obtain

(A. 1.5) J(O,O)=f

a=J(l,l)-J(O,O); f= J(0,0); b= <2 J'(l,0); c= -<2 J"(l,0); d=J'(U-l); e=-J"(l,-l)

(A.1.6)

After substituting in eq.A.1.2 we obtain eq.14 in Section 2.2.1. From eqs A. 1.5 we derive the relations between the components of the anisotropic exchange tensor (eq. 15). Appendix 2 Spin-orbit coupling matrix elements for f-orbitals and their transformations Adopting the following standard order of real f-spin-orbitals:

188 M. Atanasov, C. Daul and H. U. Giidel

(1,2,3,4,5,6,7,8,9,10,11,12,13,14)= ((Psa^spAa^sP^a.nsp^a^p^a^p^ca^ep^ca^ep) A.2.1 We have the following non-zero spin-orbit matrix elements (in units of £, the spin-orbit coupling constant):

(1,4)= JI; (l,12)=-i JI; (l,13)=il; (2,3)=- JI ; (2,11)= -i (1(2,14)=- ii; V8

V8

2

\8

V8

2

(3,6)= JI; (3,10) =-i JI; (3,11)= i; (3,14)= -i JI; (4,5)=- JI; (4,9)=-iJI \8

V8

V8

V8

V8

(4,12)=-i; (4,13)= -i JI; (5,8)= -i J I ; (5,9)=il; (5,12)=-iJI; (6,7)=-i J I ; V8

V2

2

1/8

V2

(6,10)= -il; (6,ll)=-iJI; (7,10)= -JI; (8,9)= JI; (9,12)= -JI; (10,11)= 2

V8

V2

JI;(11,14)=-JI; (12,13)= JI

V2

V8

(A.2.2)

V8

V8 V8 In eqs (A.2.2) only matrix elements (ij) with i<j have been listed; the (j,i) matrix elements can be obtained taking the complex conjugate (i j)*. The spin-orbit coupling matrix elements (A.2.2) are set up in a real forbital basis. To convert this matrix into the representation of the t 2 (0", ±1"), ti(0',±l') and a2 basis (eq.36), a two-step transformation has been performed. We first take: (-3,-2,-1,0,1,2,3)= (
2 0 —\=

° 0

0

0

0

° i ° i °

0

0

1 0

o 4= o —1=

o

o

4=

-^=

0

_&

or in short: fi = fR.T,

V2

0

V2

4-

VI °

0

0

°

o o

o o o - ) = 0

0

(A.2.3)

0

0

—\=

V2. (A.2.4)

191 -Modelling of Anisotropic Exchange Coupling 189

followed by: (-1", 0",l", - r , 0 \ l ' , a2)=(-3,-2,-l,0,l,2,3)* ~-^= 2V2 0

0

0

0

-J=

0

0

0

0

0

0

Mr 2V2 0

4=

0

0

*

2V2

0 0

V2

-£= 1/2 0

0 ~^= 2V2 0 0

0 —j= A/2

0 - ^ L L

2V2

-J* 0 0 2V2 0 1 0

0

-^L

0

00 0 -JL*

0 0

0

(A.2.5)

2V2

0 - 4 = 0

V2 0

2V2

or f=fi.T2

A.2.6) leading 2 to ( e=f-.T (A.2.7) leading to the dfor the f-orbital part of T (eq.43) and diagonal matrix elements of 1 for(A.2.7) Tf=T!.T 2 orbital part.

Appendix 3 On the relationship between the transition probability for pair excitations and the contributions to the exchange coupling constant. Let us consider two ions (a) and (b) possessing 2 and 1 unpaired electrons on orbitals ai,a2 and b, respectively. The two electrons on (a) couple to give one triplet *Fy at lower energy and one singlet *F^ at higher energy, the energy separation between the two terms being twice the exchange coupling integral between the two electrons (Fig.6). The single electron on b gives rise to a doublet, *F^. The wave functions of these terms are given by:

190 M. Atanasov, C. Daul and H. U. Giidel

*»«£}

^=^(a!a;+a;a!)U

(A3.1)

v°=-j=(a;a-2-a;a;) We assume that the triplet state on (a) couples with the doublet one on (b) to give one quartet (Q) and one doublet (D) state, 4 ^ , and 4 ^ , , respectively, while *Fj and *F^ combine to a single doublet state, 4*^,; explicitly we have

KW,

=^) = ^a+2b-

- - L . - ^ ^ a " +a;a+2)b+

(A.3.2)

V&W, =±) = -j=(a;a-2 -a;a;)b+ In addition to these states, there are charge-transfer states resulting from ai—»b, a2—>b and b-»ai and b ^ a 2 excitations; their energies and wave functions are described by: ¥,(<*, -*b) = a;b+b- ; AE(a^b) T 2 (a 2 -> 6) = a\b"b-; AE(a2->b) ^ 3 ( A ^ a 1 ) = a1X«2+; AE(b->ai) (A.3.3) «P4(6->fl2) = a,+a*aJ; AE(b->a2) In what follows we derive dipole transition matrix elements for the ^TD ~* ^SD transition (corresponding to a spin-flip local excitation) and for the exchange coupling constant using to different approaches - perturbation theory and, alternatively applying the operators H' (eq.52) and HeXC (eq 53) on the wave functions of the exchange coupled pair states (eq A.3.2). A.3.1 Perturbation theory Here we account for the first order corrections to 4 ^ , and ^ ^ (eq A.3.2)due to their mixing with the charge transfer states (eq A.3.3) and use these corrected functions to get transition dipole matrix elements; second order corrections to the energy of «p^,, E(2)( »F£), allows to account for the

Modelling of Anisotropic Exchange Coupling 191

stabilization of »p£ over the »p£ and thus for the value of the exchange coupling constant. We use the well known expressions: £<2> _ y »

I Vmn 1

• xpd) _ y

Z^ r(0) _ £<<,)

»

^

Vmn t p(o)

p(0) _ r(o) i m

•

(A.3.4)

For the matrix elements Vnm we obtain:

{v%,\h\V2(a2 - » A ) ) = ^ | A . B ; (^|*|«P 2 ( fl2 ^b)) = -^=halb (^|Aj«P,(A ->a,)) = - ^ ( ^ 1*1^(6 -»a 2 )) = ^

1 6

2 i

i ( i £ 1^3(6->«,)) = - - ^ A . , ,

(A-3.5)

J (¥£|A|^ 4 (6 -»a 2 )) = - - ^ ^ 2 4

We thus get: ro

2AE(a]^b)

2AE(6-*a,)

2AE(a2^>b)

£(2)(^) = 0 and, since (S=3/2): S.JI2=E(S-1)-E(S);

Jl2

~ 3L

( TD)

(

ro)J"

_A2 [

albLAE(a,^b)

*

a2\hE(a2-*b)

2A£(i->a 2 ) (A.3.6)

AE(b^a,)_

1

1

(A.3.7)

AE(b^>a2)

First order corrections for *F^, and W^, are given by:

tf" =£ TD

fl

^

i2AE(ai-+b)

*(«, ->b)-M Kl

^

^(a2 ->6) +

' \2AE(a2^b)

*& ipp_>«,)_ I

^

K2

y(b_>a2)

(A.3.8)

192

M. Atanasov, C. Daul and H. U. Giidel

¥£(1) =4= +

—

¥(a, -»*>)+4=

—

V(a2 ->*>) +

1 K* »F0->fl.)+ - L ^ *P(6-»fl2) V2AE(6-+a,) ' V2A£(6^a 2 )

(A-3-9>

To calculate the transition dipole *Fj^—>*F^,, the following matrix element of the dipole operator p has to be evaluated:

n ( ¥£->¥£)= («?£•> |jP^|vpW)}+(^> 1^1^")

(A.3.10)

Direct substitution from eq.A.3.2(for the zero order wave functions ) and ¥£
(A.3.11) Here we have denoted the matrix elements of the dipole moment operator p for the a2-b and a r b pairs by p^b and paib, respectively. A.3.2 Effective Hamiltonian approach In the considered example, the operator H' takes the form: H'= where

7talb E.s al s b +7t a2b E.s a2 s b

' " ^ ' " [ ^ ( o , ->b)

K*2b=-4h2bPJ

1

(A.3.12)

AE(b-*a,)]

+

a2bya2b[AE(a2^b)

l-

1

(A.3.13)

AE(b^a2)\

When applying the operator H' on the zero-order order wave functions eq A.3.2 we substitute the saisb and s^Sb spin-operators making use of the Dirac permutation operator, which replaces SjSj by (1/2)(P;J-1/2), where Py interchanges spins of electrons occupying orbitals i and j . Here we omit the constant term -(1/4) (of relevance for diagonal matrix elements only): H

' ¥ £ = H'.j=(ata-2b+ -a^a+2b+) =

Modelling of Anisotropic Exchange Coupling 193

-l( 1 - J V JtalbE.Palb +7ta2bE.Pa2b).-Wfll+a2-&+ -araJA*)^[*, ! i I.( f l , + a ! -!> t )-^,£Ka 2 t A-) + f j 2 t i . ( a I t a 2 T ) - ^ I . ( a - i ! 2 t i t ) ] (A.3.14) (*£ \H\V&) = -J3XMJZ + V s ^ J f

(A.3.15)

This is identical with eq A.3.11 obtained using perturbation theory.

References 1. R.Valiente, O.S.Wenger and H.U.Gudel (2000) Chem.Phys.Lett. 320,639. 2. R.Valiente, O.S.Wenger and H.U.Gudel (200l)Phys.Rev.B 63,165102. 3. P.Gerner, O.S.Wenger, R.Valiente and H.U.Gudel (2001) Inorg.Chem. 40, 4534. 4. CReinhard, R.Valiente and H.U.Gudel (2002) J.Phys.Chem.B 106, 10051. 5. Y.Tanabe, LMoriya and S.Sugano, (1965) Phys.Rev.Lett. 15,1023. 6. J.Ferguson, HJ. Guggenheim and Y.Tanabe (1966) J.Phys.Soc.Japan 21, 692. 7. K.-I.Gondaira and Y. Tanabe (1966) LPhys.Soc. Japan 21, 1527. 8. N.Fuchikami and Y.Tanabe (1978) J.Phys.SocJapan 45, 1559. 9. N.Fuchikami and Y. Tanabe (1979) J.Phys.Soc.Japan 47,505. 10. H.Shigi and Y.Tanabe (1982) J.Phys.SocJapan 51, 1415. 11. J.Ferguson, H.J.Guggenheim and Y.Tanabe (1965) J.Appl.Phys. 36, 1046. 12. J.Ferguson, HJ.Guggenheim and Y.Tanabe (1965) Phys.Rev.Lett. 14, 737. 13. J.Ferguson, H.J.Guggenheim and Y.Tanabe (1966) J.Chem.Phys. 45, 1134. 14. J.Ferguson, HJ.Guggenheim and Y.Tanabe (1967) Phys.Rev. 161,207. 15. J.Ferguson, H.U.Gudel and E.R.Krausz, Mol.Phys. 30,1139. 16. H.U.Gudel (1975) Chem.Phys.Lett. 36, 328. 17. H.U.Gudel (1984) Comments Inorg.Chem. 3,189. 18. H.Riesen and H.U.Gudel (1987) MoLPhys. 60, 1221. 19. M.Tachiki (1968) J.Phys.SocJapan 25, 686.

194 M. Atanasov, C. Daul and H. U. Giidel

20. P.W.Anderson (1959) Phys.Rev. 115, 2 21.0.Kahn, Molecular Magnetism (VCH Publishers, Weinheim, 1993) 22. T.Oguchi (1965) J.Phys.Soc Japan 20, 2236. 23. J.Torrance, Jr. and M.Tinkham (1968) J.appl. Phys. 39, 822. 24. W.Urland (1976) Chem.Phys. 14, 393 25. R.W.Schwartz (1977) Inorg.Chem. 16, 1694. 26J.S.Griffith, The Theory of Transition-Metal Ions (Cambridge University press, Cambridge, 1971) 27.M.Atanasov, C.A.Daul and C.Rauzy (2003) Chem.Phys.Lett. 367, 737. 28.M.Atanasov, C.A.Daul and C.Rauzy (2003) Struct.and Bonding, 106, 97. 29. We made use of the Amsterdam Density Functional (ADF) program; E.LBaerends, D.E.Ellis and P.Ros (1973) Chem.Phys. 2,42.; P.M.Boerrigter, G.te Velde and E.J.Baerends (1988) IntJ.Qunatum Chem., 33, 87; G.te Velde and EJ.Baerends (1992) ComputPhys. 99, 84. 30.S.H.Vosko, L.Wilk and M.Nusair (1980) CanJ.Phys. 58, 1200. 31J.P.Perdew, J.A.Chevary, S.H.Vosko, KAJackson, M.R.Pederson, D.J.Singh and CFiolhais (1992) Phys.Rev.B, 46,6671. 32. M.Wolfsberg and L.Hehnholz (1952) J.Chem.Phys. 20, 827. 33. P.Alemany and R.Hoffmann (1993) J.Am.Chem.Soc. 115, 8290. 34. J.des Cloizeaux (1960) Nucl.Phys. 20, 321. 35. M.A.Aebersold, H.U.Gudel, A.Hauser, A.Furrer, H.Blank and R.Kahn (1993)Phys.Rev.B,48, 12723. 36.V.S.Mironov, L.F.Chibotaru and A.Ceulemans (2003) Phys.Rev. B, 67, 14424.

Chapter 6: Is a Dihydrogen Bond a Unique Phenomenon?

