Molecular Theory of Solvation

MOLECULAR THEORY OF SOLVATION Understanding Chemical Reactivity Volume 24 Series Editor Paul G. Mezey, University of ...

Author: F. Hirata

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MOLECULAR THEORY OF SOLVATION

Understanding Chemical Reactivity Volume 24

Series Editor Paul G. Mezey, University of Saskatchewan, Saskatoon, Canada Editorial Advisory Board R. Stephen Berry, University of Chicago, IL, USA John I. Brauman, Stanford University, CA, USA A. Welford Castleman, Jr., Pennsylvania State University, PA, USA Enrico Clementi, Université Louis Pasteur, Strasbourg, France Stephen R. Langhoff, NASA Ames Research Center, Moffett Field, CA, USA K. Morokuma, Emory University, Atlanta, GA, USA Peter J. Rossky, University of Texas at Austin, TX, USA Zdenek Slanina, Czech Academy of Sciences, Prague, Czech Republic Donald G. Truhlar, University of Minnesota, Minneapolis, MN, USA Ivar Ugi, Technische Universität, München, Germany

Molecular Theory of Solvation edited by

Fumio Hirata Institute for Molecular Science, Okazaki, Japan

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: Print ISBN:

1-4020-2590-4 1-4020-1562-3

©2004 Springer Science + Business Media, Inc. Print ©2003 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America

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Contents

Preface

vii

1 Theory of molecular liquids Fumio Hirata 1 Introduction 2 Density Fluctuation in Liquids 3 Ornstein-Zernike (OZ) Equations 4 Site-SiteOZ (RISM) Equations 5 Solute-Solvent System 6 Some applications of RISM Theory 2 Electronic structure and chemical reaction in solution Hirofumi Sato 1 Introduction 2 ab initio Molecular Orbital Theory and the Solvation Effect 3 RISM-SCF/MCSCF Theory 4 Acid–base Equilibria in Aqueous Solution 5 Solvent Effects on Conformational Change of Chemical Compounds 6 Solvent Effect on Chemical Reactions 7 The NMR Chemical Shift 8 Summary Appendix: Appendix 1 Partial Charge Treatment in RISM-SCF/MCSCF 2 Variational Principle in the RISM-SCF/MCSCF method 3 Conformational stability of biomolecules in solution Masahiro Kinoshita 1 Combined RISM-MC approach for predicting peptide conformations 2 Alcohol effects on peptide conformations 3 Salt effects on solvation properties of peptides 4 Partial molar volume of amino acids and pressure effects

v

1 1 3 6 15 20 23 61 61 63 72 75 80 83 88 91 92 92 94 101 101 122 135 147

vi Appendix 1 Algorithms for solving RISM equations 4 Three-dimensional RISM theory for molecular liquids and solid-liquid interfaces Andriy Kovalenko 1 Introduction 2 3D-RISM integral equation 3 Closures for the 3D-RISM theory 4 Hydrophobic hydration 5 Potential of mean force between molecular species in solution 6 Solvation chemical potential of an ionic cluster in electrolyte solution 7 Self-consistent 3D-RISM approach 8 Combined Kohn-Sham DFT and 3D-RISM approach for a metalliquid interface 9 Hybrid 3D-RISM-SCF and ab initio MO method for solvated molecules Appendix 1 Free energy functions in the KH approximation 2 Solvation chemical potential in the SC-3D-RISM approach 3 Solvent effective potential coupling the KS-DFT and 3D-RISM equations 4 Algorithms for solving the RISM equation

155 155

169

169 175 178 184 190 202 226 240 251 257 257 258 260 261

5 Dynamical processes in solution Song-Ho Chong 1 Introductory remarks on the theory for dynamics of simple liquids 2 Interaction-site-model description of molecular-liquid dynamics 3 Collective excitations in diatomic liquids 4 Ion dynamics in diatomic liquids 5 Collective excitations and dynamics of ions in water 6 Concluding remarks

296 306 314 331 344

Index

351

277

277

Preface

In most of the past century, the study of solvent effect on chemical processes in solution has been dominated by continuum models which has been established during the 19th century. The Onsager reaction field and the Born model of ion hydration are good examples. The Onsager reaction field has been widely employed in the interpretation of solvatochromism. The Born model of ion hydration has been incorporated in the Marcus theory for electron-transfer reactions to realize the solvent reorganization energy. It has been recently generalized to explain the solvent effect on the electronic structure in solution. The generalized version of the Born model and the Poisson-Boltzmann equation have been applied also to biomolecules, i.e., protein as well as nucleic acids, in water. The model and theory are enjoying the status of standard machinery for evaluating the electrostatics which play an essential role in chemical reactions and the conformational stability of protein, which are two of the most important topics in theoretical chemistry. The success of those theories lies on rather general laws of physics: the long-range nature of the electrostatic interaction, which makes the mean-field treatment reasonably good, and the central-limiting theorem, which ensures a Gaussian character for solvent fluctuations. Owing to their generality, such models can be characterized by just a few parameters, such as dielectric constant and viscosity. However, the generality of the theory is sometimes of an unwelcome nature for chemistry, which should be able to distinguish a chemical element from other elements. The continuum theory may never be able to distinguish ethanol from acetonitrile, which have similar dielectric constants. A common maneuver to be employed in such cases to incorporate chemical specificity is to use an adjective, “effective”, e.g., “effective” radii, as boundary conditions for solving the continuum equations. Unfortunately, effective quantities so obtained for different physical processes conflict with each other, sometimes seriously, leading to vain a “religious war”. Broadly distributed values assigned to the size of ions, such as the Stokes radii of hydrodynamic equations and the Born radii of hydration free energy, are the best examples of

vii

viii the ambiguity characteristic of the continuum model. An even more serious drawback of the model shows up in some biological applications. One of the most dramatic events which proteins exhibit in solutions is so called ”cold denaturation,” that is, some protein denatures with decreasing temperature. The phenomenon is believed to be caused by a hydrophobic effect. The continuum model will never be able to explain the phenomenon even in the level of the ”effective” description. Molecular simulations, i.e., the molecular dynamics and the Monte Carlo methods, have become an invaluable tool for studying liquids and solutions during the last few decades. Those methods have been widely employed to explore the phase-space or the configuration space of the liquid system to acquire observables as an average of mechanical quantities over the space. So, it is capable of accounting for chemical specificity of the system from its most elementary level. However, this merit turns into a defect in some applications. Exploring the phase-space from the most elementary process requires a large amount of computation, and it often ends up with wandering around a rather limited region of the space, not to mention the infamous non-Ergodic trap. Naive use of the method may lead to results which are either entirely wrong or are unscientific. While the method is expected generally to provide a good account for quantities related to short wavelength and large frequency, it become more and more troublesome as the wavelength becomes greater and the frequency becomes smaller. In this book we present the third choice for describing the solvation phenomena in solutions. The method relies on the statistical mechanics applied to the liquid state of matter, especially on the RISM theory, an integral equation theory of molecular liquids. As will be described briefly in Chapter 1, the liquid state theory has been developed over the period of the past century, and has now reached the point at which almost the entire spectrum of chemistry in solution can be faithfully reproduced, at least in a qualitatively reliable manner: from ions to biomolecules, and from equilibrium to non-equilibrium. The theory is free from such adjectives as “effective”, since it is thoroughly based on the first principle, or Hamiltonian, of the system. There is no necessity for concerns about the limited sampling of the phase space, because it explores the entire phase-space in principle by means of the statistical mechanics. Of course, the theory involves some approximations, and is never be perfect, as is the case in any meaningful theory. The best part of the theory is that the approximation involved is unambiguous, which means it can be improved not by waiting for the development of the computer, but by grinding the human brain. However, the theory, by itself, is not enough to describe a variety of chemistry occurring in

PREFACE

ix

solutions. Our strategy for exploring chemistry in solutions is to combine the RISM theory with other methodologies well established in theoretical physics and chemistry. Understanding chemical reactions is the main objective of this book series. The reaction is primarily determined by changes in the electronic structure of species involved. However, if reactions are occurring in solutions, the change in the electronic energy is comparable to that of solvation energy, and the solvent effect can become a determining factor of the reaction. Such a statement can be phenomenologically understood from daily observations made by organic chemists that one solvent lets a reaction proceed, but another solvent does not. It is also important to realize that the electronic structure itself is largely influenced by a solvent, especially in polar liquids, which indicates that the electronic structure and the solvation should not be treated separately. Chapter 2 of the book is devoted to this problem, namely, electronic structure and chemical reactions in solution. The chapter deals with a theory referred to as RISM-SCF which combines the RISM theory with the ab initio molecular orbital theory (SCF). Some applications to selected topics related to chemical reactions are also presented. Biomolecules represented by protein and DNA are characterized by finite, but large, degrees of freedom which they possess, which means one has to consider a huge number of conformational “isomers”. The molecules are also distinguished from usual polymers in terms of the specificity they show in biological activities, which suggests that the theoretical treatment should be at the atomic level, just as in chemical reactions. In order to meet such a requirement, “solvent effects” should also be treated at atomic level, not by a continuum model: continuum models are never able to account for the hydrogen-bonds and hydrophobicity which are the two important interactions creating specificity of biopolymers. One question is then how to handle the effects on biomolecules of solvent molecules, the number of which is essentially infinite, without sacrificing their specificity. Chapter 3 is concerned with a methodology which projects the degrees of freedom of solvent molecules onto that of a biomolecule in the level of the site-site pair correlation functions. The method is the same in its spirit as the “influence functional” developed by Feynman. The molecular simulation technique is combined with the RISM equation to sample the conformational space of biomolecules, where atomic interactions include not only the direct intramolecular interactions but also the solvent induced interactions. The interface between a liquid and other substances or phases is the place where chemical reactions take place most actively, as is instanced by catalyses or electrodes. The problem involves the average density

x which is not constant in space but dependent on position. The interface can be treated by viewing it as a “solute” with some geometry, immersed in solutions: the “solute”-solvent density pair correlation functions mimic the local density of solvent in the interfacial region. For such a problem it is highly desired to describe the pair correlation as an explicit function of position rather than as a function projected along the radial distance. Chapter 4 is devoted to this subject, namely, to the development of the three-dimensional (3D) RISM theory for non-charged, polar, and ionic molecular species in polar molecular solvents. As a major application of the theory an electrode-solution interface is studied in which both the structures of the solution and of the metal surface are treated at the atomic level. In order to handle the electronic structure of electrodes, the 3D-RISM method is coupled in the self-consistent field loop with the Kohn-Sham type of density functional theory. The dynamical processes in solution can be treated by combining the RISM theory with the generalized Langevin theory. The conventional method of formulating dynamics in molecular liquids is based on the so called “rot-translational” model, which includes angular coordinates explicitly. The method relies on the spherical harmonic expansion in order to handle the orientational correlation of molecules, which becomes intolerably cumbersome as the asphericity of a molecule increases. The model employed in this book is the interaction-site model (ISM) which views the dynamics of a molecule as a correlated “translational”-motion of sites or atoms. The model lets us incorporate the RISM theory naturally into the generalized Langevin equation (GLE) which describes the time evolution of site density. Taking all the advantages which the RISM theory has in dealing molecular liquids, we are now able to evaluate the dynamic properties such as the velocity auto-correlation functions in realistic liquids including water. The coupled RISM-GLE treatment of liquid dynamics is presented in Chapter 5. On behalf of the authors of the book the editor is grateful to all the collaborators who have participated enthusiastically in the development of the molecular theory of solvation presented in each chapter. He also thanks Dr. T. Yamazaki for careful proof reading as well as technical help in completing the book, and Ms. R. Kawai for her assistance in word-processing the manuscript. FUMIO HIRATA

Chapter 1 THEORY OF MOLECULAR LIQUIDS Fumio Hirata Department of Theoretical Studies Institute for Molecular Science [email protected]

1.

Introduction

Increasing attention has been paid to so called “molecular liquids” from different branches of chemistry, physics, biology and their interdisciplinary fields. By “molecular liquid,” we just mean ordinary liquids such as water and alcohol, which concern our daily life and sciences. A reason why we add “molecular” to those ordinary liquids is deeply related to the development in the statistical mechanics of liquid states. Until about thirty years ago, the theory of liquids was largely limited to so called “simple liquids”, a system of spherical particle, the best example of which is hard spheres [1]. Many conceptual progresses in the physics of the liquid state have been made based on such simple models, say, density pair correlation functions, phase transition, and so on. However, such progress in the theory of liquids has not had as much influence on chemistry as it should until recently, in spite of “solvents’ being of great concern for chemists. Organic chemists have known well that a solvent can change the reactivity, the reaction rate, and yield, sometimes completely. Biochemists who want to find reasons why some proteins denature with lowering temperature will recognize it is impossible to answer the question unless the effect of water on protein stability is understood. Physical chemists are always curious about why and how a olvent changes the color of a dye molecule. The theory of simple liquids is quite useless for answering such questions. Answering such important questions in chemistry requires a characterization of liquids with molecular specificity. The chemical characteristics of a system represented just by the diameter of a molecule does not satisfy the demands of chemistry.

1

2

Molecular theory of solvation

The major breakthrough toward the liquid state theory in chemistry was made by D. Chandler and H.C. Andersen in 1971 with their theory of the reference interaction site model (RISM) [2]. The theory is a natural extension of the Ornstein-Zernike (OZ) equation to a mixture of atoms, but with strong intramolecular correlations which represent chemical bonds. The theory takes account of one of the two important chemical aspects of molecules, geometry, in terms of the intramolecular correlation. However, it does not handle the other chemical aspect of molecules, electrostatics, in its original forms. The charge distribution in a molecule, which is a classical manifestation of the electronic structure, plays a dominant role in determining the chemistry of a molecule. Therefore, without including the charge distribution in molecules, the description is incomplete in terms of chemical specificity. A complete chemical characterization of molecular liquids became possible in 1981 as a result of the appearance of the extended RISM theory (XRISM) which takes the charge distribution as well as the molecular geometry into account [3] The electrostatic interactions between atoms, a naive treatment of which causes divergence in the spatial integrals in the OZtype equations, has been handled by a renormalization technique originated by J. Mayer [4], implemented in the Ornstein-Zernike equations by Allnat, [5] and fully explored by H.L. Friedman and his coworkers [6] in their theories of the electrolyte solution. In that sense, the extended RISM theory is also an natural extension of the Mayer-Allnat-Friedman electrolyte solution theory to the molecular liquids. Applications of the theory to a variety of liquids and solutions have demonstrated its capability of describing the chemical specificity of liquids in a molecular detail. Such an application includes analysis of the structure and fluid phase behavior of water [7], [8], [9], the solvation structure and free energy of ions [10], [11], [13], the electronic structure in liquids [15],[16] chemical reactions [14], [17], [18], [19], [20], the dynamics in liquid water [21], [22], [23], the dynamic Stokes shift [24], [25], [26], interfacial liquids [27], [28], [29], the structure of water-alcohol mixture [30], the NMR chemical shift [31], the partial molar volume [32], and the stability and folding of polypeptides [33], [34]. This chapter is devoted to providing readers an outline of the modern statistical mechanics of liquid at work. The stress is put on development of physical concepts of the theory, not on the rigorous derivation of equations. Readers who are interested rather in formulation of the theory should consult with standard text books [35][36]. Many important developments, concepts, models, and mathematical approximations, after the RISM/XRISM theory appeared, are not included in the Chapter owing to the lack of space.

3

Theory of molecular liquids

2.

Density Fluctuation in Liquids

It will be instructive to begin the chapter with describing the basic concept and definition to characterize liquid states. Unlike the solid state in which molecules are making just small oscillations around their equilibrium positions, molecules in liquids are in diffusive motion, and changing their positions and orientation continuously. Owing to the thermal motion, the density of molecules is different from space to space and time to time. The most basic concept in describing the liquid state is the local density, or the density field and its fluctuation. Illustrated in Fig. 1.1 is a cartoon of the snapshot of molecules in a fluid. For pedagogical purposes, an atomic fluid with relatively low density is chosen.

3σ

g

2

2σ

σ

1

1.5 σ

0

Figure 1.1.

0

1

2

r [σ]

3

4

Illustration of density fluctuation.

Readers might have seen a similar picture from a result of a molecular dynamics simulation. If you look carefully, there are places where molecules are densely crowded, while in some places molecules are scarce. If one takes the average of the number of molecules over the entire volume of the container, N/V ≡ ρ, it is of course a constant, and it does not include useful information with respect to the structure and dynamics of the fluid. However, if one takes a product of densities at two different places r and r , and takes a thermal average over configurations, namely, ν(r)ν(r ) ≡ ρ(r, r ), it then contains ample information about the structure and dynamics of liquids. The quantity is called “densitydensity pair correlation function”. When the fluid is uniform, the quantity can be expressed by a function of only the distance between the two places, such that ρ(r, r ) → ρ(|r − r |). A mathematical definition of the pair correlation function can be made with the aide of the Dirac delta-function. The local density or the density

4


field in a snapshot of a liquid is defined as δ(r − rj ), ν(r) =

(1.1)

j

where ν(r) denotes the density of molecules at the position r, δ indicates the Dirac δ-function, and the sum is taken over all molecules in the system. How can the density field be a Dirac delta-function? Let us explain that by taking the simplest case of molecules fixed in one-dimensional space. Suppose a molecule i is fixed at a position xi in one-dimensional space, and let the density of molecules at a position x be ν(x). Now, let’s take the integration of ν(x) over some region L in the one-dimensional space. If one then integrates ν(x) over a region where the molecule is included, the result should be unity by definition. On the other hand, if the region L does not include the molecule, the answer should be zero, namely, 1 xj ∈ L, (1.2) ν(x)dx = / L. 0 xj ∈ L Now, let us focus our attention on the case the region L contains the molecule: L ν(x)dx = 1. Let us make the region L smaller and smaller, and take a limit L → 0. Since the region L still contains a molecule (rather, the center of the molecule), the result of the integral stays constant, or unity in this case. (The height of ν(x) should be infinitely large.) A function ν(r) which satisfies such a condition is nothing but the Dirac delta-function. If you have many molecules in the space, the density field is generalized to ν(x) = j δ(x − xj ), in which the xj s are the positions where molecules are fixed. If molecules are placed in a three-dimensional space, the expression should be further generalized to ν(r) = j δ(r − rj ), which is Eq. (1.1). Since the liquid suffers thermal fluctuation, we have to take a thermal average over configurations. Let us assume the Boltzmann distribution for the configuration, and take an ensemble average of the density field defined by Eq. (1.1) over configurations with the Boltzmann weight, or probability function. In uniform liquids, the first moment or the thermal average of the density field is constant, namely, ν(r) = ρ.

(1.3)

This quantity does not carry microscopic information about liquid structure. It is the second moment of the density fluctuation, defined by δν(r)δν(r ), that characterizes the microscopic structure of liquids, where δν(r) is defined as, δν(r) = ν(r) − ρ.

(1.4)

5


From Eqs. (1.1), (1.3) and (1.4) , δν(r)δν(r ) can be written as δν(r)δν(r ) = ρδ(r − r ) + ρ2 h(r, r ),

(1.5)

where ρδ(r − r ) and h(r, r ) represent the self- and intermolecularcorrelation functions, respectively, defined by, δ(r − rj )δ(r − rj ) (1.6) ρδ(r − r ) = j

δ(r − ri )δ(r − rj ) − ρ2 ρ2 h(r, r ) = i

(1.7)

j=i

Generalization of the above definitions to liquids consisting of polyatomic molecules is straightforward. Just labelling molecules and atoms by Roman and Greek letters, respectively, the density field of an atom species α can be defined as, δ(r − rαj ). (1.8) να (r) = j

The thermal average of the density field is constant, which is the average density of an atomic species α, namely, δ(r − rαj ) = ρα (1.9) να (r) = j

The second moment of the density fluctuation, δρα (r) ≡ να (r) − ρα , can be separated into two contributions, δ(r − rαj )δ(r − rγj ), (1.10) ωαγ (r, r ) ≡ ρα δ(r − r )δαγ + (1 − δαγ ) j

and

δ(r − rαi )δ(r − rγj ) − ρ2 , ρ2 hαγ (r, r ) = i

(1.11)

j=i

where δαγ denotes the Kronecker δ-function. For the uniform liquids in which the translational invariance is implied, the ω(r, r ) and h(r, r ) can be just functions of the distance between the two positions, r and r , namely, h(r, r ) → h(|r − r |)

(1.12)

ω(r, r ) → ω(|r − r |).

(1.13)

and

6

3.


Ornstein-Zernike (OZ) Equations

It is the Ornstein-Zernike equation that has been used most popularly to describe the density fluctuations in liquids [37], [35]. The equation by itself is nothing more than the defining of a correlation function called the “direct” correlation function c(r, r ) in terms of the total correlation function h(r, r ) defined in Eq. (1.7). c(r, r )ρ(r )h(r , r )dr . (1.14) h(r, r ) = c(r, r ) + V

The OZ equation can be formulated from the grand partition function by functional differentiations in a spirit of the density functional theory. Here, we give just a brief sketch of the formulation. In the present section, we confine ourselves with a simple liquid. Let us define the potential energy of the liquid system by (r1 , ..., rN ) = UN

N

ψ(ri ) + UN (r1 , ..., rN ),

(1.15)

i=1

where ψ(ri ) is the interaction potential of i-th molecule of the system with an external field, and UN (r1 , ..., rN ) is the intermolecular potential energy. Defining the local activity by z(r) = z exp[−βψ(r)], where z = exp[µ/kB T ], β = 1/kB T , µ is the chemical potential, and kB is the Boltzmann constant. Using those definitions the grand canonical partition function can be written as ∞ N 1 ... z(ri ) exp(−βUN )dr1 . . . drN . Ξ= N! N

(1.16)

i

We define the single particle and pair correlation functions of density, respectively, by

ρ(r) =

ρ(2) (r, r ) =

z(r) δΞ , Ξ δz(r)

z(r)z(r ) δ2Ξ . Ξ δz(r)δz(r )

(1.17)

(1.18)

With the aid of the mathematical theorem concerning the functional derivative, δz(r) = δ(r − r ), δz(r )

(1.19)


7

the definitions agree with the standard definitions for the density correlation functions in terms averages of the single particle of the ensemble and pair densities, i δ(r − ri ), i j=i δ(r − ri )δ(r − rj ), respectively, where · · · represents the grandcanonical ensemble average. A relation between ρ(r) and ρ(2) (r, r ) can be obtained after some manipulation as, δρ(r) = ρ(2) (r, r ) − ρ(r)ρ(r ) + δ(r − r )ρ(r). δ ln z(r )

(1.20)

We now define the direct correlation function by an inverse relation of Eq. (1.20), δ(r − r ) δ ln z(r) = − c(r, r ). δρ(r ) ρ(r )

(1.21)

Inserting Eqs. (1.20) and (1.21) into the chain-rule theorem of functional derivatives, δρ(r) δ ln z(r ) dr = δ(r − r ), (1.22) δ ln z(r ) δρ(r ) one obtains the Ornstein-Zernike equation (1.14). As can be realized by the derivation, the OZ equation is, in a sense, a differential form of the partition function, and the equation should be complemented by another equation which relates h(r, r ) and c(r, r ), in order for the OZ equation to be solved. The general relation to close the OZ equation reads c(r, r ) = exp[−βu(r, r ) + t(r, r ) + b(r, r )] − 1 − t(r, r ),

(1.23)

where t(r, r ) is defined by t(r, r ) = h(r, r ) − c(r, r ),

(1.24)

and b(r, r ) is a functional of h(r, r ), referred to as a “bridge” function. Since the equations contain two variables h and c, and we have two equations 1.14 and 1.23, we are in principle able to solve the equations. However, the bridge functions b(r, r ) include multiple integrals which are quite difficult to evaluate except for some special cases. Substantial effort has been devoted historically to developing approximate relations for the closures. Obviously the simplest approximation is just to ignore the bridge function b(r, r ), c(r, r ) = exp[−βu(r, r ) + t(r, r )] − 1 − t(r, r ).

(1.25)

8


The relation is referred to as the hypernetted-chain (HNC) approximation. Further linearizing exp t(r, r ) in 1.23, one has the Percus-Yevick (PY) approximation, c(r, r ) = exp[−βu(r, r )][1 + t(r, r )] − 1 − t(r, r ).

(1.26)

The PY closure is known to give reasonable account of the system consisting of hard spheres owing to the partial cancellation of errors coming from neglecting the bridge functions and the linearization of exp t(r, r ). However, the approximation sometimes leads to unphysical behavior for a Coulombic system: for example, the radial distribution function of such a system sometimes becomes negative at short separation. The HNC closure, on the other hand, is successful in providing reasonable account of many different types of systems including those interacting via the Coulombic force. However, the closure sometimes produces pathological results for associating liquids consisting of a mixture of spheres with highly asymmetric size ratio. It is also known to cause divergence in numerical solutions quite often, even at a position in the phase diagram where the stable solution is supposed to be. Another type of closure, which is often used for a system consisting of spheres interacting via potential functions ∞ (r ≤ σ) u(r) = (1.27) w(r) (r > σ), is the mean-spherical approximation (MSA) [38], which reads h(r) = −1 for r ≤ σ c(r) = −βw(r) for r > σ.

(1.28) (1.29)

Here, w(r) is the attractive (or repulsive) tail of the potential, σ is the diameter of spheres, and we have assumed that the fluid is uniform, therefore translational invariance is implied. The first equality in the above equation embodies the physical requirement that the center of a sphere can not penetrate the excluded volume of other spheres. The second equality is just obtained from (1.25) by linearizing the entire exponential factor. Actually, it is the asymptote of the direct correlation function at the infinite separation. The approximation is known to be superior for describing the critical phenomena. The radial distribution function, however, shows an ill-behavior for a Coulombic system, similar to those from the PY closure. Recently, a new approximation has been proposed by Kovalenko and Hirata which hybridizes the HNC with the MSA closures: [9], [28] exp[−βu(r) + t(r)] − 1 for h(r) ≤ 0, h(r) = (1.30) −βu(r) + t(r) for h(r) > 0.

9


The new closure which we call the “KH” approximation retains the advantages inherent in the HNC and MSA closures and discards their disadvantages stated above: the solution now diverges only for the thermodynamic states inside unstable regions of the phase diagram, while the total correlation satisfies the condition h(r) ≥ −1 (or the distribution function g(r) never becomes negative). The closure relations have been derived in several different ways which are roughly classified into two categories, a diagrammatic method similar to the Feynmann diagram, and a functional Taylor expansion. Here, we give just a brief outline of the method of the functional Taylor expansion, which uses Percus’ neat idea of identifying an external force exerted on a molecule in the system as the interaction force from other molecules fixed in the space [37] [35]. Let us approximate the potential energy of a liquid system consisting of N molecules by a sum of pair interactions, u(ri , rj ). (1.31) UN (r1 , r2 , ..., rN ) = i

j>i

Furthermore, we fix extra s molecules in the space, and express the interaction between the fixed molecules and one of the N molecules by, U

(s)

(r) =

s

u(r − xk ).

(1.32)

k=1

U (s) (r) can be regarded as an “external” force to the molecule at position r. This idea is called Percus’ trick. Let ρ(n) (r1 , r2 , ..., rn |U (s) ) be the n-body distribution function subject to the “external” field of the s molecules. The following relation with respect to ρ(n) (r1 , r2 , ..., rn |U (s) ) can then be proved, ρ(n+s) (x1 , x2 , ..., xs , r1 , r2 , ..., rn ) = ρ(n) (r1 , r2 , ..., rn |U (s) ), ρ(s) (x1 , x2 , ..., xs )

(1.33)

where ρ(n+s) and ρ(s) are, respectively, (n+s)-body and n-body distribution functions in the uniform system. The physical meaning of the equation is obvious, that is, (the probability of finding molecules at the (n+s) positions, ρ(s) )= (the probability of finding molecules at the s positions,x1 , x2 , ..., xs ) × (the conditional probability of finding molecules at the n positions,r1 , r2 , ..., rn provided that the s molecules are fixed at the s positions). When s = 1, the equation can be for n = 1 and n = 2, respectively, ρ(2) (x, r) = ρ(1) (r|U (1) ), ρ(1) (x)

(1.34)

10


ρ(3) (x, r, r ) = ρ(2) (r, r |U (1) ). ρ(1) (x)

(1.35)

A general idea of the density functional consists in expressing the energy as a functional of density. Let us now define a free energy of the system as a functional of the local density with a molecule fixed at a position bf x, and see the effect of perturbation on the free energy due to the fixed molecule. We define a free energy functional by, Γ[ψ] = ln(ρ(r|ψ)eψ(r)/kT = ln(ρ(r|ψ)eu(r,x)/kT ,

(1.36)

and expand Γ in a functional Taylor series with respect to ψ. Regarding the ψ itself as a functional of ρ(r|ψ), the expansion reads, δΓ {ρ(r |ψ) − ρ(r )}dr Γ[ψ] = Γ[0] + δρ(r |ψ) ψ=0 δ2 Γ + {ρ(r|ψ) − ρ(r)}{ρ(r |ψ) − ρ(r )}drdr + . . . δρ(r|ψ)δρ(r |ψ) ψ=0 (1.37) The first term in the right hand side is, Γ[0] = ln ρ(r). The functional derivative in the second term is, δ z δΓ ln ρ(r|ψ) + ln , = δρ(r |ψ) δρ(r |ψ) z(r)

(1.38)

(1.39)

which can be identified as the direct correlation function defined by Eq. (1.21). Therefore, δΓ = c(r, r |0). (1.40) δρ(r |ψ) ψ=0 From Eqs. (1.36), (1.37), (1.38), and (1.40), the following equation is obtained, c(r, r )ρ(r )h(r , x)dr . (1.41) ln{1 + h(r, x)} + u(r, x)/kT = V

The right hand side of Eq. (1.41) is h(r, x) − c(r, x) owing to the OZ equation, therefore, c(r, r ) = h(r, r ) − ln{1 + h(r, r )} − u(r, r )/kT,

(1.42)

which is nothing but the HNC approximation. The PY approximation can be obtained just by linearizing the exponential term, exp{h(r, x) − c(r, x)}, in Eq. (1.42).

