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

Hydrogen Bonding: A Theoretical Perspective

STEVE SCHEINER

Oxford University Press

HYDROGEN BONDING

TOPICS IN PHYSICAL CHEMISTRY

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

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

HYDROGEN BONDING A Theoretical Perspective

STEVE SCHEINER

New York Oxford Oxford University Press 1997

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

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

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

To my mother: You will not be forgotten

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Preface

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

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

viii

Preface

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

Preface ix

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

x

Preface

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

Contents

Abbreviations

xvii

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

14

xii Contents 1.7.2 Alternate Schemes 34 1.7.3 Perturbation Schemes 37 2

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

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

112

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

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

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

5

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

239

xiii

xiv

Contents

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

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

Contents

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

371

365

335

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Abbreviations

ACPF ANO

APT BSSE

CASSCF

CC CC cc CCD CCSD CEPA CF CPF CI

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

xviii Abbreviations

CISD CT DISP DZ DZP ECP

EFG

ES EX

GIAO GTO HF

IEPA

IGLO KM LCAO

LCCM MBPT MBS

MCSCF

MINI MO MPn

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

Abbreviations

NBO

NEDA NPAD

NQCC PES POL

QCISD SAPT SCEP SCF STO ZPVE + //

xix

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

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HYDROGEN BONDING

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I Quantum Chemical Framework

I. I Quantum Chemical Techniques

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

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

3

4

Hydrogen Bonding

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

Quantum Chemical Framework

5

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

Abbreviation

Common example

H

1st Row atoms

MBS

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

1s

6-311G

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

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

6-31G**

ls i , ls o ,P

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

DZP

lsi,lso,p

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

DZ2P

1s

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

sv

DZ

SVP

+

1si,

1s0

ls

1s

i,

6-31 + G**

o,Pi,Po

1s i ,ls o

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

a b

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

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

6

Hydrogen Bonding

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

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

Quantum Chemical Framework

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

8

Hydrogen Bonding

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

Quantum Chemical Framework

9

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

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

10

Hydrogen Bonding

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

Quantum Chemical Framework

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

1.2 Definition of a Hydrogen Bond

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

12

Hydrogen Bonding

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

Quantum Chemical Framework

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

14

Hydrogen Bonding

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

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

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

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

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

16

Hydrogen Bonding

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

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

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

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

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

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

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

Quantum Chemical Framework

17

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

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

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

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

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

The rotational entropy of each species is

18

Hydrogen Bonding

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

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

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

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

Quantum Chemical Framework

19

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

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

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

20

Hydrogen Bonding

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

Dimer

18 0.89 1.842 4.429 6.271 2 0.89 1743 3958 4109

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

14.02 0.001 34.61

10.34 0.004

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

20.93 11.02

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

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

Etrans Erot EZPVE E vib,therm

E AH Strans Srot S vib S G

-8.23

-0.89 -0.89 +2.59 +1.76

-5.66 -6.25 -32.54 0.26 11.01 -21.28 0.09

Quantum Chemical Framework

21

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

22

Hydrogen Bonding

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

1.5 Sources of Error

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

Quantum Chemical Framework

23

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

1.6 Basis Set Superposition

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

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

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

24

Hydrogen Bonding

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

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

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

Quantum Chemical Framework

25

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

26

Hydrogen Bonding

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

Quantum Chemical Framework

27

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

28

Hydrogen Bonding

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

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

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

Quantum Chemical Framework

29

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

30

Hydrogen Bonding

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

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

Quantum Chemical Framework

3I

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

32

Hydrogen Bonding

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

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

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

Quantum Chemical Framework

33

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

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

EES EEX EPL ECT EMIX

AE a

All values in kcal/mol

STO-3G

4-31G

-4.2 4.0 -0.1 -4.8 0.1 -5.1

-8.9 4.2 -0.5 -2.1 -0.3

227

-7.7

6-31G* -7.5 4.3

-0.5 -1.8 -0.1 -5.6

34

Hydrogen Bonding

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

E

-18.2

5.1 -8.2 -3.4

5.2 -19.5

3.0

-23.5 11.5 -14.6 -6.0 10.8 -21.7

2.75 -31.6 25.8 -28.5 -12.3 24.8 -21.9

2.50 -44.8 57.3 -180.9

-30.1 183.7 -16.2

a

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

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

Quantum Chemical Framework

35

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

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

BSSE

-7.8 -2.4 - 10.5 -0.6 6.2 -0.5

NEDA -7.8 -13.3 -17.8

24.8 -1.1

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

Quantum Chemical Framework

37

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

38

Hydrogen Bonding

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

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

IND EES (12) ES

(20) disp a

IX -5.6

-5.0 -18.2 0.5 -1.7

3.0

-8.8 -8.3 -23.5

0.6 -2.9

2.75 -14.8 -14.9 -31.6 0.8 -5.0

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

2.50 -27.2 -28.7 -44.8 1.0 -8.9

Quantum Chemical Framework

39

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

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

GD

S2

WTS2

SCF level ES(10) HL

exch

ind(20)

EdefSCF ESCF

-7.51 4.39 -1.6a9 -1.58 -4.70

-6.36 4.16 -1.79 -1.72 -3.92

-6.19 4.14 -1.84 -1.81 -3.86

-6.22 4.14 -1.86 -1.83 -3.92

0.54 -1.41 0.59 -0.28

0.49 -1.43 0.69 -0.26

MP2 level (I2)

ES

disp

(20)

exchange/deformation E(2) a

All values in kcal/mol255.

0.08 -0.90 0.76 -0.06

0.55 -1.17 0.62 0.01

40

Hydrogen Bonding References

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

Quantum Chemical Framework

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

48

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

Quantum Chemical Framework

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

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49

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50

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

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5I

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

2

Geometries and Energetics

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

I

52

Geometries and Energetics

53

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

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

54

Hydrogen Bonding

Figure 2.1 Equilibrium geometry of XH...ZH3.