Slawomir J. Grabowskiab and Jerzy Leszczynskib a Department

of Crystallography and Crystal Chemistry, University of Lodz, 90-236-todz, ul.Pomorska 149/153, Poland

b Computational

Center for Molecular Structure and Interactions, Department of Chemistry, Jackson State University, Jackson, MS 39217, USA

Abstract The dihydrogen bonds (DHBs) are considered. The examples of such interactions existing in crystal structures are presented. These interactions are compared with conventional hydrogen bonds. The comparison shows that DHBs are often similar in nature to O-H...O and other typical H-bonds. One can observe for DHBs the elongation of the proton donating bond due to complexation and the correlations between geometrical, energetic and topological parameters. There are different types of DHBs, such as those with 7t-electron delocalization, similar to resonance assisted hydrogen bonds (RAHBs). The insight into the nature of DHBs shows that in principle the interaction energy decomposition scheme for them is similar to that one for conventional H-bonds showing that the electrostatic energy term is the most important attractive term. The case of o-bonds as the proton acceptors is also considered. The relationships derived from the bond valence model are in agreement with the experimental neutron diffraction results as well as with ab initio and DFT calculations. The spin-spin coupling constants for dihydrogen bonded systems are analyzed.

195

196 S. J. Grabowski and J. Leszczynski

1. Introduction For a long time the nature of hydrogen bonding has been the subject of many investigations.1'2'3 This is due to the importance of H-bond interactions in chemical, physical and bio-chemical processes, especially life processes.3'4'5 Among the most pronounced examples are catalysis processes, proton transfer reactions, enzymatic reactions, cell membrane transport processes and many others which take place in the gas phase, in liquids and in solids. In addition, this is an important interaction for phase transitions and the driving force in crystal engineering.6 The criteria for the existence of hydrogen bonds are not strictly defined because this interaction is understood as a very broad phenomenon.4'7 Early works on hydrogen bonds define them as interactions between X-H donating bonds and acceptor centers Y; hence, the H-bond is usually designated as X-H...Y where X and Y should be electronegative atoms such as O, N, Cl, etc.2 Hence early works also consider H-bonds as mainly electrostatic interactions where there is excess positive charge on the H-atom and excess negative charge on the X and Y atoms. In principle, there was no problem to detect hydrogen bonds if the geometries of the complexes or interatomic contacts were known. The situation was changed drastically when it was pointed out that the C-H bond may act as a proton donor. The CH...O and C-H...N bonds were analyzed by Suttor in the crystal organic structures.8 The subject of controversy in the 60s and 70s was the issue of whether or not such interactions fulfil the criteria for the existence of hydrogen bonding. However it was commonly accepted that there are such Hbonds after the appearance of the work of Taylor and Kennard.9 The authors have performed refined statistical analyses justifying the existence of C-H... Y H-bonds in crystals. Afterwards also the other kinds of H-bonds were analysed and those with C-atom acceptors (X-H...C) or 7t-electron acceptors (X-H...71).4 Even the C-H...C hydrogen bonds were detected, for example in one of the polymorphic forms of acetylene crystals (with the Acam space group).10 Accorging to Jeffrey,5 H-bonds may be classified in the following way: those for which H-bond energies are about 15-40 kcal/mol are known as strong ones; the range of 4-15 kcal/mol is for moderate H-bonds and 1-4 kcal/mol for weak ones. The H-bonds mentioned here with C-donors or/and C-acceptors usually belong to the last group. However there are exceptions; for example the C-H...C interactions of moderate strength were analyzed theoretically, and the binding energy for H3N+-CH2'...HCCH at the MP2/6311++G(3d,3p) level of theory (BSSE included) was calculated to be 8.16 kcal/mol.11 One can mention the other kinds of unconventional H-bonds

Is a Dihydrogen Bond a Unique Phenomenon?

197

which are recently the subject of investigations: for example, blue-shifting Hbonds for which the blue shift of the stretching mode for the donating bond is observed.12 For conventional H-bonds the stretching vibration of the X-H proton donating bond changes significantly upon complexation with the acceptor molecule. The stretching mode can be shifted to the red of the spectrum by hundreds of cm"1 due to the X-H bond elongation, and the band is intensified several fold. The red shift may be treated as a measure of the hydrogen bonding strength since it correlates with the other typical descriptors such as H-bond energy as well as the proton - acceptor (H...Y) distance.13 Hobza and Havlas have studied intensively the reverse situation for which the shortening of the X-H bond is observed due to complexation. It is accompanied by a shift to the blue of the stretching mode and most often by a decrease in the intensity of the band. Such interactions with mainly C-H bonds as donors were named as "improper" or "inverse" hydrogen bonds, but currently the name blue-shifting H-bonds is commonly accepted.12 In the middle of the 90s, the X-H...H-Y bond (another kind of H-bond interaction) was reported and named as a dihydrogen bond since for such a system there is the typical conventional X-H donor, and another hydrogen atom with excess negative charge which acts as a proton acceptor and is attached to the heavier non-hydrogen atom Y.14 Recently X-H....H-Y interactions have been the subject of an increasing number of studies, both experimental as well as theoretical.15 There are few reviews of dihydrogen bonds; however, they mainly consider these interactions from an experimental point of view.15'1647 The aim of this chapter is to briefly summarise the theoretical studies of different kinds of dihydrogen bonds (DHBs) and the nature of such interactions. It seems that deeper insight into the details of DHBs will be useful for the analysis of a number of chemical and biochemical reactions.18 It was pointed out that DHBs play an important role in many biochemical processes, similar to the involvement of typical conventional X-H...Y bonds.18

2. First examples of dihydrogen bonds The first examples of the existence of dihydrogen bonds (DHBs) were described in detail by Crabtree and co-workers.16 Two such examples which prove that M-H bonds (M-metal) may act as proton acceptors were reported in 1990.19'20 The results of neutron diffraction measurements of the structure of [Ir-(PMe)3(H)(OH)]+ indicate an H...H distance of 2.4 A between the OH

198 S. J. Grabowski and J. Leszczynski

proton and the Ir-H hydride, and the authors have claimed that this probably represents a weak dipole-dipole interaction.19 The neutron diffraction crystal structure of [Fe(H)2(H2)(PEt2-Ph)4] displays the molecular hydrogen acting as a ligand and closely attached to the Fe-H group. This Fe-H...H2 interaction was termed as a "cis-effect,"20 but Morris and co-workers claimed that this may be considered to be intramolecular hydrogen bonding.21 In the middle of the 90s two groups, Morris and co-workers 21'22 and Crabtree's group,23'24'25'26 reported several examples of the metal organic crystal structures containing DHBs. For the structure of [IrH2(Me2CO)2(PPh3)2]+ the syn-orientation of the iminol OH was found which allows the OH proton to closely approach one of the iridium hydrides. This accounts for coupling between the OH and IrH protons (JHH = 2-4 Hz).23 Spectroscopic studies confirm direct H...H interactions since the v(OH) IR band of the iminol group is in the range 3310-3409 cm"1 which corresponds to lower energy than expected for free iminol OH. Similarly, the ! H NMR spectrum confirms the attractive H.. .H intramolecular interction. The cases of intermolecular DHBs were also studied in the middle of the 90s; for example the neutron diffraction solid state studies of the ReH5(PPh3)3 • indole complex14 reveal N-H...H-Re bifurcated intermolecular interactions. Fig. 1 presents the geometrical details of this type of H-bond.

/H\1.734(8)

l 6 8 3 X 1 1 8.i)\ 1 6 3 - 1 ( 6 ) /H

RB

\

/

^N 1.007(6)

\97.2(3)/ 1.683(6)\ / 2212 ( 9 ) H Fig. 1. The geometrical parameters of Re-H...H-N interactions obtained from neutron diffraction measurements, distances in A, angles in degrees.

Is a Dihydrogen Bond a Unique Phenomenon?

199

One can see that one of the H...H contacts is significantly shorter (1.73 A) than the corresponding sum of van der Waals radii, and the N-H.. .H angle for this interaction is equal to 163°, close to linearity as for the conventional intermolecular H-bonds. Fig. 2 shows the packing of molecules for another crystal structure ([ReH5(PPh3)2(imidazole)]) for which intermolecular bifurcated DHBs were detected.27 Spectacular evidence for the existence of DHBs was presented for boraneamine (H3BNH3) by Crabtree and co-workers.16 A comparison of melting points for H3BNH3 (104°C) and isoelectronic species C2H6 (-181°C) show that stronger interactions, not typical van der Waals interactions, should occur for boraneamine. H3BNH3 is a polar compound which may explain such a situation; but for non H-bonding CH3F species isoelectronic with ethane, the melting temperature amounts to -141 °C. The lack of lone electron pairs for H3BNH3 suggests the existence of DHBs. It is worth mentioning that the crystal structure of H3BNH3 was determined by the neutron diffraction method, and the existence of H...H interactions was confirmed.28 Fig. 3 displays the closest contacts between three neighbouring boraneamines. As one can see there are intermolecular bifurcated dihydrogen bonds for this crystal structure. The two H...H contacts are symmetrically equivalent; the H.. .H distance is equal to 2.02(3) A.

H2A

Fig. 2. The neighbouring molecules of [ReH5(PPh3)2(imidazole)] in the X-ray crystal structure. Reprinted with permission from ref 27. © 1995 Royal Society of Chemistry. In the initial studies of DHBs the complexes for which X-H...H-M interactions exist were considered. Hence it was pointed out that DHBs occur

200 5. J. Grabowski and J. Leszczynski

if there is the typical X-H proton donor, such as O-H or N-H, where typically there is excess positive charge on the hydrogen atom. H-M is an accepting group (M designates transition metal or boron), and the H-atom is negatively charged. However, further studies have shown that other kinds of dihydrogen bonds may exist, such as C-H...H-M (M-transition metal) 29 and even CH...H-C.30 The term "dihydrogen bond" was proposed for X-H...H-Y interactions in 1995 by Crabtree and co-workers.31 The authors of the article stated that "a referee points out that these unconventional hydrogen bonds require a new name, for which we suggest the term 'dihydrogen bond.' " This term is currently commonly accepted. Alkorta and co-workers7 claim that the "protic" or "hydride" nomenclature may be applied for hydrogen bonds. There are protic X-H^.-.Y HBs, hydric X-H^.-.Y HBs (designated by Alkorta and co-workers as inverse hydrogen bonds) and protic-hydric X-H1"8 ...'^-Y DHBs. The authors pointed out that protic HBs cover more than 99.9% of the studies of H-bonding, but the situation will change in the future since unconventional hydrogen bonds, in spite of being weak, play an important role in some cases in many chemical and biological processes. m

% I

Fig. 3. The closest N-H.. .H-B contacts for the neutron diffraction crystal structure of boraneamine. Reprinted with permission from ref 28. © 1999 American Chemical Society.

Is a Dihydrogen Bond a Unique Phenomenon?

201

3. First theoretical studies of DHBs The first theoretical calculations of dihydrogen-bonded systems were performed by Liu and Hoffmann.32 The authors considered the structure of [Ir{H(Ti1-SC5H4NH)(PCy3)}2]+ [BF4]" (where Cy = cyclohexyl) synthesized and experimentally investigated by Morris and co-workers.22 To simplify calculations Liu and Hoffman neglected the counter ion [BF4]' as not influencing the region with dihydrogen bonds. They also replaced the PCy3 group with PH3 and slightly changed the geometry to finally obtain the [Ir{H(Ti1-SC5H4NH)(PH3)}2]+ ion with C2v symmetry. The authors have applied the extended Huckel (EH) method in their study.33 The geometrical parameters for the calculations have been taken from crystallographic data since the EH method is not suitable for geometry optimizations. The considered model ion has two intramolecular symmetry equaivalent N-H.. .HIr dihydrogen bonds. The H...H distances are equal to 1.75 A, less than the corresponding sum of van der Waals radii. The calculated Mulliken net atomic charges amount to +0.22 and -0.27 for the H(N) and H(Ir) atoms, respectively, showing that an attractive electrostatic interaction should appear. The authors claimed that a positive Mulliken overlap population (OP) of 0.016 also indicates stabilization of the H.. .H interaction. Liu and Hoffman32 have also considered two model complexes (FH.. .HLi and FH...HMn(CO)5) using the RHF method. The first structure after the optimization (the 6-3 lG(d) basis set was used) has an H.. .H distance of 1.658 A with Cs symmetry and nearly linear F-H...H and Li-H...H angles. The binding energy for this complex without BSSE calculations is equal to 9.3 kcal/mol. An approximation of the electrostatic contribution using Mulliken atomic charges allowed the authors to obtain the electrostatic term of 6.7 kcal/mol. Hence they claimed that this is mainly an electrostatic interaction as observed for the other typical H-bonds. For the second FH.. .HMn(CO)5 complex calculated at the RHF/3-21G(d) level of theory, a minimum energy structure was found for the H.. .H distance of 1.683 A and the binding energy of 6.6 kcal/mol. The FH and H-Mn bonds were elongated due to complexation. Taking into account the features of these two model complexes discussed here, Liu and Hoffman claimed that they are similar in nature to the other H-bonds. Almost at the same time as the appearance of the work of Liu and Hoffman, a theoretical work of Crabtree and co-workers31 was published. In this study the H3BNH3 dimer was investigated. The authors studied the cyclic dimer of C2 symmetry with equivalent B-H.. .H-N bonds. The PCI-80/B3LYP method was applied, and the following geometrical parameters were found for

202 S. J. Grabowski and J. Leszczynski

dihydrogen bonds: the H.. .H contact is equal to 1.82 A; the N-H.. .H angle is 158.7°; and B-H..H is 98.8°. It is worth mentioning that in the scope of the PCI-80/B3LYP method the geometries were obtained after optimization with the B3LYP functional34 and the standard double X, plus polarization basis sets. The energies for the optimized structures were predicted with the use of the PCI-80 parametrized method35 which gives greatly improved results compared to unparametrized methods. The binding energy calculated for the H3BNH3 dimer amounts to 12.1 kcal/mol which corresponds to 6.1 kcal/mol for each dihydrogen bond. The Mulliken net atomic charges are equal to +0.27e and -0.09e for the H(N) and H(B) atoms, respectively. The authors claim that the existence of opposite charges at the hydrogen atoms of the H...H contact indicates that this interaction may be classified as hydrogen bonding. The same conclusion may be derived from the H...H contact energy which corresponds to the range of the typical H-bond energies (3-8 kcal/mol). The angles of the B-H...H-N system are in agreement with those found from the Cambridge Structural Database (CSD).36 Crabtree and co-workers searched CSD for B-H...H-N contacts and found 26 such species for which the H.. .H distance is less than 2.2 A. This search filter was applied since the authors expected that such a value (less than the corresponding sum of van der Waals radii at 2.4 A) is evidence of the attractive interaction. It is of course only a rough assumption since the crystal packing may be constrained more than the sum of van der Waals radii interatomic contacts. It was found that the N-H...H(B) angles vary (an average of 149°, a range of 117-171°, o=17°); they are larger than the B-H.. .H(N) angles (an average of 120°, a range of 90-171°, o=26°). Hence the results for the H3BNH3 dimer are in line with the tendencies found for the crystal structure complexes. The authors claim that the strong bending of the B-H...H(N) angle is evidence that o-bonds are acceptors of protons, not Hatoms with a negative net charge. In further studies37 they explain that the hydrogen bond concept has been extended in recent years from the typical XH.ione pair interaction to X-H...71 electron interactions, and the last case involves the interaction between the proton donating X-H bond and o-bond acceptor, the X-H...G interaction commonly known as the dihydrogen bond. Such a concept was also suggested by Cramer and Gladfelter.38 Theoretical calculations up to the MP2/6-31G(d,p) level with the BSSE correction (counterpoise method of Boys and Bernardi39) were carried out on a series of A-H...H-X dihydrogen bonds (A= B, Li, Be; X= N, C).40 It was found that the considered complexes fulfilled the commonly accepted criteria for the existence of hydrogen bonds. The systems studied were BH4...HCN,

Is a Dihydrogen Bond a Unique Phenomenon?