11


3.1

The OZ equation for Ionic Systems (Renormalization of the Coulomb Interaction)

Let us rewrite the OZ equation in terms of an infinite series of the direct correlation functions expanded in the power of density, ρ. By iteratively convoluting the h-function in the left hand side into that in the right hand side, we obtain, c(r, r1 )c(r1 , r )dr1 h(r, r ) = c(r, r ) + ρ V + ρ2 c(r, r1 )c(r1 , r2 )c(r2 , r )dr1 dr2 + . . . V

V

= c(r, r ) + ρc ∗ c + ρ2 c ∗ c ∗ c + . . . ,

(1.43)

where we have introduced a commonly employed notation for the convolution integral, c ∗ c = c(r, r1 )c(r1 , r )dr1 . It is more convenient to use a graphical expression of the equation: h(r, r ) =

c

r

c

c + ρ c

r’

c

r

+ ρ3 c r

c

s

c

s

s

c

c

s

c + ρ2 c

r’ c

c

r

c + ...

r’

s

c

s

c

c

r’

(1.44)

In the graphical expression, the filled circles (•) denote the coordinates over which integration is carried out, while the open circles (◦) depict the coordinates which are not integrated. The line (−) represents a function of the two positions. We also use the terminology, “chain sum,” to describe such a sum of a series of convolution integrals, and denote it as C[c|ρ]. The OZ equation can be written in terms of the chain sum as, h ≡ C[c|ρ].

(1.45)

Defining the indirect part of correlation function by, t = h − c = C[c|ρ] − c,

(1.46)

the closure relation can be written in a compact form, c = (1 + f )F (t) − 1 − t, where f (r) = exp[−u(r)/kB T ] − 1, and (HN C) et F (t) = 1 + t (P Y ).

(1.47)

(1.48)

12


The equations are the basis of the recursive solution of the OZ equation proposed by Dale and Friedman: [39] = (1 + f )F (tj ) − 1 − tj cj (1.49) tj+1 = C[cj |ρ] − cj . When molecules in a system bear charges, the OZ equation ∞ involves an integral of the Coulomb interaction over the entire space, 0 (1/r)4πr 2 dr, which diverges. The best example of such systems involving the Coulomb interaction is an electrolyte solution. Let us imagine ions are immersed in a continuum dielectric with a dielectric constant, . The interaction between two ions can then be most primitively expressed as a sum of short range and long range (Coulomb) interactions: uij (r) = u∗ (r)+zi zj / r, where u∗ (r) denotes the short range interaction, and zi ,zj are charges of ions. The hard sphere interaction and the Lennard-Jones interactions are those most popularly employed in the literature. Such a model is referred to as a “primitive model” of electrolyte solutions. For such a system, the asymptotic behavior of the direct correlation function at infinite separation is c(r) → −βzi zj / r, and it is the term which causes the divergence in the expansion. Let us split the direct correlation function into the long range contribution and the rest: c(r) = φ(r) + c∗ (r), where φ(r) = −βzi zj / r. Obviously, c∗ (r) has a short range character, and does not diverge upon integration over space. Therefore, if we can reorganize the graphical expansion, and convert the long range function into some converging function, then we can manage the nasty problem of divergence. The conversion to a converging function can be accomplished by a renormalization of the Coulomb interaction, which was first proposed by J. Mayer [4]. He realized that the chain sum of the Coulomb interaction in the cluster expansion of the configuration integral converges, and reorganized the cluster expansion in terms of the renormalized function. Today, the renormalization can be done much more easily with the aid of so-called hypervertices function) [40]. Let us define a hypervertex function C[A|ρ] of an arbitrary function A(r, r )by, C[A|ρ] = ρδ(r − r ) + ρ2 C[A|ρ],

(1.50)

δ(r − r )

is a Dirac delta-function, and C[A|ρ] is the chain sum of where the function A. By realizing the property of the delta function, (1.51) δ(r − r )f (r , r )dr = f (r, r ), one can readily prove the following useful relations: C[A + B|ρ] = C[A|ρ] + C[B|C[A|ρ]],

(1.52)

13


C[A + B|D] = C[A|D] + C[B|D[A|C[A|D]]].

(1.53)

Identifying the short and long range contribution of the direct correlation function as A and B, the Eq. (1.46) can be renormalized as, t = q − φ + τ,

(1.54)

where q and τ are defined in terms of the chain sum as, q = C[φ|ρ]

(1.55)

q ], τ = C [c − φ|¯

(1.56)

and

where q is the hypervertex function of q. Using the renormalized functions, Eq. (1.47) can be rewritten as c − φ = e−βu

∗ +q

F (τ ) − 1 − (τ + q), (HNC) eτ F (τ ) = 1 + τ (PY).

(1.57) (1.58)

The chain sum q can be analytically performed to give, qij (|r − r |) = C[φ|ρ] = −

zi zj exp(−κ|r − r |) ,

kT |r − r |

(1.59)

where κ is a constant stated below. It should be noted that the renormalized function q has a converging factor exp(−κr) and never diverges upon integration over the space given a finite κ. Since Eq. (1.57) includes only converging functions, u∗ , τ , and q, the equation can be solved. It will be instructive to add some explanation of how the screening factor exp(−κr) comes into the final expression by actually deriving the equation. Let us define functions φ(r) and φ (r) by φ (r) = −β

e−αr

r

(1.60)

and φ(r) = z1 z2 φ (r).

(1.61)

By taking the limit α → 0, the usual Coulomb interaction recovers. The chain sum defined in Eq. (1.55) can then be written in the matrix form, q(r) = φ + φρφ + φ ∗ ρφ ∗ ρφ + . . . ,

(1.62)

14


where q and φ are matrices with suffices indicating ion species. ρ is a diagonal matrix with the density (or concentration) of component ions. Here, the symbol ∗ denotes the convolution integral as well as the matrix product. Now, let us define the Fourier transform and its inverse of a spherical function, F (r), by F (k) = F (r) exp(−ik · r)dr (1.63) F (r) = (2π)−3

F (k) exp(ik · r)dk.

It is easy to see for a spherical function, 4π ∞ F (r)r sin(kr)dr, F (k) = k 0 −3 4π

F (r) = (2π)

(1.65)

∞

F (k)k sin(kr)dk.

r

(1.64)

(1.66)

0

Let us take the Fourier transform of Eq. (1.62). Taking the definition of φ (Eq. (1.60)) into account, the Fourier transform becomes, qij (k) = zi zj [φ (k) + Iφ (k)2 + I 2 φ (k)3 + . . . ] φ (k) , (1.67) = zi zj 1 − Iφ (k) where I = (1/2) i ρi zi2 is the ionic strength. The Fourier transform of φ (r) is known and, φ (k) = −

4πβ 1 . 2

k + α2

(1.68)

Inserting the expression into Eq. (1.67), one obtains qij (k) =

−4πβzi zj 1 , 2

k + α2 + κ2

(1.69)

where κ is the Debye screening constant defined by, κ2 =

8πβI .

(1.70)

By taking the limit α → 0, the Fourier transform of qij (r) becomes qij (k) =

1 −4πβzi zj . 2

k + κ2

(1.71)

15


The inverse Fourier transform of Eq. (1.71) is readily made, and the result is, qij (r) = −β

zi zj −κr e .

r

(1.72)

All the thermodynamic quantities including the activity coefficient for an electrolyte solution can be derived from the correlation function, which leads to the famous Debye-H¨ uckel limiting law, or the square-root law for concentration dependence. It should be noted that the origin of the square-root law is in the Debye screening constant, κ, which is proportional to the square-root of the concentration, as is clear from its definition. (see Eq. (1.70)) The procedure described above shows clearly how the long range Coulomb interaction becomes the screened Coulomb interaction as the result of the many-body effect. This is an example of the cases in which the statistical mechanics is superior to molecular simulation methods. Molecular simulations will be able to produce the limiting law to any precision they want. However, they will never be able to explain its physical origin.

4.

Site-SiteOZ (RISM) Equations

In most of the liquids consisting of polyatomic molecules, interactions between a pair of molecules depend not only on the center to center distance, but also on their mutual orientations, namely, u(12) = u(R, Ω1 , Ω2 ),

(1.73)

where (12) denotes the coordinates of molecules 1 and 2 including their orientation, R the center to center distance, and Ω1 , Ω2 the orientation of those molecules. So, we have to generalize the OZ equation so as to take account for the molecular orientation. The generalized OZ equation which includes both distance and orientation reads, ρ c(13)h(32)d(3), (1.74) h(12) = c(12) + Ω where Ω is the normalization constant for the angular coordinates. The integral d(3) is taken over the entire space as well as orientation of the third molecule. The straightforward application of the equation is to employ an interaction potential which depends not only on the center to center distance, but also on the mutual orientation of the molecules. Such a development has in fact been made, based on the spherical invariant method, but its success is largely limited by the convergence of the spherical harmonics expansion: the convergence of the expansion

16


is extremely slow when a molecule deviates from the spherical shape. Just imagine how the interaction between two protein molecules can be described by a spherical expansion. In that sense, the interaction site model gives a great advantage over the spherical invariant method in terms of applicability to molecules with complex geometry. The interaction potential between two molecules can be written in terms of the interaction site model as follows. uαγ (r), (1.75) u(1, 2) = αγ

where α, γ represent the interaction site. (Fig. 1.2)

2 1

Figure 1.2.

Interaction-site Model.

Each term in the sum depends only on the “scalar” distance between a pair of interactions sites, which simplifies the mathematical procedure to formulate the theory dramatically. The expression is nothing but that used in the molecular simulation, which adds another advantage to ISM, because theoretical results obtained using the model can be directly compared with those from molecular simulations. It is an essential step to express the pair correlation functions between two molecules in terms of those between interaction sites, which can be accomplished by averaging the functions over orientations fixing the distance between the interaction sites: −2 (1.76) d(1)d(2)δ(R1 + lα1 )δ(R2 + lγ2 − r)h(12). hαγ (r) = Ω Let us rewrite the molecular OZ (Eq. (1.74)) by iteratively inserting h in the left hand side into h in the second term of the right hand side, which

17


gives rise to an infinite series consisting of the convolution integrals of the c-functions. h(12) = C[c|ρ/Ω] ∞ C (n) [c|ρ/Ω], =

(1.77)

n=0

where we denoted the n-th term in the chain sum as C (n) (c|ρ/Ω). First few terms of the series are: C (0) (c|ρ/Ω) = c(12), C (1) (c|ρ/Ω) =

C (2) (c|ρ/Ω) =

ρ Ω

ρ 2 Ω

(1.78)

d(3)c(13)c(32),

(1.79)

d(3)d(4)c(13)c(34)c(42).

(1.80)

Now, we try to take the average of Eq. (1.77) over orientations, fixing the distance between two sites. However, it is obviously impossible to take the average analytically. The essential approximation of the RISM theory consists of representing the molecular direct correlation function, c(12), by a sum of the site-site direct correlation functions, cαγ (rα1 − rγ2 |). [2] cαγ (|rα1 − rγ2 |). (1.81) c(12) = αγ

The approximation allows one to take the average analytically. Taking advantage of the theorem about the Fourier transform of the convolution integral, the averaging of Eq. (1.77) over orientations can be accomplished as follows. (0) Iαγ = Ω−2 d(1)d(2)δ(R1 + lα1 )δ(R2 + lγ2 − r)C (0) (c|ρ/Ω) −2 d(1)d(2)δ(R1 + lα1 )δ(R2 + lγ2 − r)c(12) =Ω −2 =Ω d(1)d(2)δ(R1 + lα1 )δ(R2 + lγ2 − r) cην (|rη1 − rν3 |)

−3

= (2π)

= (2π)−3

ik·r

dke

ην

cην (k){Ω

−1

η −ik·(lα 1 −l1 )

dΩ1 e

}{Ω

−1

ν

ην

dkeik·r [ω(k)c(k)ω(k)]αγ

γ

dΩ2 e−ik·(l2 −l1 ) } (1.82)

18 (1) Iαγ


=Ω

−2

d(1)d(2)δ(R1 + lα1 )δ(R2 + lγ2 − r)C (1) (c|ρ/Ω)

= Ω−2 d(1)d(2)δ(R1 + lα1 )δ(R2 + lγ2 −r) cην (|rη1 − rν3 |)cξζ (|rξ3 − rζ2 |) = (2π)−3

ην ξζ

dkeik·r [ω(k)c(k)ρω(k)c(k)ω(k)]αγ

and so on, where ωαγ (k) = Ω

−1

(1.83)

γ

dΩ1 e−ik·(l1 −l1 ) α

= sin(k|lα1 − lγ1 |)/k|lα1 − lγ1 |,

(1.84)

or in more explicit form distinguishing the cases α = γ (same sites) and α = γ (different sites) ωαγ (k) = δαγ + (1 − δαγ ) sin(k|lα1 − lγ1 |)/k|lα1 − lγ1 |.

(1.85)

In the above expressions, quantities written in boldface letters are matrices, elements of which are site-site functions indicated by Greek subscripts. The n-th term of the series becomes, (n) −3 (1.86) dk[ωc(ρωc)n ω]αγ eik·r . Iαγ = (2π) Therefore, the right hand side of Eq. (1.76) is ∞ (n) −3 Iαγ = (2π) [ωc[1 − ρωc]−1 ω]αγ eik·r . hαγ (r) =

(1.87)

n=0

Note that the integrand of the right hand side in Eq. (1.87) is a sum ∞ ω(k)c(k)ω(k)[ρc(k)ω(k)]n , in the reciprocal of the infinite series, n=0

space. Therefore, its inverse transform is a chain sum, or an infinite series of the convolution integrals, which can be written in terms of the graphical expansion, hαγ (r) = Cαγ [c|ω] ω c ω ω c ω c ω s s c+ c s s s s c + .... α γ α γ

= c

(1.88)

This can be rewritten in the wellknown form of the RISM equation, h = ω ∗ c ∗ ω + ρω ∗ c ∗ h.

(1.89)

19


In the equation, the boldface letter signifies a matrix consisting of the site-site correlation functions, hαγ (r) and cαγ (r), and the asterisk denotes the convolution integrals as well as the matrix products. ωαγ is the Fourier inverse transform of ωαγ (k) defined by Eq. (1.10), which can be written as ωαγ (r) = δαγ δ(r) + (1 − δαγ )sαγ (r),

(1.90)

where sαγ (r) represents the intramolecular correlation functions between a site pair α and γ. For a rigid molecule, sαγ (r) =

1 δ(r − Lαγ ), 4πL2αγ

(1.91)

where Lαγ is the “bond” length or the distance constraint of the site pair α and γ. The geometry of a molecule can be defined by giving Lαγ for all pairs of sites. The recursive formula for the RISM equation with the PY and HNC closures can be written as, tαγ = hαγ − cαγ = Cαγ [c|ω] − cαγ ≡ Cαγ [c|ω],

(1.92)

cαγ = (1 + fαγ )F (tαγ ) − 1 − F (tαγ ),

(1.93)

etαγ F (tαγ ) = 1 + tαγ

4.1

(HN C) (P Y ).

(1.94)

Renormalized RISM equation

A similar procedure for renormalizing the OZ equation for the Coulomb interaction can be applied to the RISM equation [3]. Suppose the site-site interaction can be expressed by a sum of short range and Coulomb interactions as, uαγ (r) = u∗αγ (r) + zα zγ /r.

(1.95)

Let us split the direct correlation function into short and long-range contributions as, cαγ (r) = cαγ (r) − φαγ (r) + φαγ (r),

(1.96)

20


where φαγ (r) = −β(zα zγ /r) represents the long range contribution to the direct correlation function. Then, Eq. (1.92) can be expressed in terms of the chain sum, t = C [c − φ + φ|ω] = C [φ|ω] + C [c − φ|ω + ρ2 C[φ|ω].

(1.97) (1.98)

Defining new normalized functions Q and τ by, Q = C[φ|ω] = C [φ|ω] + φ,

(1.99)

τ = t − C [φ|ω] = C [c − φ|ω] + ρQρ.

(1.100)

The closure relations HNC and PY can be also renormalized as, cαγ − φαγ = exp[−βu∗ + φαγ ) + tαγ ] − 1 − tαγ − φαγ = exp[−βu∗αγ + ταγ + Qαγ ] − 1 − ταγ − Qαγ .

5.

(1.101)

Solute-Solvent System

We have so far described a statistical mechanics of molecular liquids, implying that a system includes only one chemical species. However, in ordinary chemistry, a system contains more than one component, and major and minor components in the mixture are conventionally called “solvent” and “solute”, respectively. The vanishing limit of solute concentration, or infinite dilution, is of particular interest because it purely reflects the nature of solute-solvent interactions. The word “solvation” is most commonly used for describing properties concerning solute-solvent interactions at the infinite dilution limit. Here, we provide a brief outline of the way to obtain solvation properties, solvation structure and thermodynamics, from the RISM theory described in the previous sections [3]. It is straightforward to generalize the RISM equation to a mixture of different molecular species. The equation for a mixture can be written in a matrix notation as ρhρ = ω ∗ c ∗ ω + ω ∗ c ∗ ρhρ,

(1.102)

where ∗ denotes a convolution integral as well as a matrix product, and h, c, ω are, respectively, the matrices of site-site total, direct, and intra-molecular correlation functions. The elements of those matrices , cα γ , ωα γ , where the suffix αM labels the α-th atom are hαM γM M M M M in a molecular species M . ρ is a diagonal matrix consisting of density of each molecular species, ραM γM = δαγ δM M ρM . The matrices can


21

be reorganized into those with block matrices concerning solute and solvent species. Using the conventional notation often employed for a sol“u”te-sol“v”ent system, one can rewrite the above equation as a set of equations regarding solvent-solvent (vv), solute-solvent(uv), and solutesolute (uu) correlation functions, ρv hvv ρv = ω v ∗ cvv ∗ ωv + ω v ∗ cvv ∗ ρv hvv ρv ,

(1.103)

ρu huv ρv = ω u ∗ cuv ∗ ω v + ωu ∗ cuv ∗ ρu huv ρov ,

(1.104)

ρu huu ρu = ω u ∗ cuu ∗ ω u + ωu ∗ cuu ∗ ρu huu ρu ,

(1.105)

The infinite dilutions, which is the most interesting case in the study of solvation, can be realized by the vanishing limit of solute density, i.e., ρu → 0, which gives rise to the following set of equations, hvv = wv ∗ cvv ∗wv + wv ∗ cvv ∗ρv hvv ρv ,

(1.106)

huv = wu ∗ cuv ∗wv + wu ∗ cuv ∗ ρv hvv ρv ,

(1.107)

huu = wu ∗ cuu ∗ wu + wu ∗ cuv ∗ ρv hvu ,

(1.108)

where wv = ω v /ρv and wu = ω u /ρu . Using the theory described above, one can calculate, in principle, all the observables concerning equilibrium properties of solution including the structure factor obtained from the scattering experiment as well as thermodynamic quantities. The excess chemical potential of solute, or the “solvation free energy”, at infinite dilution is of particular interest, because it is the quantity which measures the stability of solute in solvent, and because all other excess thermodynamic quantities are derived from the free energy. The excess chemical potential, which is defined as an excess from the ideal gas, can be expressed in terms of the so called Kirkwood coupling parameter. The excess chemical potential is defined as the free energy change associated with a process in which a solute molecule is coupled into solvent [41]. The coupling procedure can be expressed by, U (ξ) = Uvv + ξUuv ,

(1.109)

where U , Uvv , and Uuv denote, respectively, the interaction potential of total system, those among solvent molecules, and those between solute and solvent, and ξ = 0 (no solute coupling) or ξ = 1 (full coupling). The

22


free energy change associated with the process can be written, based on the standard statistical mechanics, as ∆µ = F (ξ = 1) − F (ξ = 0), = −kT ln Z(ξ = 1) − [−kT ln Z(ξ = 0)],

(1.110)

where Z(ξ = 1) and Z(ξ = 0) are the configuration integral of the system after and before the solute coupling, respectively. It is undergraduate course mathematics to obtain the following equation for ∆µ in terms of the pair correlation functions, which includes an integral over the Kirkwood coupling parameter from 0 to 1. ∆µ =

αv

λu

ρv

1

∞

dξ 0

0

druλu αv (r)gλu αv (r, ξ).

(1.111)

The physical meaning of the equation is obvious. Suppose a solute molecule is in solvent at a state represented by the parameter ξ. The number of solvent atoms αv within a volume element dr around solute atoms λu is ρgλu αv (r, ξ)dr. Therefore, the interaction between the solute atom and solvent atoms in dr is uλu αv (r)gλu αv (r, ξ)dr. Summing the interactions up for all atoms in and integrating over and solvent, solute ∞ the entire space, one obtains λu αv ρv 0 druλu αv (r)gλu αv (r, ξ). The integration with respect to ξ represents the solute coupling procedure. The expression given by Eq. (1.111) is quite general and useful for calculating the excess chemical potential. In fact, the equation has been employed by Chiles and Rossky to evaluate the free energy surface along a reaction coordinate in their RISM analysis of a SN 2 reaction [42]. However, the procedure requires a numerical integration with respect to ξ, which in turn requires a numerical solution of the RISM equation at each step of a finite increment of ξ. The procedure becomes intolerably tedious, sometimes formidable, if a solute molecule contains a large number of sites as is the case for protein. The problem has been solved in the case of HNC closure by Singer and Chandler [43]. They have performed the ξ integration in Eq. (1.111) analytically by extending a method originally developed for simple liquids by Morita and Hiroike to molecular liquids [44]. The equation is slightly modified by Rossky and Chiles to give more convenient form: [42] 1 1 2 hλu αv (r) − cλu αv (r) − hλu αv cλu αv (r) r 2 dr. β∆µ = 4πρ 2 2 α λu

v

(1.112)


6. 6.1

23

Some applications of RISM Theory Water

Water has been a subject of intensive researches for the past century, not only from the viewpoint of physical chemistry but also from those of biochemistry and other fields [45] [46] [47]. It has been welldocumented that the outstanding features water exhibits in thermodynamic properties, such as negative thermal expansibility below 4 C, have their origin in the unique liquid structure owed to the hydrogen-bond network. The modern structural study of water was initiated by Bernal and Fowler with the X-ray diffraction method, who first clarified the existence of the ice-like tetrahedral coordination in liquid water [48]. The negative thermal expansibility was attributed to the disruption of the bulky ice-like structure. The monumental paper motivated succeeding studies, experimental as well as theoretical, for elucidating the structure of water and its relation to thermodynamic and dynamic properties of the liquid [46]. The combined X-ray and neutron diffraction study finally established a picture concerning water structure in terms of the atom-atom pair correlation functions, an essential feature of which remains valid [49]. The pair correlation function (PCF) between oxygen and hydrogen has a distinct peak at ∼ 1.8 ˚ A, which is assigned as the hydrogen-bond. The oxygen-oxygen PCF has the second peak at the separation around 1.63 σ, where σ is the diameter of a molecule, which is characteristic of the ice-I-like tetrahedral coordination, and is regarded as the finger print of water structure. The first peak of O-O PCF is rather sharp, from which the coordination number has been determined as around 4.4 in the ambient condition. The number is very close to that in ice, four, rather than those typical to normal liquids, 10 to 11. Regarding an origin of the difference in the coordination number, 0.4, between ice and water, a variety of models and theories were proposed. The earlier theories of water assumed the existence of the tetrahedral ice-like coordination in one way or the other, and attributed the deviation in the coordination number from ice to some kind of imperfection in the tetrahedral structure due to breaking or distortion of hydrogen-bonds. Roughly speaking, three models have been proposed to explain the deviation in the coordination number: the bent hydrogen-bond model, the interstitial model, and the mixture model represented, respectively, by Pople [50], Samoilov [51], and Scheraga [52]. The bent-hydrogen bond model [50] attributes the deviation to water molecules occupying the off-lattice space of ice due to distortion of the O-H· · · O angle from 180 degrees. The interstitial model [51] explains the deviation in terms of “interstitial molecules” which spill out from the ice lattice owing to the breaking

24


of hydrogen-bonds. The mixture model [52] sees water as a mixture of two components, hydrogen-bonded clusters with ice-like structure and non-hydrogen bonded molecules, and interprets the deviation in term of the increased fraction of non-hydrogen bonded components compared to clusters. Each of those models could have explained relations between the structure characteristic to water with its thermodynamic properties to some extent. However, those theories could have never solved the more challenging problem; how is the characteristic structure of water formed from the intermolecular interactions? The problem could have been resolved only by the theories based on the first principle. One of the theoretical approaches based on the Hamiltonian model is the molecular simulations, or the molecular dynamics and Monte-Carlo methods. The methods have made great contribution for heuristic understanding of structural, dynamical as well as some thermodynamic properties of water. Since so many review articles have already been published concerning the simulation of water [53], here we will not take a trouble of reiterating them. The other theory which tries to explain structure of water from the first principle is the integral equation theories for the molecular liquids. After several attempts based on the OZ-type equations [54], [55] three different approaches have successfully achieved the goal at the same level at least qualitatively: the extended RISM (XRISM) [56], RHNC [57], and HNC with the central force model [58]. Although the three approaches could have reproduced the structural features of water experimentally observed in qualitative level, XRISM proves to have a clear advantage over the other methods in terms of applicability and extensibility to chemistry and biophysical chemistry [59]. In Fig.1.3, shown is the pair correlation functions (PCF) of water obtained from the XRISM theory using the SPCE model for the pair interaction potential. In the same figure, a typical configuration of the hydrogen-bond network in water is also illustrated by a cartoon. Comparing the peak position of the pair correlation functions with the separations of atom pairs, which are labeled a,b,c, one can see that XRISM produces well the characteristics of water structure stated above at least qualitatively: the sharp first peak (b1 ) in the O-H PCF corresponding to the hydrogen-bond, the narrow first peak (a1 ) in the O-O PCF corresponding to the small coordination number, and the discernible second peak (a2 ) in the OO PCF, which is indicative of the O-O-O configuration typical in the ice-like structure. Especially does the appearance of the second peak in the O-O PCF deserve to be highlighted, because the peak is located at the position where the first minimum appears in the case of simple liquids (cf. Fig. 1.1). The first minimum of PCF in simple liquids is

25


3

a1

O -O O -H H -H

b1 2

g αγ

b2

c2

c1

a3 b3

a2

c3

1

0

0

2

4

6

8

r [Å ]

O O H

H

H O H H

c1

H

H

c2 c3 O

H

H

H

b1 O O H O H a3 H H H b2 a2 O b3 H O H H a1 O H H O H O H H

H

O H

O

H

H

HH O

H

Figure 1.3. Site-site Pair Correlation Functions (top) and the illustration of the hydrogen- bond network (bottom) in Water. The peak positions denoted by a1 , b1 ,. . . in the top figure roughly correspond to the site-pair distance indicated by the same characters in the bottom figure. The SPCE model ([60]) with slight modification for the size of hydrogen-site is employed for the potential parameter.

26


3 T =300 K 400 K 500 K 650 K

g OO

2

1

0

0

2

4

0

2

4

r [Å ]

6

g OH

2

1

0

Figure 1.4. ter.

r [Å ]

6

Temperature Dependence of Site-Site Pair Correlation Functions in Wa-

a manifestation of the packing effect: owing to the packing effect, the second coordination shell is separated by the first coordination shell to make the second peak at r ∼ 2σ, and in between the first (r ∼ σ) and second (r ∼ 2σ) a valley appears. From such a consideration, it is tempting to view water structure as a spatial as well as temporal mixture of two weighted configurations, one typical of simple liquids and the other characteristic to ice [61]. The statistical weight itself is a function of temperature and pressure (or density). The temperature dependence of the PCF of water calculated by XRISM is shown in Fig. 1.4. The figure clearly indicates change of the statistical weight from ice-like to simple liquids as temperature increases: the second peak in the O-O PCF decreases and the third peak shifts toward the position where the second peak in the simple liquid is supposed to appear (r = 2σ ∼ 5.6σ). Realizing that the chemical potential is ensemble-invariant, the Gibbs free energy can be calculated from the excess chemical potential in the

27


G [kcal/mol]

5

G id G RISM

0

G expt ∆G RISM

-5

0

20

40

60

80

100

t [ºC] Figure 1.5. Gibbs Free Energy Calculated by theRISM Theory:∆GrmRISM, the excess free energy calculated by Eq. (1.113); Grmid , the ideal gas contribution; GrmRISM= Grmid + ∆GrmRISM ; Grmexpt, experimental results.

case of pure liquid using the equation, [62] ∆G = N ∆µ,

(1.113)

where the excess chemical potential can be obtained from Eq. (1.112). The results are plotted against temperature in Fig. 1.5 along with the experimental data.