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

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

H 3 As

HF 11.8

3.7 3.6

HC1 7.3 3.3 1.9

HBr 5.7 1.6 1.5

Geometries and Energetics

55

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

H 3 N ... HC1

11.8 15.1 2.728 2.693 0.022 0.028

11.0 3.297 3.144 0.023 0.040

- ESCF __

EMP2

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

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

H3P...HF

9.3

H3P...HCl

4.1 6.0

2.1 4.4

3.455 3.291 0.006 0.012

4.166 3.802 0.004 0.011

a

Stretch of HX bond caused by formation of complex.

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

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

-BSSE

-- ( Eelec -- BSSE)

Basis Set

SCF

MP2

SCF

MP2

SCF + MP2

SCF

MP2

SCF + MP2

6-31G**

1.07 1.21 1.48 0.69 0.35 2.43

1.16 1.21 1.59 1.23 0.79 1.87

11.79 11.85 11.08 11.31 10.42 12.27

1.48 1.71 2.28

13.27 13.56 13.37 12.94 12.08 14.20

10.72 10.64 9.61 10.62 10.07 9.83

0.32 0.51 0.69 0.40 0.87 0.07

11.04 11.15 10.30 11.02 10.94 9.90

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

1.63 1.66 1.93

56

Hydrogen Bonding

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

Geometries and Energetics

57

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

58

Hydrogen Bonding

prior to close approach of monomers

effect of ghost functions

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

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

(HF)

(NH3)

-0.022 -0.028 -0.026 0.018 -0.031 -0.021

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

(FH-NH3)

4.781 4.719 4.773 4.453 4.758 4.661

a corrb

0.918 0.932 0.860 1.099 0.838 0.897

0.862 0.888 0.965 0.998 0.967 0.982

Geometries and Energetics

59

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

where R represents the intermolecular separation, due to the ion-dipole term of the electrostatic interaction. It was found9 that this estimation of the secondary basis set superposition can be comparable to, and in many cases larger than, the primary BSSE. Whereas the primary effect is always negative, the secondary term can take either sign, belying any assumption of cancellation between the two in the general case. Of course, the energetic consequences of error in the dipole moment would be less severe when the partner is a neutral molecule, as in a typical H-bond, rather than the ionic Li+. Nonetheless, the results provide a cautionary note in calculations of this type. Clearly the issue of how to handle secondary superposition error is a thorny one. Szczesniak and Scheiner10 have demonstrated that it is possible to avoid the problem with a judicious choice of basis set. Focusing again on the strong interaction between NH3 and Li + , they demonstrated that a well-tempered basis set can not only reproduce fairly accurately the molecular properties of the subunits, but can also yield good total SCF energies. Superposition errors are also quite small. Positioning of ghost orbitals, even as close as 2 A from the NH 3 , yields a negligible change ( Cl > Br. H2O forms the strongest H-bonds; the distinction between H2S and H2Se is a small one. Any of these H2Y proton acceptors form a weaker complex than the corresponding H3Z molecule of the same row of the periodic table. Hinchliffe also found that the complexes involving H2O were of type B while those incorporating H2S or H2Se were of type A, with angles of about 110°. An additional set of SCF data emerged from calculations by Hannachi et al.15 who paired water with each of the hydrogen halides listed in Table 2.7, optimizing the geometry using a pseudopotential basis function of polarized split valence quality. The data echo the above trends of a weaker H-bond as the X atom of HX comes from a lower row of the periodic table. This progressive weakening is reflected in smaller stretches of the H—X bond. It is also worth noting that the small nonlinearity of the H-bond and the preferred angle of the proton acceptor water molecule are virtually independent of the nature of the HX molecule.

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

HF 9.0 3.5 3.4

HC1 5.4 2.4 2.0

in

HBr 4.3 1.6 1.1

Geometries and Energetics

63

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

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

H2O...HF

H2O...HC1

H 2 O ... HBr

H 2 O ... HI

8.2 2.702 0.012 3.8 135.9

4.9 3.268 0.013 2.4 140.1

4.1 3.496 0.010 2.5 139.4

2.5 3.830 0.006 2.6 139.0

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

Table 2.8 Calculated energetic (uncorrected for BSSE16J7.

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

See Reference 18.

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

H 2 O ... HF

H2O...HC1

H 2 S ... HF

H2S...HC1

7.8 9.6 2.71 2.65 2.66 0.012 0.017 140 129 3.1 4.5

4.2 6.6 3.37 3.19 3.21 0.009 0.015 140 130 2.8 0.9

3.9 6.3 3.36 3.20 3.25 0.007 0.011 100 98

2.2 5.0 4.09 3.75 3.81 0.005 0.011 101 92 1.5 1.2

1.3 -0.3

64

Hydrogen Bonding

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

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

Geometries and Energetics

65

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

SCF MP2

H 2 O ... HF

H20...HC1

H2S...HF

H S...HC1

0.13 0.49

0.15 0.33

2.96 3.65

1.59 2.52

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

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

66

Hydrogen Bonding

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

Geometries and Energetics

67

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

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

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

H2O..HBr

H2O..BrH

H2O..HI

3.496 4.10 5.04

3.228 0.22 0.59

3.830

2.47 3.35

H2O..IH

3.192 1.71 1.99

68

Hydrogen Bonding

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

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

ES

EX

POL

CT

Totala

-17.36 -9.21 -4.33 -4.49

17.67 6.94 5.16 4.50

-2.96 -1.44 -0.69 -1.35

-3.88 .-1.62 -1.47 -1.08

-6.65 -5.35 -1.37 -2.45

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

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

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

.D a , aua xx, a

au

HC1

HBr

1.094 21.1 19.6

0.819 28.5 22.3

The molecular axis is defined as the Z-axis.

HI

0.447 44.4 32.9

Geometries and Energetics

69

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

Figure 2.6 Geometry of HYH...ZH3.