203

BH4...CH4, LiH...NH4+, LiH...HCN, LiH...HCCH,BeH2... NH,+, BeH2...HCN and CH4... NHt*. It is worth mentioning that Alkorta and co-workers were the first to apply the Bader theory41 to analyse dihydrogen bonded systems. Systematic studies of simple dihydrogen-bonded complexes have been performed by Remko42 who considered electron-rich hydrogen atoms as proton acceptors (for LiH and BeH2 molecules) and as proton donors with positive H-atoms: HF, HCN, and HCCH. For these complexes full geometry optimization was performed at the MP2(full)/6-31G(d) level. Such obtained geometries were used to carry out the Gl and G2 energetic calculations. Generally, the obtained results show that the dihydrogen bonds are energetically similar to the typical, conventional H-bonds. The H...H distances are in the range of 1.575 A (for LiH...HF) to 2.271 A (for H-BeH...HF). Orlova and Scheiner have studied43 intermolecular H...H interactions between HF, H2O and H3O+ donating species and metalloorganic compounds such as Mo(H)(CO)2(L)2(L')2 (where cis-ligand L=PH3 and NH3, while L' trans-ligand =NO,C1,H) and W(H)(CO)2(NO)(PH3)2 using the HF and DFT methods. The results show that the energy of the H...H contact for these complexes surpasses 10 kcal/mol. The same authors investigated the specific role of the H...H bonded complexes in reactions of transition metal hydrides with the proton donors. The ruthenium and rhenium compounds were considered, and the calculations were performed at the DFT level using nonlocal correction. The B3PW91 and LANL2DZ basis sets were used.44 Rozas and co-workers have investigated the problem of the influence of field effects on dihydrogen bonded systems.18 They found, using the HF/631G(d,p) level of theory, that a coordinated H2 molecule within the Li+...H2...F" system may react yielding the dihydrogen bonded system LiH...H-F. And a dihydrogen bonded complex may yield two molecules coordinated to the H2 molecule as in the case of the following reaction: H3lSrH...H-BH3" = NH3...H2...BH3. The other early calculations of dihydrogen-bonded complexes include particularly small systems where acceptor molecules involve main group elements, eg., LiH, BeH2, BH3 and A1H3.3M5'46'47

4. Geometrical and energetic features of DHBs The equilibrium geometry of molecules may be treated as a source of chemical information. Especially, geometrical parameters describe the nature of intra- and intermolecular interactions.3'4 Some of them are sensitive to

204 S. J. Grabowski and J. Leszczynski

environmental effect; terminal bonds are usually elongated due to the H-bond formation. For example this was observed for O-H...O=C systems.48 For the neutron diffraction results taken from the Cambridge Structural Database,36'49 it was revealed that not only the proton donating bonds but also the accepting C=O bonds are elongated.48 The statistical approach has showed that for shorter H...O contacts the elongation of O-H and C=O bonds is greater. The proton donating bond elongation is usually considered as strong evidence of the existence of hydrogen bonding3'4 and it was many times detected for different types of hydrogen bonds. There are also good correlations between the H...Y distance and the X-H proton donating bond length for X-H...Y Hbonds.3'5 Such correlations were observed for O-H...O systems,50'51 for NH...O, N-H...N52 and even for species with C-H...O.53 The only criterion that should be fulfilled for the investigated samples is that they must consist of related compounds; in other words the samples have to be homogeneous.54 Fig. 4 shows the relationship between the H...O distance and the O-H bond length for O-H...O bonds taken from accurate crystal structure neutron diffraction results.55 The open circles relate to the O-H...O systems, while the full ones correspond to deuterated crystals (O-D...O systems). The solid line represents the relationship between H...0 and O-H obtained from the Bond Valence Model (BV model - see Appendix). It is worth mentioning that Fig. 4 shows the correlation obtained for the sample of O-H...O systems taken from different crystal structures, and in spite of the heterogenity of the sample the relationship is fulfilled. Similar relationships for O-H...O species were obtained from the results of ab initio and DFT calculations.55'56 The presented dependencies are well fulfilled for typical H-bonds. For example the elongation of the proton donating bond is not a criterion for the existence of such interactions for blueshifting H-bonds for which the shortening of the X-H bond is observed due to the H-bond formation.12 Similarities between dihydrogen bonds and conventional O-H.. .O bonds were investigated,57 and a relationship between the elongation of the proton donating bond and the H...H distance was found for samples of simple complexes (Fig. 5). The following samples were taken into account. The first one consists of XH4..HF complexes (where X designates the elements of the fourth group of the periodic table of elements: C, Si, Ge, Sn and hydrogen fluoride was chosen as relatively strong proton donor). The CrH2...HF complex is also included in the first sample to show the proton accepting abilities of transition metal hydrate since the first crystal structures for which DHBs were observed represent metal organic compounds of transition metals. This sample is designated by circles in Fig. 5, and the related complexes were

Is a Dihydrogen Bond a Unique Phenomenon?

205

calculated within the B3LYP/Lanl2DZ level of theory. For two remaining samples there is the same proton donor molecule (HF) and the following acceptors: CH4, S1H4, BeH2, MgH2, LiH and NaH. For one of the samples the optimization of complexes was performed at the MP2/6-31G(d,p) level of theory (triangles in Fig. 5); for another sample optimization was performed at the MP2/6-311++G(d,p) level (squares). All calculations were carried out with the use of the Gaussian98 suite of programs.58 One can see good correlation between the H...H distance and the proton donating bond length, similarly as for typical H-bonds. The greatest elongation of the H-F bond and the shortest H...H contact for the first sample is revealed for the CrH2...HF complex; for the remaining samples the greatest elongations occur for LiH...HF and NaH...HF. Fig. 5 also displays the relationship between the H...H distance and the elongation of the H-F bond obtained from the Bond Valence model (continuous line - see Appendix). The results of a simple BV model are in good agreement with the MP2 and DFT calculations. The presented analysis of DHBs was based on the samples for which there is the same proton donor (hydrogen fluoride molecule) and different acceptors. , i

1.26 -j

o

°#

|

1.020.94-I 1.1

i

°^L 1 1.4

1 1.7

1 2

i 2.3

H..0 distance

Fig. 4. The relationship between the H...O distance and the O-H bond length (both parameters in A). Reprinted with permission from ref 55. © 2000 Elsevier Science B.V.

206 S. J. Grabowski and J. Leszczynski 2.8 I

— — —

~

2.4 -1

i

\ 1.6-

*

1.2 J

0

^ ^

•

1

•

0.01

0.02

0.03

•

1

~

0.04

0.05

elongation of H-F Fig. 5. The dependence between the elongation of the H-F bond and the H...H distance (both values in A). Reprinted with permission from ref 57. © 1999 Elsevier Science B.V.

A similar analysis based on MP2 results of DHBs was performed for the sample where there is the same proton acceptor (the BeH2 molecule), and there are different donors such as HF, HC1, HBr, HCN, HCCH, HNNN, HBF2 and HOF.59 For such a sample, there is no correlation either between the elongation of the Be-H donating bond and the H.. .H distance or between this elongation and the H-bond energy. Practically the Be-H bond length is constant; there is only one exception: the BeH2.. .HOF complex for which the H-bond is strongest within the sample (4.8 kcal/mol) and for which a slight elongation of Be-H is observed due to the H-bond formation (of about 0.03

A).

The following geometrical criteria for the existence of hydrogen bonds are commonly applied and accepted by crystallographers:9 the distance between the proton and the acceptor atom is shorter than the sum of their van der Waals radii;

Is a Dihydrogen Bond a Unique Phenomenon?

207

the conformation of the X-H... Y system should be close to linearity, at least for intermolecular H-bonds because for intramolecular H-bonds the strain effects may force greater deviations from linearity; commonly the systems for which the X-H.. .Y angles are less than 90° are not considered as Hbonds; - sometimes the next geometrical criterion is included which requires that the proton donating bond is elongated due to the H-bond formation.3'53 However these criteria are controversial since the hydrogen bonding interaction is mainly electrostatic, and there is the electrostatic attraction far beyond the van der Waals cut-off.4'51 It is also difficult to estimate accurate positions of hydrogen atoms from X-ray crystal structure measurements. Hence the neutron diffraction results should be applied for a geometrical analysis of H-bonds in crystals.3 However the number of neutron diffraction crystal structures collected in CSD is less than 0.5% (about 1000 crystal structures)60 of the whole database (the remaining are X-ray crystal structures), and sometimes for the statistical analysis of a required class of compounds there is not a sufficient number of neutron diffraction results. For the weaker C-H...Y bonds the elongation of the proton donating bond is not detectable or is negligible.60 Additionally, as was mentioned before, there are blue-shifting H-bonds for which the third criterion cannot be applied. Recently few crystal structures were analyzed where there are repulsive interactions existing due to crystal packing and fulfilling the H-bond geometrical criteria.61 In spite of the reservations given above these criteria may be applied for DHBs shown in Fig. 5. These ab initio results accurately reveal the H-atom positions, and elongation of HF bonds is detectable. For some cases the H.. .H distance is much smaller than the sum of van der Waals radii. For example for the NaH...HF complex the H...H distance is equal to 1.371 A, and the H-F bond length amounts to 0.958 A (the MP2/6-311++G(d,p) level), while for HF not involved in H-bond interactions the bond length is equal to 0.917 A. The analyzed systems have features typical of other conventional H-bonds, a meaningful difference between the energy of the complex on one hand and the energies of isolated monomers on the other hand. The binding energy is often treated as the H-bond energy of intermolecular connections. It amounts to -13.8 and -12.6 kcal/mol for the NaH...HF and LiR.HF complexes (the MP2-6-311++G(d,p) level). One can see that selected DHBs are stronger than the typical conventional H-bond of the linear-trans water dimer (about -5 kcal/mol). And for all DHB systems presented here there are the opposite net

-

208 S. J. Grabowski and J. Leszczynski

atomic charges on connected H-atoms, positive on H(F) and negative on the second H-atom. The only exception is the CH4...HF dimer for which both charges are positive and which is not energetically stable; the binding energy is positive (repulsive) at the MP2/6-311++G(d,p) level (0.39 kcal/mol). All XH.. .H systems included in Fig. 5 are linear or are close to linearity. The relationships mentioned above were confirmed by the use of high level ab initio calculations performed at the MP4(SDQ)/6-311++G(d,p), OCISD/6-311++G(d,p) and QCISD(T)/6-311++G(d,p) levels of theory.62 For example, the H-bond energy of the NaH...HF complex calculated at the latter level of theory amounts to -12.66 kcal/mol (correction for BSSE included), and the corresponding H.. .H distance is equal to 1.426 A. In this high level ab initio study the Bader theory41 was also applied to analyse DHBs (see next sections).

5. Application of the "atoms in molecules" theory to study DHBs The electron density, p(r), defines a structure as well as ground-state properties of many-electron systems.63 The topological "atoms in molecules" (AIM) theory of Bader41 allows quantitative structural information from electron density to be obtained. The AIM theory is often applied to locate and characterize critical points. The following relation, which requires that the gradient of p(r) vanish, stands for the definition of the critical points (CPs): (1)

Vp(rJ = 0

CPs are classified according to the number of negative eigenvalues of the matrix of partial second derivatives of p with respect to {x,y,z} (Hessian matrix). h

=

_a^n

( r

)

P.q=x,y,z

(2)

There are four types of stable critical points: maxima in p(r) correspond to attractors usually attributed to nuclei, and minima correspond to cage, bond or ring critical points. The characteristics of bond critical points (BCPs) are often used in studies of inter-atomic interactions. For example, the negative value of Laplacian for BCP means that there is an increase in electron density in the inter-atomic region; such a situation corresponds to the covalent bond. The

Is a Dihydrogen Bond a Unique Phenomenon?