28


χ [ 1 0 -5 / B a r ]

5 .0

4 .8

4 .6

4 .4 0

20

40

60

80

100

t [ºC ] Figure 1.6. Isothermal Compressibility Calculated by the RISM Theory: the solid line, theoretical results; the asterisk, experimental data.

The isothermal compressibility calculated from XRISM is plotted against the temperature in Fig. 1.6 along with the experimental data. Apparently, the theory overestimates the experiment. However, the qualitative nature of the experiment is well reproduced: especially, the appearance of a minimum. The appearance of a minimum in the temperature dependence of the compressibility is a non-trivial feature which distinguishes water from other liquids. In ordinary liquids, the compressibility increases monotonically with increasing temperature, because compressible space enhances owing to thermal expansion. The compressibility of water, too, shows the ordinary behavior at the higher temperatures. However, in the lower temperatures, water exhibits the opposite dependency. The abnormal behavior can be explained in terms of the cross-over of the statistical weight of two configurations characteristic of ice and of simple liquids with increasing temperature. As temperature increases, the statistical weight shifts toward simple liquids, or a more packed state, as is clarified in the interpretation of the pair correlation functions, and reduces compressible space.

6.2

Hydration free energy of ions

Ion hydration is the most elementary process which is related to chemical reactions in solution. First, let us outline in the most primitive fashion how a continuum model treats the problem [63]. In the electromagnetic theory, the energy density of the electrostatic field E(r) is

29


expressed by E 2 /8π, where E is the magnitude of the field. Suppose charge of an ion produces a spherically symmetric field, the electrostatic field intensity becomes, E(r) =

z r , r2 r

(1.114)

where z is the charge of the ion, and r is the radial distance from the ion. The energy density of the field therefore is, ∞ 2 1 z2 z E 2 (r) 2 dr = , (1.115) 4πr dr = G= 8π 8π R r2 2R where R is the radius of the ion. When the ion is placed in a continuum medium with dielectric constant , the field strength is reduced by a factor of , G =

z2 . 2 R

(1.116)

The solvation free energy, which is the difference between the free energy of an ion in solvent and that in vacuum is,

z2 1 −1 . (1.117) ∆G = G − G = 2R

This is the famous equation of solvation free energy of an ion proposed first by M. Born [64]. However, the theory turns out to be very poor in reproducing experimental results of the solvation free energy and its derivatives assuming Pauling’s crystal radii for the ionic radius R. The agreement with experiments was dramatically improved later by Latimer et. al. introducing empirical corrections for R, and who proposed the following equation.

z2 1 −1 , (1.118) ∆G = G − G = 2 (R + ∆)

where ∆ is the correction for ionic radii, and 0.1 ˚ A and 0.8 ˚ A are assigned, respectively, to cations and anions. However, the physical meaning of the correction was not clear at all especially, from a microscopic point of view. The theory of electrostatics based on the continuum model has been developed dramatically since the first work by Born, especially in two important fields in chemistry: chemical reactions and protein stability [65] [66]. The mathematical formulation has been elaborated much more so that it can be applied to molecules with complex shape, and the parameters representing the boundary conditions are refined so

30


as to reproduce the experimental data even quantitatively. In spite of those latest developments, the essential problems which the continuum model has originally had remains unsolved. The main questions to be answered are why such corrections are necessary, and why corrections are asymmetrical for cations and anions. The continuum model will never be able to answer the questions. The ion radii in the continuum model were introduced as a boundary condition, and such ambiguous corrections are an unavoidable consequence associated with the use of continuum solvent models no matter how well it is refined.

Figure 1.7. Hydration Free Energy of Alkali-Halide Ions: symbols, RISM theory; lines, Eq. 1.77; and —, cations; and - - -, anions.

Let us try to answer the questions based on the RISM theory [10]. The solvation free energy of ions can be obtained from Eq. (1.112). Shown in Fig. 1.7 is the free energy of alkali and halide ions in aqueous solutions at infinite dilution. Plotted also in the figure are corresponding results due to Eq. (1.118), which are supposed to reproduce the experimental


31

results well. A half of the Lennard-Jones σ parameter is used for the ionic radii R. As may be readily seen, both plots are in qualitative accord. One can deduce two important conclusions from the results. First, if one uses a microscopic model for solvent, no such corrections ∆ in Eq. (1.118) for the ionic radii are necessary. Second, the asymmetry in the correction originates entirely from some kind of asymmetry intrinsic to solvent, not to ions. A water molecule has a charge distribution in which positive charges are localized at the positions of proton, while electrons or negative charges are delocalized in the molecule with its center almost at the oxygen position. Such charge distribution in a water molecule can be well described by the point charge model commonly employed in molecular simulation studies, such as TIPS and SPC [67] [68]. A feature common to all those models is an asymmetry in the charge distribution: the two positive charges are localized near the surface of a sphere, while the negative charges are located either right at center or near center of the sphere. The asymmetry of charge distribution in a water molecule is the physical origin of the asymmetrical response to the electric field of cations and anions.

6.3

Non-equilibrium Free Energy Profile for the Electron Transfer Reactions

In the previous section, the hydration free energy of ions was obtained from the RISM theory. By “free energy,” we meant the free energy in an “equilibrium” system. Namely, the solvent around an ion is equilibrated with the electrostatic field arising from the solute. However, the density of solvent around an ion is in continuous thermal fluctuations, and the solvent polarization is not necessarily in equilibrium with the electrostatic field of the solute. The greater the fluctuation, the larger the free energy becomes. The free energy profile plotted against the fluctuating polarization takes a parabola-like form, mainly, owing to the central limiting theorem, and the minimum of the free energy profile is identified as the equilibrium by definition. The solvent polarization at equilibrium depends highly on the charge state (or charge distribution) of the redox pair, as one can imagine from the previous section: for example, the solvent polarization must be quite small for a neutral pair, while it can be large for an ionic pair. A solvent configuration with small polarization, corresponding to a neutral pair, is a state of large solvent fluctuation for an ionic pair, and vice versa. Therefore, the solvent polarization, or equivalently the electrostatic field due to solvent at a redox pair can be regarded as an order parameter which characterizes the degree of solvent fluctuation. The order parameter can be

32


also regarded as a “reaction coordinate” which characterizes a transition of the redox pair from one state to the other. If one draws the free energy profile of a redox pair along the order parameter, one obtains two parabola-like curves with their minima (equilibria) located at two different values of the parameter, corresponding to the redox states of the pair, which intersect each other at a position corresponding to the same solvent polarization, or the solvation energy (Fig. 1.8). The free energy corresponding to the same polarization is higher than that corresponding to either state of the redox pair; therefore the electron transfer reaction, in general, becomes a barrier crossing process. Marcus [69] has expressed the rate of electron transfer reaction in polar solvent relying on a transition state theory as −∆G‡ , (1.119) ke = ν exp kB T where ν is the transmission coefficient concerning the quantum tunneling, and ∆G‡ denotes the activation free energy expressed in terms of the solvent reorganization energy λ as follows, ∆G‡ =

(λ + ∆G0 )2 . 4λ

(1.120)

In this expression, ∆G0 is the free energy difference between the reactant and product states in equilibrium; λ is the non-equilibrium free energy stated above (Fig. 1.8). An electron has to overcome the nonequilibrium free energy in order to change from one redox state to the other. So, it is essential to evaluate the non-equilibrium free energy to predict the reaction rate. But, how can the non-equilibrium free energy be evaluated? Marcus has devised a succinct idea for realizing the nonequilibrium free energy using a thermodynamic cycle illustrated in Fig. 1.9 and the Born free energy formula (Eq. 1.117).

33


R ∆GR(X)

P ∆Gp(X)

λ ‡

∆G ∆G0 Figure 1.8.

Free energy profile in the electron transfer reaction.

D0

∆F1

A0

∆F0

∆FZ

A0

DZ+

D0

∆F

DZ+

R

AZ–

∆F2+∆F3 D0

Figure 1.9.

AZ–

A0

Thermodynamic Cycle.

0 ,z 0 ), Suppose a donor-acceptor (DA) pair is in the charge state (zD A 0 0 which is labelled as D A in the figure. The unit of the charges is the electronic charge |e|. Solvent around the solute pair will have an equilibrium polarization corresponding to the electrostatic field from the solute. Solvent polarization is represented by the orientation of the thick arrows. The solvent polarization will be caused to deviate from

34


the equilibrium value occasionally, although infrequently, by thermal motions. The free energy increase ∆F R associated with the spontaneous polarization corresponds to the non-equilibrium free energy. The non-equilibrium state can be achieved through an alternative thermodynamic path illustrated in Fig.1.9. Employing the thermodynamic cycle, we can calculate the non-equilibrium free energy associated with solvent reorganization, ∆F R , as ∆F0 + ∆F R = ∆F1 + ∆Fz + ∆F2 + ∆F3 .

(1.121)

Suppose the non-equilibrium solvent polarization around the D0 A0 pair can be produced as an “equilibrium” response to a DA state with “hy0 + z and z 0 − z, which is labelled by Dz+ Az− . By pothetical charges” zD A “hypothetical charges”, we mean the charges hypothetically put on the redox pair in order to produce a solvent polarization which is equivalent to the solvent fluctuation, or non-equilibrium polarization around the solute. In that sense, the hypothetical charges serve as an order parameter, or a reaction coordinate. It is convenient for an electron transfer process to define another reaction coordinate called “solvent coordinate” by ∆Ez = (VA − VD ),

(1.122)

where VD (or VA ) is the electric potential at the donor (or acceptor) site produced by a fluctuating polarization of solvent corresponding to a hypothetical charge z. ∆F0 and ∆Fz is the (equilibrium) solvation free energy, or excess chemical potential, of the redox pair in the state D0 A0 and in the hypothetical state Dz+ Az− , respectively. Those free energies can be evaluated essentially by some standard methods, such as Born’s method described in the previous subsection, or a (equilibrium) statistical mechanical method. The change in free energy ∆F1 of the DA pair in vacuum from D0 A0 to Dz+ Az− states is essentially the energy associated with electronic reorganization, and can be evaluated by a method of quantum chemistry. The free energies associated with the process from the Dz+ Az− state to the D0 A0 state in solution consist of two contributions, ∆F2 and ∆F3 . ∆F2 denotes the reorganization energy of electrons from the Dz+ Az− state to the D0 A0 state, which has approximately same magnitude but opposite sign from ∆F1 , namely, ∆F2 ≈ −∆F1 . It is only “approximate,” because ∆F1 is the electron reorganization energy in the gas phase, while ∆F2 is that in solution. Actually, ∆F0 as well as ∆Fz described above have similar components originated from the reorganization of electronic state due to the solvent effect, which nearly cancel out. Contributions from those components can be significant when a redox pair is polyatomic solutes, and it requires

35


careful treatment based on quantum chemistry. Such a case will be presented in the following chapter, where solvent effects on electronic state of a molecule in solution are treated by means of the RISM-SCF method. In the present chapter, the solvent effect on the electronic structure is disregarded completely for making the explanation of the solvent reorganization simpler. ∆F3 is just an ion-solvent interaction energy with fixed solvent polarization, which can be readily evaluated by, ∆F3 = z∆Ez .

(1.123)

Taking everything stated above into account, the solvent reorganization energy corresponding to the reactant and products can be, respectively, written as, ∆F R (∆Ez ) = ∆Fz − ∆F0 + z∆Ez

(1.124)

∆F P (∆Ez ) = ∆Fz − ∆F1 + (1 − z)∆Ez .

(1.125)

and

By inspecting the thermodynamic cycle illustrated in Fig. 1.9, the total free energy of the reactant and product states at the reaction coordinate ∆E become, respectively, F R (∆E) = Ermel, 0 + ∆F0 + ∆FR (∆E)

(1.126)

F P (∆E) = Ermel, 1 + ∆F1 + ∆F P (∆E),

(1.127)

and

where Ermel, 0 and Ermel, 1 are the electronic energy of the reactant and product in the gas phase, respectively. From Eqs. 1.124) through (1.126) we obtain a rather general relation between the free energy profiles and the reaction coordinate [70][71]. F R (∆E) − F P (∆E) = ∆E + ∆ Ermel,

(1.128)

where ∆Eel is the difference in the electronic structure between the product and reactant defined by, ∆Ermel,= Ermel, 0 −Ermel, 1 .

(1.129)

The reaction coordinate at the transition state can be identified as ∆E ‡ = −∆ Ermel, since F R (∆E ‡ ) = F P (∆E ‡ ) at the transition state by definition. The solvent reorganization energy λ, which is the difference

36


in the free energy between the reactant and product states at the reaction coordinate ∆E0 defining the (equilibrium) reactant state, can be obtained from the Eq. (1.128) as λ = F R (∆E0 ) − F P (∆E0 ) = ∆E0 + ∆ Ermel.

(1.130)

The popular expression for (λ) of Marcus can be readily obtained from Eq.(1.130) employing the Born formula (Eq.1.117) for solvation free energy of a redox pair assuming ∆Ermel = 0, z2 λ= 2

1 1−

1 1 2 + − RD RA L

,

(1.131)

where RD and RA denote the radii of donor and acceptor ions, and L is the separation between the two ions. In the expression, we have also ignored the molecular polarization of solvent, namely, 1 − 1/ rmop= 0, where rmop is the dielectric constant owed to molecular polarization. The non-equilibrium free energy profile along the reaction coordinate can be realized by the XRISM theory as follows. To calculate ∆E in Eq. (1.124), we use the following equation, VD = 4πρ

s

r 2 dr

zs uv g (r), r Ds

(1.132)

uv is the site-site pair correlation function between a donor site where gDs D and a solvent site s. The expression for VA is given by replacing D in Eq.(1.132) with A. Then ∆Ez can be calculated from Eq.(1.122) with the solute charge distribution {z0D + z, z0A − z}. The equilibrium solvation free energy ∆Fz at infinite dilution can be calculated from the RISM equation with the HNC closure using the Singer-Chandler formula, Eq. (1.112), introduced in the previous section. Note that the solvation free energy calculated from Eq.(1.112) is the excess chemical potential taking pure solvent as the reference state. However, the terms associated with the pure solvent in ∆Fz and ∆F0 cancels out in Eq.(1.124). In Fig.1.10, plotted is the free energy profile calculated from the extended RISM theory for an exchange reaction compared with the simulation results of Zichi. Excellent agreement between the theory and the computer experiment demonstrates that the theoretical method is capable of reproducing the non-equilibrium free energy surface along the reaction coordinate.

37


9

Free energy [kcal/mol]

8

o

o o

7 6

o

o

5

o

o

4 3

o

o o

o

2 o

1

o o

0

o

-40

o

o o

o

-60

o

o

-20 0 20 Delta E [kcal/mol]

40

60

Figure 1.10. Non-equilibrium free energy surface calculated by RISM (solid line) and by molecular simulation (circle)

Unlike the original Marcus theory, which uses the continuum model for solvent, the method described above can provide a microscopic picture for the solvent fluctuation. It will be of great interest to explore the chemistry of the electron transfer reaction, including the specific dependence of the rate constant on the variety of solute and solvent.

6.4

Hydration Structure

Two closely related models for a microscopic picture of hydration have been proposed in the 1957 issue of the Discussion of the Faraday Society. Frank and Wen proposed a model based on the solvation thermodynamics, according to which a small ion such as Li+ is surrounded by three concentric regions [72]. The innermost region, A, is the region where water molecules are firmly bound to the ion, the second region, B, is one in which water is less “ice-like,” i.e. more random in organization, than “normal water”, and the third region, C, contains normal water polarized in the ordinary way by the ionic field which has become relatively weak. They presumed that the cause of the structure-breaking effect is an approximate balance in the region B between two competing orientational effects which exert on water molecules: one of those is the hydrogen-bonding with neighboring water molecules, and the other is the orienting influence upon the dipole of the spherically-symmetric

38


ionic field. All three regions appear around a small ion such as the Li+ ion, while only B and C regions can be seen in the case of larger ions such as the Cl− ion, since the A-region is hidden by the atomic core. (Fig.1.11)

C B +

Figure 1.11.

A

Frank-Wen model of Ion-hydration.

39


+

Figure 1.12. crossing.

+

Exchange of water molecules around an ion viewed as activated barrier

In the same issue, Samoilov proposed essentially the same picture for ion Hydration, but from a different point of view [73]. Samoilov could have abstracted activation energies (Eiw ) associated with the activated jump of water molecules from the immediate vicinity of an ion to the bulk water from experimental data of the diffusion constant and the viscosity. Essentially the same quantity, but with a water molecule in place of an ion, can be obtained in a similar manner, and it is denoted E0 . (E0 is the barrier height of the diffusion process in bulk water viewed as an activated jump.) Samoilov found that Eiw > E0 for small ions such as Li+ and Na+ while Eiw < E0 for large ions such as K+ and Cl− ions. Samoilov referred to the two cases, ∆Ei (= Eiw − E0 ) > 0 and ∆Ei < 0 as “positive” and “negative” hydration, respectively. Those two pictures of ion hydration proposed more than four decades ago have had a great influence on interpreting experiments in the field of solution chemistry, and they still do [74]. However, it is still ambiguous what those concepts like “negative hydration” and “structure breaking” really mean in terms of the microscopic picture of solvent around the ions. It is of interest to see how the phenomena can be characterized in terms of the integral equation method [13].

40


2

P M F [k c al/m o l]

1 0 w ater

-1

+ + Cs N a+ K

L i+

3 2 1 0 -1 -2

F-

1

C l- B r

2

I-

3

4

5

r [Å ] Figure 1.13.

Potential of mean force

The logarithm of PCF multiplied by −kB T is the potential of mean force (W (r) = −kT log g(r)), which can be regarded as a solvent averaged potential surface along the reaction coordinate of an exchange reaction. The transition of a water molecule from the first to second minima of W (r) is considered as an activated barrier crossing process. The energy difference between the two extrema defines the activation energy corresponding to the Samoilov picture of ion hydration, namely, Eiw ≡ Wiw (rmax ) − Wiw (rmin ), where Wiw (rmax ) and Wiw (rmin ) denote the first maximum and the first minimum of the site-site potential of mean force between an ion (i; C, A) and a water site (w; O,H), respectively. Since the water site primarily attracted by a cation is the oxygen-site (O), and that by an anion is the hydrogen-site (H), we have chosen those site pairs for evaluating Eiw , namely, ECO and EAH . For bulk water, we have chosen the hydrogen bonding site pair (O and H) for evaluating the activation barrier, namely, E0 = EOH . In Fig.1.13, shown are the potential of mean force for those site pairs. The activation energy difference ∆Ei = Eiw − E0 are plotted in Fig.1.14. It can be readily seen that the picture for the ion hydration proposed by Samoilov is qualitatively reproduced by the theory. (The reader may have a question about the result for Na+ , which is negative rather than positive, in contradiction to our previous calculation [13]. However, the ambiguity is due to the slight difference in the potential parameters employed in

41


the present and previous calculations. The results should be understood as ∆Ei ≈ 0 for Na+ . Namely, the mobility of a water molecule in the vicinity of Na+ is not very different from that in bulk water. ) The residence time calculated from a simple formula of the transition state theory,

kB T ∆Ei exp , (1.133) τi = h kB T shows reasonable agreement with those obtained by Samoilov. All those observations indicate that the old concepts for ion hydration have some theoretical basis which can be interpreted in terms of the potential of means.

2 ∆E i [k cal/m o l]

F-

1

L i+

0

C l-

N a+

B r-

K+

-1

I-

C s+

1

2

3

4

5

σi [Å ] Figure 1.14. Samoilov’s model of Ion-hydration: ∆Ei > 0, positive hydration; ∆Ei < 0, negative hydration.

The next question then to be asked is “Is it possible to characterize the structural modification of solvent around the ions?”

42

Molecular theory of solvation 3 2.5 2 1.5 1 0.5 0 -0.5 -1 0

1

2

3

4 r [A]

5

6

7

8

1

2

3

4 r [A]

5

6

7

8

2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 0

Figure 1.15. Li+ -density derivative of water-water PCF: solid line, water-water PCF; dotted line, the density derivative (or perturbation due to Li+ ). Top panel, the O-O pair; bottom panel, the O-H pair.

43

Theory of molecular liquids 4 3.5 3 2.5 2 1.5 1 0.5 0 -0.5 -1 0

1

2

3

4 r [A]

5

6

7

8

1

2

3

4 r [A]

5

6

7

8

2 1 0 -1 -2 0

Figure 1.16. Cl− -density derivative of water-water PCF: solid line, water-water PCF; dotted line, the density derivative (or perturbation due to Cl− ). Top panel, the O-O pair; bottom panel, the O-H pair.

44


gww αγ

3

(δρu gww αγ )ρtot

O-O

2 1

gww gww αγ , δρu αγ

0 2

O-H

1 0

H-H

1 0 0

2

4

6

8

r/Å

Figure 1.17. Methane-density derivative of water-water PCF: solid lines, water-water PCF; dashed lines, the density derivative (or perturbation due to methane). Top panel, the O-O pair; middle panel, the O-H pair; bottom panel, the H-H pair.

In Figs.1.15 and 1.16 plotted are the solvent-solvent radial distribution function for water between O-O atoms and between O-H atoms, and the perturbation due to the existence of ions calculated from the RISM equation. The perturbation owed to ions or the density derivative of the PCF are calculated using the method developed by Yu, Roux,


45

and Karplus [75]. Only the results for Li+ and Cl− are shown, which represents, respectively, the ions of positive and negative hydrations. As has been described in the previous section, the structural characteristic of water is manifested in two features in the PCFs: the sharp peak around 1.4 ˚ A in the O-H PCF and the small dent around 4.2 ˚ A. The first feature is due to the hydrogen-bond between a pair of molecules, and the second one is a manifestation of the three-body correlation in the hydrogen-bond network in water, which has an ice-like tetrahedral coordination. The way in which those features of water structure are modified characterizes the ion perturbation. As can be clearly seen, the peak around 1.4 ˚ A in the O-H PCF is disrupted in either case owing to the ion perturbation. It is not surprising because in both cases the solvent-solvent hydrogen-bond is weakened as a result of the orientational effect of the ionic field. What then distinguishes the negative from positive hydrations? This question can be answered by examining the O-O PCF. The comparative inspection of the density derivative curves for Li+ and Cl− indicates that the first peak of the O-O PCF shifts outward for Li+ while it shifts inward for Cl− . What physical significance can be attached to the observation? When an ion is positively hydrated, as in the case of Li+ , water molecules around the ion tend to align the dipole moments in parallel with the ion field. The water molecules will then repel each other due to the repulsive electrostatic interactions between O-O and between H-H, which will manifest as the inward shift of the O-O PCF. On the other hand, when the negative hydration occurs as in case of Cl− , the structural equilibrium of water shifts toward the structure with “less hydrogen-bonding” or “more distorted hydrogenbonds”, which is denser than the structure with more hydrogen-bonds. The phenomenon is analogous to the thermal perturbation which causes the maximum density at 4 C. The combination of the shift in the first peak of the O-O correlation and the sign of the change in the first peak of the O-H correlation as well as the second peak of the O-O correlation characterizes the modification of the solvent structure around ions. It is interesting to see from such a view point how the water-water correlation function is modified around a hydrophobic solute. In Fig.1.17, plotted are such correlation functions for methane as the solute. The first peak of the O-H correlation increases around the hydrophobic solute indicating that the hydrogen-bond is strengthened. On the other hand the first peak of the O-O correlation function shift inward indicating that the water density is increased around the solute. The observation implies that the water structure is in fact increased. However, the modified structure is not ice-like but some structure which has higher density than that characteristic of the tetrahedral coordination.

46

6.5


Fluid phase transitions of associating fluids

More than a century ago, van der Waals [76] explained the essence of the molecular forces driving the liquid-gas phase transition: attraction at larger distances and repulsion as molecular cores overlap. With increasing density, the ideal gas pressure is increasingly compensated by the contribution of molecular attraction, which is then overcome by the rise of pressure in close packing. At a temperature T below the critical temperature, the P −V isotherm acquires a wiggle and thus admits the existence of two fluid phases of lower and higher density ρvap and ρliq in mechanical equilibrium, P (ρvap , T ) = P (ρliq , T ).

(1.134)

The condition of chemical equilibrium, µ(ρvap , T ) = µ(ρliq , T ),

(1.135)

is established by means of the so-called Maxwell construction [77] which replaces the intermediate portion of the wiggle with a horizontal segment representing the coexisting phases at the corresponding volume fraction. It ensures that the chemical potential of a particle passing from one phase to the other along the isotherm wiggle does not change if the two areas vap V dP = 0. bound by the segment and wiggle are equal, µvap −µliq = liq

The Helmholtz free energy of a fluid can be broken up into the ideal part and the contributions from the core repulsion and the longer-range attraction, A = Aid +Arep +Aattr , where Aid = ρkB T (ln ρ+3 ln Λ−1) is the free energy per unit volume of an ideal monatomic gas with the de Broglie thermal wavelength Λ = (2π2 /(mkB T ))1/2 . (The rotational, vibrational, and internal electronic terms are omitted here for simplicity.) The classical van der Waals (vdW) theory estimates the attraction contribution as Aattr = −αρ2 /2, where the positive constant α =

druattr (r)

represents the energy of a fluid molecule in the mean field of attraction of other molecules assumed to be random. The thermodynamic

∂(A/ρ) 2 leads to the pressure P = P0 − αρ2 /2, relation P = ρ ∂ρ T where P0 = Pid + Prep is the pressure of a molecular fluid interacting only by the core repulsion. It is approximated simply by the pressure P0 = ρkB T /(1 − ρvrep ) of the ideal gas occupying the effective space reduced by the molecular excluded volume. The latter is estimated in view of binary collisions as vrep = 2πσ 3 /3 per molecular repulsive core.