70

Hydrogen Bonding

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

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

MP2

6-31G**

+VPS

+ VPs(2d)s

+VPM

3.050 0.008 2.1 100.5 116.2

3.074 0.008 3.7 100.8 116.0

3.096 0.007 4.6 101.1 116.1

2.942 0.013 4.8 101.8 117.7

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

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

- ESCF _ E MP2

6-31G**

+VPS

+VPs(2d)s

5.61 6.35

5.10 5.92

4.59 5.72

Geometries and Energetics

71

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

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

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

3.015 -6.5 -5.3 -26.7 2.7

MP2

2.902 -9.1 -7.8 0.1

72

Hydrogen Bonding

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

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

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

(degs)

(degs)

2.687 2.788 2.81 2.82 2.83

8.1 7.9 9 7.2 6.0

2.768 2.762 2.759 2.72

6.4 6.9 5.5 10±6

- Eelec (kcal/mol)

Reference

124.1 117 115 116.5 123.2

8.0 4.7 4.7 3.7a 3.8

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

120.1 — 112 117±6

4.6 5.7 5.0

[56J [571 [57] [58J

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

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

Corrected for BSSE

74

Hydrogen Bonding

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

SCF

MP2

MP3

MP4

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

5.97 3.98 5.87 3.75 5.06 3.71

7.45 4.69 7.58 4.61 6.22 4.66

7.02 4.62 7.13 4.61 5.82 4.64

7.34 4.71 7.52 4.66 6.15 4.71

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

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

SCF MP2 MP4 CCSD CCSD(T)

no cc 2.821 2.737 2.735 2.745 2.732

cc 2.824 2.753 2.749 2.759 2.745

(degs)

(degs)

6.8 6.4 6.6 6.7 6.7

119.7 111.6 110.4 112.1 110.8

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

cc 3.66 4.38 4.44 4.31 4.49

Geometries and Energetics

75

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

(degs)

(degs)

- E

(kcal/mol)

Reference

SCF

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

3.986 3.96 4.156 4.111 4.210

7 2.6 14.6 10.3 11.3

102 83.0 — 97.3 90.3

2.1 3.6a 1.1 1.0" 0.5a

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

— — 91.1 91.4 90.0

1.4a 1.6a 1.7 1.7 2.0

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

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

3.876 3.838 3.912 3.887 3.78 3.80

— — 6.6 6.1 8.0

a

Corrected for BSSE. Bond functions added.

b

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

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

3.465 3.294 7.4 93.2 1.77 2.39

HC1...HF 3.499 3.365 8.2 119.7 1.93 2.45

76

Hydrogen Bonding

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

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

Dav2 Dav2(s)

Pople

(HF),

(HC1),

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

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

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

Geometries and Energetics

7777

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

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

R(A)

(degs)

(degs)

4.11 4.22 4.49 4.38 4.48 4.76 4.77 4.87 5.14

13.4 7.8 3.2 16.0 8.2 3.2 18.2 8.0 3.3

90.9 90.4 92.2 88.3 89.8 92.1 85.7 89.8 91.7

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

Counterpoise corrected.

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

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

78

Hydrogen Bonding

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

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

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

- E elec (kcal/mol)

(degs)

SCF

MP2

SCF

MP2

SCF

MP2

2.740 2.797 2.971 2.964 2.999 3.035 3.026 2.98 = 0.01

2.802 2.913 2.901 2.910 2.911 —

124.0 124.6 117.5 130.3 143.1 130.8 133.2 123±10

107.9 102.7 128.9 135.8 123.2 —

5.9 10.9 5.6 5.4 4.8 4.1 4.0

12.6 7.4 7.1 6.1 5.4 —

Geometries and Energetics

79

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

80

Hydrogen Bonding

Figure 2.9 Proposed structures of H2Y dimer.

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

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

SCF MP2

R(S-S) (A)

(degs)

4.600 4.161

6.8 3.2

(degs) 97.1 89.6

Eelec

(kcal/mol) -0.8 -2.3

(

E+ZPVE)

(kcal/mol) 0.1 -1.1

Geometries and Energetics

81

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

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

HOH-OH2

HOH-SH2

HSH--OH2

HSH-SH2

2.922 5 117

3.811 0.2 100

3.622 5 131

4.489 4 104

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

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

~AE

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

5.54 5.42 5.85 5.52 5.42 5.44 5.66

82

Hydrogen Bonding

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

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

EclccMP2 H° G°

HOH-OH2

HOH-OHSiH3

-4.70 -2.96 2.97

-4.56 -2.94 4.97

SiH3OH-OH2 -6.25 -4.64 3.13

Geometries and Energetics

83

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

- EelecSCF - EelecMP2 - H (0K)

SiH3OH--OHCH3

SiH3OH-OHSiH3

CH3OH-OHCH3

CH3OH OHSiH3

5.69 6.60 5.19

4.59 5.59 4.30

4.21 5.09 3.90

3.61 4.30 3.20

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

84

Hydrogen Bonding

Figure 2.10 Syn and anti conformers of H2O + HOC1.

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

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

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

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

R(NN) (A)

- Eelec .

BSSE

3.08 3.31 3.44 3.44 3.44 3.54

3.8 4.1 2.9

3.05 1.97 0.53 0.55 1.04 0.68

3.1 2.4 2.4

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

(NH3) (D)

1.66 2.28 1.93 1.87 1.84 1.50

Geometries and Energetics

85

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

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

86

Hydrogen Bonding

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

Geometries and Energetics

87

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

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

Linear

Cyclic

1.67 2.86 2.78 2.66 2.85

1.56 2.88 2.78 2.67 2.87

88

Hydrogen Bonding

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

Geometries and Energetics

89

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

2.7 Carbonyl Group

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

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

90

Hydrogen Bonding

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

r(O2H1)

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

1.88 1.97 2.04 2.12

r(CO2)

r(O1H1)

0.002 0.006 0.005 0.005

-0.001 0.002 0.004 0.004

(O1H1O2)

- E

177.9 141.0 139.4 146.3

3.38 9.14 6.69 5.25

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

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

Geometries and Energetics

91

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

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

I

II

3.042 2.096 176.7 0.004 0.003

2.945 2.107 146.7 0.005 0.004

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

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

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

I

II

4.6 5.6 4.0 4.7 4.6 4.5 2.9 3.9

5.2 6.7 3.9 5.0 4.9 4.8 3.0 4.6

92

Hydrogen Bonding

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

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

R(Cl--O)

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

-( E + ZPVE)