209

positive Laplacian value shows that there is the depletion of the electron density, and this corresponds to the interaction of closed-shell systems: ionic interaction, hydrogen bonding, and the van der Waals interaction.64 The characteristics of the bond critical points may be useful to estimate the H-bond strength. ' ' The correlations between different topological parameters and H-bond energy were found for various samples of complexes. It is worth mentioning that the AIM theory provides new measures of H-bond strength. From a geometrical point of view the proton.. ..acceptor distance is a rough descriptor of H-bond strength, although it does not always correlate with H-bond energy.3 The elongation of the proton donating bond may also reveal the H-bond strength, and for stronger H-bonds the X-H...Y system should be linear. There are three geometrical parameters often used by crystallographers; however, they are not accurate indicators of H-bond strength. The characteristics of BCPs such as the electron density at H...Y (Y designates acceptor) BCP (PH...Y), its Laplacian (V2pH...vX the electron density at the proton donating bond length (PXH) and its Laplacian (V2pXH) are often applied to characterize H-bond interactions. All of these descriptors reveal the H-bond strength and correlate with the H-bond energy if the homogeneous sample of complexes is taken into account.54 One can observe that for typical H-bonds there is the elongation of the proton donating bond due to complexation; the pXH value decreases in comparison with the free donating molecule; V2pXH increases. For stronger H-bonds H...Y contacts are shorter, and hence the PH...Y and V2pH...y values are greater.68 Similar observations were revealed for dihydrogen bonds. Fig. 6 shows the relief map of the electron density for the Li-H...H-F system. One can see the differences between the electron densities of hydrogen atoms. For both of them the densities are much lower than those of the F and Li atoms. However, the hydrogen atom connected with Li is separated from Li (left side of the figure), while the second hydrogen atom is strongly connected with the fluorine atom (right side of the figure).

210 S. J. Grabowski and J. Leszczynski

Fig. 6. The relief map of the electron density of the Li-H...H-F complex; the large "mounds" correspond to the Li (left) and F (right) attractors, small ones to hydrogen atoms. The first rough correlation between H-bond energy and pH...y for DHBs was found by Alkorta and co-workers (Y in such a case denotes an H-atom acceptor). Table 1 presents the topological parameters obtained in subsequent studies from MP2/6-311++G(d,p) wave functions for homogeneous samples of dihydrogen bonded complexes.57'62 For all DHB complexes hydrogen fluoride represents the proton donating molecule. Hence other systems with an HF donor, with conventional H-bonds and calculated at the same level of theory, are included in Table 1 for comparison. The translinear conformation of the water dimer is also included as the most often applied reference system in studies of hydrogen bonding.69

Is a Dihydrogen Bond a Unique Phenomenon? 211

Table 1. The topological parameters (in au) and H-bond energies (EHB, corrected for BSSE and in kcal/mol) for dihydrogen-bonded systems and for other conventional Hbonds at the MP2/6-311++G(d,p) level of theory. For all cases (except for the translinear dimer of water) the hydrogen fluoride molecule is a donor. pXH

V 2 p XH

PH...Y

V 2 p H ... Y

EHB

0.323

-2.323

0.042

0.057

-12.62

NaHa

0.314

-2.205

0.046

0.048

-13.81

BeH2a

0.361

-2.763

0.016

0.049

3!94

MgH 2a

0.348

-2.624

0.026

0.060

^02

CH/

0.370

-2.830

0.002

0.005

+0.39

SiH,"

0.367

-2.810

0.009

0.028

^085

CH 2 O a a

0.359

-2.787

0.022

0.107

-5.43

H2Oaa

0.347

-2.652

0.037

0.142

^54

NH 3 a a

0.325

-2.365

0.050

0.120

-11.18

F&aa

0.174

-0.349

0.174

-0.349

-39.87

H 2 O" & * aa

0.356

-2.512

0.023

0.091

A45

Acceptor LiH

a

&(F...H...F)"

system liner-trans water dimer arefs 57,59 aaref68 &&

The results presented in Table 1 show the tendencies discussed above that the topological parameters correlate with H-bond energy. For the DHB complexes the linear correlation coefficient between P H . H and H-bond energy is equal to 0.995, while it decreases to 0.991 for all systems presented in the table. This indicates that P H . Y topological parameter may be applied not only for homogeneous samples but also for heterogeneous samples. This was also revealed for the other H-bonded systems. Fig. 7 shows the details of the aforementioned relationships.

212

S. J. Grabowski and J. Leszczynski

5-

.

0*b

c 0 * n 0.05 -5<*o |-15-

,

,

,

0.1

0.15

0.2

*

•S - 2 0 | X

-25

-30 -35-40-

O

-45electron density at H...Y BCP

Fig. 7. The relationship between PH...Y and the H-bond energy for the complexes collected in Table 1: foil circles correspond to DHBs, while empty circles correspond to the other conventional H-bonds. There is also good linear correlation for DHBs between PXH and EHB (R, the linear correlation coefficient, amounts to 0.997). For the relation between V2pXH and EHB, R is equal to 0.991; only the correlation between V2pH...y and EHB is poor (R = 0.687). Similar relationships between parameters describing DHB systems were observed in other studies. For example, for the neutral and charged dihydrogen-bonded complexes with the LiH molecule as a proton acceptor, good correlations between H-bond energy and H...H distance and H-bond energy vs. PH...H were found.71 The authors also observed that there is a slight shortening of the Li-H proton accepting bond for neutral complexes, while for positive charged complexes there is the elongation of the Li-H bond. A detailed analysis of the (BH3NH3)2 dimer based on the AIM theory was performed by Popelier.72 The optimization of the geometry of the dimer was carried out at the HF and MP2 levels of theory using the 6-31G(d,p) basis set. In this case a topological analysis based on appropriate wave functions was performed using the program MORPHY97.73 The positions of the critical

Is a Dihydrogen Bond a Unique Phenomenon?

213

points were found using the eigenvector method.74 There are three B-H...H-N contacts for the optimized (BH3NH3)2 dimer; because of its symmetry (Cs symmetry) two of them are equivalent due to the presence of a mirror plane. The geometrical parameters of these DHB connections for both levels of theory are presented in Table 2. One can see that the N-H...H-B connections of the (BH3NH3)2 dimer roughly fulfil the geometrical criteria for the existence of H-bonds; H...H distances are less than the corresponding sum of van der Waals radii; N-H and B-H bonds are elongated due to complexation; N-H...H and B-H...H angles are greater than 90°. Also the latter angles are smaller than N-H...H angles which is in accordance with the results known for crystal structures where DHBs exist. Table 2. The geometrical parameters of B-H...H-N connections (in A and degrees) for the (BH3NH3)2 dimer (resultsfromref. 72). Parameter

First N-H...H-B bond

I

HF/6-31G(d,p) I MP2/6-31G(d,p)

&

Second N-H...H-B bond & HF/6-31G(d,p)

I MP2/6-3 lG(d,p)

H...H

L914

1/726

2.324

2.149

N-H...H

1631

168?7

128/7

BOO

B-H...H

144.6

139.2

113.8

112.6

AN-H &&

0.004

0.008

0.001

0.002

AB-H**

0.011

0.011

0.007

0.007

there are two such symmetrically equivalent systems the elongation of the bond due to complexation

44

For the (BH3NH3)2 complex Popelier has also investigated the topological criteria for the existence of hydrogen bonds72 which were proposed earlier.75 There are eight topological criteria based on the AIM theory: - there is the bond critical point and the bond path for the H...Y interaction; - the electron density at this BCP ranges from 0.002 to 0.035 au; - the Laplacian of this electron density ranges from 0.024 to 0.139 au; - there is the mutual penetration of the hydrogen and the acceptor atoms; - an increase in net charge of the hydrogen atom is observed; - energetic destabilization of the hydrogen atom; - decrease of dipolar polarization of the hydrogen atom;

214

S. J. Grabowski and J. Leszczynski

- decrease of the hydrogen atomic volume. For the H.. .H contacts of (BH3NH3)2 there are BCPs and bond paths. The HF/6-31G(d,p) level pH...H values are equal to 0.0126 and 0.0067 au (0.0190 and 0.0096 calculated at the MP2/6-31G(d,p) level). The corresponding values of Laplacians (V2pH...H) amount to 0.0353 and 0.0247 au (the HF level) and 0.0464 and 0.0347 au (the MP2 level). One can see that the electron densities and Laplacians are within the ranges proposed by Koch and Popelier. Usually for studies of hydrogen bonds with the use of the AIM theory only the first three criteria are investigated. However, for the (BH3NH3)2 dimer all eight criteria are fulfilled.

6.rc-electron delocalization-assisted intramolecular dihydrogen bonds First experimental works on DHBs have included the metalloorganic crystal structures for which mainly the intramolecular interactions of this type exist. Similarly the first theoretical calculations have been often performed for such systems. For example in the work of Liu and Hoffman32 the intramolecular NH...H-Ir dihydrogen bonds have been analyzed. In this section the cases of intramolecular DHBs for which the effect of TT-electron delocalization occurs are analyzed. The influence of this effect on the strength and other characteristics of dihydrogen bonds are presented. The resonance-assisted hydrogen bonds (RAHBs) are very well known as those whose properties are strongly influenced by 7t-electron delocalization.51'76 For example, malonaldehyde (Fig. 8) is an example of a simple system in which intramolecular resonance-assisted hydrogen bond exists.77 For malonaldehyde and its simple derivatives (different R-substituents Fig. 8) there is equalization of dl and d4 CO bonds on one hand and equalization of the d2 and d3 bonds on the other hand. For such intramolecular RAHBs the Q-parameter was introduced as a descriptor of the delocalization.51 Q = dl-d4 + d3-d2

(3)

Is a Dihydrogen Bond a Unique Phenomenon?

R1

d2

215

C

I R2 Fig. 8. Malonaldehyde (R1=R2=R3=H) is an example of a system with intramolecular H-bonds assisted by Ji-electron delocalization. Reprinted with permissionfromref 77. © 2001 Elsevier Science B.V. The greater is the Jt-electron delocalization, the smaller is the Qparameter, and also the H-bonds are stronger. Such dependencies were observed for experimental RAHBs in organic crystal structures51 as well as were confirmed theoretically for malonaldehyde and its fluoro- and chloroderivatives77 at the MP2/6-311++G(d,p) level of theory. For the border case of delocalization, when there is full equalization of conjugated CO and CC bonds, the proton is exactly in the middle of the 0....0 distance. H-bonds in such a case are very strong, and the H-bond energy amounts to about 30-40 kcal/mol.51 Such systems are often known as low barrier hydrogen bonds.78 It was also pointed out that the electron withdrawing Rl-substituents and the electron donating R3-substituents (Fig. 8) should cause greater 7C-electron delocalization, greater bond length equalization and stronger RAHBs.51 This conclusion was confirmed by MP2/6-311++G(d,p) calculations.77 The influence of 7t-electron delocalization on intramolecular dihydrogen bonds was also studied for (lZ)-2-borylethen-l-ol (Fig. 9)79 and its derivative containing simple R-substituents causing an increase in H-bond energy (R1=C1, R3=Na). For these molecules ab initio calculations were performed up to the MP2/6-311++G(3d,3p) level of theory. The H-bond energies were roughly estimated as the difference between the closed conformations (such as the one

216 S. J. Grabowski and J. Leszczynski

presented in Fig. 8) and the open conformations. The open conformation may be obtained from the closed form after the rotation of the O-H bond by 180° around the dl (C-O) bond. The molecular geometries are optimized before and after rotation. Such an approach in the evaluation of intramolecular Hbond energy is often applied.77'80 However, it is a rough estimation of H-bond energy since other effects come into play such as the unfavourable approach of the lone pairs of the two oxygen atoms after the changing of conformations.80

T<->

TH

d1l R1

^

J B<M [

C •

<*

R3

R2

Fig. 9. (lZ)-2-borylethen-l-ol (R1=R2=R3=H) as an example of intramolecular dihydrogen bonding. Reprinted with permission from ref 79. © 2000 Elsevier Science B.V.

It is worth mentioning that in the case of typical intramolecular RAHBs such as malonaldehyde the 7t-electron delocalization occurs not only for the closed conformation but also for the open one.77 However, for the closed conformation the delocalization and bond equalization should be greater. For (lZ)-2-borylethen-l-ol an elongation of the C=C (d2) and B-H (d4) bonds and a shortening of the C-O (dl) and C-B (d3) bonds can be observed due to the transformation from the open to the closed conformation. Table 3 partly supports this, showing the meaningful influence of the formation of an intramolecular dihydrogen bond on the geometrical parameters of the molecules. The only exception is for the d2 bond length of the A molecule.

Is a Dihydrogen Bond a Unique Phenomenon?

217

Table 3. Geometrical parameters (in A) of (lZ)-2-borylethen-l-ol (A) and its derivative, R1=C1, R2=Na (B), at the MP2/6-311++G(3d,3p) level of theory (results taken from ref. 79). Molecule

I

dl

I

d2

I

d3

I

d4

A closed conformation

1.340

1.346

1.516

1.200

A open conformation

1.352

1.355

1.528

1.189

B closed conformation

1.333

1.367

1.534

1.225

B open conformation

1.343

1.356

1.547

1.202

The results collected in Table 3 also indicate that the delocalization is greater for the B molecule where there are Cl and Na substituents which make this effect stronger. The H...H intramolecular distances amount to 1.933 and 1.647 A for the A and B closed conformations, respectively (the MP2/6311++G(3d,3p) level), while the H-bond energies roughly estimated in the manner described previously are equal to 4.6 and 6.5 kcal/mol. Similarly, the values of the topological parameters show that the intramolecular DHBs presented here are of medium strength. The PH.H values are equal to 0.016 and 0.027 au for A and B, respectively, and the V2pH...H values are equal to 0.049 and 0.059 au for A and B (the MP2/6-311++G(2d,2p) level). A more extended sample of intramolecular dihydrogen bonds was also considered to investigate if the dependencies known for typical H-bonds are fulfilled for DHBs.81 Systems similar to (lZ)-2-borylethen-l-ol were considered, but the -BH2 group was replaced by -BH3\ All possible fluoro derivatives were included in this study. The calculations have been performed at the MP2/6-311++G(d,p) level of theory. Again, the well known approach of roughly estimating H-bond strength as a difference in energy between the closed and open conformations was applied. However, in this case this difference does not correlate with the other parameters describing H-bond strength, not with H...H distance or with the PH...H values. It seems that for this sample with negative charged species, other effects than H...H interactions strongly disturb the H-bond interaction. Hence another approach was applied81 to estimate the H-bond strength. It was found that the local potential energy density at the proton...acceptor contact, V(rcp), for H-bonds correlates with the H-bond energy.67 For H-bonds taken from crystal structures and for their electron densities, the local energy densities, V(rCp), were calculated, and the following relationship was proposed: E H B = V* V(rcp).67 The H-bond energies were calculated from the local energy densities

218 S. J. Grabowski and J. Leszczynski

for the intramolecular DHBs mentioned above, and good correlations between such energy and the other parameters were found. The linear correlation coefficient for the relationship between the calculated H-bond energy and H.. .H distance is equal to 0.995 (Fig. 10 shows this dependence). _

•on

16

>. -6

2> g-7J


HLHdstsnoe

1

1

1

1.65

1.7

1.75

1

1.8

R=Q995

Fig. 10. The relationship between the H-bond energy (kcal/mol) calculated from the local potential energy density and H...H distance (in A) for intramolecular DHBs. Reprinted with permissionfromref 81. © 2003 Elsevier Science B.V. Similarly, the linear correlation coefficient between this energy and the topological parameter pH...H is equal to 0.988. It is worth mentioning that the linear correlation coefficient for the relationship between the energy calculated from the local energy density, and the difference in energy between the open and closed conformations amounts to 0.226. For this sample of ionized intramolecular DHBs, the properties of the ring critical point were also investigated.81 The ring critical point (RCP) is a point of minimum electron density within the ring surface and a maximum on the ring line.64 For O-H...O intramolecular H-bonds it was found previously that electron density at RCP correlates with H-bond strength.82 Similar relationships were found for O-H.. .N and N-H.. .O intramolecular H-bonds83 and also for double intermolecular H-bonds for which the closed rings of the covalent bonds and the H...Y contacts are formed.84 For the sample of BH3'-CH-CH-OH and its fluoro derivatives a good polynomial correlation (the polynomial of the

Is a Dihydrogen Bond a Unique Phenomenon?