47


The equation of state (EOS) resulting from the vdW consideration, P =

αρ2 ρkB T , − 1 − ρvrep 2

(1.136)

provides a very simple and physically transparent semi-phenomenological picture of the underlying fluid phase behavior. It was shown rigorously that for a general class of potentials, the van der Waals-Maxwell theory of phase transitions results directly from the classical partition function, with the Maxwell rule following from the statistical-mechanical requirement of the free energy being a convex function of density [78]. Many generalizations of van der Waals’ ideas have been proposed for improving the prediction of the fluid phase behavior in a wide range of thermodynamic states and to extend the description to molecular fluids and mixtures [79]. Longuet-Higgins and Widom [80] suggested that the repulsive term P0 be replaced with the accurate expressions for the pressure of hard spheres elaborated in the theory of liquids [81], such as the Carnahan-Starling approximation,[82] P0CS = ρkB T

1 + η + η2 − η3 , (1 − η)3

(1.137)

where η = πσρ/6 is the packing fraction of the hard sphere fluid. The Longuet-Higgins and Widom version, as well as the vdW theory, calculates the attractive mean field in the assumption of the molecules to be random, analogously to the mean-field theory of the Ising model [81]. The attraction term of the Redlich-Kwong EOS [83] is empirically modified so as to account for the change of the effective attraction constant with temperature and density. It is combined with the hard-sphere repulsion term in the Carnahan-Starling-Redlich-Kwong EOS [84] which can be written as P = ρkB T

1 + η + η 2 − η 3 αρ2 (T ∗ /T )1/2 − , (1 − η)3 2 1 + ρvrep

(1.138)

where T ∗ = α/(2kB vrep ) is the Boyle temperature at which the second virial coefficient turns intOZero. Barker and Henderson have developed a thermodynamic perturbation theory of simple liquids [81]. It splits the free energy into that of the reference fluid (hard spheres) and the perturbation potential (attraction) integrated with the radial distribution function of the reference fluid, which can be regarded as a generalization of the vdW approach on the statistical-mechanical basis. This scheme proved to be highly successful for atomic fluids and mixtures interacting via the 12-6 Lennard-Jones potential [81]. The perturbation series

48


becomes slowly convergent and thus questionable to truncate for more complex fluids, such as dipolar hard spheres. Scott and van Konynenburg [85] generalized the vdW EOS to binary fluid mixtures and showed that it is able to reproduce most of the phase behavior of binary fluids. The classification scheme of van Konynenburg and Scott subdivides the phase equilibria of binary fluid mixtures into types I to V, according to the critical lines behavior. Gubbins and Twu [86] later added type VI phase behavior for liquid mixtures exhibiting closed loop immiscibility, not found for the vdW equation. Beret and Prausnitz proposed the perturbed hard-chain theory [87] and Donohue and Prausnitz extended it to mixtures [88], yielding an EOS for nonpolar chain-like molecules of industrial interest. They constructed the repulsive term based on the Carnahan-Starling expression, and using Prigogine’s parameter [89] to take into account molecular rotations and vibrations. The simplified perturbed hard-chain theory [90] with a simplified form of the attractive term [91] has been used to calculate and classify global phase diagrams for binary mixtures of nonpolar molecules with various sizes [79]. For precise recapitulation of the behavior of a molecular fluid or mixture in a given thermodynamic range, one can always elaborate a semiempirical EOS with numerous adjustable parameters. For instance, Saul and Wagner’s EOS for water [92] provides accurate approximation in the region from the melting line to 1237 K at pressures up to 250 kbar. An empirical non-cubic EOS with 13 adjustable parameters was devised for reproduction of the critical point of pure substances, and fitted to the critical isotherms of methane, propane, and hydrogen [93]. Apart of these practical tasks, of fundamental importance is the development of a microscopic theory yielding the EOS of a molecular fluid taking account of its chemical specificity from the first principles on a statisticalmechanical basis. An ab initio description of the phase diagram of such hydrogen-bonding fluids as water and methanol constitutes a significant challenge because of the strong association effects, particularly increasing at lower densities. The extended RISM theory can readily describe ion-molecular solutions of polyatomic species at liquid density, with proper account of their chemical specificity [59]. However, the RISM/HNC equations strongly overestimate clustering of charged species in a vapor phase and near critical lines. The HNC solutions pathologically diverge close to the coexistence curve [94, 95]. They also become divergent as association of fluid molecules increasingly strengthens in the gas phase. For instance, Lue and Blankschtein [96] described the vapor-liquid coexistence of water by employing the RISM as well as Chandler-SilbeyLadanyi integral equations [35] with the HNC closure corrected by the


49

bridge diagrams that contribute at zeroth order in the density. However, they treated the water vapor branch just by the second virial coefficient. Even with the bridge correction to the HNC closure, they encountered divergency because of the bridge diagrams becoming huge owing to the strong hydrogen-bonding attraction between water molecules in the vapor phase. The mean spherical approximation (1.28) is known to be appropriate for the description of critical fluids, their phase equilibria, and critical lines [97, 98]. It can be shown that the direct correlation function has the long-range asymptotics of the interaction potential between the particles, cαγ (r) → −βuαγ (r) as r → ∞. The MSA closure enforces the direct correlation function outside the repulsive core to be equal to the interaction potential. (Its run inside the core is determined from the physical condition of the distribution function there to be equal tOZero.) At some thermodynamic condition (T, ρ) the determinant D(k) = det[1 − ω(k)c(k)ρ] acquires the first-order zero D(k → 0) = 0, resulting in a corresponding singularity of the compressibility and the distribution functions, g(r) ∼ 1 + r −1 exp(−r/ζ) with correlation length ζ → ∞. This properly reproduces spinodal critical lines, and hence coexistence lines based on the conditions of chemical and mechanical equilibrium. On the other hand, the MSA and its extensions have a substantial drawback of producing nonphysical negative regions of the distribution functions for high density of liquid and/or strong intermolecular attraction of associating species. The above disadvantage is eliminated by complementing the RISM equation (1.89) with the closure, using the recently proposed [9, 28] KH approximation (1.30). It describes such features of the distribution functions as long-range enhancement tails in the critical regime and high peaks of strong association by using the MSA. On the other hand, the depleted density regions, in particular the repulsive core, are treated in the HNC approximation which ensures the physical condition g > 0 and thus prevents negative peaks of the distributions produced by the bare MSA. The distribution and its first derivative are continuous at the joint point g(r) = 1 by construction. Both the HNC and MSA, and hence the closure (1.30), provide the correct long-range asymptotic of the direct correlation functions. Notice that the KH approximation (1.30) is distinct from other mixed HNC-MSA schemes [99, 100, 101] which switch smoothly from the HNC closure at small interparticle separation r to the MSA at a larger distance. They adjust the parameter of the switching function by requiring that the virial and compressibility equations lead to the same isothermal compressibility. Successful for spherical species, these approaches cannot be readily implemented

50


for polyatomic molecules because a switching function appropriate for a more complex short-range structure could not have a simple form with the parameters to be determined from the few conditions of thermodynamic consistency. In contrast, the KH approximation switches from the HNC to MSA automatically at all the distances corresponding to association peaks and critical enhancement tales. In addition, the KH closure ensures existence of the solutions for systems with strong attraction between species, such as a metal electrode-water interface where the HNC closure would lead to nonphysical divergence (see Chapter 4). Much as for the RISM/HNC equations [43], the free energy function in the KH approximation has an exact differential [9, 28], and so the free energy and such derivatives of it as the chemical potential and pressure are expressed in a closed analytical form in terms of the correlation functions (see Appendix of Chapter 4). The existence of an exact differential is a non-trivial result. For comparison, a closed form is not available for the MSA or PY closures, since the solvation chemical potential depends on a path of thermodynamic integration in the process of “switching on” the interactions between fluid particles. This results from the contribution of the repulsive core edge spreading in the switching-on process. Notice, however, that an analytical expression for the chemical potential can indeed be obtained for the PY closure in the switching-on procedure involving the assumption of unique functional dependence of the bridge function on the indirect correlation function [102]. In the KH closure, the repulsive core region is obtained in the HNC approximation possessing the exact differential, whereas the MSA regions are smoothly joint and hence have it too. With the chemical potential and pressure obtained in the form of the closed expressions (4.A.9) and (4.A.11) in Chapter 4, the phase coexistence envelope can be localized directly by solving the mechanical and chemical equilibrium conditions (1.134) and (1.135) for the vapor and liquid phase densities, ρvap and ρliq , whether or not the solution exists for all intermediate densities. Provided the isotherm is continuous across all the region of vapor-liquid phase coexistence, Eqs.(1.134) and (1.135) are exactly equivalent to the Maxwell construction on either pressure or chemical potential isotherm. This stems from the fact that the RISM/KH theory yields an exact differential for the free energy function (4.A.10) in Chapter 4, which thus does not depend on a path of thermodynamic integration. As a numerical illustration, the RISM/KH theory has been applied [9, 29] to calculate the vapor-liquid coexistence lines for associating molecular fluids with different structures forming due to strong hydrogen bonding: water, methanol, hydrogen fluoride, and dimethyl sulfoxide

51

Theory of molecular liquids 12

T = 615 K 600 K 585 K 565 K 540 K

P (MPa)

8

4

0.0

0.2

0.4

0.6

0.8

1.0

v (l/mol) Figure 1.18. Pressure isotherms of SPC water (solid lines) obtained by the RISM/KH theory supplemented with the Maxwell construction. Metastable and unstable states (thin solid and short-dash portions, respectively). Binodal and spinodal (bold solid and dashed lines, respectively).

(DMSO). They are described by interaction site molecular models with rigid geometry and the site-site interaction potentials comprising the Coulomb and 12-6 Lennard-Jones terms. Water is represented by the SPC model [68], and the parameters for methanol are taken from the OPLS-UA force field [103]. Among simple models of hydrogen fluoride, the three-site model of Cournoyer and Jorgensen [104] is best suited to reproduce its vapor-liquid coexistence curve [105]. The calculation results for the DMSO model of van Gunsteren [106] best fit the experimental data for the DMSO liquid-vapor coexistence curve. Figure 1.18 illustrates the water pressure isotherms obtained from the RISM/KH theory, supplemented with Maxwell’s horizontal segments for the liquidvapor coexistence. Alternatively, the Maxwell construction can be applied to the chemical potential isotherms ρ(µ). The theory predicts a reasonable value of T = 366 K for the boiling temperature of water at atmospheric pressure P = 0.1 MPa. The spinodal line separating metastable states from the mechanical instability region is determined by the wiggle extrema on the isotherms. One should note that the RISM/KH theory yields a qualitative estimate to the spinodal line rather than a full description of the metastable states. A true theory of metastable states would require proper account of dynamics and inhomogeneity effects, such as cavitation and nucleation. The critical exponents following from the free energy (1.113) have the classical values, as expected for the theory involving the MSA.

52


600

600

500

500

T (K)

T (K)

SPC water

SPC water

OPLS methanol

400

400

OPLS methanol 300

300

0.0001

0.001

0.01

ρ (g/cm3)

0.1

1

0.2

0.4

0.6

ρ (g/cm3)

0.8

1

Figure 1.19. Liquid-vapor coexistence line of SPC water and OPLS methanol (solid and dashed lines) following from the RISM/KH theory versus the simulation data (open circles and squares, respectively) and critical point estimates (closed symbols). Logarithmic and linear scales are used to resolve the density in gas and liquid state.

Figure 1.19 draws a comparison between the predictions of the RISM/KH approach and the simulation data for the liquid-vapor coexistence envelopes of SPC water [107] and OPLS-UA methanol [108]. The theory qualitatively reproduces the phase diagrams. Remarkable is a good fit for their vapor branches. The predicted critical temperature of water, Tc = 600 K, is quite close to the simulation result of 587 K. The predicted critical density, ρc = 0.12 g/cm3 , differs considerably from the value of 0.27 g/cm3 following from the simulations. The underestimation of the binodal density decreases to a factor of 1.3 at room temperature. This discrepancy for the liquid branch of water originates from the well-known drawback of the HNC approximation underestimating the molecular core repulsion, and the strong directionality of the hydrogen bonding which is underestimated by a local closure not including multi-site correlations. The former can be obviated by introducing bridge corrections to the KH closure (1.30) to ensure consistency between the virial and compressibility routes to the isothermal compressibility. The latter drawback can be corrected by proceeding to the three-dimensional RISM approach. The RISM/KH theory recapitulates the substantial shift of the vaporliquid coexistence envelope of methanol towards lower temperatures (Tc = 503 K versus 500 K in the simulation) as compared to water. Unlike the tetrahedral hydrogen bonding of water, methanol molecules associate in zigzag-type chains by hydrogen bonding between their hydroxyl groups. The strength of association into a chain with a methanol mo-

53

Theory of molecular liquids 600

T (K)

500

400

300

200 0

0.2

0.4

0.6

0.8

1

1.2

1.4

ρ (g/cm3)

Figure 1.20. Liquid-vapor coexistence curve of the hydrogen fluoride model [104], following from the RISM/KH theory (solid line), the molecularOZ/RHNC approach [109] (dashed line), and the SAFT method [105] (dash-dotted line) versus the molecular simulation [105] (filled squares) and experimental data from Refs. [105, 109] (open circles).

lecule involved in two hydrogen bonds is decreased compared to four hydrogen-bonded neighbors in water. The decreased strength of intermolecular attraction results in the vdW wiggle of methanol smaller in width as well as in depth than that of water. That is, the density difference between the vapor and liquid states at a given temperature is less for methanol than for water, and the vapor-liquid phase coexistence curve of methanol is shifted to lower temperatures. This can be understood also from the principle of corresponding states for the methanol and water isotherms locally approximated with the vdW ones. The effective excluded volume parameter vrep determining the critical density ρc = 1/(2vrep ) appears to be quite close for water and methanol in spite of the smaller repulsive core of the water molecule, because of voids in the water structure tending to form the hydrogen-bonding network. On the other hand, the critical temperature Tc = α/(4vrep kB ) appears to be lower for methanol owing to the smaller value of its intermolecular attraction parameter α. Hydrogen fluoride presents another example of a fluid in which hydrogen bonding is crucial for the structural and thermodynamic properties. Figure 1.20 draws the HF vapor-liquid coexistence curve obtained from the RISM/KH theory in comparison with that following from the molecular OZ/RHNC approach [109] and the simulation data

54


700

DMSO

600

T (K)

water 500 hydrogen fluoride 400

methanol

300 0

0.2

0.4

0.6

ρ (g/cm3)

0.8

1

1.2

Figure 1.21. Comparison of the RISM/KH theory results for the coexistence curves of hydrogen fluoride (solid line), methanol (short-dashed line), water (dashed line), and DMSO (dash-dotted line) against experiment for their critical and ambient states (closed and open triangles, squares, circles, and diamonds, respectively.

[105]. Hydrogen fluoride exhibits an extremely high degree of association to which one may ascribe the breakdown of the molecular OZ/RHNC theory at low densities and its failure to yield a good estimate of the HF vapor-liquid coexistence [109]. The coexistence curve becomes significantly distorted, although the molecular OZ/RHNC treatment correctly reproduces the HF structure in the liquid phase, including the hydrogen bonding. For illustration, the HF coexistence curve obtained by using the statistical associating fluid theory (SAFT) approach is also shown. It was obtained with the SAFT hard-sphere reference system, association, and mean field parameters deduced directly from the HF interaction potential rather than adjusted to fit experiment [105]. The significant discrepancy between the SAFT prediction and the simulation data can be attributed to the coarse-graining of the Cournoyer-Jorgensen potential within SAFT [105]. Adjustment of the parameters supposed to be done in SAFT yields a much better agreement with the simulation [105]. This, however, renders SAFT an engineering rather than ab initio method. As distinct from the molecular OZ/RHNC and SAFT approaches, the RISM/KH theory yields the HF coexistence curve in much better agreement with the simulations. It qualitatively reproduces the change of the hydrogen fluoride structure as its density varies from gas to liquid, which is determined to the most extent by hydrogen bonding, especially strong in the gas phase.

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55

Figure 1.21 shows the change of the critical and ambient temperature states of liquid-vapor coexistence for the considered fluids (HF, water, methanol, and DMSO) obtained by the RISM/KH calculation versus experimental data. The ambient state density as well as the critical temperature decrease in the order: DMSO–water–methanol. The critical temperature of hydrogen fluoride is even somewhat lower but quite close to that of methanol. However, the HF hydrogen bonding becomes extremely strong in the liquid phase. To balance the hydrogen bonding attraction by the repulsion of molecular cores, the packing of molecules in HF liquid has to be very high. This results in the high number density of HF liquid. Its physical density is even higher in view of the low molar weight of HF compared with the other fluids, and is close to that of water. The RISM/KH theory well reproduces the decrease sequence of the critical temperatures, yielding the coexistence curves of DMSO, water, and methanol subsequently shifted to lower temperatures and narrowed. On the other hand, it also predicts the larger ambient state density of hydrogen fluoride resulting in the HF coexistence curve stretched in density.

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[106] H. Liu, F. M¨ uller-Plathe, and W. F. van Gunsteren, J. Am. Chem. Soc. 117, 4363 (1995). [107] J. J. de Pablo, J. M. Prausnitz, H. J. Strauch, and P. T. Cummings, J. Chem. Phys. 93, 7355 (1990). [108] M. E. van Leeuwen and B. Smit, J. Phys. Chem. 99, 1831 (1995). [109] C. Martín, M. Lombardero, J. A. Anta, and E. Lomba, J. Chem. Phys. 114, 355 (2001). 6964 (1994).

Chapter 2 ELECTRONIC STRUCTURE AND CHEMICAL REACTION IN SOLUTION Hirofumi Sato Department of Molecular Engineering Kyoto University [email protected]

Keywords: Electronic Structure in Solution, Chemical Reactions, RISM-SCF/MCSCF

1.

Introduction

Chemical reaction, which is undoubtedly one of the most important issues in the field of chemistry, has been studied extensively by means of ab initio molecular orbital (MO) theory since it is the electronic structure of molecules that primarily governs a chemical reaction. However, for a reaction in solution, it is also well regarded that solvent plays an essential role in determining the direction (what the product is), efficiency (how much the yield), and rate (how fast it is) of a reaction. In other words, the solvent effect in chemical reactions manifests itself not only through solute–solvent interactions and solvent reorganizations but also through intra-molecular processes which are always associated with changes in electronic structure. Such effects manifested in the effective charge of atoms amount to 10–40% sometimes, and it cannot be neglected. In this regard theoretical investigations of chemical processes in solutions are inevitably coupled with studies of quantum and statistical mechanics. [1] The ab initio approach of molecular dynamics (MD) , which was originated by Car and Parrinello, or combined methods of standard MO theory and molecular simulation are the most straightforward realization of such couplings. However, the treatment has serious limitations or difficulties in terms of system size: it will be virtually impossible to treat even a small system of a size typical of classical molecular simula-

61

62


tion, say ∼ 1000 molecules, by an ab initio quantum chemical method. A logical and practical solution of such a problem would be to employ a hybrid approach in which a chemical species of interest (solute) is treated by quantum chemistry whilst the surroundings (solvent) is handled classically. In the past two decades several types of quantum chemical methods for computing wave functions of solvated molecules have been proposed. They can be roughly classified into three categories, i.e., continuum model methods, the QM/MM method, and the RISMSCF/MCSCF method. The continuum model, in which solvent is regarded as a dielectric continuum, has been used for a long time to study solvent effects [2]. Solvation energies can be primarily approximated by a reaction field owed to polarization of the dielectric continuum as solvent, and other contributions such as dispersion interactions, which must be explicitly considered for non-polar solvent systems, have usually been treated with an empirical quantity such as the macroscopic surface tension of the solvent. An obvious advantage of the method is its handiness, whilst its disadvantage is an artifact introduced at the boundary between the solute and solvent. Agreement between experiment and theory is considerably governed by the boundary conditions. QM/MM (quantum mechanical molecular mechanical simulation method) is a typical hybrid approach. [3] The total energy of the liquid system at an instantaneous configuration, generated by a Monte Carlo or molecular dynamics procedure, is evaluated, and the modified Schr¨ odinger equations are solved repeatedly until sufficient sampling is accumulated. Since millions of electronic structure calculations are needed for sufficient sampling, the ab initio MO method is usually too slow to be practical in the simulation of chemical systems in solution. The QM/MM methods have their own disadvantages, the obvious one being the computational load added to the already complex calculation of the electronic structure. The RISM-SCF/MCSCF [4] which will be elucidated in this chapter is ‘the third’ method that overcomes above mentioned difficulties. The method combines the two theoretical elements: the ab initio MO method (quantum chemistry) and the RISM (statistical mechanics of liquids), and elicits the nature of chemical system in solution with a moderate computational demanding. It this chapter, after reviewing ab initio MO theory, the RISM-SCF / MCSCF method and its several applications, including acid–base equilibria, isomerization, and reactions are presented. The theory for NMR chemical shift of solvated molecules based on the RISM-SCF/MCSCF method is also described.

63

Electronic structure andchemical reaction in solution

2.

ab initio Molecular Orbital Theory and the

Solvation Effect 2.1

Isolated Molecular System

We shall begin with a brief sketch of the standard ab initio molecular orbital (MO) theory that deals with an isolated molecule, since this is a fundamental part of chemical reaction analysis. [5] A molecule consists of M nuclei and n electrons. Applying the Born–Oppenheimer approximation, in which electrons are moving around the spatially fixed nuclei, electronic hamiltonian of the system (Helec ) expressed in atomic unit is given by Helec =

n i=1

1 ZA 1 − ∇2i + + , 2 rij riA n

i>j

M

(2.1)

i=1 A=1

where ZA is atomic number and the denominators represent riA = |xi − RA |, rij = |xi − xj |.

(2.2)

Shr¨ odinger equation for this hamiltonian, Helec Ψelec = Eelec Ψelec ,

(2.3)

is the equation we would like to consider. The electronic energy is given by dx1 dx2 · · · dxN Eelec = × Ψ∗elec (x1 , x2 , · · · , xN )Helec Ψelec(x1 , x2 , · · · , xN ).(2.4) Sum of Eelec and the contribution from the repulsive energy between the nuclei is the total energy of the system. Etotal = Eelec + Enuc = Eelec + A>B

ZA ZB . |RA − RB |

(2.5)

It is noteworthy that Eelec parametericallly depends on the nuclear positions ({RA }). Let us consider the simplest case, a closed shell system. A way to describe the electronic wave function (Ψelec ) is to use the Slater determ-

64


inant that satisfies the Pauli’s principle. Ψelec (x1 , x2 , x3 , · · · , xN ) ψ1 (x1 ) ψ2 (x1 ) · · · 1 ψ1 (x2 ) ψ2 (x2 ) · · · = √ .. . N · ψ1 (xN ) ψ2 (xN ) · · ·

, · ψN (xN ) ψN (x1 ) ψN (x2 )

(2.6)

where ψi in the determinant is the one particle orbital. The problem is now how to determine the orbital ψi . According to the variational principle the best wave function should give the lowest energy. Imposing the constrains to the orthnormality of one particle orbital, we defined the following trial function, ∗

ij ψi (x)ψj (x)dx − δij . (2.7) L(ψ1 , ψ2 · · · ψN ) = Etotal − ij

Variations with respect to the functions yield δL = 0 N dx1 δψi∗ (x1 ) = i=1

⎫ ⎤ ⎡⎧ M N ⎬ ⎨ 1 ZA + (Jj − Kj ) ψi (x1 ) −

ij ψj (x1 )⎦ × ⎣ − ∇21 + ⎭ ⎩ 2 r1A j

A=1

j

+ complex conjugate. The Coulombic and exchange operators are defined as follows, 1 dx2 ψj∗ (x2 ) ψj (x2 )ψi (x1 ), Jj ψi (x1 ) = r12 1 dx2 ψj∗ (x2 ) ψi (x2 )ψj (x1 ). Kj ψi (x1 ) = r12 The Hartree–Fock equation is obtained with diagonalized . ⎫⎤ ⎡⎧ M N ⎬ ⎨ 1 Z A ⎣ − ∇21 + + (Jj − Kj ) ⎦ ψi (x1 ) = i ψi (x1 ), ⎭ ⎩ 2 r1A A=1

(2.8)

(2.9)

(2.10)

j

or Fψi (x1 ) = i ψi (x1 ),

(2.11)


65

by using the Fock operator ZA 1 + (Jj − Kj ) . F = − ∇21 + 2 r1A M

N

(2.12)

j

A=1

This is the fundamental equation in ab initio MO theory. In the practical procedure, the one particle orbital is expanded as a linear combination of proper functions (atomic orbital) whose center is located at the nuclei. (Linear Combination of Atomic Orbital; LCAO approximation) Ciµ χµ (x). (2.13) ψi (x) = µ

In this case solving the Hartree–Fock equation is equivalent to determine the coefficient matrix C (Ciµ ). FC = SC ,

(2.14)

where F, S and are matrices whose elements, for example, are calculated by dxχ∗µ (x)Fχν (x), (Fock matrix) Fµν = dxχ∗µ (x)χν (x). (overlap matrix) Sµν = (2.15) Eq.(2.14) is called the Roothaan equation and widely used as the basic equation in many computational program codes in quantum chemistry. Improvement of the approximation beyond the Hartree–Fock method introduced here, say taking account of the electron correlation effect, are discussed in many textbooks.

2.2

Dielectric Continuum Model

The most primitive model that takes solvent effect into account is (dielectric) continuum model. A treatment combined with a quantum chemical calculations including semi-empirical methods began in early 1970’s. In this section we summarize the continuum methods applied to quantum chemical calculations.

2.2.1 General aspect of the continuum model. The solute molecule with electronic distribution ρM , which is a function of r, can

66


be written as the sum of contribution from nuclear ρnuc and that from electron ρel . ρM (r : {R}) = ρnuc (r : {R}) + ρel (r : {R}),

(2.16)

where {R} indicate positions of nuclei that govern electron distribution parametrically. The contribution from the nuclei is given by Zα δ(r − Rα ). (2.17) ρnuc (r : {R}) = α

The contribution from electrons can be computed by using the wave function. We consider a vacuum hole (a cavity) in the dielectric continuum with an arbitrary shape. The solute molecule represented in the electric distribution ρM is immersed and located in this cavity. The basic equation for dielectric continuum model is Poisson–Laplace equation in the electromagnetics. The electrostatic field in the cavity Φin and outside one Φout can be obtained by solving the following equations. ∇2 Φin (r) = −4πρM (r), ∇2 Φout (r) = 0,

(2.18)

with boundary conditions: Φin (r∗ ) = Φout (r∗ ), ∂ ∂ Φin (r∗ ) = ε Φout (r∗ ), ∂n ∂n

(2.19)

where n is a vector normal to the surface of the cavity and r∗ indicates the cavity’s boundary. ε is the dielectric constant of the medium. Only this quantity represents character of solvent. The electronic potential in the cavity can be divided into the direct contribution from the solute molecule Φρ and the effect from the continuum medium Φσ called the reaction field. Φin = Φρ + Φσ .

(2.20)

The electrostatic interaction energy WM S between the solute molecule and the continuum is given by (2.21) WM S = dτ ρM Φσ . The solvation free energy is work of moving infinitely small charges; 1 ∆Gsol = WMS . 2

(2.22)


67

When we adopt a ‘classically’ described molecular model, that is, an assembly of points with fixed charge, as the solute the solvation energies are readily evaluated. However, in the case of a quantum chemical description of the solute molecules the evaluation is not trivial, since the solute’s electronic structure and the reaction field obviously depend on each other. This is a typical non-linear problem and has to be solved in an iterative manner. For practical purposes several approaches have been proposed for solving these problems. All of them are based on the modification of the abovementioned electronic Hamiltonian. [Helec + Vint ]Ψ = EΨ,

(2.23)

where Helec is the usual Hamiltonian for an isolated molecule and Vint is the operator representing the interaction between the solute and the continuum medium, Ψ|Vint |Ψ = WMS .

(2.24)

In the case of the Hartree–Fock method for a closed shell system the new Fock matrix is written as (0) = Fµν + µ|Vint |ν, Fµν

(2.25)

where F (0) is the Fock matrix of isolated molecule. In constructing the practical Fock matrix two points should be clarified. One of them is the shape and size of the cavity, and the other is the description of the ρM . The spatial character of an ideal cavity should be the inclusion of the whole charge distribution of the solutes and the exclusion of empty space where solvent can intrude. A molecular shaped cavity is feasible for fitting such a purpose, but expression of the interaction energy is in a complex form in general. In the case of a regularly shaped cavity, e.g., sphere, ellipsoid, and cylinder, the expression is usually given in analytical form. Note that an expression for the perturbing matrix elements (Vint ) are an important issue in evaluating various properties of molecules. For example, it is difficult to derive the energy gradient with respect to the solute molecular nuclear coordinates in a numerically expressed Vint system. If one has an analytical form of Vint the evaluation of the gradient is straightforward.

2.2.2 Several methods based on the continuum model. Self Consistent Reaction Field (SCRF) This is the simplest and oldest approach of all. The electronic distributions of the solute molecules are expressed in terms of the multipole expansions.

68


First, we consider a spherically shaped cavity with radius a. The reaction field by multipoles located at the center of the cavity can be written in Legendre polynominals. Φσ =

∞ l=0

l (l + 1)(ε − 1) r l+1 − Blm Ylm (θ, φ). l + (l + 1)ε a2l+1

(2.26)

m=−l

The most simple, but important, case only has the term l = 0, corresponding to Born’s equations. The solvation free energies can be written as ∆Gsol = −

ε−1q , 2ε a

(2.27)

where q is net charge in the cavity. For a neutral species the first contributing term is the dipolar term (l = 1) ∆Gsol = −

ε − 1 µ2 . 2ε + 1 a3

(2.28)

The application to quantum chemical calculations is achieved with a new Fock matrix, m,m (0) + Mlm fl,l µ|Mlm (2.29) Fµν = Fµν |ν l,l m,m (0)

where Fµν is the Fock matrices for isolated molecule, Mlm is the mulm,m is a constant defined by the cavity’s shape. tipole operator, and fl,l The interaction energy of this method can be written as, WM S

=

l ∞

Mlm Rlm

l=0 m=−l m,m = Mlm fl,l Mlm .

(2.30)

l,l m,m

The Fock operator in the simplest case, just taking the dipolar term, [6] is (0)

Fij = Fij −

2ε − 1 µ i|ˆ µ|j. ε + 1 a3

(2.31)

It is obvious that all these methods are too crude for describing a molecular system, although they are convenient to handle. Ambiguity in defining the cavity’s radius, center, or the orientation of 3-axis for an ellipsoidal cavity are also problems of these methods. In this sense


69

quantitative values obtained by the SCRF may lead to serious error. Poor convergence in the multipole expansion (l) has been also pointed out. Polarizable Continuum Model (PCM) This method was developed by Tomasi’s group in 1981 and many applications have been proposed [2]. The most distinctive feature of this method is to be able to treat a molecular shaped cavity. Applications not only to Hartree– Fock methods, but to UHF, MCSCF, MBPT, CASSCF, MR-SDCI and DFT etc. have been reported. They also proposed a extension to nonequilibrium solvation problems. The basic concept of their method is that the reaction potential may be described in terms of an apparent charge distribution on the cavity boundary’s surface. The charge distributions σ and the potential from them can be evaluated as σ = −P · n =

ε−1 ε−1 ∇Φout · n, = ∇Φin · n 4π 4πε

ε−1 ∂ (ΦM − Φσ ), 4πε ∂n σ(s) ds2 , Φσ (r) = |r − s| =

(2.32)

where P is the polarization vector. The Fock matrices of these methods are σ(s) (0) ds2 |ν. (2.33) Fµν = Fµν + µ| |r − s| In order to solve these equations, the surface is divided into an appropriate number of tesserae qk = ∆Sk σ(rk ). qk . Φσ (r) = |r − sk |

(2.34)

k

The tessellation is derived from the union of van der Waals spheres centered on the solute’s atoms. The pentakisdodecahedron (which has about 60 faces for one sphere) are adopted for the tesserae. Their method is essentially based on a discrete representation of the reaction field and very similar to numerical methods such as finite differential methods. Image Charge (IC) The fundamental idea of this method was proposed by Friedman [7]. The image charges with appropriate position and value are used so as to describe the reaction potential by dielectric is the image operator which generates the image charge of media. Im

70


the charge distribution. ρM (r ) a M (r) = − ε − 1 dr , (2.35) Imρ ε+1 |r − N| |r − a2 (r − N)/|r − N|3 | where N and a, respectively, indicate the center and radius of the cavity. The solvation free energy for the system is given by M (r). (2.36) WM S = drρM (r)Imρ Note that the energy is reduced to a simple form in the case of a point charge in a spherical cavity ε − 1 q2 . ε + 1 2a

(2.37)

ε − 1 µ2 . ε + 1 2a3

(2.38)

∆Gsol = − For the case of dipole moment ∆Gsol = −

The Fock operator in this model is represented by 1 1 (0) Im ˆ ˆ Dλσ (µν|Im|λσ) + (λσ|Im|µν) , (2.39) Fµν = Fµν + Hµν + 2 2 λσ

where H Im includes the interaction of distributions between real electron and image nuclei, and between image electron and real nuclei. The first applications to some chemical system were reported by Karlström. [8] The advantage of this method is that all the multipole expansions are implicitly included. This leads that expression of derivative with respect to the solute nuclear coordinates are rather simple. [9] The application to the semi-empirical Hamiltonian was also done by Klamt and Sch¨ uu ¨rmann named COSMO [10], although it can also be regarded as an intermediate between IC and PCM.