MP2

CPF

DZP

TZ2P

DZP

TZ2P

DZP

TZ2P

3.356 0.003 0.008 172.7 132. 1 4.3 2.9

3.375 0.003 0.008 169.1 124.2 3.6 2.2

3.166 0.004 0.018 168.0 113.7 6.2 4.3

3.124 0.005 0.021 164.4 107.3 5.7 3.8

3.246 0.004 0.011 167.8 115.7 5.3 3.6

3.208 0.004 0.014 163.2 109.0 4.8 3.1

Geometries and Energetics

93

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

94

Hydrogen Bonding

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

1.867 1.871 2.087 2.088

r(HX)

r(C=O)

0.007 0.007 0.010 0.009

0.006 0.010 0.005 0.009

Eelec -4.57 -4.12 -3.45 -3.22

Eelec +

ZPVE

-2.52 -1.84 -1.85 -1.60

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

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

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

HC1

I

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

ESCF

_AE MP2

II

I

II

4-31G

6-31G**

4-31G

6-31G**

4-31G

6-31G**

4-31G

6-31G**

2.586 0.019 0.014 -0.020 97.5 24.7 14.52

2.636 0.015 0.013 -0.016 98.3 24.6 11.75 14.72

2.618 0.009 -0.005 0.018 112.8 22.8

2.740 0.004 -0.003 0.012 110.3 27.5 5.03 6.70

3.180 0.011 0.008 -0.011 105.9 24.7 6.98

3.308 0.009 0.006 -0.009 114.3 17.2 5.08 6.64

3.262 0.008 -0.004 0.011 126.0 2.0 4.06

3.435 0.004 -0.003 0.008 129.1 1.7 2.53 3.86

8.60

96

Hydrogen Bonding

Figure 2.14 Possible configurations of HCOOH + HX.

more effective proton acceptor than does the hydroxyl oxygen, and will be the preferred site in most cases. The preference of site I over II is confirmed by observation of the HCOOH---HF complex in solid Ar170. The interaction energies in the last row of Table 2.36 would probably be reduced by 1 or 2 kcal/mol were BSSE corrected. One may hence consider the electronic contribution of the binding energy of HF to formic acid to be perhaps 12-13 kcal/mol, stronger than the case where formaldehyde is the acceptor molecule. Likewise, the interaction of formic acid with HC1 would be some 5 kcal/mol, only slightly stronger than the comparable data for formaldehyde in Table 2.34. Other indicators of the stronger interaction with the carbonyl oxygen emerge from the perturbations of the internal geometries. Considerably larger stretches of the HF or HC1 bond are seen for site I, as much as 0.015 A for HF with the larger basis. As in the case of formaldehyde, H-bonding to the carbonyl oxygen stretches the C=O bond. Interestingly, in the case of formic acid, a concomitant contraction of the C—O bond to the hydroxyl oxygen is observed as well. This contraction is larger in magnitude than the C=O stretch. Analogously, formation of a H-bond to the hydroxyl oxygen stretches the C—O bond and shrinks C=O. There is an interesting difference in angle of approach amongst the various types of Hbonds. First, the 9 (X OC) angles are smaller for site I than for II. (These angles are insensitive to basis set choice.) The specific halogen atom makes a difference in that the angles are appreciably larger for HC1 than for HF. Another intriguing distinction is the greater deviation from linearity for HF, as indicated by the larger values of the (HX O) angle. Zheng and Merz48 have paired HCOOH with water which serves as the proton donor. The geometry optimized with a 6-31G* basis set is like geometry I for HCOOH plus HX, and is illustrated in Fig. 2.15 as the syn geometry. Table 2.37 indicates that correlation strengthens the H-bond and shortens the interoxygen distance by some 0.07 A. Comparison with Table 2.36 indicates the interaction of HCOOH with water is slightly weaker than with HF, but certainly stronger than with HC1. The last row indicates that HOH and HCOOH are bound with respect to one another, that is, G 2, again at the correlated level. One can see that neglect of anharmonicity yields F—L frequencies that are too high, for all F—L including FLi. The effects are much smaller on the F..N stretches.

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

..

F-L F..N

..

FD NH3

FH NH3

SCF

MP2

SCF

3773 240

3331(3514) 263(265)

2752 236

MP2

2427(2527) 255(262)

FLi NH3

SCF

MP2

932 292

947(1010) 281(272)

154

Hydrogen Bonding

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

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

01

00

03

00 00

04 05

00

10

02

00 00

11

00 00

13 14 15

00

12

FH NH3

FD..NH3

FLi..NH3

263 509 739

255 494 717 931

281 557 829

1096

1141 2427 2735

1272

958 1172 3331 3644 3921 4180 4425 4658

2995 3239 3470 3691

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

947

1551 1822

Vibrational Spectra

155

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

- Vz//Vzz°

0.618 0.196

0,17 0.06

..

C1H NH3 C1H..PH3

-

2 1/2




12.7 18.8

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

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

..

C1H NH3 C1H- PH a

SCF

ACPF(+CC) a

Vzz/ R SCF

0.924 0.340

0.862(0.848) 0.285 (0.282)

0.83 0.30

With full counterpoise correction.

-

z

SCF

ACPF

0.486 0.304

0.499 0.279

156

Hydrogen Bonding

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

3.5 H2Y...HX

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

Vibrational Spectra

157

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

MP2

SCF TZ2P

DZP

Mode

mon

v

TZ2P

DZP

Vmon

Av

4471

-264

Vmon

V Vmon

vmon

V

HF

4511 4166 1752 4289

V1 V2 V3

-259

H2O -9 0 -10

4128 1760 4228

-7 5 -10

-340

4221

3913 1671 4059

2 -3 -18

4154 3858 1657 3980

-363 -10 -3 -20

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

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

Amon_

Raman (A4/amu)

A A dim /A mon

Smon_

Sdim/Smon

HF

147

4.6

26

1.6

70 4 29

1.0 0.7 1.0

H2O v1 v2 v3

14 92 70

5.5 1.0 1.6

158

Hydrogen Bonding

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

v S C F (cm - 1 )

vMPZ(cm-1)

A IR (km mol - 1 )

220 182 234 644 786

270 232 252 742 862

87 155 3 226 194

SRamen

(A4/amu) 0.4 0.8 2 1 0.2

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

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

V mon

V

A mon

Adim/Amon

HC1 3141 v1 v2 v3

4139 1759 4244

-105 H2O -4 1 -3

56

6.4

19 103 91

1.9 0.9 1.3

Vibrational Spectra

159

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

Frequency ( c m - 1 )