219

second degree) was found with a correlation coefficient of 0.973. Fig. 11 displays this relationship which confirms that, for intramolecular dihydrogen bonds, the topological properties of RCP may be used as descriptors of Hbond strength as for the other conventional intramolecular hydrogen bonds. These molecules with intramolecular DHBs were chosen as model systems because of their simplicity and the fact that they are convenient species for higher level ab initio calculations. However, similar but usually much more complex molecules exist in the liquid or solid phase. For example, for crystals of 2,2,2-tris((Boranato(diphenyl)phosphonio)methyl)ethanol (YIBTAE refcode in the Cambridge Crystal Structural Database36) intramolecular B-H'5... +5H-O hydrogen bonds were detected (the BH3" group as a proton acceptor) with an H...H distance of 2.230 A. Single point ab initio calculations of the moiety of 2,2,2-tris((Boranato(diphenyl)phosphonio)methyl)ethanol (geometry taken from the crystal structure) have been performed.81 Because of the complexity and the large size of the moiety the HF/3-21G* level of theory was chosen. The AIM calculations yielded the value of the electron density of the H"8../5!! bond critical point (pH...H = 0.0060 au) and the value of the electron density at the ring critical point (pRCp = 0.0055 au). The corresponding Laplacian values, V2pH..H and V2pRCp, amount to 0.0340 and 0.0282 au, respectively. These results suggest that for this crystal structure intramolecular dihydrogen bonds may exist since the topological criteria are fulfilled.75 0.030-I

0.028 r

0.026-

1=1

0.0240.0220.020 I +^—r 0.015 0.016

Jt *

/

j(* ^ ^ ^ 1 0.017

1 0.018

1 0.019

1 0.020

PRCP

Fig. 11. The relationship between electron density at H.. .H BCP and electron density at RCP (values in au). Reprinted with permission from ref 81. © 2003 Elsevier Science B.V.

220 S. J. Grabowski and J. Leszczynski

7. The nature of DHBs - energy decomposition analysis It is possible to obtain more detailed insight into the nature of interactions by decomposition of the total interaction energy into different components. There are different approaches to perform such a decomposition, but the scheme which is the most often applied is that of Morokuma and co-workers.85 Briefly speaking, the electrostatic (ES) term represents the Coulombic interaction between the charge distributions of the two subunits of the dimer, and the exchange energy (EX) corresponds roughly to steric repulsion between the two charge clouds. When the monomers are allowed to influence the charge clouds between themselves, it is possible to extract polarization (POL) energy connected with the shifts of electron density occurring within monomers and charge transfer (CT) energy connected with the density shifts from one monomer to the other. The effects of electron correlation are included in the CORR term. Such a decomposition scheme was applied recently to different complexes containing dihydrogen bonds.86 Among them the (H3BNH3)2 and (H2BNH2)2 complexes were considered where for the former four equivalent H...H contacts exist, and for the latter there are two equivalent H....H interactions. Their binding energies calculated at the MP2/aug-cc-pVDZ level of theory (BSSE included) are equal to 14.1 and 2.1 kcal/mol, respectively. For the (H3BNH3)2 complex for each equivalent B(sp3)-H...H-N(sp3) dihydrogen bond, the H-bond energy amounts to 3.5 kcal/mol, and for each B(sp2)-H.. .HN(sp2) interaction the energy is 1.1 kcal/mol for (H2BNH2)2. This is indicative of the influence of hybridization on the strength of the dihydrogen bond. For these and other complexes the Morokuma partitioning scheme was applied.85 The results are presented in Table 4 and compared with the other complexes connected through conventional and unconventional H-bonds.87'88'89 The results presented in Table 4 show that for conventional H-bonds in trans-linear dimers of water, the electrostatic attractive term is the most important; it overweighs the repulsive exchange term. The polarization and charge-transfer terms are much smaller than the electrostatic term, similar to the electron correlation energy term. For the unconventional C-H.. .0 bond of the CH4...OH2 complex, the electrostatic energy is compensated by the exchange energy, but due to the small attractive remaining terms, the total interaction for this complex is slightly attractive. The substitution of a methane-donating molecule by electronegative fluorine atoms causes an increase in the donating characteristics of the C-H bond. Hence, the percentage contributions of the energy terms for the F3CH.. .OH2 complex are similar to those of the water dimer. For two complexes with 7t-electrons of the

Is a Dihydrogen Bond a Unique Phenomenon? 221

acetylene molecule as proton acceptors (FH...C2H2 and HCCH...C2H2) the electrostatic term is compensated by the exchange term, and the remaining attractive contributions cause stabilization of the complexes. The polarization and charge-transfer contributions are greater in the case of the FH...C2H2 complex. There is a similar situation for HOH...C6H5OH; however, in such a case the correlation term is significant. Table 4. Morokuma's partitioning of Interaction Energy (kcal/mol). Complex/ref

I

ES

I

EX

I

POL

I

CT

I CORRd

(H 2 O) 2 /87 a

^6

42

4J7

4$

^03

CH 4 ...OH 2 /87 a

~^0A

O4

50

~4A

^Ol

F 3 CH...OH 2 /87 a

I7l

4~\

^O7

To

^03

FH...C2H2/88b

4A

63

T5

22

^O4

HCCH...C2H2/88b

^2

Ti

^03

^O5

^O4

HOH...C 6 H 5 OH/89 a

^

32

^09

^07

2~A

(H3BNH3)2 /86 c

^9

4\4

3?7

T6

4l

(H2BNH2)2 /86 c

A3

12

^5

^O8

4 l

"6-31+G** basis set b 6-311++G** basis set c aug-cc-pVDZ basis set d CORR=AE(MP2)- AE(HF)

What is a unique feature of the two dihydrogen bonded complexes presented in Table 4? For (H2BNH2)2 the binding energy and H-bond interactions are weaker than for the (H3BNH3)2 complex. For the first complex the exchange energy overweighs the electrostatic term, but finally the total interaction is attractive due to the remaining attractive terms, the correlation term being the most important. For the second complex, (H3BNH3)2, the electrostatic term is slightly greater than the exchange term, and the remaining terms contribute significantly to the total interaction energy. Hence one can see that for stronger dihydrogen bonds such as in the (H3BNH3)2 complex the decomposition components are slightly similar to typical H-bonds; the electrostatic contribution is slightly greater than the exchange contribution, and the other attractive terms play an important role.

222 S. J. Grabowski and J. Leszczynski

Another decomposition scheme was applied recently for the sample of complexes with dihydrogen bonds.90 The interaction energy for the IJH...H2, LiH.. .CH4, LiH.. .C2H6 and LiH.. .C2H2 complexes has been partitioned using the IMPPT scheme.91'92'93'94 The IMPPT energy terms are designated by e(ij) where i and j are the orders of the intermolecular interaction operator and the intramolecular correction operator, respectively. All e(i j ) terms are calculated in the basis of full complex to avoid the basis set superposition error. The total interaction energy at the all-electron MP2 level is decomposed into a HartreeFock (SCF) contribution and a correlation term: AEMP2 = AE SCF +AE CORR

(4)

The Hartree-Fock term is further decomposed: AESCF = A E SCF d e f + A E HL

(5)

Where AESCFdef is the deformation energy, and AEHL is a Heitler-London term. The first term describes an effect due to the relaxation of orbitals in the Coulomb field of the partner with the restriction imposed by the Pauli principle. The second-order approximation to AESCFdef considers this term as the induction response term e ^ O ) ^ . The Heitler-London contribution is further decomposed into electrostatic and exchange energies: AEHL = e(l,0) els + eHLexch (6) The correlation energy term is also decomposed into: AECORR = AE(2)exch + 8(2,0)^+ e(l,2)els,r

(7)

Where AE(2)exCh is the second-order exchange correlation correction; e(2,0)disP is the dispersion correlation correction; and E(l,2)eisr is the secondorder electrostatic correlation correction. The complexes mentioned above have been studied at the MP2 level of theory using the aug-cc-pVDZ basis set. For smaller complexes the aug-ccpVTZ basis set was also used. Table 5 shows the energy contributions for the complexes. Only results of MP2/aug-cc-pVDZ are given in order to compare all complexes. The results obtained within the same level and the same scheme of energy decomposition for the water dimer are also presented for comparison. Emphasis is put only on the selected terms.

Is a Dihydrogen Bond a Unique Phenomenon?

223

Table 5. Decomposition of the MP2 interaction energy (in kcal/mol) of the LiH...H2, L1H...CH4, LiH...C2H2, LiH...C2H6 and H2O...H2O complexes; the aug-cc-pVDZ basis set was used (results taken from ref. 90). Term 8(1,0)*

I LiH...H2 I

H

I LiH...CH4 I LiH...C2H6 I LiH...C2H2 I H2O...H2O Tl

A2

^2

TS

O

T3

22

6M

7^

AE^drf

: O6

5?7

5H

5?7

53

lOOW

5?7

TI

T5

53

42

£(2,1)^

51

51

51

51

51

e(l,2)elv

51

00

00

O6

51

~^~

A brief analysis of the results presented in Table 5 indicates that only for one of dihydrogen bonded complexes (LiH.. .C2H2) the first order electrostatic term (e(l,0) e i s ) is greater than the exchange term, similarly as for the water dimer. For the remaining dihydrogen bonded complexes the dispersion term (£(2,0YjjsP) is comparable with the electrostatic term. The authors of this paper explain that, for LiH...C 2 H 2 , the dihydrogen bond may be classified as hydrogen bond interaction; however, the remaining species should be characterized as van der Waals complexes.

8. Spin-spin coupling constants across dihydrogen bonds Vibrational frequencies and NMR shielding constants have been applied for a long time to describe hydrogen bonds and other intermolecular interactions. However, in recent years data concerning spin-spin coupling constants has become a new tool in the analysis of interactions, especially hydrogen bonds. A review of ab initio and DFT scalar coupling constants has appeared recently, and the usefulness of this new method is herein explained in detail. The scalar coupling can be defined as the mixed second derivative of the energy of the molecule regarding the magnetic moments of two nuclei. It can be divided into four components: Fermi contact (FC), paramagnetic spin-orbit (PSO), diamagnetic spin-orbit (DSO) and spin-dipolar (SD). The nJ designation is usually used for nuclear spin-spin coupling constants through n bonds, where bond means not only the typical covalent bond but also the

224 S. J. Grabowski and J. Leszczynski

inter- or intramolecular interaction such as, for example, the H-bond. Parentheses or subscripts may be also used to designate the coupled nuclei. Couplings through hydrogen bonds are usually designated as ***]. Hence the hydrogen-hydrogen coupling between atoms of two different water molecules connected through the O-H...O hydrogen bond is designated as 2hJ if one of the hydrogens belongs to the proton donating O-H bond and as 4hJ if none of the coupled hydrogen atoms belongs to the O-H.. .O system. A number of studies dealing with calculations of J-values have been performed in recent years. For example, the J-values for the water monomer and for the water dimer were calculated and compared with experimental results. The scalar coupling constants were analyzed for X-H.. .0 H-bonds in the following complexes: CH2O...H2O, C2H2...H2O, CH3OH...H2O and (HCOOH)2.97 The calculations were performed at the MCSCF level, and the results show that lhJoH intermolecular coupling correlates with the H-bond strength and is dominated by the FC term. Similarly the 2hJxy couplings through the X-H...Y hydrogen bonds are dominated by the FC component. Complexes with N-H...O=C and N-H...N=C systems were also studied, and exponential decay of 2hJxY upon increasing X... Y distance was observed. Different kinds of other H-bonds and van der Waals complexes95 were analyzed in terms of1*! values. However in the latter case only the theoretical results are available, and the experimental results are not available for verification. Among ^J values related to this review those concerning dihydrogen bonds are the most interesting. Recently ab initio calculations have been performed to determine onebond IhJHH and three-bond 3hJHH spin-spin couplings across X-H...H-M dihydrogen bonds for complexes with 13C-'H, 15N-[H and 17O-!H protondonor groups and 7Li-!H and 23Na-!H proton-acceptor metal hydrides. Coupling constants have been calculated using the equation-of-motion coupled cluster singles and doubles method (EOM-CCSD) employing Cl-like approximation.99>10"ll01'102The lhJm couplings containing the PSO, DSO, FC and SD components mentioned above have been calculated for linear CTOV dihydrogen bonded complexes of NCH...HLi, NCH...HNa, CNH...HLi, CNH...HNa and LiNCH+...HLi and for nonlinear Q complexes of HOH.. .HLi and HOH.. .HNa. The authors have observed that the SD term is negligible (ranging from -0.04 to -0.20 Hz). The relationships between the remaining terms and the H...H distance for each of the complexes were investigated. The FC term decreases exponentially with an increase in the distance of H.. .H. PSO and DSO depend linearly on the H.. .H intermolecular distance but have opposite signs and tend to cancel each other. Hence the

Is a Dihydrogen Bond a Unique Phenomenon?