2.2.3 Summary of the continuum model. The method described in this section is the first application to calculating a solvated molecule’s properties in a quantum chemical manner. Qualitative considerations seem to be reasonable in spite of their considerable simplicity. PCM can be regarded as a final achievement amongst the series of ab initio MO–dielectric continuum combined methods, since all the method is based on the Poisson–Laplace equation. However, it is important to pay much attention to the serious disadvantage in all these

71


method, i.e., how to define the cavity. Actually the solvation energies are very sensitive to the radius and shape of the cavity. Even if an evaluated energy looks reasonable, we must be very careful about its physical origins. It should be also noted that we do not obtain any information on the solvent, e.g., solvation structure with these methods.

2.3

QM/MM Method

The dielectric continuum models such as the self-consistent reaction field method, are efficient in taking account of long range solute–solvent electrostatic interaction and the effect of solvent polarization, but these can not describe local solute–solvent interaction such as hydrogen bonding. Other methods are therefore desired which can describe a short range solute–solvent interaction in a proper manner. An alternative which can overcome the weak points of the dielectric continuum model is the QM/MM method. The idea introduced here is simple. We consider a system composed of one solute molecule and several hundreds of solvent molecules. Since full quantum chemical treatment of molecules is computationally too demanding, the electronic structure of the solute molecule is only treated quantum mechanically (QM) whilst the molecular mechanics (MM) is introduced to describe the solvent molecule. Interaction between the solute and solvent molecule is usually expressed in the following manner. HQM/MM

S N qs =− , riS s=1

(2.40)

i

where S is number of solvent molecule surrounding the solute molecule, qs is the point charge assigned to an atom in solvent, and riS is the distance between the atom in solvent and the solute. The interaction Hamiltonian is incorporated into the electronic Hamiltonian QM/MM QM/MM QM/MM = Eelec Ψelec , (2.41) Helec + HQM/MM Ψelec QM/MM

) is the solute electronic wavefunction affected by the solvent (Ψ elec obtained by solving this modified equation. Since the electron exchangeexclusion interaction between the solute and solvent is not treated in this Hamiltonian, a contribution from repulsive energy, such as the van der Waals interaction, is usually added to the total energy, which is evaluated at an instanteous solvent configuration generated by molecular dynamics or the Monte Carlo procedure. To compute appropriate thermodynamic properties of the system the quantum chemical calculations are repeated and averaged over more than millions of geometrical configurations.

72


The QM/MM method is becoming popular for studying chemical reactions not only in systems of fluids, but in a biological system such as enzyme. It is obvious that a serious bottleneck of the method is its computational demands.

3.

RISM-SCF/MCSCF Theory

A new method referred to as RISM-SCF/MCSCF is based on the ab initio electronic structure theory and RISM. The original RISM-SCF method was proposed by Ten-no, Hirata, and Kato in 1993. [11] The basic idea of the method is to replace the reaction field in the continuum models by a microscopic expression in terms of the site–site radial distribution functions between solute and solvent, which can be calculated from the RISM theory. Exploiting the microscopic reaction field, the Fock operator of a molecule in solution can be expressed as Vλ bλ , (2.42) Fisolv = Fi − fi λ∈solute where the first term in the right hand side is the operator for an isolated molecule, and the second term represents the solvent effect. Vλ represents the electrostatic potential of the reaction field at solute atom λ produced by solvent molecules and bλ is a population operator of solute atoms. The potential Vλ takes a microscopic expression in terms of the site–site radial distribution functions between solute and solvent: qα gλα (r)dr, (2.43) Vλ∈solute = (V)λ∈solute = ρ r α∈solvent where qα is the partial charge on the site α in solvent, ρ is the bulk density of solvent, and gλα is the site–site radial distribution function (RDF) or pair correlation function (PCF). The practical procedure for evaluating the matrix elements will be found in the Appendix. In the RISM-SCF theory the statistical solvent distribution around solute is determined by the electronic structure of the solute, whilst the electronic structure of solute is influenced by the surrounding solvent distribution. Therefore the ab initio MO calculation and the RISM equation must be solved in a self-consistent manner. It is noted that the word ‘SCF’ applies not only to the electronic structure calculation but to the whole system, e.g., a self-consistent treatment of electronic structure and solvent distribution. Essentially, the RISM and extended RISM theories can provide information equivalent to those obtained from simulation techniques, namely, thermodynamic properties, microscopic liquid structure, and so on. But

73


it is noteworthy that the computational cost is drastically reduced by these analytical treatment of theories, which allow it to be combined with computationally expensive ab initio MO theory. Thanks to this advantage the MO part in the method can be readily extended to the more sophisticated levels beyond Hartree–Fock (HF), such as CI (configuration interaction) and coupled cluster. This is a remarkable merit compared to the other hybrid methods. In 1996 Sato, Hirata, and Kato reported that the solvated Fock operator can be naturally derived from the variational principle [12] defining the Helmholtz free energy of the system (A) by ˆ + ∆µ|Ψ ˆ A = Ψ|H + Enuc .

(2.44)

Here Enuc is the nuclear repulsion energy and 1 2 1 ρ ˆ dr cαs (r) − hαs (r) + hαs (r)cαs (r) , Ψ|∆µ|Ψ = ∆µ = − β αs 2 2 (2.45) is the modified version of solvation free energy originally defined by Singer and Chandler [13]. This is a functional of the total correlation function hαs (r), the direct correlation function cαs (r), and the solute wave function |Ψ. Energy of the solute molecule Esolute is defined as follows ˆ + Enuc . Esolute = Ψ|H|Ψ

(2.46)

Note this is also the functional of hαs (r), cαs (r) and |Ψ. Imposing constraints concerning the orthonormality of the configuration state function(C) and one particle orbital(φi ) to the equation, one can derive the Fock operator from A based on the variational principle; δ(A[c, h, t, v, C] − [constraints to orthonormality]) = 0, and Fijsolv

= Fij − γij

λ∈solute

∂ bλ ∂qλ

(2.47)

ρ dre[−βuαs (r)+hαs (r)−cαs (r)] − β αs

.

(2.48) If classical coulombic interactions are assumed amongst point charges for electrostatic interactions between solute and solvent, and the term for the CI coefficients (C) are omitted, the solvated Fock operator is reduced

74


to Eq. (2.42). (For the details, see the Appendix) The significance of this definition of the Fock operator from a variational principle is that it enables the analytical first derivative of the free energy with respect to the nuclear coordinate of the solute molecule Ra , ∂ω ˆ αγ (k) ∂Enuc 1 ∂A = − cγs (k) χ ˆss (k) dkˆ cαs (k)ˆ ∂Ra ∂Ra 2(2π)3 β ∂Ra α,γ,s,s

1 γij haij + Γijkl (φi φj |φk φl )a + 2 i,j i,j,k,l a −Vt qa − εij Sij .

i

(2.49)

j

The second term of the right hand side in Eq. (2.48) corresponds to the change of solute–solvent distribution function owed to the modification of the intra-molecular correlation function ω. The fifth term represents the effective charges change caused by the change of solute geometry. Other notations used here have their usual meanings. It has been well recognized that the energy gradient technique in the ab initio electronic structure theory is a powerful tool to investigate the mechanism of chemical reactions of polyatomic system and it opens up a variety of application to the actual chemical processes in solution: carrying out the geometry optimization of reactant, transition state, and product in solvated molecular system, constructing the free energy surfaces along the proper reaction coordinates, computation of the vibrational frequencies and modes, and so on. On analyzing the computational results, two important factors are essentially considered as the solvent effect; the energy change associated with the solute reorganization (Ereorg ) and the solvation free energy (∆µ) upon solvation. Ereorg = Esolute − Eisolate ,

(2.50)

where Eisolate is the total energy of the solute molecule in an isolated condition, and Esolute is the energy of solute molecule defined above. The quantity Ereorg represents the reorganization energy associated with the relaxation or distortion of the electronic cloud and molecular geometry in solution. The solvation free energy in turn consists of the solute– solvent interaction energy and the free energy change associated with the solvent reorganization whose analytical form of this quantity is given in Eq. (2.45). Thus total free energy (Atotal ) of the system involving solute and solvent molecules consists of four terms (Fig. 2.1) Atotal = Aid + Eisolated + Ereorg + ∆µ,

(2.51)


75

where the Aid is solute kinetic contribution to the free energy, which is evaluated from the elementary statistical mechanics of ideal systems. Total Energy of the Solute Eisolated

Solvation Free Energy

Kinetic Contribution Aid

Relaxation of Electronic Structure and Geometry Ereorg

related to energy change in reaction

Figure 2.1.

∆µ

Energy components associated with the RISM-SCF framework.

Now we have got tools to tackle various problems in solvated molecules. In the following sections, we present our recent efforts to explore such a phenomena by means of the RISM-SCF/MCSCF method.

4.

Acid–base Equilibria in Aqueous Solution

The acidity and basicity are fundamental and familiar concepts in chemistry and biochemistry. Quantum chemistry has provided a theoretical understanding of the phenomena as far as the gas phase in concerned. However, in solution the reactivity is seriously affected by solvents. In this regard the acid–base equilibria is a very good example to see the ‘inevitable coupling’ of quantum and statistical mechanics in solution, which often governs chemical observations.

4.1

The Basicity of the Methylamines in Aqueous Solution

It has been experimentally known that the basicity of the methyl amines increases monotonically with successive methyl substitutions in the gas phase, NH3 < (CH3 )NH2 < (CH3 )2 NH < (CH3 )3 N, and that the order reverses at the trimethylamine in aqueous environment, NH3 < (CH3 )NH2 < (CH3 )2 NH > (CH3 )3 N.

76


The monotonic increase of the basicity in the gas phase has been explained in terms of the ‘negative induction’ or the polarization effect due to the methyl groups. How can we understand the irregular behavior in aqueous solution? Let us define the respective basicity (free energy difference upon protonation) by −∆Gg in the gas phase and −∆Gs in aqueous solution. [14] For the discussions concerning the relative strength in basicity of a series of methylamines, it is sufficient that only the relative magnitude of these quantities are considered. Thus the free energy changes associated with the protonation of the methylamines relative to those of ammonia are defined as, ∆∆Gi298 [(CH3 )n NH3−n ] = ∆Gi298 [(CH3 )n NH3−n ] − ∆Gi298 [NH3 ] (n = 1, 2, 3). (2.52) The subscript i is either g or s, representing the quantity in the gas phase or in aqueous solution, respectively. ∆Gs298 can be derived as a sum of two terms, ∆Gs298 (solute) and ∆Gs298 (solvent): The former includes the electronic contributions and kinetic contribution, while the latter corresponds to solvation energy (∆µ). Computed values are plotted in Figure 2.2 against the increasing number of the methyl groups.

∆∆G298i / kcal mol-1

10

0

-10 ∆∆Gs298(solute) ∆∆Gs298(solvent)

-20

∆∆Gs298 (total) ∆∆Gs298(explt.)

∆∆Gg298(computed) ∆∆Gg298(explt.)

-30 0

Figure 2.2.

1 2 Number of CH3

3

Free energy changes on protonation referred to NH3 .

The contribution from solute itself (∆∆Gs298 (solute)) exhibits similar monotonic behavior with the gas phase result (∆∆Gg298 ) which is in good agreement with the experimental data. The difference between ∆∆Gs298 (solute) and ∆∆Gg298 is owed essentially to the electron reorganization energy. The solvation free energy ∆∆Gs298 (solvent) shows the monotonic increase with the successive methyl substitution. The sum of the two contributions produces the inversion in the overall free energy

77


change ∆∆Gs298 , which is in qualitative accord with the experimental result.

4.2

Acid strength of the hydrogen halides in aqueous solution

The order of acidity of hydrogen halides (HX, where X=F, Cl, Br, I) in the gas phase can be successfully predicted by quantum chemical considerations, F < Cl < Br < I. However, in aqueous solution, while hydrogen chloride, bromide, and iodide completely dissociate in aqueous solutions, hydrogen fluoride shows a small dissociation constant. This unexpected observation can be understood by studying the chemical equilibrium [15] HX + H2 O X− + H3 O+ , in solution phase for a series of the hydrogen halides (HX). 2.0

H X – H

H

1.5

O

water B H

O H 1.0

water A

0.5

HF (F-H) HCl (Cl-H) HBr (Br-H) HI (I-H)

0.0 0

2

4

6

8

R/Å

Figure 2.3. Pair correlation function (radial distribution function) between the halogen atom in HX and the hydrogen in solvent water.

Figure 2.3 shows the PCF between a halogen site in HX and a hydrogen site in solvent water. In the case of hydrogen fluoride a distinct peak is found at 1.82 ˚ A, whilst there is no corresponding peak in the other X–H correlation functions. From other PCFs and the geometrical consideration, one can conclude that hydrogen bonds between solute hydrogen - solvent oxygen are found in all the hydrogen halides (H2 O: B in the figure), whilst only hydrogen fluoride forms a distinct F–H (solvent water) hydrogen bonding (H2 O: A in the figure). It is expected that the liquid structure around HF is markedly different from those around the other hydrogen halides. The characteristic behavior of hydrogen fluoride as a weak acid is explained in terms of the enhanced stability of the non-

78


dissociated form of the molecule in aqueous solution due to hydrogenbonding. The free energy difference in aqueous solution (∆G) can be obtained in terms of the free energy difference associated with the reaction in vacuo and solvation free energy, ∆G = Gaq (X− ) + Gaq (H3 O+ ) − {Gaq (HX) + Gaq (H2 O)} . (2.53)

Energy / kcal mol -1

100 50

∆G / kcal mol

150

-1

10

200

explt. RISM-SCF

5 0 HX X-5 -10 -15 F

Cl

Br

I Br Cl F

I

0 -50 -100 process I process III -150

process IV process II

∆G

HX+H2O X–+H3O+ aqueous solution HX+H2O

gas phase

X–+H3O+

standard state: 1 H2+ 1 X2+H2O 2 2

Figure 2.4.

Energy components of the proton dissociation in hydrogen halides.

∆G can be further decomposed into four contributions associated with thermodynamic cycle.(Fig. 2.4) Two formation free energies measured from the standard state (Process I and II for HX+H2 O and X− +H3 O+ , respectively) and two solvation free energies for these chemical species (Process III and IV for HX and X− , respectively). One can see that the two energy contributions concerned with halide anions compensate each other (Process II and IV), and the final free energy difference is mainly governed by the subtle balance of the two energetic components; formation energies of hydrogen halides (Process I) and solvation energies of the halide anions (Process IV). The final result is in very good accord with the experimental data. It is of great interesting that the molecularlevel microscopic character governs the macroscopic properties.

4.3

Auto-ionization of water

A water molecule has amphoteric character. This means it can act as both an acid and a base. The auto-ionization equilibrium process in water, H2 O + H2 O H3 O+ + OH− ,

(2.54)

is one of the most important and fundamental reactions in a variety of fields in chemistry, biology, and biochemistry. The ionic product (Kw )


79

and its logarithm defined by Kw = [H3 O+ ][OH− ],

pKw = −logKw ,

(2.55)

are measures of the autoionization. The quantity can be related to the free energy change (∆Gaq ) associated with the reaction of Eq. (2.54) by the standard thermodynamic relation. ∆Gaq = 2.303RT pKw .

(2.56)

It is experimentally known that the value of pKw shows significant temperature dependence, i.e., it decreases with increasing temperature. However, there is no easy explanation of this phenomenon even from the phenomenological point of view. The free energy change associated with the reaction in Eq. (2.54), can be written [16] in terms of the energy change associated with the reaction in vaccuo (∆Gvac ) and the free energy change of the reacting species due to solvation as, ∆Gaq = ∆Gvac + δG(H3 O+ ) + δG(OH− ) − 2δG(H2 O),

(2.57)

where δG(H3 O+ ), δG(OH− ) and δG(H2 O) are, respectively, the free energy changes of H3 O+ , HO− and H2 O upon solvation. It is noted that the species concerning the reaction, H2 O, OH− , and H3 O+ are regarded as ‘solute’ in the infinitely dilute solution in RISM-SCF treatment. ∆Gaq is also possible to be decomposed into intra- and intermolecular contributions as vac + ∆δGkin + ∆δGreorg + ∆δµ, ∆Gaq = ∆Eelec

(2.58)

vac , ∆δG where ∆Eelec kin , ∆δGreorg are the electronic energy in vacuo, the kinetic term and the electronic reorganization energies, respectively, which are intra-molecular contributions. ∆δµ is the solvation free energy change. (We use ∆ for changes of quantities associated with the chemical reaction and δ for changes owed to solvation.) Relative value of pK w at temperature T to that at T = 273.15K given by

∆T pK w (T ) = pK w (T ) − pK w (273.15),

(2.59)

is further decomposed into the four contributions corresponding to the free energy components. vac + ∆T pKw,kin(T ) ∆T pKw (T ) = ∆T pKw,elec +∆T pKw,reorg (T ) + ∆T pKw,solv (T ).

(2.60)

80

Molecular theory of solvation 5.0

∆pKw,solv 4.0

60.0 40.0

∆pKw,solv

20.0 0.0

3.0

-20.0

2.0

-40.0 -60.0

∆pKw,elec 0

20

40

1.0

60

80

100

120

T / ˚C

(∆pKw,solv+∆pKw,elec)

0.0

∆pKw,kin

∆pKw

-1.0

(computed)

∆pKw (explt.)

-2.0

∆pKw,elec

∆pKw,reorg

-3.0 0

20

40

60

80

100

120

T / ˚C Figure 2.5. Temperature dependence of pKw : computed pKw and its components (solid and dashed line) and the experimental result (dotted line) are shown.

The resultant ∆T pKw (T ) and their components are plotted in Figure 2.5. As shown in the Figure, contributions from ∆T pKw,solv and vac are very large, but they compensate each other. The final ∆T pKw,elec temperature dependence of pKw is determined by an interplay of several contributions with different physical origins. It is also interesting that the temperature dependence is dominated by ∆T pKw,reorg after the compensation of the two largest contributions. The theoretical results for temperature dependence of the ionic product show fairly good agreement with experiments and also demonstrate the importance of polarization effects. How is the property of pKw altered along the change of higher temperature or density? We refer reader who are interested in the detail to the original paper. [16]

5.

Solvent Effects on Conformational Change of Chemical Compounds

Acidity of organic acids (and/or basicity of organic bases) such as carboxylic acid and amines is governed by many factors; solvent, substitution, conformation and so forth. Amongst those factors the effect of conformational change is of special interest in terms of its significance


81

in the biological system. In a biomolecular system such as protein the acidity or basicity of related functional groups depends sensitively upon the molecular conformations: owing to such sensitivity the property is sometimes exploited to detect the conformational change of protein.

5.1

Conformational equilibrium

A prototype of the abovementioned phenomena can already be seen in the simplest carboxylic acid, acetic acid (CH3 CHOOH). [17] Acidity is determined by the energy or free energy difference between the dissociated and non-dissociated form, whose energetics usually depends significantly upon its conformation, e.g. the syn-/anti-conformational change of carboxylate group in the compound substantially affects the acid/base equilibrium. 20.0

20.0 E(gas) ∆G(aq.)

10.0

C(methyl) H(methyl) ∆µ+Ereorg

0.0

0.0 O

H(-O) -10.0

-10.0

∆µ O(=C)

-20.0

-30.0

H O–H

-20.0

O

CH3–C

-40.0

CH3–C O

syn

anti

Contributions to the solvation energy of the solute atoms

Energy Change / kcal mol-1

10.0

C(carbonyl) Ereorg

O

syn -50.0

-150 -100

-50

0

50

100

150

Torsional Angle (H-O-C-C) θ/ deg.

Figure 2.6. Potential and Free energy curve around the torsional angle of θ (CH3 – C–O–H) in acetic acid.

Potential energy profiles in the gas phase and potential of mean force insolution along the torsional angle θ (H–O–C–C) are shown in Figure 2.6. In the gas phase the syn-acetic acid (θ = ±180) is more stable than the anti-conformer by 6.9 kcal/mol and the barrier height of rotation between these conformer is estimated as 13.2 kcal/mol. In aqueous solution the calculated free energy difference is significantly reduced to 1.7 kcal/mol. The rotational barrier also become lower than that in the gas phase by 10.3 kcal/mol. The reduction of the free energy gap indicates that the pKa difference between the two conformers is drastically reduced from 5.1 to 1.2 on transferring from the gas phase to aqueous solution at room temperature. Stabilization of syn-conformer in the gas phase is explained rather intuitively in terms of the increased interaction between the H atom in the OH group and the O atom in C=O group. As

82


one can find in the Figure, the extra stabilization in the anti-conformer in aqueous solution arises from the solvation energy, especially at the carbonyl oxygen site. The electronic redistribution of the solute upon transferring from the gas phase to aqueous solution is another interesting issue. The Mulliken charge population analysis demonstrates substantial change of effective charge on the hydrogen and oxygen in the OH group: for anti-conformer, 0.355(H) and −0.428(O) are altered to 0.377(H) and −0.434(O) from gas to aqueous solution phase, and for syn-conformer, 0.373(H) and −0.463(O) to 0.395(H) and −0.476(O), respectively. One can notice a character of proton increases on the hydrogen in OH group by being immersed into aqueous solution.

5.2

Tautomerization of Acetylacetone

The keto-enol tautomerization of acetylacetone (CH3 -CO-CH2 -COCH3 ), a prototype β-diketone, has been extensively studied experimentally, and attention has been paid to its solvent effect. Although the enol form is more stable than the keto in the gas phase owing to the intramolecular hydrogen bonding, the equilibrium is known to shift toward the keto in solution as the solvent polarity increases. The tautomerization in various types of solution, which includes H2 O, dimethyl sulfoxide (DMSO), and carbon tetrachloride (CCl4 ), was examined by means of RISM-SCF method. [18] For the keto tautomer the optimized geometry with a larger dihedral angle τ (O=C-(CH2 )-C=O) is found in CCl4 (140.4◦ ). But the geometry of the stable conformer undergoes a drastic change in polar solvents and the torsional angle, τ , significantly decreases to 56.4◦ in H2 O and to 75.2◦ in DMSO. In the enol tautomer, in contrast, the geometric change owed to the solvation is smaller than that in the keto. All the atoms except for the methyl hydrogens lie in the same plane in all the solvents as in the gas phase. Detailed analysis of the thermodynamic property reveals the origin of a large geometric change of the keto form in polar solvents: such a geometric change is driven by the enthalpy term for aqueous solution while the entropy term is important in DMSO. Fig. 2.7 shows the calculated free energy differences between the keto and enol tautomers along with the available experimental values. In H2 O the free energy of the keto form was calculated to be lower than that of the enol by −1.37 kcal/mol, which is in good agreement with the experiment, −1.18 kcal/mol. For DMSO solution the stability of the two tautomers became comparable to each other, 0.14 kcal/mol, as in the experiment. The free energies were also very close in CCl4 , which is


83

2.0

∆G (keto-enol) / kcal mol-1

O

τ

C H3 C

1.0

O C

CH2

CH3

0.0

RISM-SCF explt. (1) explt.(2)

-1.0

CCl4

DMSO

H2O

Figure 2.7. The keto–enol free energy difference of acetyl acetone in various solutions (CCl4 , DMSO and H2 O).

different from the experiment where the enol form is more stable than the keto by 1.76 kcal/mol. Considering that the solute geometries are similar to those in the gas phase, and that the electric field is weak in CCl4 , this discrepancy in the free energy difference comes from the underestimation of solute electronic energy difference between the tautomers.

6.

Solvent Effect on Chemical Reactions

What is generally wished for in the quantum chemical computations is a reaction profile. As we have seen in this chapter, the electronic structure is remarkably affected by surrounding solvent molecules, which means that reactions are strongly governed (and controlled) by the environment as well. Three chemical reactions, S N 2 type reaction, Diels– Alder reaction and redox process, are described in this section.

6.1

SN 2 Reaction

Menshutkin type of SN 2 reaction in aqueous solution, NH3 + CH3 Cl → NH3 CH3 + + Cl− , is regarded as a typical system for a benchmark of a theroy of chemical reaction in solution environment, and has been extensively studied by the various methods including QM/MM simulation and several dielectric continuum approaches. The RISM-SCF method has been applied to the same reaction[19] to provide information on solvation structures for use in interpreting the solvent effects on the reaction mechanism and to compare the free energy profile with those obtained by other methods.

84


The potential and free-energy profiles along the reaction coordinate calculated both by the standard ab initio MO and the RISM-SCF in the Hartree–Fock level are shown in Fig. 2.8. Although this reaction is endothermic in the gas phase by 106.3 kcal/mol with the HF method, it becomes exothermic in aqueous solution by 27.8 kcal/mol at the RISMHF level. The barrier height was calculated to be 17.7 kcal/mol. As seen in Fig. 2.8, there is a very shallow potential well around 1.9 ˚ A, + − which corresponds to a contact ion pair of NH3 CH3 and Cl formed in aqueous solution. The Cl–H RDF depicted in Fig. 2.9 clearly demon80 gas phase aqeous solution

Energy / kcal mol-1

60 40 20

H3 N+CH3 Cl

H3 NCH3 ++Cl-

0 -20 -40 -4

-2 0 2 Reaction Coordinate / Å

4

Figure 2.8. Energy profile of SN 2 reaction in the gas phase (dotted line) and in aqueous solution (solid line)

strates the progress of hydrogen bonding along the reaction coordinate. At the reactant the peak corresponding to the Cl–H interaction is not observed. The distinct first peak around R = 2.2 ˚ A grows up monotonically and slightly shifts toward inside as the reaction proceeds. The other H atom of H2 O in the first solvation shell corresponds to the second peak around 4.2 ˚ A. The number of H2 O molecules hydrogen bonded to Cl was estimated by integrating the first peak of the RDFs and was found to be 4.3 and 6.6 at the TS and product, respectively, which is comparable to the QM/MM result, 3.1 and 6.6, respectively. The breaking of the hydrogen bond between the ammonia N and water H atoms is observed in Fig. 2.9, where the first peak around 2.0 ˚ A in the reactant disappears as the reaction progresses. Although the lone pair electron on the N atom forms hydrogen bonds with H2 O in the reactant region, it participates in the formation of a new chemical bond with the C atom of CH3 Cl in the reaction process. The importance of the formation of Cl–H hydrogen bonds and the breaking of N–H bonds was demonstrated in characterizing the reaction mechanism as in the previous QM/MM calculations. It is noted, however, that there are large discrepancies in the C1–O and N–O RDFs between the RISM-SCF and QM/MM results.

85

Electronic structure andchemical reaction in solution 4.0

2.0

PCF( N-H )

PCF(Cl-H)

3.5

3.0

1.5

2.5

1.0

2.0

1.5

0.5

1.0

0.5 0.0

0.0 0

2

4

6

8

0

10

2

4

R/Å

6

8

10

R/Å

2.0

4.0

3.5

PCF( N-H )

1.5

PCF(Cl-H)

3.0

2.5

1.0

2.0

1.5

0.5

1.0

0.5 0.0

0.0 0

2

4

6

8

10

0

2

4

R/Å

6

8

10

R/Å

2.0

4.0

PCF( N-H )

PCF(Cl-H)

3.5

1.5

3.0

2.5

1.0

2.0

1.5

0.5

1.0

0.5 0.0

0.0 0

2

4

6

8

10

R/Å

0

2

4

6

8

10

R/Å

Figure 2.9. Solvation structure change along the SN 2 reaction in aqueous solution: reactant (top panel), transition state (middle), and product (bottom).