Intensity (km mol - 1 )

118 459 351 143 94

77 38 33 28

3

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

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

C1H..OH2 Harmonic approximation 3035 150 155 Anharmonic data

BrH..OH2

IH..OH2

2656 129 120

2390 59 98

XH - vs ..

xo

4243 321 235

XH - vs

4019 370

2847 257

2501 209

2307 76

X..O

226

146

121

97

160

Hydrogen Bonding

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

frequency, cm-1

intensity, km mol-1

107 507 399 10 35 -170

3 37 51 4 0

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

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

0.431 0.290

-Vzz/Vzz° 0.12 0.08

-z

z

0.281 0.235

< 2 > 1/2 13.8 17.8

Vibrational Spectra

161

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

ClH..OH2 ClH..SH2 a

-

vzz / R a

SCF

ACPF (+CC)

0.546 0.394

0.499 (0.488) 0.361 (0.359)

z

SCF

SCF

ACPF

0.50

0.319 0.293

0.324 0.295

0.30

With full counterpoise correction.

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

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

162

Hydrogen Bonding

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

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

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

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

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

Vibrational Spectra

163

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

mean 2 xx yy zz

xy

Noninteracting

Interacting

16.48 1.48 16.63 16.26 16.54 -0.68

16.60 8.20 18.09 15.67 16.04 -0.02

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

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

( ')2

mean'

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

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

a

mean'

Units:

in aoe2Eh- 1

-1/2

S

P

mono

nonint

ilnt

mono

nonint

int

mono

nonint

int

mono

nonint

int

4.75 4.75 0.0 0.0 -0.24 -0.24

5.84 2.60 -2.08 0.0 -0.18 -0.27

6.28 2.33 2.04 0.0 -0.09 -0.22

26.9 26.9 51.6 51.6 1.78 1.78

18.0 38.8 48.5 51.6 2.05 1.50

49.8 34.2 54.9 50.6 1.88 2.76

0.07 0.07 0.75 0.75 0.55 0.55

0.03 0.25 0.37 0.75 0.64 0.48

0.08 0.27 0.40 0.75 0.71 0.63

704 704 211 211 8.8 8.8

970 337 312 211 9.2 8.1

1240 282 334 207 7.9 12.5

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

-1

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

Vibrational Spectra

165

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

(H2O)2

sym stretch

bend asym stretch a

V mon

vd

va

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

-45 27 -25

-5 -1 -8

v

vd

va

—5 4 -3

1 -2 1

vmon

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

Taken from reference 53.

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

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

(H20)2 Amon

Ad

Aa

Amon mon

1.71 1.06 1.52

6.7 5.0

sym stretch

17(2.2)

10.41

bend asym stretch

97 (54) 58 (45)

0.91 1.79

a

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

1.8

Ad

Aa

3.17 0.94 0.54

0.39 1.09 0.56

166

Hydrogen Bonding

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

Intensity (km/mol)

(H2O)2

(H2S)2

(H2O)2

605 375 175 145 142 121

217 130 43 67 62 40

185 78 127 64 237 142

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

(H2S)2 20 21 0.7 9.5 48 34

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

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

asym stretch

mon

Vd

Va

V mon

Vd

Va

4146 3865 3878 3838 3657 3628

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

-7

4247 3986 3986 3947 3756 3724

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

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

0

-8 -9 -57 -3

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

3894 3861 3943 3845 3638

asym stretch

vd

va

-51 -95 -17 -72 -64

-5 -12 +46 -8 -4

vmon

4238 3983 4042 3951 3733

bend

vd

va

vmon

vd

va

-22 -31 +31 -26 -24

-10 -17 +44 -13 -7

1764 1663 1698 1679 1596

2 2 14 1 3

23 29 41 28 21

168

Hydrogen Bonding

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

sym stretch bend asym stretch

V mon

vd

2842 1332 2857

-1 +4 +2

MP2 va

+1 -1 - 1

v

mon

2694 1233 2720

vd

vd

-35 12 -10

-5 -3 -5

Vibrational Spectra

169

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

gas phase Ne matrix Ar matrix Kr matrix ArN cluster

free OH

bridging OH

3730 3734 3709 3701 3714

3601 3590 3574 3569 3576

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

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

FA

SCF

MP2

obsd

SCF

-2 -3 -7

-8 -5 -23

-16 -4 -14

-50 + 28 -28

12 -72

33 -111

14 -128

-16 -0

H2O V1 V2

CH3OH vCO vOH

170

Hydrogen Bonding

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

normal decoupled

CPa anharmonic

CPa

(CH3OH)2

(SiH3OH)2

SCF

MP2

SCF

MP2

SCF

MP2

-74 -76 -73 -89 -87

-133 -136 -123 -164 -150

-71 -72 -71 -86 -85

-156 -142 -191 -177

-111 -112 -111 -136 -135

-208 -196 -259 -245

_

a

With counterpoise corrections.