225

relationship between the total coupling and H...H distance is practically the same as between FC and H.. .H distance. And hence the FC value may be an indication of the strength of the hydrogen bond. Similar dependencies were observed between the 3hJHH spin-spin coupling and the C...M (M = Li, Na) distance for each of the complexes. However, in the case of three-bond spinspin couplings, three terms, PSO, DSO and SD, are negligible and this is the reason why almost the whole contribution to the total coupling comes from the FC term. An analysis of the intermolecular coupling constants have been performed very recently for the following complexes: LiH...H2, LiH.-.CtL,, LiH...C2H6 and LiH...C2H2.90 The intermolecular lhJHH and 3hJXY (3hJuc) constants have been plotted as functions of internuclear distance. The ^JRH values have been calculated at the DFT (B3LYP) and CCSD levels of theory using the aug-ccpVTZ basis set. The obtained DFT and CCSD curves are similar, and the following conclusions were derived. 3hJLjC decreases exponentially with an increase in the Li...C distance for all four investigated complexes. For three weaker complexes (LiH...H2, L1H...CH4 and LiH...C2H6) lhJHH is negative and tends to approach zero when the H...H distance increases. A different behaviour was noticed for the LiH...C2H2 complex. In this case lhJHH is negative for larger distances, is positive for shorter H...H contacts and is similar to the other typical covalent bonds. It is worth mentioning that the authors have classified the LiH...C2H2 complex as dihydrogen-bonded, and the three remaining complexes as van der Waals complexes since the exchange energy terms overweight the electrostatic contributions (see previous section).

9. Borders and limitations - summary It is difficult to conclude that there is something specific for dihydrogen bonds in comparison with the other conventional H-bonds. It is certain that at least one feature is unique: in DHB complexes another hydrogen atom is the proton acceptor. It was pointed out that such an atom should be negatively charged as was revealed for some metal or boron hydrides. However even in early studies of DHBs it was indicated that such interactions may be treated in the another way, ' namely that a bonds are the proton acceptors following the trend: conventional X-H....Y hydrogen bonds, H-bonds with Tt-electrons as acceptor (X-H...Jt) and X-H...0 H-bonds. However Crabtree et al. also claim that early examples of interactions termed as dihydrogen bonds were

226 S. J. Grabowski and J. Leszczynski

studied for transition metal hydrides. For such cases one cannot be sure if the M-H a bond is an acceptor since transition metals also possess nonbonding d^-electrons that could act as alternative H-bond acceptors. The strength of DHBs is usually similar as predicted for typical H-bonds. DHBs are usually considered to be medium to strong H-bonds. The results of the decomposition energy analysis do not indicate unique properties of DHBs. For stronger DHBs such as for LiH...C2H2 the energy terms are similar to those of the typical H-bonds. The most important is the electrostatic term on which is greater than the exchange energy term (Table 5). For other systems such as LiH...H2, IJH...CH4 and LiH...C2H2, the exchange term overweighs the electrostatic contribution. These complexes are weak, and the dispersion term is important for them (Table 5). Such systems may be classified as van der Waals complexes. The clusters of ammonia with hydrogen molecules (NH4+(H2)n) could be considered an important example of the continuously changing characteristic of the DHB complex. These clusters have been studied where the number of H2 ligands ranges from one to eight. Geometry optimizations have been performed at the MP2/6-311G(d,p) level of theory, and the dissociation energies were calculated within the MP2 theory using the single point calculations in the extended aug-cc-pVDZ basis set. ' 5 Corrections for the basis set superposition error according to the scheme of Boys and Bernardi were applied. For the studied systems no symmetry constraints were imposed during the geometry optimization. The energetically lowest structure for the NH»+ + H2 complex is the vertex bond moiety with the H2 molecule perpendicular to the line of the N-H bond (Fig. 12). The binding energy for such a complex amounts to 2.56 kcal/mol, and the molecular arrangement supports the idea of the proton acceptor abilities of o bonds. It is worth mentioning that for the other arrangement of molecules for N H / + H2 (for the face bonded molecules where the H2 molecule is parallel to the HHH plane of NH/) the transition state is predicted, and such a complex is higher in energy (by 0.98 kcal/mol) than the equilibrium complex. It was indicated, using the interaction energy decomposition scheme, that for NH4+(H2)n complexes the exchange energy usually overweighs the electrostatic term, and the dispersion energy significantly contributes to the total energy. The situation is similar to that observed for the LiH...H2, IJH...CH4 and LiH...C2H2 complexes; however, for NH41" H2 the binding energy is greater. Therefore, it seems reasonable for the latter case to be considered the charge-assisted (N-H... o)+ interaction.

Is a Dihydrogen Bond a Unique Phenomenon?

227

Fig. 12. The structure of the NH/ complex of the lowest energy (distances in A, angles in degrees). Reprinted with permissionfromref 103. © 2001 Elsevier Science B.V. Very recently, a comprehensive study of C-H.. .H-C dihydrogen bonds existing in crystal structures was published. This kind of interaction in crystals is very important since for a large number of metal organic and organic crystals the H.. .H contacts are common. Until now they were usually treated as typical van der Waals contacts. Robertson et al.30 considered the crystal structures of the tetraphenylborates for which DHBs exist as well as the other H-bonds such as N-H.. .N, C-H.. .N, N-H.. .Ph and C-H.. .Ph. These interactions have been characterized using the Bader theory after multipole refinement of the structures. Different dependencies were studied; the authors found that the difference between the Mulliken charges on H-atoms in the uncomplexed X-H and H-Y components correlates well with some of the parameters of the X-H...H-Y dihydrogen-bonded system. They also found that the H...H intermolecular distance is within the range of 2.18 - 2.57 A; the electron density at H...H BCP is within 0.05-0.08 au, and the binding energy is about 1.5 kcal/mol. The range of the electron density values shows that the systems analyzed are within the criterion of Koch and Popelier.75 The authors also pointed out that there is a detectable border between the H2

228 S. J. Grabowski and J. Leszczynski

connections and the H...H contacts. The border is connected with the acute change of the geometrical and topological parameters. The correlation curves have characteristic gaps indicating these borders. And there are no discontinuities in the considered features between the dihydrogen bonds and the van der Waals complexes. Interestingly, early studies of the methane dimer show that the H.. .H interaction is repulsive and that this system is not stable. However, the later, more accurate post-HF calculations indicate that the methane dimer is bound in all six examined configurations: H3CH...H2CH2 (Cs), H2CH2...H2CH2 (D2h and D2d), HCH3...H3CH (D3h and D3d) and H3CH...HCH3 (D3h). It is worth mentioning that for such weak H...H interactions the analyzed structures are very sensitive to the level of the calculations. For example, the minimum (no imaginary frequencies) D3<j symmetry H3CH...HCH3 dimer not included in the analysis of Novoa et al.106 was obtained at the MP2/6-311++G(3d,3p) level of theory.107 The other lower basis sets indicate that such a configuration, H3CH...HCH3 (D3d), is a transition state. Fig. 13 shows a molecular graph of this D3d symmetry methane dimer obtained within the AIM theory.

Fig. 13. Molecular graph of the H3CH...HCH3 (D3(i) methane dimer (the MP2/6311++G(3d,3p) level); big circles correspond to attractors attributed to nuclei, and small circles correspond to bond critical points. The electron density at H...H BCP for this dimer is equal to 0.0035 au and is within the range of Koch and Popelier which suggests that this interaction may be classified as an H-bond. The examples presented here show that there are problems with distinguishing between van der Waals interactions and DHBs. The only not-

Is a Dihydrogen Bond a Unique Phenomenon?

229

so-sharp difference is in the nature of the interactions; for the former species the dispersion term is the most important; for the latter it is the electrostatic term that should overweight the exchange contribution.

10. Appendix - the Bond Valence Model Previously, in 1947, for metal crystal structures, L. Pauling introduced the concept of the bond number n (Pauling, 1947). rn-r0 = Ar = -clogn

(Al)

In the above equation, rn is the considered interatomic distance for which the bond number is equal to n; c is the constant; and r0 is the reference distance for which the bond number is usually equal to unity. The bond number may be understood as the number of electron pairs attributed to the considered pair of atoms, and it correlates with the strength of the interaction. The number of applications for this idea exists such as for example the analysis of chemical reactions and the construction of reaction paths (Burgi, 1975; Dunitz, 1979). This example concerns the proton transfer reactions. This concept is also useful for studies of intra- and intermolecular interactions (Grabowski, Krygowski, 2002). This idea could be apply to studies of hydrogen bond interactions. For example, for the O-H bond not involved in any intermolecular interactions, the bond number should be equal to unity. When the O-H bond is involved in the O-H...O interaction, the O-H bond is elongated, and n-value decreases according to eq. (Al). But the decrease of n is compensated by the H...0 contact. The sum of bond numbers of O-H and H...0 should be equal to unity. It is known as the bond number conservation rule (Dunitz, 1979). A similar and more often applied idea of bond valence was developed by Brown (Brown, 1978; Brown, 1992) within the consistent and simple model known as the Bond Valence (BV) model. According to the BV model, the bond valences s are usually related to bond distances in the following way: siJ = expf(ro-riJ)/BJ

(A2)

Sy = (ro/riJf

(A3)

or

230 5. J. Grabowski and J. Leszczynski

Where, sg designates that the valence bond concerns the pair of i-th and j th atoms; r0 is the reference (usually single) bond length. B and N are constants which may be easily determined if, for the partial bond, both values S;J and ry are known. For example, for the symmetrical linear system of O-H.. . 0 with the hydrogen atom in the middle of the O.. .0 distance, the sy value of H.. .O is equal to 0.5, and the H.. .O distance is equal to 1.22 A (Dunitz, 1979) or even less, 1.2 A (Gilli et al., 1994). Such an attitude is often applied to evaluations of constant values. However, Brown proposed to treat all constant values, B, N and r0, iteratively. This means that all constants may be determined from the large sets of systems taken from the results of crystal structures (Brown and Altermat, 1985) and by applying the least squares or one of the other procedures. One of the most important statements of the BV model is that the atomic valence may be expressed in terms of the bond valences. This is the so-called valence sum rule (VSR):



r.-zs,

The valence V; of the i-th atom is assumed to be shared between the bonds that it forms. According to the Bond Valence Model the bond between the i-th and j-th atoms may be the typical covalent bond or the intermolecular contact. The atomic valences usually correspond to the oxidation states. For the hydrogen atom within O-H...O system the valence sum rule takes the form: exp[(r0OH - rOH)/BoHl + exp[(r0OH - rH...o)/BOHI

=1

(A5)

This equation simply corresponds to the bond number conservation rule mentioned above. In other words the valence sum rule is more general than the bond number conservation rule. If the constant values r0 and B are known it is possible to derive the relationship between the rOH bond length and the H...0 distance rH...o (Grabowski, 1988). This is a theoretical relationship based on the simple BV model, and it agrees excellently with the experimental results (Gilli et al., 1994). It is worth mentioning that the r0 and B constant values are attributed to pairs of atoms; thus they are the same for O-H covalent bonds and the H...0 contacts.

Is a Dihydrogen Bond a Unique Phenomenon?

231

Similar relationships as the one expressed by eq. (A5) may be constructed for any atom in the solid, liquid or gas phase. For example, for the proton acceptor oxygen atom in C=O...H-O systems, the following valence sum rule equation may be written: exp[(roc'° - rc=o)/Bc=o] + exp[(r0OH - rK,.o)/Bo«]

=2

(A6)

The relationship between the C=O bond length and H.. .O contact distance obtained from eq. (A6) corresponds well to the experimental neutron diffraction results (Grabowski, 1998). The VSR for the +8H-atom within the F-lT8...'5!! dihydrogen bonded system may be written in the following way: VH = exp[(ro"-F - rH.F)/BHF]

+ exp[(roH"H - rH_jJ/BH..jJ

(A7)

Where, rH.F and rH...H are the H-F bond length and the H...H distance within F-H...H system; roHF ,BHF , r0H'H and BH...H are constants. The dependencies mentioned above obtained from the VSR often correspond to experimental results and to the relationships obtained from ab initio or DFT calculations (see the main text of this chapter). They also show the proper dependencies between the geometrical parameters and are indicative of the changes in the geometrical parameters during chemical reactions. Brown, I.D. Chem. Soc. Rev. 1978,7,359. Brown, I.D. Acta Cryst. 1992, B48,553. Brown, I.D. Altermat, D. Acta Cryst. 1985, B41,244. Burgi, H.-B. Angew. Chem. Internat. Edit. 1975,14,460. Dunitz, J.D. X-ray Analysis and the Structure of Organic Molecules, Cornell University Press, Ithaca, 1979. Gilli, P., Bertolasi, V., Ferretti, V., Gilli, G. J. Am. Chem. Soc. 1994,116,909. Grabowski, S.J. Croat. Chem. Acta 1988,61, 815. Grabowski, S.J. Tetrahedron 1998,54,10153. Grabowski, S.J., Krygowski, T.M. Molecular Geometry as a Source of Chemical Information - Application of the Bond Valence-Bond Number Models, in Advances in Quantitative Structure Property Relationships, Eds. B.Charton & M.Charton, Vol. 3, pages 27-66, Elsevier Science B.V. 2002. Pauling, L. J. Am. Chem. Soc. 1947, 69, 542.