6.2

Diels–Alder Reaction

The Diels–Alder reaction is a cyclo-addition reaction, which is reported to be greatly accelerated by aqueous solvent. Recent experiments indicate also that the rate and yield of the reaction increase dramatically by bringing the solvent to supercritical conditions. (a) Why and how is the reaction accelerated when the system is transferred into aqueous environment from gas phase? (b) Is the physical cause of the increase in reaction rate in the supercritical condition the same as that in ambient water? (c) Why does the reaction yield increase in supercritical conditions? To answer these questions the reaction concerning methylvinyl ketone (MVK) and cyclopentadiene (CP) in ambient water (AW) and subcritical condition (SCW) were investigated based on the RISM-SCF method. [20] One of the advantages of the method is that it enables us to change easily thermodynamic conditions of the system, such as temperature and density. Thus it is suitable for investigating the chemical reaction in solvent, including SCW, which requires numerical analyses in wide range of temperature and pressure. The key results are shown in Figure 2.10, which exhibits the free energy diagram concerning the reaction through the endo-cis transition structure in gas phase, in AW (298K, 1.0 g/cm3 ), and in SCW (873K, 0.6

86 Relative Free Energy (kcal/mol)

Molecular theory of solvation 80

gas phase (873K) SCW

60

gas phase (298K)

40

AW COCH3

+

20 COCH3

0

Reactant

TS

Product

Reaction Coordinate

Figure 2.10. Free energy profile of the Diels–Alder reaction in ambient water (AW), in subcritical water (SCW), and in the gas phase.

g/cm3 ). All the energy levels are determined by regarding the reactants at corresponding temperature as a standard. The activation free energy ‡ ) is 52.5 kcal/mol, while the computed in gas phase at 278K (∆F298 ‡ ∆FAW is calculated to be 45.6 kcal/mol: it is reduced by 6.9 kcal/mol ‡ , from gas phase to AW. The activation free energy in SCW, ∆FSCW is found to decrease by 2.7 kcal/mol compared with that in gas phase, ‡ . (54.3 kcal/mol) Together with the RDF analysis, it is found that ∆F873 the solvent effect in SCW becomes weaker than that in AW, although the substantial effect of solvent on the chemical reactions largely remains in SCW. These results seem quite reasonable if one considers that the density of water molecules in SCW is low and that the solvent effect in SCW is less than in AW. From these values the relative rate constant in SCW compared to that in AW is estimated to be 1.27× 1013 ; the rate constant in SCW is extremely increased. Although the solvent effect for the rate constant is apparently decreased in SCW, the rate is dramatically increased because of the thermal activation rather than the solvent effect. The determination of the reaction yield is somewhat similar. In spite of the increasing endothermicity in SCW, the yield is approximately 667 times higher than in AW. This is because the increased solubility of the reactants (CP and MVK) in SCW provides an advantage that is enough to overcome the energetical disadvantage. Now we can answer the aforementioned questions: The reaction in SCW is accelerated owing to increased thermal activation, not by the same cause in AW. The reaction yield is increased dramatically in SCW owing to enhanced solubility compared to that in AW.

87


6.3

Charge Transfer Reaction

Electron transfer (ET) reactions have been extensively studied because of the essential roles they play in a variety of chemical, physical, and biological processes. Among various types of organometalic complexes, a ruthenium complex has been widely chosen as a model to study the oxidation-reduction chemistry of metalloproteins. More fundamental phenomena, ET reactions between the ruthenium complex in aqueous solution environment, [Ru(NH3 )6 ]2+ + [Ru(NH3 )6 ]3+ [Ru(NH3 )6 ]3+ + [Ru(NH3 )6 ]2+ , have been also investigated by many researchers. The rates and mechanisms of the ET reactions are of great interest from both experimental and theoretical view points. It is well recognized that Marcus type of diabatic free energy surface is very useful for understanding ET processes. Such a diabatic free energy profile in polar solvent can be realized within the framework of the RISM theory by Chong’s method (Chapter I) [21], in which the hypothetical charge distribution is introduced. Although the method was originally developed for a redox pair, extension to a polyatomic system is straightforward. [22] The free energy profile around the dication ([Ru(NH3 )6 ]2+ with a charge set of q2+ ) is given by, ∆F 2+ = ∆µs − ∆µ2+ − sVs · (q3+ − q2+ ).

(2.61)

The profile for the product state ([Ru(NH3 )6 ]3+ with a charge set of q3+ can be obtained similarly. ∆F 3+ = ∆µs − ∆µ3+ − (s − 1)Vs · (q3+ − q2+ ).

(2.62)

In both equations a state with the hypothetical charges (q = (1 − s) + sq3+ ) is introduced to compute the solvation free energy (∆µs ) and electrostatic potential on the solute site (Vs ). s (r) j gij . (2.63) qsolvent dr4πr 2 (Vs )i = ρ r j

The diabatic free energy profile obtained by the method is plotted in Figures 2.11. The most notable feature of the figures is that it is essentially parabolic with respect to the reaction coordinate. This is nothing but the manifestation of the central limiting theorem, and the results indicate that the response of solvent fluctuation to the electrostatic filed is linear. The behavior, however, is not trivial, because the free energy calculated via Eqs. (2.61) and (2.62) are inherently non-linear.

88


Figure 2.11.

Marcus type of free energy curve for [Ru(NH3 )6 ]2+ and [Ru(NH3 )6 ]3+ .

Since the reaction is a symmetric one, the activation free energy of the present system (∆G‡ ) can be estimated with the relation,

1 1 + ∆F 3+ , ∆G‡ ∼ ∆F 2+ 2 2 which gives 24 kcal mol−1 . Unfortunately, experimental data directly corresponding to the theoretical result is not available at the moment. Here, we just refer to the experimental value, ∼ 14 kcal mol−1 , obtained for the same metal complex in the different solvent condition (involving CF3 SO3 H and counter ions). The present model treats these metal complexes in pure aqueous solution and the screening from counter ions as well as CF3 SO3 H is completely disregarded. The screening effect in general weakens the ion–ion interactions, in this case, repulsive ones, and the effect very likely lowers ∆G‡ .

7.

The NMR Chemical Shift

Over the past decades the NMR chemical shift measurement in solutions has been applied to a vast range of problems in chemistry and biochemistry, and has revealed itself to be an invaluable microscopic probe. It has played an especially important role in the structural understanding of protein owing to its great sensitivity to the environment in which the probing atom is situated. Theories of the chemical shift in solution, on the other hand, have not been well developed owing to the lack of a theory for describing the electronic structure of a solvated molecule. In 2000 Yamazaki et al. proposed a successful theory [23] for computing the NMR chemical shift within the framework of the RISM-SCF

89


method. The nuclear magnetic shielding tensor σ X of a nucleus X can be expressed as mixed second derivative of the free energy A with respect to the magnetic field B and the nuclear magnetic moment mX : 2A ∂ X σαβ = , (2.64) ∂Bα ∂mX β X Y B,m ,m ···=0

X , where Bα and mX β are the Cartesian components of B and of m respectively, and A is a sum of the electronic energy and the excess chemical potential defined in the beginning of this chapter (Eq. (2.44)). The gauge-invariant atomic orbital (GIAO) method is adopted to solve the gauge problem in the calculation of nuclear magnetic shieldings. Thus, each molecular orbital (φi ) is expressed as a linear combination of GIAO χν : ciν (B, mX )χν (B), (2.65) φi = ν

where i χν (B) = exp − (B × Rν ) · r . 2

(2.66)

Final expression of the nuclear magnetic shielding tensor can be obtained from these equations

(0,0) (1,1) X (1,0) (0,1) X X + Pνλ Pνλ Hνλ Hνλ , (2.67) σαβ = αβ

νλ (l,m)

α

β

(l,m)

where Pνλ and Hνλ are the (ν, λ)–element of the derivatives of the density and one electron matrices, respectively, with respect to the external magnetic field and the magnetic moment of nucleus,

∂Pνλ (1,0) = , (2.68) Pνλ ∂Bα B,mX =0 α

X 2H ∂ (1,1) νλ = . (2.69) Hνλ αβ ∂Bα ∂mX β X B,m =0

(1,1) X are calculated from the usual RISM-SCF method, whilst Hνλ αβ

(0,1) X and Hνλ can be obtained by the standard ab initio calculation in (0,0)

Pνλ

β

90


Proton chemical shift of water (ppm)

2.0 H2O 1.5

1.0 CH3CH3CO 0.5 CHCl3 CCl4

0.0 0

10

20

30

40

50

60

Temperature (°C)

Figure 2.12. Temperature dependence of the chemical shift of a water molecule in various solvent. All the shifts are given relative to the isolated water molecule in the gas phase.

(1,0) the gas phase. For Pνλ the solution of the first-order coupled– α perturbed Hartree–Fock equation, in which the RISM-SCF type of solvated Fock operator is incorporated, is employed. The proton chemical shifts in various solvents, obtained by the theory, are plotted as a function of temperature in Fig.2.12. The present method qualitatively reproduces the two observations in the recent experiment by Nakahara et al. with a modern NMR technique. [24] First, the proton chemical shifts of the solute water molecule are in the sequence over the temperature range studied, in water > in acetone > in chloroform > in carbon tetrachloride. Secondly, the proton chemical shifts decrease with increasing temperature. These results clearly demonstrate the capability of the theory to predict experimental results. Analysis on the radial distribution function provides molecular level information on solvation structure and its relation to NMR chemical shifts. The study of chemical shift reveals a serious drawback inherent in the classical–quantum hybrid approach. The solvent effect on the oxygen chemical shift showed temperature dependence opposite to corresponding experimental results. An ab initio analysis with a small cluster suggested stongly that the ill behavior is originated from the lack of electron exchange between solute and solvent. Therefore an obvious direction of improvement of the RISM-SCF theory is to take the electron exchange into account.


8.

91

Summary

In this chapter we have reviewed the RISM-SCF/MCSCF method, a theory which combines the ab initio electronic structure theory and RISM. Its capability of treating extensive problems in chemistry is demonstrated. The methodology has numerous advantages over the other related methods such as dielectric continuum models and QM/MM. The RISM-SCF method provides the molecular level information on the solvation structure including hydrogen bonding, whic can never be made by the continuum model. Much less computational demand by the RISM theory compared to the MM and MD approaches makes possible to combine the theroy with highly sophisticated ab initio MO theory which by itself requires heavy computational load. All the useful knowledges and techniques developed in the gas phase quantum chemistry as well as in the statistical mechanics theory (RISM) can be utilized in the combined theory. There are several directions to extend the RISM-SCF/MCSCF method that are not described in the present chapter. One of such directions is a combination of ab initio MO theory with 3D-RISM, which is explained in chapter 4. The site–site treatment of the solute–solvent correlations involving the approximation of radial averaging constitutes a bottleneck of the RISM-SCF method, and thus lacks a 3D picture of the solvation structure for complex solutes. The SCF theory combined with the 3D-RISM is free from such a bottleneck. The test computation on the carbon monoxide in water provides a detailed hydration structure of water solvent as well as polarized CO electronic structure. [25] It is also found that the results from the original RISM-SCF/MCSCF method are in reasonable accord with those following from the 3D-RISM-SCF approach after reduction of the orientational dependence. This shows the RISM-SCF/MCSCF approach gives a proper picture for a solvation process. The electronic structure plays a primary role in determining structure of a molecule. However, changes of the electronic energy associated with a chemical process are comparable, in many cases, with those owed to solvation in solution. This subtle balance between changes in the electronic energy and the solvation free energy sometime causes an ‘irregularity’ of the order in stability among substituents as we have seen in the several examples. The solvent effect even reverses the equilibrium between a reactant and a product. In this regards, both the electronic structure theory (quantum mechanics) and RISM (statistical mechanics) contribute equally to describe the molecular processes in solution.

92


A self-consistent treatment and natural coupling between those methods are essential to reveal the nature of chemical reactions in solution.

Appendix: Appendix 1.

Partial Charge Treatment in RISM-SCF/MCSCF

In practical RISM-SCF/MCSCF computations partial charges assigned on interaction sites are the bridge between the RISM and ab initio MO theory. It is well known that the electrostatic potential generated by electron clouds of a molecule is very different from that by Mulliken gross charge. An appropriate set of partial charges is usually determined so as to reproduce the electrostatic potential around the solute with the least squares fitting procedure. In this appendix the detailed procedure is described by following the original paper by Ten-no et al. [4] The electrostatic potential at arbitrary point (r) near the solute can be divided into two different contributions, namely, the potential from the nucleus and that from the electron clouds. U (r) = UN (r) + Ue (r),

(2.A.1)

and UN (r) =

N A=1

ZA , |r − RA |

Ue (r) = −tr (PA(r)) ,

(2.A.2)

where P is the density matrix calculated from MO coefficients, N is the number of atoms in the solute and A(r) is three center one-electron integrals, ∗ χµ (r )χν (r ) dr , (2.A.3) (A(r))µν = |r − r | where χµ (r ) is a set of basis functions. Other notations used here have usual mean(N) (e) ings. The set of partial charge ({qα }, {qα }), which we would like to determine, induces electrostatic potential, Ñ (r) = U

n α=1

(N)

qα , |r − Rα |

˜e (r) = U

n α=1

(e)

qα , |r − Rα |

(2.A.4)

where n is the number of interaction sites. Note that n is not necessary to be the (N) (e) same as N . {qα }, {qα } are determined with the standard procedure minimizing target functions under the constraints of total charge conservation. ! l n (N) 2 ∂ ˜ wk UN (rk ) − UN (rk ) + 2λN qj − (constant) = 0, (2.A.5) (N) ∂qi j=1 k=1 ! l n (e) 2 ∂ ˜ wk Ue (rk ) − Ue (rk ) + 2λe qj − (constant) = 0, (2.A.6) (e) ∂qi j=1 k=1 where l is the numbers of grid points (rk ) where the electrostatic potential is evaluated (N) in, wk is proper weight and λ are Lagrange multipliers. The nuclear part, {qα }, is

93

Electronic structure andchemical reaction in solution given by ⎛ ⎜ ⎜ q(N) = ⎜ ⎜ ⎝

⎞

(N)

q1 (N) q2 .. . (N) qn

⎟ ⎟ ⎟ = a−1 a Z − λN a−1 1, ⎟ ⎠

(2.A.7)

where ⎛ l ( ) −1 −1 wk rkA rki , a Ai =

(a)ij =

k=1

l

−1 −1 wk rki rkj ,

and

k=1

⎜ ⎜ 1=⎜ ⎝

1 1 .. . 1

⎞ ⎟ ⎟ ⎟. ⎠

(2.A.8)

By using the total nuclear charge (NN = 1t · q(N) = 1t · Z) conservation, λN may be written as λN =

1t a−1 a Z − NN , 1t a−1 1

(2.A.9)

and q(N) = a−1 a Z −

1t a−1 a Z − NN −1 a 1. 1t a−1 1

(2.A.10)

It is noted that when the set of interaction sites is identical to that of nucleus, i.e. a = a , the equation is immediately reduced to the trivial equation, q(N) = Z. Similarly, the electronic part can be evaluated by q(e) = −a−1 tr (PB) −

−1t a−1 tr (PB) − Ne −1 a 1. 1t a−1 1

(2.A.11)

B is defined by (B)µ,ν,i =

l

−1 wk rki Aµν (Rk ),

(2.A.12)

k=1

and Ne = −tr (PS) is the total numbers of electrons. The modified fock operator (Eq.2.42) includes the interaction between the effective charges derived here (q(e) ) and the electrostatic potential. (V in Eq. 2.43) Thus the one-electron part of the solvated Fock matrix is given by,

a−1 1 T −1 solv t −1 1 a B−S , Hc = Hc − V · a B − T −1 (2.A.13) 1 a 1 where Hc is the one-electron integrals of the isolated solute molecule and S is overlap matrix. In the present RISM-SCF framework the free energy described above (Eq.(2.44)) is chosen to derive the solvated Fock operator (Eq.(2.A.13)). One might notice that other choices of the energy expression and its Fock operator are possible. The averaged potential energy is chosen instead of A in Eq. 2.44, for instance, Etotal = Eisolated + V · q(e) + ELennard−Jones ,

(2.A.14)

94


the one-electron part of solvated Fock matrix corresponding to the energy is given by,

∂V a−1 1 T −1 T (e) T −1 · a 1 = H − V + q · B − a B − S Hsolv . (2.A.15) c c ∂q 1T a−1 1 This is another type of solvated Fock operator in the combination method of RISM and ab initio MO theory. It should be noted that the first order derivatives of radial distribution functions with respect to the effective charges are required to construct the Fock matrix.

2. 2.1

Variational Principle in the RISM-SCF/MCSCF method RISM-MCSCF equation

We defined the total Helmholtz free energy A as the sum of the electronic energy of the solute molecule Esolute and the excess chemical potential coming from solute– solvent interaction ∆µ; A = Esolute + ∆µ.

(2.A.16)

Esolute can be estimated by ab initio electronic structure methods such as MCSCF ; Esolute

= = =

Enuc + Eel Za Zb 1 + γij hij + Γijkl (φi φj |φk φl ) Rab 2 ijkl ij a>b

Za Zb 1 + γij hij + Γijkl φi |gkl |φj , R 2 ab ij a>b ijkl

where γij and Γijkl are the vector coupling coefficients, IJ CI CJ γij , γij =

(2.A.17)

(2.A.18)

I,J

Γijkl

=

CI CJ ΓIJ ijkl ,

(2.A.19)

I,J

with the CI coefficient CI . hij and φi |gkl |φj are the one and two electron integrals defined in terms of the one particle orbital. Other notations used here have usual meanings. For the chemical potential term we have adopted the free energy derived from the hyper–netted chain (HNC) closure relation by Singer and Chandler [13],

1 1 ρ dr cαs − h2αs + hαs cαs ∆µ = − β α,s 2 2 ρ 1 = − dr e[−βuαs (r)+tαs (r)] − 1 − tαs (r) − hαs (r)tαs (r) + h2αs (r) β α,s 2 1 ˆ αs (k) cˆαs (k)ρh dk − − 3 (2π) β αs ⎫ ⎬ 1 cˆαs (k)ˆ cγs (k)ˆ ωαγ (k)χ ˆss (k) , + (2.A.20) ⎭ 2 α,γ,s,s

95


where the Greek subscripts refer to the interaction sites of solute molecules, and the Roman to the sites on the solvent. cαs and hαs are the direct and total correlation functions, respectively. χss is the pure solvent site density pair correlation functions and ωαγ the intramolecular correlation function containing the structural information of the solute. The solute–solvent interaction potentials uαs (r) are given as the sum of Coulombic and Lennard-Johns terms; σαs 12 σαs 6 qα qs uαs (r) = − + 4εαs . (2.A.21) r r r Note that the effective charges qα assigned on the solute sites depend on the solute electronic wave function in the present case; γij φi |bα |φj (2.A.22) qα (C, v, R) = ij

where bα is the population operator generates the charge qα on the site α. Although there can be several possibilities to define the operator bλ , one of the promising ways is the least squares fitting procedure to the electrostatic potential originated by the solute electron distribution described in the previous section. The quantity A can be regarded as a functional of the correlation functions hαs , cαs and tαs , as well as the MO coefficient vri and the CI coefficient CI . Imposing the constrains to the orthnormality of configuration state functions and one particle orbital, we have defined the following Lagrangian, 2 CI − 1 − εim (Sim − δim ). (2.A.23) L = A[c, h, t, v, C] − E i

I

m

Variations with respect to the functions yield + * ρ δL = − dr e[−βuαs (r)+tαs (r)] − 1 − hαs (r) δtαs (r) β α,s + {−tαs (r) + hαs (r) − cαs (r)} δhαs (r) ⎫ ⎧ ⎬ ⎨ −1 ωαγ ∗ cγs ∗ χss (r) δcαs (r) + −hαs (r) + ρ ⎭ ⎩ γ,s ! +2 HIJ CJ − ECI δCI I

+2

J

δφi |γij h +

i,j

− γij

λ

−εij |φj ,

∂ bλ ∂qλ

kl

Γijkl gkl

ρ dre[−βuαs (r)+tαs (r)] − β α,s

(2.A.24)

where all the quantities were taken to be real. The second curly bracket with the first one gives the HNC closure relation and the third one is referred to the RISM Ornstein–Zernike (OZ) equation. Now, we introduced the column vector V. The λth element of this vector (V)λ is the averaged

96


electrostatic potential acting on the solute site λ which comes from all the solvent charges; gλs (r) ρ ∂ [−βuαs (r)+tαs (r)] dr ≡ (V)λ . (2.A.25) dr = ρ qs e − ∂qλ β α,s r s The εim is the Lagrange mutiplier related to the MCSCF Fock operator Fij , εim = φi |γmj h + Γmjkl gkl − γmj Vt · b|φj j

=

kl

φi |Fmj |φj

(2.A.26)

j

and the CI matrix element is written as IJ 1 IJ γij φi |h − Vt · b|φj + Γijkl φi |gkl |φj , HIJ = 2 ij

(2.A.27)

ijkl

where b is the column vector with the component bλ . Thus we find that the variational condition for the free energy functional leads to the HNC approximation, the RISM equations and the modified MCSCF equations involving the solvent effect. Equation (2.A.23) is easily reduced to the case of CI and HF wavefunctions. If the variation with respect to the one particle orbital φi is dropped, the equations for the CI method are given. By omitting the term for the CI coefficients, the equation is reduced to the HF approximation. In this case the same solvated Fock operator introduced in the original RISM-SCF method is obtained.

2.2

Energy gradient

The first derivative of the free energy with respect to the nuclear coordinate of the solute molecule Ra is written as ∂Enuc ρ dhαs dtαs ∂A = − + {definition of t} dr[{HNC equation} ∂Ra ∂Ra β α,s dRa dRa dcαs +{RISM equation} ] dRa 1 ∂ω ˆ αγ (k) − cγs (k) χ ˆss (k) dkˆ cαs (k)ˆ 2(2π)3 β ∂Ra α,γ,s,s

1 ∂ φi |γij h + Γijkl gkl − γij Vt · b|φj . + ∂Ra ij 2 kl

(2.A.28)

Using the RISM-MCSCF variational conditions, Eq. (2.A.23), it is easy to show that the free energy gradient is reduced to ∂Enuc 1 ∂ω ˆ αγ (k) ∂A = − cγs (k) χ ˆss (k) dkˆ cαs (k)ˆ ∂Ra ∂Ra 2(2π)3 β ∂Ra α,γ,s,s

1 + + Γijkl (φi φj |φk φl )a 2 i,j i,j,k,l t a a −V q − εij Sij ,

γij haij

i

j

(2.A.29)

97

REFERENCES

where the second term of the right hand side corresponds to the change of solute– solvent distribution function due to the modification of intramolecular correlation. a are calculated by transforming the derivatives The MO integrals haij , (ij|kl)a and Sij of corresponding atomic orbital integrals. The derivatives of effective charges qα are defined as qa =

γij φi |

ij

∂b |φj . ∂Ra

(2.A.30)

With the least squares fitting procedure to the solute electrostatic potential, the electronic contribution q(e) to the solute effective charges can be written as q(e) = −a−1 tr(PB) − λe a−1 1,

(2.A.31)

where the matrices a and B and the Lagrange multiplier λe are defined in [4]. P is the density matrix with the element; IJ Prs = CI CJ γij vri vsj . (2.A.32) I,J

i,j

The fifth term appeared in Eq.(2.A.28) can be rewritten in the following form; t −1 ∂B ∂a−1 Va 1 t t 1 a−1 tr(P − V )− tr(PB) V t qa = t −1 1a 1 ∂Ra ∂Ra

t −1 1 a tr(PB) − tr(PS) ∂a−1 1 − ∂Ra 1t a−1 1 −

Vt a−1 1 ∂S ). tr(P 1t a−1 1 ∂Ra

(2.A.33)

The implementation of the RISM-MCSCF gradient method is rather simple. We only need to add the second and fifth terms to ordinary gradient packages in the electronic structure calculations.

References [1] ACS symposium series, 712 Combined Quantum Mechanical and Molecular Mechanical Methods; J. Gao and M. A. Thompson, Eds.; ACS, Washington DC, 1998 ; M. Orozco and F. J. Luque, Chem. Rev., 100, 4187 (2000). [2] J. Tomasi and M. Persico, Chem. Rev., 94, 2027 (1994); C. J. Cramer and D. G. Truhlar, Chem. Rev., 99, 2161 (1999). [3] J. Gao, Acc. Chem. Res., 29, 298 (1996). [4] F. Hirata, H. Sato, S. Ten-no and S. Kato, in [1] ; F. Hirata, H. Sato, S. Ten-no and S. Kato, ‘The RISM-SCF/MCSCF Approach for the Chemical Processes in Solutions’, in Computational Biochemistry and Biophysics; Oren M. Becker, Alexander D. MacKerell, Jr., ˆ Roux, Masakatsu Watanabe, Eds.; Marcel Dekker Inc., New Benoit York, 2001 .

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[5] for example, see A. Szabo and N. S. Ostlund, ‘Modern Quantum Chemistry,’ Macmillan Publishing Co. Inc., 1982. [6] M. W. Wong, M. J. Frish and K. B. Wiberg, J. Am. Chem. Soc., 113 (4776,1991). [7] H. L. Friedman, Mol. Phys., 29, 1533 (1974). [8] M. Andersson and G. Karlstr¨ om, J. Phys. Chem., 89, 4957 (1985). [9] H. Sato and S. Kato, J. Mol. Struct.(THEOCHEM), 310, 67 (1994). [10] A. Klamt and G. Sch¨ uu ¨rmann, J. Chem. Soc., Perkin Trans. II, , 799 (1993). [11] S. Ten-no, F. Hirata and S. Kato, Chem. Phys. Lett., 214, 391 (1993); S. Ten-no, F. Hirata and S. Kato, J. Chem. Phys., 100, 7443 (1994). [12] H. Sato, F. Hirata and S. Kato, J. Chem. Phys., 105, 1546 (1996). [13] S. J. Singer and D. Chandler, Mol. Phys., 55, 621 (1985). [14] M Kawata, S Ten-no, S Kato and F Hirata, J. Am. Chem. Soc., 117, 1638 (1995); M Kawata, S Ten-no, S Kato and F Hirata, Chem. Phys., 203, 53 (1996). [15] H. Sato and F. Hirata, J. Am. Chem. Soc., 121, 3460 (1999). [16] H. Sato and F. Hirata, J. Phys. Chem. A, 102, 2603 (1998); H. Sato and F. Hirata, J. Phys. Chem. B, 103, 6596 (1999). [17] H. Sato and F. Hirata, J. Mol. Struct.(THEOCHEM), 461-462, 113 (1999). [18] T. Ishida, F. Hirata and S. Kato, J. Chem. Phys., 110, 3938 (1999). [19] K. Naka, H. Sato, A. Morita, F. Hirata and S. Kato, Theor. Chem. Acc., 102, 165 (1999). [20] Y. Harano, H. Sato and F. Hirata, J. Am. Chem. Soc., 122, 2289 (2000); Y. Harano, H. Sato and F. Hirata, Chem. Phys., 258, 151 (2000). [21] S.-H. Chong, S. Miura, G. Basu, F. Hirata, J. Phys. Chem., 99, 10526 (1995); S.-H. Chong, F. Hirata, Mol. Simulation, 16, 3 (1996); S.-H. Chong, F. Hirata, J. Chem. Phys., 106, 5225 (1997); S.-H. Chong, F. Hirata, Chem. Phys. Lett., 293, 119 (1998); R. Akiyama, M. Kinoshita, F. Hirata, Chem. Phys. Lett., 305, 251 (1999). [22] H. Sato and F. Hirata, J. Phys. Chem. A, 106, 2300 (2002). [23] T. Yamazaki, H. Sato and F. Hirata, Chem. Phys., 325, 668 (2000); T. Yamazaki, H. Sato and F. Hirata, J. Chem. Phys., 115, 8949 (2001). [24] M. Nakahara and C. Wakai,Chem. Lett.,809 (1992); M. Nakahara, DENKI KAGAKU (in Japanese), 62, 108 (1994).

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[25] H. Sato, A. Kovalenko and F. Hirata, J. Chem. Phys., 112, 9463 (2000).

Chapter 2 ELECTRONIC STRUCTURE AND CHEMICAL REACTION IN SOLUTION Hirofumi Sato Department of Molecular Engineering Kyoto University [email protected]

Keywords: Electronic Structure in Solution, Chemical Reactions, RISM-SCF/MCSCF

1.

Introduction

Chemical reaction, which is undoubtedly one of the most important issues in the field of chemistry, has been studied extensively by means of ab initio molecular orbital (MO) theory since it is the electronic structure of molecules that primarily governs a chemical reaction. However, for a reaction in solution, it is also well regarded that solvent plays an essential role in determining the direction (what the product is), efficiency (how much the yield), and rate (how fast it is) of a reaction. In other words, the solvent effect in chemical reactions manifests itself not only through solute–solvent interactions and solvent reorganizations but also through intra-molecular processes which are always associated with changes in electronic structure. Such effects manifested in the effective charge of atoms amount to 10–40% sometimes, and it cannot be neglected. In this regard theoretical investigations of chemical processes in solutions are inevitably coupled with studies of quantum and statistical mechanics. [1] The ab initio approach of molecular dynamics (MD) , which was originated by Car and Parrinello, or combined methods of standard MO theory and molecular simulation are the most straightforward realization of such couplings. However, the treatment has serious limitations or difficulties in terms of system size: it will be virtually impossible to treat even a small system of a size typical of classical molecular simula-

61

62


tion, say ∼ 1000 molecules, by an ab initio quantum chemical method. A logical and practical solution of such a problem would be to employ a hybrid approach in which a chemical species of interest (solute) is treated by quantum chemistry whilst the surroundings (solvent) is handled classically. In the past two decades several types of quantum chemical methods for computing wave functions of solvated molecules have been proposed. They can be roughly classified into three categories, i.e., continuum model methods, the QM/MM method, and the RISMSCF/MCSCF method. The continuum model, in which solvent is regarded as a dielectric continuum, has been used for a long time to study solvent effects [2]. Solvation energies can be primarily approximated by a reaction field owed to polarization of the dielectric continuum as solvent, and other contributions such as dispersion interactions, which must be explicitly considered for non-polar solvent systems, have usually been treated with an empirical quantity such as the macroscopic surface tension of the solvent. An obvious advantage of the method is its handiness, whilst its disadvantage is an artifact introduced at the boundary between the solute and solvent. Agreement between experiment and theory is considerably governed by the boundary conditions. QM/MM (quantum mechanical molecular mechanical simulation method) is a typical hybrid approach. [3] The total energy of the liquid system at an instantaneous configuration, generated by a Monte Carlo or molecular dynamics procedure, is evaluated, and the modified Schr¨ odinger equations are solved repeatedly until sufficient sampling is accumulated. Since millions of electronic structure calculations are needed for sufficient sampling, the ab initio MO method is usually too slow to be practical in the simulation of chemical systems in solution. The QM/MM methods have their own disadvantages, the obvious one being the computational load added to the already complex calculation of the electronic structure. The RISM-SCF/MCSCF [4] which will be elucidated in this chapter is ‘the third’ method that overcomes above mentioned difficulties. The method combines the two theoretical elements: the ab initio MO method (quantum chemistry) and the RISM (statistical mechanics of liquids), and elicits the nature of chemical system in solution with a moderate computational demanding. It this chapter, after reviewing ab initio MO theory, the RISM-SCF / MCSCF method and its several applications, including acid–base equilibria, isomerization, and reactions are presented. The theory for NMR chemical shift of solvated molecules based on the RISM-SCF/MCSCF method is also described.