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

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

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

Calc

Expt

158 17 30 55 70 90

162 22 35

55 65 91

Vibrational Spectra

171

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

3.7 Expected Accuracies

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

172

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

A (HF)2

(HF)2

Basis set

HF

A

D

MINI-1 3-21G 4-31G

4183 4064 4124 4103 4440 4494 4314 4485 4488 4452

4171 4041 4102 4070 4397 4453 4279 4449 4448 4418

4131

DZ

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

3958 4047 4021 4349 4406 4242 4401 4394 4366

A

D

23 79

279 371

77

110

395

133

182

374

171 168

170 167

450 459

HF

8.6 33

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

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

+VP (2d) Extended

Ad/Ain

A

D

A

D

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

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

2.7 2.4 1.4

32 11 5.1

1.4

2.8

1.0 1.0

2.6 2.7

Vibrational Spectra

I 73

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

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

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

D

A

v1

v2

V3

3897 3811 3960 4149 4071 4128 4130

1816 1799 1767 1770 1797 1746 1747

4127 3945 4098 4267 4190 4235 4231

v1

V2

3900 3817 3979 4144 4068

1806 1785 1771 1767 1806

4123

1752

V3

v1

V2

V3

4115 3945 4121 4258 4181 4226

3843 3712 3907 4102 4028 4076

1852 1845 1813 1798 1826 1767

4071 3891 4085 4240 4165 4213

Vibrational Spectra

175

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

A

Basis set

V1

V2

V3

V1

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

3 6 19 -5 -3 -5

-10 -14 4 -3 9 6

-12 0 23 -9 -9 -9

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

V2

36 46 46 28 29 21

V3

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

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

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

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

Basis set 3-21G

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

v1

A

v2

v3

v1

v2

v3

v1

98 172 112 110

45 131 89 103

80

9

5

4 16

125 105

54

8 26

14

91

0.05

58 81

D

22

v2

v3

282

98

283

138

52 110

184 191

99 65

104 127

176

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

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

D

V1

V2

V3

V1

V2

V3

100 1.9 1.6 1.6

1.2 1.4 1.1 1.2

5 2.4 1.5 1.3

5640 71 11 14

1.2 1.1 0.9 0.7

6 2.0 1.8 1.6

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

V

A

mon

Mode

SCF

SCF

v1

-103

V2

4129 1727

V3

4242

MP2

Adim/Amon

n mo

SCF

SCF

MP2

25 94 85

14 1.1 1.2

88 0.4 1.5

0.4

2.5

6

1.9

17 17 46 0.9 1.3 3.7

H2O

Vstr Vstr

37 -16

-167

60

-45 NH3

3709 3847

Vbend

1096

Vbend

1794

1 4 -15 83 0 -2

3

2 0

2.0

36

242

0.9

-4 -13

29

0.5 0.8

Vibrational Spectra

177

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

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

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

A(km mol - 1 ) a

Mode

calc

expt

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

115 669 443 181 219 50

202 662 430 180 411 20

a

See reference 74.

calc 3.3 215 124 59 37 0.5

178

Hydrogen Bonding

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

Mode

calc

expt

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

165 305 242 64 31 37

162

62

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

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

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

vmon proton donor 3929 3898 3738 3710 1866 1830

822 proton acceptor 3926 3922 1836 1829

817

v

A

-28 -59 -22 -50 45 11 200

0.09 1.78 0.01 1.52 0.83 0.77 10.07

-31 -35 15 8 195

0.36 0.38 1.15 1.17 15.71

intermolecular Mode a', v

a' a', bend a", bend

a' a", bend

V

112 134 381 314 112 imaginary

A 0.76 1.16 1.71 1.20 0.76

Vibrational Spectra

I 79

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

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

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

477 255 159 96 93

v

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

A

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

0 2.06 0 0.61 5.55

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

Carbonyl Oxygen

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

180

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

a1 bend cbend

linear

cyclic

linear

cyclic

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

-11 -7 -12

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

75 — 3 — 87 — 2.2 — 1.9 — 1.1

41 -7

3.10.1 Relationship between

A dimer /A monomer

-2

-3 -12 39

15 8 23 11 -18

E and v

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

Vibrational Spectra

181

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

SCF/TZ2P(f,d) Mode

V

mon

v1, stretch V 2 , stretch v3, bend

4073 1780

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

3173 3125 1980 1645 1343 1341

a

In solid Ar.

4261

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

expta v

v

4023 3802 1722

-61 15 -20

-58 21 -24

3055 3004 1790 1550 1259 1214

25 -14 -3 7 42 7

16 -6 1 5 16

mon V

2

182

Hydrogen Bonding

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

A (km m o l - ' )

SCF/TZ2P(f,d)

CISD/TZ2P

CCSD/TZ2P

151 329 65 457 148 39

179 359 95 501 171 29

180 354 101 493 173 20

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

expt

a

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

250,261 435,439 68

a

ln solid Ar.

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

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

Mode

Vmon

SCF/TZ2P

V

Vmon

MP2/DZP

V

Vmon

CPF/DZP

V

Vmon

V

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

3226 3149 2009

vv CH2 scissor

1659

v6, CH2rock v10, CH2wag

1370 1338

30 22 -10

-1 0 6

3168 3096 1992

28 19 -11

3128 3040 1774

1655

0

1572

1373 1339

2 7

1284 1217

46 32 -11

-4 1 4

3089 3012 1796

1566 1281 1197

43 30 -11

-3 2 7

Vibrational Spectra

183

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

SCF/DZP

SCF/TZ2P

MP2/DZP

CPF/DZP

127 394

120 391 36 384 122

170 554 54 521 186

155 478 45 460 158

38 399 112

184

Hydrogen Bonding

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

A dim

v3, stretch

412

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

85 57 211 21 17 1.7

A

TZ2P dim/Amon

HC1 8.4 H2CO 0.7 0.8 1.3 1.5 0.9 0.9

A

dim

A

dim /A mon

388

7.4

70 41 186 26 19 2.3

0.7 0.7 1.2 1.5 0.9 1.0

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

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

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

DZP

TZ2P

16 53 12 53 0.1

22 43 12 4) 0.0

Vibrational Spectra

185

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

vmon

Av(HOH)

Av(CH3OH)

CH 2 =NH V

OH

V

CN

V

NH

1866 3686

-45 -5 21

-173 -5 20

-49 -4 25

-202 -3 25

cis-CH 3CH=NH V

OH

V

CN

V

NH

1884 3659

trans-CH3CH=NH V

OH V

CN

-y

NH

1896 3699

-51 -6 14

-210 -5 14

-46 -2

-177 -1

-52 -5

-213 -3

CH2=NCH3 V

OH

V

CN

V

OH

V

CN

1901 CH3CH=NCH3

1932

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

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

186

Hydrogen Bonding

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

V

HF

V

HC

V

CN V

bend a

V

DZP

mon

V

4117

-127

3695

2384 911

-13 12 26

Vmon

4511

3638 2406 861

expta

TZ2P v

Vmon

v

Vmon

HF -189

4471

-171

4138

-251

3600 2408 869

-7

3442 2129 727

24

HCN 0

31 17

19 12

v

Harmonic frequencies.