Acknowledgement Support for this research is provided by grant 505/675 2003 (University of L6dz), grant No. 3 T09A 138 26 (State Committee for Scientific Research),

232 S. J. Grabowski and J. Leszczynski

NSF CREST HRD-0318519, NSF EPSCoR 02-01-0067-08/MSU, and ONR Grant N00034-03-1-0116.

11. References 1 Pauling, L., The nature of the chemical bond. Cornell University Press, Ithaca NY,

1939. Pimentel, G.C., McClellan, A.L. The Hydrogen Bond. Freeman, San Francisco, CA, 1960. 3 Jeffrey, G.A., Saenger, W. Hydrogen Bonding in Biological Structures. SpringerVerlag: Berlin, 1991. 4 Desiraju, G.R., Steiner, T. The weak hydrogen bond in structural chemistry and biology. Oxford University Press Inc., New York, 1999. 5 Jeffrey, G.A. An Introduction to Hydrogen Bonding. Oxford University Press, New York, 1997. 6 Desiraju, G.R. Crystal Engineering - The Design of Organic Solids. Elsevier, Amsterdam, 1989. 7 Alkorta, I., Rozas, I., Elguero, J. Chem.Soc.Rev. 1998, 27, 163 (and references cited therein). 8 Suttor, D. J. J. Chem. Soc. 1963, 1105. * Taylor, R., Kennard, O. J. Am. Chem. Soc. 1982, 104, 5063. Grabowski, S.J. Polish J. Chem. 1995, 69,223. 11 Platts, J. A., Howard, S.T., Wozniak, K. Chem.Commun. 1996, 63. 12 Hobza, P., Havlas, Z. Chem.Rev. 2000, 100,4253 (and references cited therein). 13 Scheiner, S. Hydrogen Bonding: A Theoretical Perspective. Oxford University Press: New York, 1997. 14 Wessel, J., Lee, J. C , Peris, E., Yap, G. P. A., Fortin, J. B., Ricci, J. S., Sini, G., Albinati, A., Koetzle, T. F., Eisenstein, O., Rheingold, A. L., Crabtree, R. H. Angew. Chem. Int. Ed. Engl. 1995,34,2507. 15 Custelcean, R., Jackson, J. E. Chem. Rev. 2001, 101,1963. 16 Crabtree, R. H., Siegbahn, P. E. M., Eisenstein, O., Rheingold, A. L., Koetzle, T. F. Ace. Chem. Res. 1996,29, 348. 17 Epstein, L. M., Shubina, E. S. Coord. Chem. Rev. 2002, 231, 165 (and references cited therein). 18 Rozas, I., Alkorta, I., Elguero, J. Chem.Phys.Lett. 1997, 275, 423 (and references cited therein). 19 Stevens, R. C , Bau, R., Milstein, D., Blum, O., Koetzle, T. F. J.Chem.Soc, Dalton 2Q Trans. 1990, 1429. Van der Sluys, L. S., Eckert, J., Eisenstein, O., Hall, J. H., Huffman, J. C , Jackson, S. A., Koetzle, T. F., Kubas, G. J., Vergamini, P. J., Caulton, K. G. J. Am. Chem. 21 Soc. 1990,112,4831. Park, S., Ramachandran, R., Lough, A. J., Morris, R. H. J. Chem. Soc. Chem. Commun. 1994,2201. 2

Is a Dihydrogen Bond a Unique Phenomenon?

233

Lough, A. J., Park, S., Ramachandran, R., Morris, R. H. J. Am. Chem. Soc. 1994, 116,8356. Lee, J. C , Jr., Rheingold, A. L., Miiller, B., Pregosin, P. S., Crabtree, R. H. J. Chem. Soc. Chem. Commun. 1994,1021. 24Lee, Jr., J. C , Peris, E., Rheingold, A. L., Crabtree, R. H. J. Am. Chem. Soc. 1994, 116,11014. " Peris, E., Lee, Jr., J. C , Crabtree, R. H. J. Chem. Soc. Chem. Commun. 1994,2573. Peris, E., Lee, Jr., J. C , Rambo, J., Eisenstein, O., Crabtree, R. H. J. Am. Chem. Soc. 1995,117,3485. 27 Peris, E., Wessel, J., Patel, B. P., Crabtree, R. H. J. Chem. Soc. Chem. Commun. 1995,2175. 28 Kloosrer, W. T., Koetzle, T. F., Siegbahn, P. E. M., Richardson, T. B., Crabtree, R. H. J. Am. Chem. Soc. 1999,121, 6337. Z* Richardson, T. B., Koetzle, T. F., Crabtree, R. H. Inorg. Chim. Ada 1996,250,69. Robertson, K. N., Knop, O., Cameron, T. S. Can. J. Chem. 2003,81,727. 31 Richardson, T. B., de Gala, S., Crabtree, R. H., Siegbahn, P. E. M. J. Am. Chem. Soc. 1995,117, 12875. I Liu, Q., Hoffinan, R. J. Am. Chem. Soc; 1995, 117,10108. Hoffinan, R. J. Chem. Phys. 1963,39,1397. 34 Becke, A. D. Phys. Rev. 1988, A38,3098. 35 Siegbahn, P. E. M., Blomberg, M. R. A., Svensson, M. Chem. Phys. Lett. 1994, ™ 223,35. Allen, F. H., Kennard, O. Chem. Des. Autom. News 1993, 8,31. 37 Klooster, W. T., Koetzle, T. F., Siegbahn, P. E. M., Richardson, T. B., Crabtree, R. 38 H. J. Am. Chem. Soc. 1999,121, 6337. Cramer, C. J., Gladfelter, W. L. Inorg. Chem. 1997,36, 5358. 39 Boys, S. F., Bernardi, F. Mol. Phys. 1970,19, 553. 40 Alkorta, I., Elguero, J., Foces-Foces, C. Chem. Commun., 1996,1633 41 Bader, R. F. W., (1990) Atoms in Molecules. A Quantum Theory. Oxford University Press, New York. f Remko, M. Molecular Physics, 1998,94,839. Orlova, G., Scheiner, S. /. Phys. Chem. 1998, 102, 260. */ Orlova, G., Scheiner, S. J. Phys. Chem. 1998,102,4813. * Kulkarni, S. A. J. Phys. Chem. A 1998,102, 7704. *! Kulkami, S. A. J. Phys. Chem. A 1999,103,9330. Kulkarni, S. A., Srivastava, A. K. /. Phys. Chem. A 1999,103, 2836. 48 Grabowski, S. J., Tetrahedron 1998, 54, 10153 (and references cited therein). 49 Allen, F. H., Davies, J. E., Galloy, J. E., Johnson, J. J., Kennard, O., Macrave, C. F., Mitchel, E.M., Smith, J. M., Watson, D. G. J. Chem. Inf. Comput. Sci. 1991, 22

,,

50

31, 187. 3 1 ' 1 8 7 '

51 Ichikawa, M. Acta Cryst. 1978, B34,2074. 52 Gilli, P., Bertolasi, V., Ferretti, V., Gilli, G. J. Am. Chem. Soc. 1994,116,909. " Steiner, T. J. Phys. Chem. 1998, 102,7041. 54 Steiner, T. Chem. Soc. Perkin Trans. 2, 1996, 1315. Grabowski, S. J. J. Phys. Org. Chem. 17,2004,18.

234 5. J. Grabowski and J. Leszczynski

Grabowski, S. J. J. Mol. Struct. 2000, 552, 153. Grabowski, S. J., Krygowski, T. M. Chem. Phys. Letters, 1999,305,247. "Grabowski, S. J., Chem. Phys. Lett. 1999,312, 542. Frisch, M. J., Trucks, G. W., Schlegel, H. B., Scuseria, G. E., Robb, M. A., Cheeseman, J. R., Zakrzewski, V. G., Montgomery, J. A., Stratmann, R. E., Burant, J. C , Dapprich, S., Millam, J. M., Daniels, A. D., Kudin, K. N., Strain, M. C , Farkas, O., Tomasi, J., Barone, V., Cossi, M., Cammi, R., Mennucci, B., Pomelli, C , Adamo, C , Clifford, S., Ochterski, J., Petersson, G. A., Ayala, P. Y., Cui, Q., Morokuma, K., Malick, D. K., Rabuck, A. D., Raghavachari, K., Foresman, J. B., Cioslowski, J., Ortiz, J. V., Stefanov, B. B., Liu, G., Liashenko, A., Piskorz, P., Komaromi, I., Gomperts, R., Martin, L. R., Fox, D. J., Keith, T., Al.-Laham, M. A., Peng, C. Y., Nanayakkara, A., Gonzalez, G., Challacombe, M., Gill, P. M. W., Johnson, B., Chen, W., Wong, M. W., Andres, J. L., Gonzalez, C , Head-Gordon, M., Replogle, E. S., Pople, J. A. Gaussian 98; Revision A.6; Gaussian, Inc.: Pittsburgh, PA, 1998. 5 Grabowski, S. J. J. Mol. Struct. 2000, 553,151. 60 Grabowski, S. J. "Simulations of Hydrogen Bonds in Crystals and Their Comparison with Neutron Diffraction Results, in Neutrons and Numerical Methods - N2M," Grenoble, France December 1998, AIP Conference Proceedings 479, (eds.) M. R. Johnson, G. J. Kearley and H. G. Buttner, American Institute of Physics 1999. 61 Novoa, J. J., Nobeli, I., Grepioni, F., Braga, D. NewJ. Chem. 2000,24, 5. "Grabowski, S. J. J. Phys. Chem. A2000,104, 5551. £ Hohenberg P., Kohn, W. Phys. Rev. 1964, B864, 136. Popelier, P. Atoms in Molecules. An Introduction, Prentice Hall, Pearson Education Limited 2000. 65 M6,0., Yanez, M. Elguero, J. /. Chem. Phys. 1992,97,6628. ^ Mo, O., Yanez, M. Elguero, J. J. Mol. Struct. (Theochem) 1994,314,73. Espinosa, E., Molins, E., Lecomte, C. Chem. Phys. Lett. 1998,285,170. * Grabowski, S. J. J. Phys. Chem. A, 2001, 105,10739. Scheiner, S. Annu. Rev. Phys. Chem. 1994,45,23. 70Grabowski, S. J. J. Mol. Struct. 2002,615,239. 71 Alkorta, I., Elguero, J., M6, O., Yanez, M., del Bene, J. E. J. Phys. Chem. 2002, „ 106,9325. Popelier, P. L. A. J. Phys. Chem. A 1998,102, 1873. 73 MORPHY97, a program written by P. L. A. Popelier with a contribution from R. G.A. Bone, UMIST, Manchester, U.K., EU, 1997. 74Popelier, P. L. A. Chem. Phys. Lett. 1994,228,160. "Koch, U. Popelier, P. L. A. J. Phys. Chem. A 1995,99,9747. Madsen, G. K. H., Iversen, B. B., Larsen, F. K., Kapon, M., Reisner, G. M., Herbstein, F. H. J. Am. Chem. Soc. 1998,120,10040. 77 Grabowski, S. J. J. Mol. Struct. 2001, 562,137. 78 Garcia-Viloca, M., Gelabert, R., Gonzalez-Lafont, A., Moreno, M., Lluch, J. M. J. Phys. Chem. A 1997,101,8727 (and references cited therein). 79 Grabowski, S. J. Chem. Phys. Lett. 2000,327,203. 55

Is a Dihydrogen Bond a Unique Phenomenon?

235

", Cuma, M., Scheiner, S., Kar, T. J. Mol. Struct. (Theochem), 1999,467, 137. Wojtulewski, S., Grabowski, S. J. J. Mol. Struct. 2003,645,287. 82 Grabowski, S. J, Monatsh.Jur Chetnie, 2002,133,1373. 83 Rybarczyk-Pirek, A. J., Grabowski, S. J., Malecka, M., Nawrot-Modranka, J. J. 84 Phys. Chem. A 2002,106,11956. Dubis, A., Grabowski, S. J, Romanowska, D. B., Misiaszek, T., Leszczynski, J. /. g5 Phys. Chem. A 2002,106,10613. Morokuma, K., Kitaura, K. in Molecular Interactions, vol. 1, W. J. Orville-Thomas (ed.), Wiley, New York, 1980, page 21. I Kar, T., Scheiner, S. J. Chem. Phys. 2003, 119,1473. Gu, Y., Kar, S., Scheiner, S. J. Am. Chem. Soc. 1999, 121,9411. * Scheiner, S., Grabowski, S. J. J. Mol. Struct. 2002,615,209. f Scheiner, S., Kar, T., Pattanayak, J. J. Am. Chem. Soc. 2002,124, 13257. Cybulski, H., Pecul, M., Sadlej, J. J. Chem. Phys. 2003,119, 5094. I* Chalasinski, G., Sczeiniak, M. Mol. Phys. 1988,63,205. Moszyhski, R., Rybak, S., Cybulski, S. M., Chalasinski, G. Chem. Phys. Lett. 1990, „ 166,609. Cybulski, S. M., Chaiasinski, G., Moszyhski, R. J. Chem. Phys. 1990,92,4357. 9 Chalasinski, G., Sczfiniak, M. Chem. Rev. 1994,94, 1723. Alkorta, I., Elguero, J. Int. J. Mol. Sc. 2003,4, 64. : ! Pecul, M., Sadlej, J. Chem. Phys. Lett. 1999,308,486. 9J Pecul, M., Leszczynski, J., Sadlej, J. J. Chem. Phys. 2000,112, 7930. * Pecul, M., Leszczynski, J., Sadlej, J. J. Chem. Phys. 2000,104, 8105. i Perera, S. A., Sekino, H., Bartlett, R. J. J. Chem. Phys. 1994,101,2186. °, Perera, S.A., Nooijen, M., Bartlett, R. J. J. Chem. Phys. 1996,104,3290. j°, Perera, S. A., Bartlett, R. J. J. Am. Chem. Soc. 1995, 117,8476. ~T Perera, S. A., Bartlett, R. J. J. Am. Chem. Soc. 1996, 118, 7849. !°: Urban, J., Roszak, S., Leszczynski, J. Chem. Phys. Lett. 2001, 346,512. r* Dunning, Jr., T. H. J. Chem. Phys. 1989, 90, 1007. Kendall, R. A., Dunning, Jr., T. H., Harrison, R. J. J. Chem. Phys. 1992, 96, 6769. ^ Novoa, J. J., Whangbo, M.-H., Williams, J. M. J. Chem. Phys. 1991,94,4835. Grabowski, S. J., Leszczynski, J. unpublished results.