63


2.

ab initio Molecular Orbital Theory and the

Solvation Effect 2.1

Isolated Molecular System

We shall begin with a brief sketch of the standard ab initio molecular orbital (MO) theory that deals with an isolated molecule, since this is a fundamental part of chemical reaction analysis. [5] A molecule consists of M nuclei and n electrons. Applying the Born–Oppenheimer approximation, in which electrons are moving around the spatially fixed nuclei, electronic hamiltonian of the system (Helec ) expressed in atomic unit is given by Helec =

n i=1

1 ZA 1 − ∇2i + + , 2 rij riA n

i>j

M

(2.1)

i=1 A=1

where ZA is atomic number and the denominators represent riA = |xi − RA |, rij = |xi − xj |.

(2.2)

Shr¨ odinger equation for this hamiltonian, Helec Ψelec = Eelec Ψelec ,

(2.3)

is the equation we would like to consider. The electronic energy is given by dx1 dx2 · · · dxN Eelec = × Ψ∗elec (x1 , x2 , · · · , xN )Helec Ψelec(x1 , x2 , · · · , xN ).(2.4) Sum of Eelec and the contribution from the repulsive energy between the nuclei is the total energy of the system. Etotal = Eelec + Enuc = Eelec + A>B

ZA ZB . |RA − RB |

(2.5)

It is noteworthy that Eelec parametericallly depends on the nuclear positions ({RA }). Let us consider the simplest case, a closed shell system. A way to describe the electronic wave function (Ψelec ) is to use the Slater determ-

64


inant that satisfies the Pauli’s principle. Ψelec (x1 , x2 , x3 , · · · , xN ) ψ1 (x1 ) ψ2 (x1 ) · · · 1 ψ1 (x2 ) ψ2 (x2 ) · · · = √ .. . N · ψ1 (xN ) ψ2 (xN ) · · ·

, · ψN (xN ) ψN (x1 ) ψN (x2 )

(2.6)

where ψi in the determinant is the one particle orbital. The problem is now how to determine the orbital ψi . According to the variational principle the best wave function should give the lowest energy. Imposing the constrains to the orthnormality of one particle orbital, we defined the following trial function, ∗

ij ψi (x)ψj (x)dx − δij . (2.7) L(ψ1 , ψ2 · · · ψN ) = Etotal − ij

Variations with respect to the functions yield δL = 0 N dx1 δψi∗ (x1 ) = i=1

⎫ ⎤ ⎡⎧ M N ⎬ ⎨ 1 ZA + (Jj − Kj ) ψi (x1 ) −

ij ψj (x1 )⎦ × ⎣ − ∇21 + ⎭ ⎩ 2 r1A j

A=1

j

+ complex conjugate. The Coulombic and exchange operators are defined as follows, 1 dx2 ψj∗ (x2 ) ψj (x2 )ψi (x1 ), Jj ψi (x1 ) = r12 1 dx2 ψj∗ (x2 ) ψi (x2 )ψj (x1 ). Kj ψi (x1 ) = r12 The Hartree–Fock equation is obtained with diagonalized . ⎫⎤ ⎡⎧ M N ⎬ ⎨ 1 Z A ⎣ − ∇21 + + (Jj − Kj ) ⎦ ψi (x1 ) = i ψi (x1 ), ⎭ ⎩ 2 r1A A=1

(2.8)

(2.9)

(2.10)

j

or Fψi (x1 ) = i ψi (x1 ),

(2.11)


65

by using the Fock operator ZA 1 + (Jj − Kj ) . F = − ∇21 + 2 r1A M

N

(2.12)

j

A=1

This is the fundamental equation in ab initio MO theory. In the practical procedure, the one particle orbital is expanded as a linear combination of proper functions (atomic orbital) whose center is located at the nuclei. (Linear Combination of Atomic Orbital; LCAO approximation) Ciµ χµ (x). (2.13) ψi (x) = µ

In this case solving the Hartree–Fock equation is equivalent to determine the coefficient matrix C (Ciµ ). FC = SC ,

(2.14)

where F, S and are matrices whose elements, for example, are calculated by dxχ∗µ (x)Fχν (x), (Fock matrix) Fµν = dxχ∗µ (x)χν (x). (overlap matrix) Sµν = (2.15) Eq.(2.14) is called the Roothaan equation and widely used as the basic equation in many computational program codes in quantum chemistry. Improvement of the approximation beyond the Hartree–Fock method introduced here, say taking account of the electron correlation effect, are discussed in many textbooks.

2.2

Dielectric Continuum Model

The most primitive model that takes solvent effect into account is (dielectric) continuum model. A treatment combined with a quantum chemical calculations including semi-empirical methods began in early 1970’s. In this section we summarize the continuum methods applied to quantum chemical calculations.

2.2.1 General aspect of the continuum model. The solute molecule with electronic distribution ρM , which is a function of r, can

66


be written as the sum of contribution from nuclear ρnuc and that from electron ρel . ρM (r : {R}) = ρnuc (r : {R}) + ρel (r : {R}),

(2.16)

where {R} indicate positions of nuclei that govern electron distribution parametrically. The contribution from the nuclei is given by Zα δ(r − Rα ). (2.17) ρnuc (r : {R}) = α

The contribution from electrons can be computed by using the wave function. We consider a vacuum hole (a cavity) in the dielectric continuum with an arbitrary shape. The solute molecule represented in the electric distribution ρM is immersed and located in this cavity. The basic equation for dielectric continuum model is Poisson–Laplace equation in the electromagnetics. The electrostatic field in the cavity Φin and outside one Φout can be obtained by solving the following equations. ∇2 Φin (r) = −4πρM (r), ∇2 Φout (r) = 0,

(2.18)

with boundary conditions: Φin (r∗ ) = Φout (r∗ ), ∂ ∂ Φin (r∗ ) = ε Φout (r∗ ), ∂n ∂n

(2.19)

where n is a vector normal to the surface of the cavity and r∗ indicates the cavity’s boundary. ε is the dielectric constant of the medium. Only this quantity represents character of solvent. The electronic potential in the cavity can be divided into the direct contribution from the solute molecule Φρ and the effect from the continuum medium Φσ called the reaction field. Φin = Φρ + Φσ .

(2.20)

The electrostatic interaction energy WM S between the solute molecule and the continuum is given by (2.21) WM S = dτ ρM Φσ . The solvation free energy is work of moving infinitely small charges; 1 ∆Gsol = WMS . 2

(2.22)


67

When we adopt a ‘classically’ described molecular model, that is, an assembly of points with fixed charge, as the solute the solvation energies are readily evaluated. However, in the case of a quantum chemical description of the solute molecules the evaluation is not trivial, since the solute’s electronic structure and the reaction field obviously depend on each other. This is a typical non-linear problem and has to be solved in an iterative manner. For practical purposes several approaches have been proposed for solving these problems. All of them are based on the modification of the abovementioned electronic Hamiltonian. [Helec + Vint ]Ψ = EΨ,

(2.23)

where Helec is the usual Hamiltonian for an isolated molecule and Vint is the operator representing the interaction between the solute and the continuum medium, Ψ|Vint |Ψ = WMS .

(2.24)

In the case of the Hartree–Fock method for a closed shell system the new Fock matrix is written as (0) = Fµν + µ|Vint |ν, Fµν

(2.25)

where F (0) is the Fock matrix of isolated molecule. In constructing the practical Fock matrix two points should be clarified. One of them is the shape and size of the cavity, and the other is the description of the ρM . The spatial character of an ideal cavity should be the inclusion of the whole charge distribution of the solutes and the exclusion of empty space where solvent can intrude. A molecular shaped cavity is feasible for fitting such a purpose, but expression of the interaction energy is in a complex form in general. In the case of a regularly shaped cavity, e.g., sphere, ellipsoid, and cylinder, the expression is usually given in analytical form. Note that an expression for the perturbing matrix elements (Vint ) are an important issue in evaluating various properties of molecules. For example, it is difficult to derive the energy gradient with respect to the solute molecular nuclear coordinates in a numerically expressed Vint system. If one has an analytical form of Vint the evaluation of the gradient is straightforward.

2.2.2 Several methods based on the continuum model. Self Consistent Reaction Field (SCRF) This is the simplest and oldest approach of all. The electronic distributions of the solute molecules are expressed in terms of the multipole expansions.

68


First, we consider a spherically shaped cavity with radius a. The reaction field by multipoles located at the center of the cavity can be written in Legendre polynominals. Φσ =

∞ l=0

l (l + 1)(ε − 1) r l+1 − Blm Ylm (θ, φ). l + (l + 1)ε a2l+1

(2.26)

m=−l

The most simple, but important, case only has the term l = 0, corresponding to Born’s equations. The solvation free energies can be written as ∆Gsol = −

ε−1q , 2ε a

(2.27)

where q is net charge in the cavity. For a neutral species the first contributing term is the dipolar term (l = 1) ∆Gsol = −

ε − 1 µ2 . 2ε + 1 a3

(2.28)

The application to quantum chemical calculations is achieved with a new Fock matrix, m,m (0) + Mlm fl,l µ|Mlm (2.29) Fµν = Fµν |ν l,l m,m (0)

where Fµν is the Fock matrices for isolated molecule, Mlm is the mulm,m is a constant defined by the cavity’s shape. tipole operator, and fl,l The interaction energy of this method can be written as, WM S

=

l ∞

Mlm Rlm

l=0 m=−l m,m = Mlm fl,l Mlm .

(2.30)

l,l m,m

The Fock operator in the simplest case, just taking the dipolar term, [6] is (0)

Fij = Fij −

2ε − 1 µ i|ˆ µ|j. ε + 1 a3

(2.31)

It is obvious that all these methods are too crude for describing a molecular system, although they are convenient to handle. Ambiguity in defining the cavity’s radius, center, or the orientation of 3-axis for an ellipsoidal cavity are also problems of these methods. In this sense


69

quantitative values obtained by the SCRF may lead to serious error. Poor convergence in the multipole expansion (l) has been also pointed out. Polarizable Continuum Model (PCM) This method was developed by Tomasi’s group in 1981 and many applications have been proposed [2]. The most distinctive feature of this method is to be able to treat a molecular shaped cavity. Applications not only to Hartree– Fock methods, but to UHF, MCSCF, MBPT, CASSCF, MR-SDCI and DFT etc. have been reported. They also proposed a extension to nonequilibrium solvation problems. The basic concept of their method is that the reaction potential may be described in terms of an apparent charge distribution on the cavity boundary’s surface. The charge distributions σ and the potential from them can be evaluated as σ = −P · n =

ε−1 ε−1 ∇Φout · n, = ∇Φin · n 4π 4πε

ε−1 ∂ (ΦM − Φσ ), 4πε ∂n σ(s) ds2 , Φσ (r) = |r − s| =

(2.32)

where P is the polarization vector. The Fock matrices of these methods are σ(s) (0) ds2 |ν. (2.33) Fµν = Fµν + µ| |r − s| In order to solve these equations, the surface is divided into an appropriate number of tesserae qk = ∆Sk σ(rk ). qk . Φσ (r) = |r − sk |

(2.34)

k

The tessellation is derived from the union of van der Waals spheres centered on the solute’s atoms. The pentakisdodecahedron (which has about 60 faces for one sphere) are adopted for the tesserae. Their method is essentially based on a discrete representation of the reaction field and very similar to numerical methods such as finite differential methods. Image Charge (IC) The fundamental idea of this method was proposed by Friedman [7]. The image charges with appropriate position and value are used so as to describe the reaction potential by dielectric is the image operator which generates the image charge of media. Im

70


the charge distribution. ρM (r ) a M (r) = − ε − 1 dr , (2.35) Imρ ε+1 |r − N| |r − a2 (r − N)/|r − N|3 | where N and a, respectively, indicate the center and radius of the cavity. The solvation free energy for the system is given by M (r). (2.36) WM S = drρM (r)Imρ Note that the energy is reduced to a simple form in the case of a point charge in a spherical cavity ε − 1 q2 . ε + 1 2a

(2.37)

ε − 1 µ2 . ε + 1 2a3

(2.38)

∆Gsol = − For the case of dipole moment ∆Gsol = −

The Fock operator in this model is represented by 1 1 (0) Im ˆ ˆ Dλσ (µν|Im|λσ) + (λσ|Im|µν) , (2.39) Fµν = Fµν + Hµν + 2 2 λσ

where H Im includes the interaction of distributions between real electron and image nuclei, and between image electron and real nuclei. The first applications to some chemical system were reported by Karlström. [8] The advantage of this method is that all the multipole expansions are implicitly included. This leads that expression of derivative with respect to the solute nuclear coordinates are rather simple. [9] The application to the semi-empirical Hamiltonian was also done by Klamt and Sch¨ uu ¨rmann named COSMO [10], although it can also be regarded as an intermediate between IC and PCM.

2.2.3 Summary of the continuum model. The method described in this section is the first application to calculating a solvated molecule’s properties in a quantum chemical manner. Qualitative considerations seem to be reasonable in spite of their considerable simplicity. PCM can be regarded as a final achievement amongst the series of ab initio MO–dielectric continuum combined methods, since all the method is based on the Poisson–Laplace equation. However, it is important to pay much attention to the serious disadvantage in all these

71


method, i.e., how to define the cavity. Actually the solvation energies are very sensitive to the radius and shape of the cavity. Even if an evaluated energy looks reasonable, we must be very careful about its physical origins. It should be also noted that we do not obtain any information on the solvent, e.g., solvation structure with these methods.

2.3

QM/MM Method

The dielectric continuum models such as the self-consistent reaction field method, are efficient in taking account of long range solute–solvent electrostatic interaction and the effect of solvent polarization, but these can not describe local solute–solvent interaction such as hydrogen bonding. Other methods are therefore desired which can describe a short range solute–solvent interaction in a proper manner. An alternative which can overcome the weak points of the dielectric continuum model is the QM/MM method. The idea introduced here is simple. We consider a system composed of one solute molecule and several hundreds of solvent molecules. Since full quantum chemical treatment of molecules is computationally too demanding, the electronic structure of the solute molecule is only treated quantum mechanically (QM) whilst the molecular mechanics (MM) is introduced to describe the solvent molecule. Interaction between the solute and solvent molecule is usually expressed in the following manner. HQM/MM

S N qs =− , riS s=1

(2.40)

i

where S is number of solvent molecule surrounding the solute molecule, qs is the point charge assigned to an atom in solvent, and riS is the distance between the atom in solvent and the solute. The interaction Hamiltonian is incorporated into the electronic Hamiltonian QM/MM QM/MM QM/MM = Eelec Ψelec , (2.41) Helec + HQM/MM Ψelec QM/MM

) is the solute electronic wavefunction affected by the solvent (Ψ elec obtained by solving this modified equation. Since the electron exchangeexclusion interaction between the solute and solvent is not treated in this Hamiltonian, a contribution from repulsive energy, such as the van der Waals interaction, is usually added to the total energy, which is evaluated at an instanteous solvent configuration generated by molecular dynamics or the Monte Carlo procedure. To compute appropriate thermodynamic properties of the system the quantum chemical calculations are repeated and averaged over more than millions of geometrical configurations.

72


The QM/MM method is becoming popular for studying chemical reactions not only in systems of fluids, but in a biological system such as enzyme. It is obvious that a serious bottleneck of the method is its computational demands.

3.

RISM-SCF/MCSCF Theory

A new method referred to as RISM-SCF/MCSCF is based on the ab initio electronic structure theory and RISM. The original RISM-SCF method was proposed by Ten-no, Hirata, and Kato in 1993. [11] The basic idea of the method is to replace the reaction field in the continuum models by a microscopic expression in terms of the site–site radial distribution functions between solute and solvent, which can be calculated from the RISM theory. Exploiting the microscopic reaction field, the Fock operator of a molecule in solution can be expressed as Vλ bλ , (2.42) Fisolv = Fi − fi λ∈solute where the first term in the right hand side is the operator for an isolated molecule, and the second term represents the solvent effect. Vλ represents the electrostatic potential of the reaction field at solute atom λ produced by solvent molecules and bλ is a population operator of solute atoms. The potential Vλ takes a microscopic expression in terms of the site–site radial distribution functions between solute and solvent: qα gλα (r)dr, (2.43) Vλ∈solute = (V)λ∈solute = ρ r α∈solvent where qα is the partial charge on the site α in solvent, ρ is the bulk density of solvent, and gλα is the site–site radial distribution function (RDF) or pair correlation function (PCF). The practical procedure for evaluating the matrix elements will be found in the Appendix. In the RISM-SCF theory the statistical solvent distribution around solute is determined by the electronic structure of the solute, whilst the electronic structure of solute is influenced by the surrounding solvent distribution. Therefore the ab initio MO calculation and the RISM equation must be solved in a self-consistent manner. It is noted that the word ‘SCF’ applies not only to the electronic structure calculation but to the whole system, e.g., a self-consistent treatment of electronic structure and solvent distribution. Essentially, the RISM and extended RISM theories can provide information equivalent to those obtained from simulation techniques, namely, thermodynamic properties, microscopic liquid structure, and so on. But

73


it is noteworthy that the computational cost is drastically reduced by these analytical treatment of theories, which allow it to be combined with computationally expensive ab initio MO theory. Thanks to this advantage the MO part in the method can be readily extended to the more sophisticated levels beyond Hartree–Fock (HF), such as CI (configuration interaction) and coupled cluster. This is a remarkable merit compared to the other hybrid methods. In 1996 Sato, Hirata, and Kato reported that the solvated Fock operator can be naturally derived from the variational principle [12] defining the Helmholtz free energy of the system (A) by ˆ + ∆µ|Ψ ˆ A = Ψ|H + Enuc .

(2.44)

Here Enuc is the nuclear repulsion energy and 1 2 1 ρ ˆ dr cαs (r) − hαs (r) + hαs (r)cαs (r) , Ψ|∆µ|Ψ = ∆µ = − β αs 2 2 (2.45) is the modified version of solvation free energy originally defined by Singer and Chandler [13]. This is a functional of the total correlation function hαs (r), the direct correlation function cαs (r), and the solute wave function |Ψ. Energy of the solute molecule Esolute is defined as follows ˆ + Enuc . Esolute = Ψ|H|Ψ

(2.46)

Note this is also the functional of hαs (r), cαs (r) and |Ψ. Imposing constraints concerning the orthonormality of the configuration state function(C) and one particle orbital(φi ) to the equation, one can derive the Fock operator from A based on the variational principle; δ(A[c, h, t, v, C] − [constraints to orthonormality]) = 0, and Fijsolv

= Fij − γij

λ∈solute

∂ bλ ∂qλ

(2.47)

ρ dre[−βuαs (r)+hαs (r)−cαs (r)] − β αs

.

(2.48) If classical coulombic interactions are assumed amongst point charges for electrostatic interactions between solute and solvent, and the term for the CI coefficients (C) are omitted, the solvated Fock operator is reduced

74


to Eq. (2.42). (For the details, see the Appendix) The significance of this definition of the Fock operator from a variational principle is that it enables the analytical first derivative of the free energy with respect to the nuclear coordinate of the solute molecule Ra , ∂ω ˆ αγ (k) ∂Enuc 1 ∂A = − cγs (k) χ ˆss (k) dkˆ cαs (k)ˆ ∂Ra ∂Ra 2(2π)3 β ∂Ra α,γ,s,s

1 γij haij + Γijkl (φi φj |φk φl )a + 2 i,j i,j,k,l a −Vt qa − εij Sij .

i

(2.49)

j

The second term of the right hand side in Eq. (2.48) corresponds to the change of solute–solvent distribution function owed to the modification of the intra-molecular correlation function ω. The fifth term represents the effective charges change caused by the change of solute geometry. Other notations used here have their usual meanings. It has been well recognized that the energy gradient technique in the ab initio electronic structure theory is a powerful tool to investigate the mechanism of chemical reactions of polyatomic system and it opens up a variety of application to the actual chemical processes in solution: carrying out the geometry optimization of reactant, transition state, and product in solvated molecular system, constructing the free energy surfaces along the proper reaction coordinates, computation of the vibrational frequencies and modes, and so on. On analyzing the computational results, two important factors are essentially considered as the solvent effect; the energy change associated with the solute reorganization (Ereorg ) and the solvation free energy (∆µ) upon solvation. Ereorg = Esolute − Eisolate ,

(2.50)

where Eisolate is the total energy of the solute molecule in an isolated condition, and Esolute is the energy of solute molecule defined above. The quantity Ereorg represents the reorganization energy associated with the relaxation or distortion of the electronic cloud and molecular geometry in solution. The solvation free energy in turn consists of the solute– solvent interaction energy and the free energy change associated with the solvent reorganization whose analytical form of this quantity is given in Eq. (2.45). Thus total free energy (Atotal ) of the system involving solute and solvent molecules consists of four terms (Fig. 2.1) Atotal = Aid + Eisolated + Ereorg + ∆µ,

(2.51)


75

where the Aid is solute kinetic contribution to the free energy, which is evaluated from the elementary statistical mechanics of ideal systems. Total Energy of the Solute Eisolated

Solvation Free Energy

Kinetic Contribution Aid

Relaxation of Electronic Structure and Geometry Ereorg

related to energy change in reaction

Figure 2.1.

∆µ

Energy components associated with the RISM-SCF framework.

Now we have got tools to tackle various problems in solvated molecules. In the following sections, we present our recent efforts to explore such a phenomena by means of the RISM-SCF/MCSCF method.

4.

Acid–base Equilibria in Aqueous Solution

The acidity and basicity are fundamental and familiar concepts in chemistry and biochemistry. Quantum chemistry has provided a theoretical understanding of the phenomena as far as the gas phase in concerned. However, in solution the reactivity is seriously affected by solvents. In this regard the acid–base equilibria is a very good example to see the ‘inevitable coupling’ of quantum and statistical mechanics in solution, which often governs chemical observations.

4.1

The Basicity of the Methylamines in Aqueous Solution

It has been experimentally known that the basicity of the methyl amines increases monotonically with successive methyl substitutions in the gas phase, NH3 < (CH3 )NH2 < (CH3 )2 NH < (CH3 )3 N, and that the order reverses at the trimethylamine in aqueous environment, NH3 < (CH3 )NH2 < (CH3 )2 NH > (CH3 )3 N.

76


The monotonic increase of the basicity in the gas phase has been explained in terms of the ‘negative induction’ or the polarization effect due to the methyl groups. How can we understand the irregular behavior in aqueous solution? Let us define the respective basicity (free energy difference upon protonation) by −∆Gg in the gas phase and −∆Gs in aqueous solution. [14] For the discussions concerning the relative strength in basicity of a series of methylamines, it is sufficient that only the relative magnitude of these quantities are considered. Thus the free energy changes associated with the protonation of the methylamines relative to those of ammonia are defined as, ∆∆Gi298 [(CH3 )n NH3−n ] = ∆Gi298 [(CH3 )n NH3−n ] − ∆Gi298 [NH3 ] (n = 1, 2, 3). (2.52) The subscript i is either g or s, representing the quantity in the gas phase or in aqueous solution, respectively. ∆Gs298 can be derived as a sum of two terms, ∆Gs298 (solute) and ∆Gs298 (solvent): The former includes the electronic contributions and kinetic contribution, while the latter corresponds to solvation energy (∆µ). Computed values are plotted in Figure 2.2 against the increasing number of the methyl groups.

∆∆G298i / kcal mol-1

10

0

-10 ∆∆Gs298(solute) ∆∆Gs298(solvent)

-20

∆∆Gs298 (total) ∆∆Gs298(explt.)

∆∆Gg298(computed) ∆∆Gg298(explt.)

-30 0

Figure 2.2.

1 2 Number of CH3

3

Free energy changes on protonation referred to NH3 .

The contribution from solute itself (∆∆Gs298 (solute)) exhibits similar monotonic behavior with the gas phase result (∆∆Gg298 ) which is in good agreement with the experimental data. The difference between ∆∆Gs298 (solute) and ∆∆Gg298 is owed essentially to the electron reorganization energy. The solvation free energy ∆∆Gs298 (solvent) shows the monotonic increase with the successive methyl substitution. The sum of the two contributions produces the inversion in the overall free energy

77


change ∆∆Gs298 , which is in qualitative accord with the experimental result.

4.2

Acid strength of the hydrogen halides in aqueous solution

The order of acidity of hydrogen halides (HX, where X=F, Cl, Br, I) in the gas phase can be successfully predicted by quantum chemical considerations, F < Cl < Br < I. However, in aqueous solution, while hydrogen chloride, bromide, and iodide completely dissociate in aqueous solutions, hydrogen fluoride shows a small dissociation constant. This unexpected observation can be understood by studying the chemical equilibrium [15] HX + H2 O X− + H3 O+ , in solution phase for a series of the hydrogen halides (HX). 2.0

H X – H

H

1.5

O

water B H

O H 1.0

water A

0.5

HF (F-H) HCl (Cl-H) HBr (Br-H) HI (I-H)

0.0 0

2

4

6

8

R/Å

Figure 2.3. Pair correlation function (radial distribution function) between the halogen atom in HX and the hydrogen in solvent water.

Figure 2.3 shows the PCF between a halogen site in HX and a hydrogen site in solvent water. In the case of hydrogen fluoride a distinct peak is found at 1.82 ˚ A, whilst there is no corresponding peak in the other X–H correlation functions. From other PCFs and the geometrical consideration, one can conclude that hydrogen bonds between solute hydrogen - solvent oxygen are found in all the hydrogen halides (H2 O: B in the figure), whilst only hydrogen fluoride forms a distinct F–H (solvent water) hydrogen bonding (H2 O: A in the figure). It is expected that the liquid structure around HF is markedly different from those around the other hydrogen halides. The characteristic behavior of hydrogen fluoride as a weak acid is explained in terms of the enhanced stability of the non-

78


dissociated form of the molecule in aqueous solution due to hydrogenbonding. The free energy difference in aqueous solution (∆G) can be obtained in terms of the free energy difference associated with the reaction in vacuo and solvation free energy, ∆G = Gaq (X− ) + Gaq (H3 O+ ) − {Gaq (HX) + Gaq (H2 O)} . (2.53)

Energy / kcal mol -1

100 50

∆G / kcal mol

150

-1

10

200

explt. RISM-SCF

5 0 HX X-5 -10 -15 F

Cl

Br

I Br Cl F

I

0 -50 -100 process I process III -150

process IV process II

∆G

HX+H2O X–+H3O+ aqueous solution HX+H2O

gas phase

X–+H3O+

standard state: 1 H2+ 1 X2+H2O 2 2

Figure 2.4.

Energy components of the proton dissociation in hydrogen halides.

∆G can be further decomposed into four contributions associated with thermodynamic cycle.(Fig. 2.4) Two formation free energies measured from the standard state (Process I and II for HX+H2 O and X− +H3 O+ , respectively) and two solvation free energies for these chemical species (Process III and IV for HX and X− , respectively). One can see that the two energy contributions concerned with halide anions compensate each other (Process II and IV), and the final free energy difference is mainly governed by the subtle balance of the two energetic components; formation energies of hydrogen halides (Process I) and solvation energies of the halide anions (Process IV). The final result is in very good accord with the experimental data. It is of great interesting that the molecularlevel microscopic character governs the macroscopic properties.

4.3

Auto-ionization of water

A water molecule has amphoteric character. This means it can act as both an acid and a base. The auto-ionization equilibrium process in water, H2 O + H2 O H3 O+ + OH− ,

(2.54)

is one of the most important and fundamental reactions in a variety of fields in chemistry, biology, and biochemistry. The ionic product (Kw )


79

and its logarithm defined by Kw = [H3 O+ ][OH− ],

pKw = −logKw ,

(2.55)

are measures of the autoionization. The quantity can be related to the free energy change (∆Gaq ) associated with the reaction of Eq. (2.54) by the standard thermodynamic relation. ∆Gaq = 2.303RT pKw .