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

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

4-31G

DZP

TZ2P

expt

193 561 86

176 645 108

159 581 84

155±1O 55±3 7()±24

Vibrational Spectra

! 87

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

vmon

CEPA-1

expt

v

vmon

v

-164

4160 3982

-168 -245

3439 3325 2155 2128

-3 1 28 27

vmon

v

HF V

HF

4458 4295

harm anharm

HCN V

harm

V

anharm harm

HC

CN

anharm

3604 3505 2407 2385

-242

-2 0

21 19

3716

-245

3310

1

2121

24

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

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

188

Hydrogen Bonding

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

Mode

V

mon

MP2 V

V

mon

Anharmonic V

expt

v

Intramolecular

HF V

4322

HF

-189

3955

-238

-226

HCN V

HC

V

CN

3638 2437 878

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

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

5 45 8

2 53 0 101 169 539

72 ±4

163 550

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

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

A 147

V

ITC

V

V

CN

bend

76 9.7 66

Raman (A4/amu)

A..

/A

HF 4.8 HCN 1.1 2.6 0.9

S

S,. /S

26

2.5

14.1 48 1.8

1.1 1.1 1.1

Vibrational Spectra

189

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

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

SRaman (A4/amu) 0 4 6

3 346 30

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

SCF SCF+BSSE MP2 MP2+BSSF

Anharmonic

v(FH)

..

v(F N)

v(FH)

v(F .. N)

4202 4202 3785 3797

167 164 201 201

4010 4009 3604 3636

168 151 201 171

190

Hydrogen Bonding

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

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

SCF + BSSE

MP2

MP2 + BSSE

100> 100> 100> 100>

101> 102> 103> 104>

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

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

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

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

100> 100>

110> lll>

4028 4217(189)

4019 4209 (190)

3611 3838 (227)

3642 3839 (197)

101 101> 101

110> lll> 112>

3864 4053 (189) 4216(163)

3869 4059 (190) 4211(152)

3410 3638 (228) 3841 (203)

3471 3669 (198) 3843(174)

102> 102> 102> I02>

110> 111> 112> 113>

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

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

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

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

a

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

Vibrational Spectra

191

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

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

TZ2P

expta

-189

-162

-162

-29 153

-26 131

-11

DZP

HCN V

HC

V

CN

V

bend

V

a

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

vbend(e) a

Harmonic frequencies.

NH3 -9 -18 93 0

-7 -12 68 1

192

Hydrogen Bonding

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

V V

V

Raman

dim/smon s

HCN 4.7 7.3 1.2 NH3

HC

CN

bcnd

0.7 8.3 1.8

100a

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

0.9 1.0 0.2 0.9

3.4 1 1.1

vbend(e) a

IR A dim /A mon

Intensity in monomer too small for ratio to be meaningful.

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

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

MP2b

Mode

DZP

TZ2P

6-3 1G*

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

expt

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

160 323 120

146 302 113

190 308 123

158 283 132

141 ± 3

a b

Data from Reference 6. Data from Reference 97.

Vibrational Spectra

193

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

Mode

AIR(km mol-1)

SRaman (A4/amu)

2 158 4

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

0 0 10

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

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

V

HC

V

CN

V,bend,

V

HC

V

CN

V

bend

a

Harmonic frequencies.

calc

-69 -11 76 -2 16 13

expta HCN (donor) -65 —9

HCN (acceptor) 11 8

Intensity factor (Adim/Amon) calc

expt

4.1 4.8 1.3

1 2

0.8 4.0 1.1

1.0 3

194

Hydrogen Bonding

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

A(kmmol-1)

Mode

calc

expt

calc

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

122 159 60

119

2 122 10

40a

a

See references 104 and 105.

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

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

V

CH

V

CN

V

bend

V

CH

V

CN

V

a

bend

V mon

-64 -11

868

60

3618 2418

-7 9 13

Harmonic freqilencies 99 . See Reference 100.

b

v

3618 2418

868

expta

CPF V mon

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

v

V mon

v

-56 -3 76

3441a 2129a 721b

-65 -6 77

-1 13 15

344 la 2129a 721b

11 8 13

Vibrational Spectra

195

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

SCF

Mode

mon A

dim/Amon

Amon

expta

Adtm/Amon

dim/Amon

4.7 64.3 1.03

30±10

1.03 10.2 0.84

30±10

HCN (donor) V

CH

V

CN

V

bend

V

CH

V

3.7

75.2 10.3 83.0

0.93 3.1 0.88

62.3 0.12 86.7

3.9 1.0

HCN (acceptor)

CN

V

75.2 10.3 83.0

bend

62.3 0.12 86.7

a

See reference 99.

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

3.13

Amide

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

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

A(km mol - 1 )

Mode

SCF

CPF

SCF

CPF

H-bond stretch (v ) bend bend

106 139 48

116 125 38

1.3 128 13

2.8 84 16

196

Hydrogen Bonding

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

vvmon

donor

V

acccptor

in-plane(a')

(NCO) r(NH2) v(CN)

(CH) (NH2)

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

(CH)

618 1154 1368 1547 1773 1966 3193

14 30 21 —2

19 -11 24 -30 -29

3835

3982 out-of plane(a") 252

664 1175

253 78 6

4 11 18 1 3 -27 -4 -4

88 15 5

a

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

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

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

4-31G

6-31G**

6-31 + +G**

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

144 21 43

122 23 59

114 21 48

torsion bend strain

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

28; 28 72

15; 21 81

Vibrational Spectra

! 97

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

3.14

Summary

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

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

CN CO NH a

MP2

k

ka

k

ka

8.98 14.68 7.71

0.76 -1.10 -0.75

8.45 12.26 6.67

0.89 -0.82 -1.18

Difference between force constant in dimer versus isolated monomer.