INDEX cr-bond acceptor 202, 225, 226 7r-electron delocalization 214-219 ab initio 85, 88, 89, 105 adhesion 61 aglycone bond 68 amylose 67, 70, 71 Anderson's theory 154-157 angle bending 105-107 angular overlap model 166 anisotropic exchange 154, 155, 157, 163, 165, 187 atomic force microscope (AFM) 47-51, 53-62, 64, 66-71 atomic valence 230 atoms in molecules (AIM) theory 208-214 ATPase 56, 65, 66 basis set superposition error 222, 226 Bethe lattice 7, 8, 11, 38 biased molecular dynamics 54, 62, 64, 71 bifurcated dihydrogen bonds 198, 199 binding energy 196, 201, 202, 207, 208, 220, 221, 226, 227 blue-shifting H-bonds 197, 204, 207 bond critical point (BCP) 208, 209, 213, 219, 228 bond number 229-231 bond number conservation rule 229, 230 bond valence 204, 205, 229, 230 bond valence model (BV model) 204, 205, 229-231 Cambridge structural database (CSD) 202, 204 carbon nanotube (CNT) 31-33, 35, 37 chair-boat transition 50, 67 charge transfer 157, 182, 190 charge transfer energy 220 CHARMM 67 conduction channel 3, 4, 19 conventional H-bonds 197, 199, 203, 204, 207, 210-212, 219, 220, 225 correlation energy term 220, 222 coupled electron-pair approximation (CEPA) 121-124, 126-132, 136, 138, 139, 148

237

238

Index

coupling constant 153, 156, 166, 170, 173, 183-185, 188-191 coupling integral 189 criteria for the existence of hydrogen bonds 196, 202, 206, 207, 213, 214, 219 crystal (ligand) field 158, 160 crystal engineering 196 density functional theory (DFT) 5, 6, 9, 13, 14, 27, 28, 53, 70, 169 dextran 67, 68 dihedral 109, 110, 112-115 dihydrogen bond 195-235 dipole moment 88, 114, 116, 179, 180, 192 Dzyaloshinski-Moriya 163 elastin 59 electric field 179, 180 electronic hopping 159 electronic transition 178 electrostatic energy term 201, 207, 220-223, 225, 226, 229 energy decomposition analysis 220-223 enzymatic catalysis 53 EPR-spectra 163, 165 exchange energy 220-223, 225, 226, 229 exchange interaction 154, 155, 157, 158, 163 exchange operator 153, 156, 180, 186 excited state 121, 125, 126, 133, 134, 136, 138-140, 142-144, 145, 148 exclusion principle violating (EPV) terms 123, 124, 129, 130, 132 ferromagnetic 154, 163, 164, 177, 178, 186 fibronectin 50, 56, 58 force probe molecular dynamics 60 force probe spectroscopy 49 free energy 53, 55, 59, 62, 63 frequency spectrum 92, 93, 101, 103, 110 fullerene 1, 24, 26-28, 30 Gaussian Embedded Cluster Method (GECM) GAUSSIAN98/03 1, 69 Go-like models 65 Green's function 7, 9, 10, 40, 41 g-tensor 163 H... H interactions 202, 203, 212

6, 25, 26, 31

Index 239

H-bond energy 197, 206-212, 215-218, 220 H-bond strength 209, 217-219, 224 Heisenberg-Dirac-van Vleck Hamiltonian 157 heterogeneity of the sample 204 HF calculations 67 homogeneity of the sample 204, 209-211 hydrogen bonding 195-235 implicit solvent 59, 62 IMPPT energy terms 222 intramolecular dihydrogen bonds 198, 201, 214-219 intruders 121, 124, 125, 127, 131-133, 139, 143, 145, 148 Jarzynski 63, 64 Keldysh formalism 13 kinetic exchange 153, 156, 182, 185 Landauer's formalism, formula 2-5, 25, 37 Laplacian of the electron density 208, 209, 213, 214, 219 LDA 169 Lennard-Jones 105 Li2 121, 143 linear response theory (LRT) 125, 126, 133 local energy density 217, 218 localized magnetic orbitals 154, 156 low barrier hydrogen bonds 215 magnetic exchange 153, 166 magnetic field 163, 165 magnetic orbitals 154-156, 186 MBPT 124 mechanically controllable break junction (MCBJ) 2, 17 methane dimer 228 molecular bridge 2, 7, 22 molecular device 86, 88, 89, 91, 101, 103, 115 molecular dynamics 47, 52, 54, 56, 60, 69, 85, 88, 89, 92, 93, 114, 115 molecular junction 2, 6 MRCC 124-126 MRCEPA 124-126, 138-140 MR-CISD 123, 124, 128, 130 MscL channel 60

240

Index

Mulliken atomic charges 201, 202, 227 NAMD2 code 68 nanocell 85, 115 nanocontact 2, 4, 6, 10-12, 14-25 NDR 86, 89, 91 negative differential resistance 86, 89, 91 optomechanical 66 orbital momentum 168 P4 121, 139-143, 145, 146, 148 pectin 67 perturbation theory 156, 164, 182, 190, 193 polarization energy 220, 221 polysaccharides 50, 51, 66, 67 potential energy surface (PES) 121-125, 139-141, 143-148 potential exchange 156, 185 potential of mean force 55, 56, 59, 62 protein folding 51-53 proton acceptors 197, 202-204, 206, 212, 219, 221, 224-226, 231 proton donors 196, 197, 200, 202-205, 207, 209, 210, 224 pustulan 67-69 PW96 169 pyranose ring 67, 68 quantum molecular dynamics 70 radiation field 179 random expulsion molecular dynamics (REMD) rare earth 153, 186 reaction coordinate 55, 61, 62 reaction kinetics 53 resonance assisted hydrogen bonds 214-219 resonant bond 93, 98, 99, 101, 112-114 ring critical point (RCP) 208, 218, 219 scanning tunneling microscope (STM) SCC-DFTB 70 Schrodinger equation 3, 4 size-extensivity 123-125, 127 size-intensivity 138

64, 65

2, 17, 24, 25, 31

Index 241

spectrin 60 spin operator 159, 160, 168 spin-forbidden transition 179 spin-orbit coupling 153, 157, 160, 166, 167, 170, 172, 185, 187, 188 spin-orbit interaction 158 spin-spin coupling constants 223-225 S-S bridge 58 SS-MRCC 122, 125-128, 133, 134, 136, 146 streptavidin-biotin 54 targeted molecular dynamics (TMD) 54, 60, 66, 72 temperature 85, 89, 96, 98, 103, 105, 107, 110, 112, 113, 115, 116 the electron density 208-210, 213, 218-220, 227, 228 titin 50, 51, 56-59, 61, 65 topological parameters 209-211, 217, 218, 228 transition metal 153, 154, 181 unconventional H-bonds 196, 200, 220 up-conversion 153-155, 178, 183, 185 valence sum rule (VSR) 230, 231 van der Waals interaction 199, 209, 225-228 van der Waals radii 199, 201, 202, 206, 207, 213 van der Waals 93 vertical excitation energies 145 vibrational modes 87, 88, 92, 98, 114, 115 water dimer 207, 210, 211, 220, 222-224 wave function 210, 212 WHAM 56, 64

CONTENT INDEX Volume 1 Relativistic Many-Body Calculations on Atoms and Molecules (Y. Ishikawa & U. Kaldor) Modern Developments in Hartree-Fock Theory: Fast Methods for Computing the Coulomb Matrix (M. Challacombe, E. Schwegler & J. Almlof) Local Shape Analysis of Macromolecular Electron Densities (P. G. Mezey) Liquid-State Quantum Chemistry: Computational Applications of the Polarizable Continuum Models (J.-L. Rivail & D. Rinaldi) Elemental Boron Route to Stuffed Fullerenes (E. D. Jemmis & B. Kiran) Interactions of DNA Bases and the Structure of DNA. A Nonempirical Ab Initio Study with Inclusion of Electron Correlation (J. Sponer, P. Hobza & J. Leszczynski) Computational Approaches to the Design of Safer Drugs and Their Molecular Properties (N. Bodor & M.-J. Huang) Volume 2 The Electron Propagator Picture of Molecular Electronic Structure (J. V. Ortiz) SAC-CI Method: Theoretical Aspects and Some Recent Topics (H. Nakatsuji) Quantum Monte Carlo and Electronic Structure (R. N. Barnett & W. A. Lester, Jr.) Molecular Structure and Infrared Spectra of the DNA Bases and Their Derivatives: Theory and Experiment (M. J. Nowak, L. Lapinski, J. S. Kwiatkowski & J. Leszczynski) Derivation and Assessment of a New Set of Ab Initio Potentials and Its Application to Molecular Dynamics Simulations of Biological Molecules in vacuo, in Crystal and in Aqueous Solution (M. Aida) Practical Exercises in Ab Initio Quantum Chemistry — the World Wide Web as a Teaching Environment (H. P. Lu'thi, G. Vacek, A. Hilger & W. Klopper) Volume 3 Multireference Brillouin-Wigner Coupled-Cluster Theory (/. Hubac, J. Masik, P. Mach, J. Urban & P. Babinec)

243

244 Content Index Model Core Potentials: Theory and Applications (M. Klobukowski, S. Huzinaga & Y. Sakai) Stratospheric Bromine Chemistry: Insights from Computational Studies (5. Guha & J. S. Francisco) Bonding in Gas-Phase Sulfur Radicals (D. C. Young & M. L. McKee) Chemistry of the Liquid State: Current Trends in Quantum-Chemical Modeling (L. Gorb & J. Leszczynski) Chemistry and the Internet (K. Flurchick, M. Hurley, J. K. Labanowski, G. H. Lushington & T. L. Windus) Volume 4 Topography of Atomic and Molecular Scalar Fields (5. R. Gadre) The Ab Initio Model Potential Method: A Common Strategy for Effective Core Potential and Embedded Cluster Calculations (L. Seijo & Z. Barandiarari) Continuum Models of Macromolecular Association in Aqueous Solution (M. A. Olson) Interactions of Nucleic Acid Bases: The Role of Solvent (M. Orozco, E. Cubero, B. Hernandez, J. M. Lopez & F. J. Luque) Recent Advances in Multireference M0ller-Plesset K. Nakayama, T. Nakajima & H. Nakano)

Method

(K. Hirao,

Detonation Initiation and Sensitivity in Energetic Compounds: Some Computational Treatments (P. Politzer & H. E. Alper) Volume 5 In Search of the Relationship between Multiple Solutions Characterizing CoupledCluster Theories (P. Piecuch & K. Kowalski) Computational Time-Dependent Two-Electron Theory and Long-Time Propagators (C. A. Weatherford) Self-Consistent Field Theory of Weakly Bonded Systems (E. Gianinetti, I. Vandoni, A. Famulari & M. Raimondi) Aromatic DNA Base Stacking and H-Bonding (J. Sponer, P. Hobza & J. Leszczynski) Direct Ab Initio Dynamics Methodology for Modeling Kinetics of Biological Systems (T. N. Truong & D. K. Maity)

Content Index 245

Molecular Structure and Vibrational IR Spectra of Fluoro, Chloro and Bromosubstituted Methanes, Silanes and Germanes: An Ab Anitio Approach (J. S. Kwiatkowski & J. Leszczynski) Volume 6 Relativistic Multireference M0ller-Plesset Perturbation Theory (Y. Ishikawa & M. J. Vilkas)

15 Years of Car-Parrinello Simulations in Physics, Chemistry and Biology (U. Rothlisberger)

Methods of Combined Quantum/Classical (QM/MM) Modeling for Large Organometallic and Metallobiochemical Systems (/. B. Bersuker) A Review of Ab Initio Calculations on Proton Transfer in Zeolites (M. Allavena & D. White) Ionic Clusters with Weakly Interacting Components-Magic Numbers Rationalized by the Shell Structure (S. Roszak & J. Leszczynski) Turning Point Quantization and Scalet-Wavelet Analysis ( C R. Handy) Volume 7 Molecules as Components in Electronic Devices: A First-Principles Study (M. Di Ventra) Tackling DNA with Density Functional Theory: Development and Application of Parallel and Order-N DFT Methods (C. F. Guerra, F. M. Bickelhaupt, E. J. Baerends & J. G. Snijders) Low-Scaling Methods for Electron Correlation (5. Saeb0) Iterative and Non-Iterative Inclusion of Connected Triple Excitations in CoupledCluster Methods: Theory and Numerical Comparisons for Some Difficult Examples (J. D. Watts) Explicitly Correlated Coupled Cluster R12 Calculations (J. Noga & P. Valiron) Ab Initio Direct Molecular Dynamics Studies of Atmospheric Reactions: Interconversion of Nitronium Ions and Nitric Acid in Small Clusters (Y. Ishikawa & R. C. Binning, Jr.) Volume 8 Computational Modelling of the Solvent Effects on Molecular Properties: An Overview of the Polarizable Continuum Model Approach (R. Cammi, B. Mennucci & J. Tomasi)

246

Content Index

Electronic and Nonlinear Optical Properties of 2-Methyl-4-Nitroaniline Clusters (M. Guillaume, B. Champagne, F. Castet & L. Ducasse) Ab Initio Calculations of the Intermolecular Nuclear Spin-Spin Coupling Constants (M. Pecul & J. Sadlej) Base Polyad Motifs in Nucleic Acids - Biological Significance, Occurrence in Three-Dimensional Experimental Structures and Computational Studies (M. Meyer & J. Siihnel) Model Calculations of Radiation Induced Damage in DNA Constituents Using Density Functional Theory (D. M. Close) Excited States of Nucleic Acid Bases (M. K. Shukla & J. Leszczynski)