198

Hydrogen Bonding

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

k(v ), mdyne/A — Eelec, kcal/mol

MINI-1

MID1-1

4-31G

6-31G**

MP2/6-31G**

0.27 16.0

0.17 20.3

0.15 17.2

0.12 13.4

0.15 17.4

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

Vibrational Spectra

199

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

200

Hydrogen Bonding

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

Vibrational Spectra

201

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

202

Hydrogen Bonding

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

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203

32. Karpfen, A., Bunker, P. R., and Jensen, P., An ab initio study of the hydrogen chloride dimer: the potential energy surface and the characterization of the stationary points, Chem. Phys. 149, 299-309 (1991). 33. Hannachi, Y., and Angyan, J. G., The role of induction forces in infra-red matrix shifts: Quantum chemical calculations with reaction field model Hamiltonian, J. Mol. Struct. (Theochem) 232,97-110(1991). 34. Tapia, O., and Goscinski, O., Self-consistent reaction field theory of solvent effects, Mol. Phys. 29, 1653-1661 (1975). 35. Person, W. B., and Zerbi, G., ed., Vibrational intensities in infrared and Raman spectroscopy; Elsevier, Amsterdam (1982). 36. Zilles, B. A., and Person, W. B., Interpretation of infrared intensity changes on molecular complex formation. I. Water dimer, J. Chem. Phys. 79, 65-77 (1983). 37. Gussoni, M., Castiglioni, C., and Zerbi, G., Charge distribution for infrared intensities: Charges on hydrogen atoms and hydrogen bond, J. Chem. Phys. 80, 1377-1381 (1984). 38. Schemer, S., Proton transfers in hydrogen bonded systems, 6. Electronic redistributions in (N2H7)+ and(O2H5)+, J. Chem. Phys. 75, 5791-5801 (1981). 39. Janoschek, R., Weidemann, E. G., Pfeiffer, H., and Zundel, G., Extremely high polarizability of hydrogen bonds, J. Am. Chem. Soc. 94, 2387-2396 (1972). 40. Swanton, D. J., Bacskay, G. B., and Hush, N. S., An ab initio SCF calculation of the dipole-moment derivatives and infrared-absorption intensities of the water-dimer molecule, Chem. Phys. 82,303-315(1983). 41. Bouteiller, Y., Mijoule, C., Karpfen, A., Lischka, H., and Schuster, P., Theoretical Vibrational investigation of hydrogen-bonded complexes: Application to CIH'-NH^, CIH"NH2CH3, and BrH-NH3, J. Phys. Chem. 91, 4464-4466 (1987). 42. Bouteiller, Y., Latajka, Z., Ratajczak, H., and Scheiner, S., Theoretical vibrational study of FX-NH3 (X=H,D,Li) complexes, J. Chem. Phys. 94, 2956-2960 (1991). 43. Del Bene, J. E., Person, W. B., and Szczepaniak, K., Ab initio theoretical and matrix isolation experimental studies of hydrogen bonding: Vibrational consequences of proton position in 1:1 complexes ofHCl and 4-X-pyridines, Chem. Phys. Lett. 247, 89-94 (1995). 44. Amos, R. D., Gaw, J. E, Handy, N. C., Simandiras, E. D., and Somasundram, K., Hydrogenbonded complexes involving HF and HCl: the effects of electron correlation and anharmonicity, Theor. Chim. Acta 71, 41-57 (1987). 45. Latajka, Z., and Scheiner, S., Structure, energetics and vibrational spectrum of H2O—HCl, J. Chem. Phys. 87, 5928-5936 (1987). 46. Hannachi, Y., Silvi, B., and Bouteiller, Y., Structure and vibrational properties of water hydrogen halide complexes, J. Chem. Phys. 94, 2915-2922 (1991). 47. Millen, D. J., and Schrems, O., Comparative infrared study of hydrogen-bonded heterodimers formed by HCl, DCl, HF and DF with (CH3)2O, CH3OH, and (CH3)3COH in the gas phase. Assignment of vibrational band structure in (CH3)2O-HCl, Chem. Phys. Lett. 101,320-325 (1983). 48. Barnes, A. J., and Wright, M. P., Molecular complexes of hydrogen halides with ethers and sulphides studied by matrix isolation vibrational spectroscopy, J. Mol. Struct. (Theochem) 135, 21-30 (1986). 49. Curtiss, L. A., and Pople, J. A., Ab initio calculation of the vibrational force field of the water dimer, J. Mol. Spectrosc. 55, 1-14 (1975). 50. van Duijneveldt-van de Rijdt, J. G. C. M., van Duijneveldt, F. B., Kanters, J. A., and Williams, D. R., Calculations on vibrational properties of'H-bonded OH groups, as a function ofH-bond geometry, J. Mol. Struct. (Theochem) 109, 351-366 (1984). 51. Swanton, D. J., Bacskay, G. B., and Hush, N. S., An ab initio SCF calculation of the polarizability tensor, polarizability derivatives and Raman scattering activities of the water-dimer molecule, Chem. Phys. 83, 69-75 (1984). 52. Amos, R. D., Structures, harmonic frequencies and infrared intensities of the dimers ofHJO and H2S, Chem. Phys. 104, 145-151 (1986).

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

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

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

4

Extended Regions of Potential Energy Surface

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

I

207

208

Hydrogen Bonding

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

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

Extended Regions of Potential Energy Surface

209

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

(HX)2

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

210

Hydrogen Bonding

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

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

Extended Regions of Potential Energy Surface

21 I

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

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

212

Hydrogen Bonding

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

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

Extended Regions of Potential Energy Surface

21 3

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

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

214

Hydrogen Bonding

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

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

Extended Regions of Potential Energy Surface

21 5

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

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Hydrogen Bonding

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

linear cyclic bifurcated a

ESXa

CT

POL

DISP

Total

-2.68 -2.91 -1.96

-1.28 -0.87 -0.37

-0.90 -0.47 -0.27

-0.32 -0.42 -0.28

-5.19 -4.67 -2.87

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

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

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

218 Hydrogen Bonding

Figure 4.7 Trifurcated type of arrangement of the water dimer.

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

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219

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

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

220

Hydrogen Bonding

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

Figure 4.9 Geometry of the water dimer equilibrium structure.

Extended Regions of Potential Energy Surface

221

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

222

Hydrogen Bonding

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

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

Extended Regions of Potential Energy Surface

223

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

4.5 Carbonyl Group

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

224

Hydrogen Bonding

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

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

Extended Regions of Potential Energy Surface

225

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

4.6 Amines

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

226

Hydrogen Bonding

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

4.7 Summary

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

Extended Regions of Potential Energy Surface

227

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

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