(2.56)

It is experimentally known that the value of pKw shows significant temperature dependence, i.e., it decreases with increasing temperature. However, there is no easy explanation of this phenomenon even from the phenomenological point of view. The free energy change associated with the reaction in Eq. (2.54), can be written [16] in terms of the energy change associated with the reaction in vaccuo (∆Gvac ) and the free energy change of the reacting species due to solvation as, ∆Gaq = ∆Gvac + δG(H3 O+ ) + δG(OH− ) − 2δG(H2 O),

(2.57)

where δG(H3 O+ ), δG(OH− ) and δG(H2 O) are, respectively, the free energy changes of H3 O+ , HO− and H2 O upon solvation. It is noted that the species concerning the reaction, H2 O, OH− , and H3 O+ are regarded as ‘solute’ in the infinitely dilute solution in RISM-SCF treatment. ∆Gaq is also possible to be decomposed into intra- and intermolecular contributions as vac + ∆δGkin + ∆δGreorg + ∆δµ, ∆Gaq = ∆Eelec

(2.58)

vac , ∆δG where ∆Eelec kin , ∆δGreorg are the electronic energy in vacuo, the kinetic term and the electronic reorganization energies, respectively, which are intra-molecular contributions. ∆δµ is the solvation free energy change. (We use ∆ for changes of quantities associated with the chemical reaction and δ for changes owed to solvation.) Relative value of pK w at temperature T to that at T = 273.15K given by

∆T pK w (T ) = pK w (T ) − pK w (273.15),

(2.59)

is further decomposed into the four contributions corresponding to the free energy components. vac + ∆T pKw,kin(T ) ∆T pKw (T ) = ∆T pKw,elec +∆T pKw,reorg (T ) + ∆T pKw,solv (T ).

(2.60)

80


∆pKw,solv 4.0

60.0 40.0

∆pKw,solv

20.0 0.0

3.0

-20.0

2.0

-40.0 -60.0

∆pKw,elec 0

20

40

1.0

60

80

100

120

T / ˚C

(∆pKw,solv+∆pKw,elec)

0.0

∆pKw,kin

∆pKw

-1.0

(computed)

∆pKw (explt.)

-2.0

∆pKw,elec

∆pKw,reorg

-3.0 0

20

40

60

80

100

120

T / ˚C Figure 2.5. Temperature dependence of pKw : computed pKw and its components (solid and dashed line) and the experimental result (dotted line) are shown.

The resultant ∆T pKw (T ) and their components are plotted in Figure 2.5. As shown in the Figure, contributions from ∆T pKw,solv and vac are very large, but they compensate each other. The final ∆T pKw,elec temperature dependence of pKw is determined by an interplay of several contributions with different physical origins. It is also interesting that the temperature dependence is dominated by ∆T pKw,reorg after the compensation of the two largest contributions. The theoretical results for temperature dependence of the ionic product show fairly good agreement with experiments and also demonstrate the importance of polarization effects. How is the property of pKw altered along the change of higher temperature or density? We refer reader who are interested in the detail to the original paper. [16]

5.

Solvent Effects on Conformational Change of Chemical Compounds

Acidity of organic acids (and/or basicity of organic bases) such as carboxylic acid and amines is governed by many factors; solvent, substitution, conformation and so forth. Amongst those factors the effect of conformational change is of special interest in terms of its significance


81

in the biological system. In a biomolecular system such as protein the acidity or basicity of related functional groups depends sensitively upon the molecular conformations: owing to such sensitivity the property is sometimes exploited to detect the conformational change of protein.

5.1

Conformational equilibrium

A prototype of the abovementioned phenomena can already be seen in the simplest carboxylic acid, acetic acid (CH3 CHOOH). [17] Acidity is determined by the energy or free energy difference between the dissociated and non-dissociated form, whose energetics usually depends significantly upon its conformation, e.g. the syn-/anti-conformational change of carboxylate group in the compound substantially affects the acid/base equilibrium. 20.0

20.0 E(gas) ∆G(aq.)

10.0

C(methyl) H(methyl) ∆µ+Ereorg

0.0

0.0 O

H(-O) -10.0

-10.0

∆µ O(=C)

-20.0

-30.0

H O–H

-20.0

O

CH3–C

-40.0

CH3–C O

syn

anti

Contributions to the solvation energy of the solute atoms

Energy Change / kcal mol-1

10.0

C(carbonyl) Ereorg

O

syn -50.0

-150 -100

-50

0

50

100

150

Torsional Angle (H-O-C-C) θ/ deg.

Figure 2.6. Potential and Free energy curve around the torsional angle of θ (CH3 – C–O–H) in acetic acid.

Potential energy profiles in the gas phase and potential of mean force insolution along the torsional angle θ (H–O–C–C) are shown in Figure 2.6. In the gas phase the syn-acetic acid (θ = ±180) is more stable than the anti-conformer by 6.9 kcal/mol and the barrier height of rotation between these conformer is estimated as 13.2 kcal/mol. In aqueous solution the calculated free energy difference is significantly reduced to 1.7 kcal/mol. The rotational barrier also become lower than that in the gas phase by 10.3 kcal/mol. The reduction of the free energy gap indicates that the pKa difference between the two conformers is drastically reduced from 5.1 to 1.2 on transferring from the gas phase to aqueous solution at room temperature. Stabilization of syn-conformer in the gas phase is explained rather intuitively in terms of the increased interaction between the H atom in the OH group and the O atom in C=O group. As

82


one can find in the Figure, the extra stabilization in the anti-conformer in aqueous solution arises from the solvation energy, especially at the carbonyl oxygen site. The electronic redistribution of the solute upon transferring from the gas phase to aqueous solution is another interesting issue. The Mulliken charge population analysis demonstrates substantial change of effective charge on the hydrogen and oxygen in the OH group: for anti-conformer, 0.355(H) and −0.428(O) are altered to 0.377(H) and −0.434(O) from gas to aqueous solution phase, and for syn-conformer, 0.373(H) and −0.463(O) to 0.395(H) and −0.476(O), respectively. One can notice a character of proton increases on the hydrogen in OH group by being immersed into aqueous solution.

5.2

Tautomerization of Acetylacetone

The keto-enol tautomerization of acetylacetone (CH3 -CO-CH2 -COCH3 ), a prototype β-diketone, has been extensively studied experimentally, and attention has been paid to its solvent effect. Although the enol form is more stable than the keto in the gas phase owing to the intramolecular hydrogen bonding, the equilibrium is known to shift toward the keto in solution as the solvent polarity increases. The tautomerization in various types of solution, which includes H2 O, dimethyl sulfoxide (DMSO), and carbon tetrachloride (CCl4 ), was examined by means of RISM-SCF method. [18] For the keto tautomer the optimized geometry with a larger dihedral angle τ (O=C-(CH2 )-C=O) is found in CCl4 (140.4◦ ). But the geometry of the stable conformer undergoes a drastic change in polar solvents and the torsional angle, τ , significantly decreases to 56.4◦ in H2 O and to 75.2◦ in DMSO. In the enol tautomer, in contrast, the geometric change owed to the solvation is smaller than that in the keto. All the atoms except for the methyl hydrogens lie in the same plane in all the solvents as in the gas phase. Detailed analysis of the thermodynamic property reveals the origin of a large geometric change of the keto form in polar solvents: such a geometric change is driven by the enthalpy term for aqueous solution while the entropy term is important in DMSO. Fig. 2.7 shows the calculated free energy differences between the keto and enol tautomers along with the available experimental values. In H2 O the free energy of the keto form was calculated to be lower than that of the enol by −1.37 kcal/mol, which is in good agreement with the experiment, −1.18 kcal/mol. For DMSO solution the stability of the two tautomers became comparable to each other, 0.14 kcal/mol, as in the experiment. The free energies were also very close in CCl4 , which is


83

2.0

∆G (keto-enol) / kcal mol-1

O

τ

C H3 C

1.0

O C

CH2

CH3

0.0

RISM-SCF explt. (1) explt.(2)

-1.0

CCl4

DMSO

H2O

Figure 2.7. The keto–enol free energy difference of acetyl acetone in various solutions (CCl4 , DMSO and H2 O).

different from the experiment where the enol form is more stable than the keto by 1.76 kcal/mol. Considering that the solute geometries are similar to those in the gas phase, and that the electric field is weak in CCl4 , this discrepancy in the free energy difference comes from the underestimation of solute electronic energy difference between the tautomers.

6.

Solvent Effect on Chemical Reactions

What is generally wished for in the quantum chemical computations is a reaction profile. As we have seen in this chapter, the electronic structure is remarkably affected by surrounding solvent molecules, which means that reactions are strongly governed (and controlled) by the environment as well. Three chemical reactions, S N 2 type reaction, Diels– Alder reaction and redox process, are described in this section.

6.1

SN 2 Reaction

Menshutkin type of SN 2 reaction in aqueous solution, NH3 + CH3 Cl → NH3 CH3 + + Cl− , is regarded as a typical system for a benchmark of a theroy of chemical reaction in solution environment, and has been extensively studied by the various methods including QM/MM simulation and several dielectric continuum approaches. The RISM-SCF method has been applied to the same reaction[19] to provide information on solvation structures for use in interpreting the solvent effects on the reaction mechanism and to compare the free energy profile with those obtained by other methods.

84


The potential and free-energy profiles along the reaction coordinate calculated both by the standard ab initio MO and the RISM-SCF in the Hartree–Fock level are shown in Fig. 2.8. Although this reaction is endothermic in the gas phase by 106.3 kcal/mol with the HF method, it becomes exothermic in aqueous solution by 27.8 kcal/mol at the RISMHF level. The barrier height was calculated to be 17.7 kcal/mol. As seen in Fig. 2.8, there is a very shallow potential well around 1.9 ˚ A, + − which corresponds to a contact ion pair of NH3 CH3 and Cl formed in aqueous solution. The Cl–H RDF depicted in Fig. 2.9 clearly demon80 gas phase aqeous solution

Energy / kcal mol-1

60 40 20

H3 N+CH3 Cl

H3 NCH3 ++Cl-

0 -20 -40 -4

-2 0 2 Reaction Coordinate / Å

4

Figure 2.8. Energy profile of SN 2 reaction in the gas phase (dotted line) and in aqueous solution (solid line)

strates the progress of hydrogen bonding along the reaction coordinate. At the reactant the peak corresponding to the Cl–H interaction is not observed. The distinct first peak around R = 2.2 ˚ A grows up monotonically and slightly shifts toward inside as the reaction proceeds. The other H atom of H2 O in the first solvation shell corresponds to the second peak around 4.2 ˚ A. The number of H2 O molecules hydrogen bonded to Cl was estimated by integrating the first peak of the RDFs and was found to be 4.3 and 6.6 at the TS and product, respectively, which is comparable to the QM/MM result, 3.1 and 6.6, respectively. The breaking of the hydrogen bond between the ammonia N and water H atoms is observed in Fig. 2.9, where the first peak around 2.0 ˚ A in the reactant disappears as the reaction progresses. Although the lone pair electron on the N atom forms hydrogen bonds with H2 O in the reactant region, it participates in the formation of a new chemical bond with the C atom of CH3 Cl in the reaction process. The importance of the formation of Cl–H hydrogen bonds and the breaking of N–H bonds was demonstrated in characterizing the reaction mechanism as in the previous QM/MM calculations. It is noted, however, that there are large discrepancies in the C1–O and N–O RDFs between the RISM-SCF and QM/MM results.

85

Electronic structure andchemical reaction in solution 4.0

2.0

PCF( N-H )

PCF(Cl-H)

3.5

3.0

1.5

2.5

1.0

2.0

1.5

0.5

1.0

0.5 0.0

0.0 0

2

4

6

8

0

10

2

4

R/Å

6

8

10

R/Å

2.0

4.0

3.5

PCF( N-H )

1.5

PCF(Cl-H)

3.0

2.5

1.0

2.0

1.5

0.5

1.0

0.5 0.0

0.0 0

2

4

6

8

10

0

2

4

R/Å

6

8

10

R/Å

2.0

4.0

PCF( N-H )

PCF(Cl-H)

3.5

1.5

3.0

2.5

1.0

2.0

1.5

0.5

1.0

0.5 0.0

0.0 0

2

4

6

8

10

R/Å

0

2

4

6

8

10

R/Å

Figure 2.9. Solvation structure change along the SN 2 reaction in aqueous solution: reactant (top panel), transition state (middle), and product (bottom).

6.2

Diels–Alder Reaction

The Diels–Alder reaction is a cyclo-addition reaction, which is reported to be greatly accelerated by aqueous solvent. Recent experiments indicate also that the rate and yield of the reaction increase dramatically by bringing the solvent to supercritical conditions. (a) Why and how is the reaction accelerated when the system is transferred into aqueous environment from gas phase? (b) Is the physical cause of the increase in reaction rate in the supercritical condition the same as that in ambient water? (c) Why does the reaction yield increase in supercritical conditions? To answer these questions the reaction concerning methylvinyl ketone (MVK) and cyclopentadiene (CP) in ambient water (AW) and subcritical condition (SCW) were investigated based on the RISM-SCF method. [20] One of the advantages of the method is that it enables us to change easily thermodynamic conditions of the system, such as temperature and density. Thus it is suitable for investigating the chemical reaction in solvent, including SCW, which requires numerical analyses in wide range of temperature and pressure. The key results are shown in Figure 2.10, which exhibits the free energy diagram concerning the reaction through the endo-cis transition structure in gas phase, in AW (298K, 1.0 g/cm3 ), and in SCW (873K, 0.6

86 Relative Free Energy (kcal/mol)


gas phase (873K) SCW

60

gas phase (298K)

40

AW COCH3

+

20 COCH3

0

Reactant

TS

Product

Reaction Coordinate

Figure 2.10. Free energy profile of the Diels–Alder reaction in ambient water (AW), in subcritical water (SCW), and in the gas phase.

g/cm3 ). All the energy levels are determined by regarding the reactants at corresponding temperature as a standard. The activation free energy ‡ ) is 52.5 kcal/mol, while the computed in gas phase at 278K (∆F298 ‡ ∆FAW is calculated to be 45.6 kcal/mol: it is reduced by 6.9 kcal/mol ‡ , from gas phase to AW. The activation free energy in SCW, ∆FSCW is found to decrease by 2.7 kcal/mol compared with that in gas phase, ‡ . (54.3 kcal/mol) Together with the RDF analysis, it is found that ∆F873 the solvent effect in SCW becomes weaker than that in AW, although the substantial effect of solvent on the chemical reactions largely remains in SCW. These results seem quite reasonable if one considers that the density of water molecules in SCW is low and that the solvent effect in SCW is less than in AW. From these values the relative rate constant in SCW compared to that in AW is estimated to be 1.27× 1013 ; the rate constant in SCW is extremely increased. Although the solvent effect for the rate constant is apparently decreased in SCW, the rate is dramatically increased because of the thermal activation rather than the solvent effect. The determination of the reaction yield is somewhat similar. In spite of the increasing endothermicity in SCW, the yield is approximately 667 times higher than in AW. This is because the increased solubility of the reactants (CP and MVK) in SCW provides an advantage that is enough to overcome the energetical disadvantage. Now we can answer the aforementioned questions: The reaction in SCW is accelerated owing to increased thermal activation, not by the same cause in AW. The reaction yield is increased dramatically in SCW owing to enhanced solubility compared to that in AW.

87


6.3

Charge Transfer Reaction

Electron transfer (ET) reactions have been extensively studied because of the essential roles they play in a variety of chemical, physical, and biological processes. Among various types of organometalic complexes, a ruthenium complex has been widely chosen as a model to study the oxidation-reduction chemistry of metalloproteins. More fundamental phenomena, ET reactions between the ruthenium complex in aqueous solution environment, [Ru(NH3 )6 ]2+ + [Ru(NH3 )6 ]3+ [Ru(NH3 )6 ]3+ + [Ru(NH3 )6 ]2+ , have been also investigated by many researchers. The rates and mechanisms of the ET reactions are of great interest from both experimental and theoretical view points. It is well recognized that Marcus type of diabatic free energy surface is very useful for understanding ET processes. Such a diabatic free energy profile in polar solvent can be realized within the framework of the RISM theory by Chong’s method (Chapter I) [21], in which the hypothetical charge distribution is introduced. Although the method was originally developed for a redox pair, extension to a polyatomic system is straightforward. [22] The free energy profile around the dication ([Ru(NH3 )6 ]2+ with a charge set of q2+ ) is given by, ∆F 2+ = ∆µs − ∆µ2+ − sVs · (q3+ − q2+ ).

(2.61)

The profile for the product state ([Ru(NH3 )6 ]3+ with a charge set of q3+ can be obtained similarly. ∆F 3+ = ∆µs − ∆µ3+ − (s − 1)Vs · (q3+ − q2+ ).

(2.62)

In both equations a state with the hypothetical charges (q = (1 − s) + sq3+ ) is introduced to compute the solvation free energy (∆µs ) and electrostatic potential on the solute site (Vs ). s (r) j gij . (2.63) qsolvent dr4πr 2 (Vs )i = ρ r j

The diabatic free energy profile obtained by the method is plotted in Figures 2.11. The most notable feature of the figures is that it is essentially parabolic with respect to the reaction coordinate. This is nothing but the manifestation of the central limiting theorem, and the results indicate that the response of solvent fluctuation to the electrostatic filed is linear. The behavior, however, is not trivial, because the free energy calculated via Eqs. (2.61) and (2.62) are inherently non-linear.

88


Figure 2.11.

Marcus type of free energy curve for [Ru(NH3 )6 ]2+ and [Ru(NH3 )6 ]3+ .

Since the reaction is a symmetric one, the activation free energy of the present system (∆G‡ ) can be estimated with the relation,

1 1 + ∆F 3+ , ∆G‡ ∼ ∆F 2+ 2 2 which gives 24 kcal mol−1 . Unfortunately, experimental data directly corresponding to the theoretical result is not available at the moment. Here, we just refer to the experimental value, ∼ 14 kcal mol−1 , obtained for the same metal complex in the different solvent condition (involving CF3 SO3 H and counter ions). The present model treats these metal complexes in pure aqueous solution and the screening from counter ions as well as CF3 SO3 H is completely disregarded. The screening effect in general weakens the ion–ion interactions, in this case, repulsive ones, and the effect very likely lowers ∆G‡ .

7.

The NMR Chemical Shift

Over the past decades the NMR chemical shift measurement in solutions has been applied to a vast range of problems in chemistry and biochemistry, and has revealed itself to be an invaluable microscopic probe. It has played an especially important role in the structural understanding of protein owing to its great sensitivity to the environment in which the probing atom is situated. Theories of the chemical shift in solution, on the other hand, have not been well developed owing to the lack of a theory for describing the electronic structure of a solvated molecule. In 2000 Yamazaki et al. proposed a successful theory [23] for computing the NMR chemical shift within the framework of the RISM-SCF

89


method. The nuclear magnetic shielding tensor σ X of a nucleus X can be expressed as mixed second derivative of the free energy A with respect to the magnetic field B and the nuclear magnetic moment mX : 2A ∂ X σαβ = , (2.64) ∂Bα ∂mX β X Y B,m ,m ···=0

X , where Bα and mX β are the Cartesian components of B and of m respectively, and A is a sum of the electronic energy and the excess chemical potential defined in the beginning of this chapter (Eq. (2.44)). The gauge-invariant atomic orbital (GIAO) method is adopted to solve the gauge problem in the calculation of nuclear magnetic shieldings. Thus, each molecular orbital (φi ) is expressed as a linear combination of GIAO χν : ciν (B, mX )χν (B), (2.65) φi = ν

where i χν (B) = exp − (B × Rν ) · r . 2

(2.66)

Final expression of the nuclear magnetic shielding tensor can be obtained from these equations

(0,0) (1,1) X (1,0) (0,1) X X + Pνλ Pνλ Hνλ Hνλ , (2.67) σαβ = αβ

νλ (l,m)

α

β

(l,m)

where Pνλ and Hνλ are the (ν, λ)–element of the derivatives of the density and one electron matrices, respectively, with respect to the external magnetic field and the magnetic moment of nucleus,

∂Pνλ (1,0) = , (2.68) Pνλ ∂Bα B,mX =0 α

X 2H ∂ (1,1) νλ = . (2.69) Hνλ αβ ∂Bα ∂mX β X B,m =0

(1,1) X are calculated from the usual RISM-SCF method, whilst Hνλ αβ

(0,1) X and Hνλ can be obtained by the standard ab initio calculation in (0,0)

Pνλ

β

90


Proton chemical shift of water (ppm)

2.0 H2O 1.5

1.0 CH3CH3CO 0.5 CHCl3 CCl4

0.0 0

10

20

30

40

50

60

Temperature (°C)

Figure 2.12. Temperature dependence of the chemical shift of a water molecule in various solvent. All the shifts are given relative to the isolated water molecule in the gas phase.

(1,0) the gas phase. For Pνλ the solution of the first-order coupled– α perturbed Hartree–Fock equation, in which the RISM-SCF type of solvated Fock operator is incorporated, is employed. The proton chemical shifts in various solvents, obtained by the theory, are plotted as a function of temperature in Fig.2.12. The present method qualitatively reproduces the two observations in the recent experiment by Nakahara et al. with a modern NMR technique. [24] First, the proton chemical shifts of the solute water molecule are in the sequence over the temperature range studied, in water > in acetone > in chloroform > in carbon tetrachloride. Secondly, the proton chemical shifts decrease with increasing temperature. These results clearly demonstrate the capability of the theory to predict experimental results. Analysis on the radial distribution function provides molecular level information on solvation structure and its relation to NMR chemical shifts. The study of chemical shift reveals a serious drawback inherent in the classical–quantum hybrid approach. The solvent effect on the oxygen chemical shift showed temperature dependence opposite to corresponding experimental results. An ab initio analysis with a small cluster suggested stongly that the ill behavior is originated from the lack of electron exchange between solute and solvent. Therefore an obvious direction of improvement of the RISM-SCF theory is to take the electron exchange into account.


8.

91

Summary

In this chapter we have reviewed the RISM-SCF/MCSCF method, a theory which combines the ab initio electronic structure theory and RISM. Its capability of treating extensive problems in chemistry is demonstrated. The methodology has numerous advantages over the other related methods such as dielectric continuum models and QM/MM. The RISM-SCF method provides the molecular level information on the solvation structure including hydrogen bonding, whic can never be made by the continuum model. Much less computational demand by the RISM theory compared to the MM and MD approaches makes possible to combine the theroy with highly sophisticated ab initio MO theory which by itself requires heavy computational load. All the useful knowledges and techniques developed in the gas phase quantum chemistry as well as in the statistical mechanics theory (RISM) can be utilized in the combined theory. There are several directions to extend the RISM-SCF/MCSCF method that are not described in the present chapter. One of such directions is a combination of ab initio MO theory with 3D-RISM, which is explained in chapter 4. The site–site treatment of the solute–solvent correlations involving the approximation of radial averaging constitutes a bottleneck of the RISM-SCF method, and thus lacks a 3D picture of the solvation structure for complex solutes. The SCF theory combined with the 3D-RISM is free from such a bottleneck. The test computation on the carbon monoxide in water provides a detailed hydration structure of water solvent as well as polarized CO electronic structure. [25] It is also found that the results from the original RISM-SCF/MCSCF method are in reasonable accord with those following from the 3D-RISM-SCF approach after reduction of the orientational dependence. This shows the RISM-SCF/MCSCF approach gives a proper picture for a solvation process. The electronic structure plays a primary role in determining structure of a molecule. However, changes of the electronic energy associated with a chemical process are comparable, in many cases, with those owed to solvation in solution. This subtle balance between changes in the electronic energy and the solvation free energy sometime causes an ‘irregularity’ of the order in stability among substituents as we have seen in the several examples. The solvent effect even reverses the equilibrium between a reactant and a product. In this regards, both the electronic structure theory (quantum mechanics) and RISM (statistical mechanics) contribute equally to describe the molecular processes in solution.

92


A self-consistent treatment and natural coupling between those methods are essential to reveal the nature of chemical reactions in solution.

Appendix: Appendix 1.

Partial Charge Treatment in RISM-SCF/MCSCF

In practical RISM-SCF/MCSCF computations partial charges assigned on interaction sites are the bridge between the RISM and ab initio MO theory. It is well known that the electrostatic potential generated by electron clouds of a molecule is very different from that by Mulliken gross charge. An appropriate set of partial charges is usually determined so as to reproduce the electrostatic potential around the solute with the least squares fitting procedure. In this appendix the detailed procedure is described by following the original paper by Ten-no et al. [4] The electrostatic potential at arbitrary point (r) near the solute can be divided into two different contributions, namely, the potential from the nucleus and that from the electron clouds. U (r) = UN (r) + Ue (r),

(2.A.1)

and UN (r) =

N A=1

ZA , |r − RA |

Ue (r) = −tr (PA(r)) ,

(2.A.2)

where P is the density matrix calculated from MO coefficients, N is the number of atoms in the solute and A(r) is three center one-electron integrals, ∗ χµ (r )χν (r ) dr , (2.A.3) (A(r))µν = |r − r | where χµ (r ) is a set of basis functions. Other notations used here have usual mean(N) (e) ings. The set of partial charge ({qα }, {qα }), which we would like to determine, induces electrostatic potential, Ñ (r) = U

n α=1

(N)

qα , |r − Rα |

˜e (r) = U

n α=1

(e)

qα , |r − Rα |

(2.A.4)

where n is the number of interaction sites. Note that n is not necessary to be the (N) (e) same as N . {qα }, {qα } are determined with the standard procedure minimizing target functions under the constraints of total charge conservation. ! l n (N) 2 ∂ ˜ wk UN (rk ) − UN (rk ) + 2λN qj − (constant) = 0, (2.A.5) (N) ∂qi j=1 k=1 ! l n (e) 2 ∂ ˜ wk Ue (rk ) − Ue (rk ) + 2λe qj − (constant) = 0, (2.A.6) (e) ∂qi j=1 k=1 where l is the numbers of grid points (rk ) where the electrostatic potential is evaluated (N) in, wk is proper weight and λ are Lagrange multipliers. The nuclear part, {qα }, is

93

Electronic structure andchemical reaction in solution given by ⎛ ⎜ ⎜ q(N) = ⎜ ⎜ ⎝

⎞

(N)

q1 (N) q2 .. . (N) qn

⎟ ⎟ ⎟ = a−1 a Z − λN a−1 1, ⎟ ⎠

(2.A.7)

where ⎛ l ( ) −1 −1 wk rkA rki , a Ai =

(a)ij =

k=1

l

−1 −1 wk rki rkj ,

and

k=1

⎜ ⎜ 1=⎜ ⎝

1 1 .. . 1

⎞ ⎟ ⎟ ⎟. ⎠

(2.A.8)

By using the total nuclear charge (NN = 1t · q(N) = 1t · Z) conservation, λN may be written as λN =

1t a−1 a Z − NN , 1t a−1 1

(2.A.9)

and q(N) = a−1 a Z −

1t a−1 a Z − NN −1 a 1. 1t a−1 1

(2.A.10)

It is noted that when the set of interaction sites is identical to that of nucleus, i.e. a = a , the equation is immediately reduced to the trivial equation, q(N) = Z. Similarly, the electronic part can be evaluated by q(e) = −a−1 tr (PB) −

−1t a−1 tr (PB) − Ne −1 a 1. 1t a−1 1

(2.A.11)

B is defined by (B)µ,ν,i =

l

−1 wk rki Aµν (Rk ),

(2.A.12)

k=1

and Ne = −tr (PS) is the total numbers of electrons. The modified fock operator (Eq.2.42) includes the interaction between the effective charges derived here (q(e) ) and the electrostatic potential. (V in Eq. 2.43) Thus the one-electron part of the solvated Fock matrix is given by,

a−1 1 T −1 solv t −1 1 a B−S , Hc = Hc − V · a B − T −1 (2.A.13) 1 a 1 where Hc is the one-electron integrals of the isolated solute molecule and S is overlap matrix. In the present RISM-SCF framework the free energy described above (Eq.(2.44)) is chosen to derive the solvated Fock operator (Eq.(2.A.13)). One might notice that other choices of the energy expression and its Fock operator are possible. The averaged potential energy is chosen instead of A in Eq. 2.44, for instance, Etotal = Eisolated + V · q(e) + ELennard−Jones ,

(2.A.14)

94


the one-electron part of solvated Fock matrix corresponding to the energy is given by,

∂V a−1 1 T −1 T (e) T −1 · a 1 = H − V + q · B − a B − S Hsolv . (2.A.15) c c ∂q 1T a−1 1 This is another type of solvated Fock operator in the combination method of RISM and ab initio MO theory. It should be noted that the first order derivatives of radial distribution functions with respect to the effective charges are required to construct the Fock matrix.

2. 2.1

Variational Principle in the RISM-SCF/MCSCF method RISM-MCSCF equation

We defined the total Helmholtz free energy A as the sum of the electronic energy of the solute molecule Esolute and the excess chemical potential coming from solute– solvent interaction ∆µ; A = Esolute + ∆µ.

(2.A.16)

Esolute can be estimated by ab initio electronic structure methods such as MCSCF ; Esolute

= = =

Enuc + Eel Za Zb 1 + γij hij + Γijkl (φi φj |φk φl ) Rab 2 ijkl ij a>b

Za Zb 1 + γij hij + Γijkl φi |gkl |φj , R 2 ab ij a>b ijkl

where γij and Γijkl are the vector coupling coefficients, IJ CI CJ γij , γij =

(2.A.17)

(2.A.18)

I,J

Γijkl

=

CI CJ ΓIJ ijkl ,

(2.A.19)

I,J

with the CI coefficient CI . hij and φi |gkl |φj are the one and two electron integrals defined in terms of the one particle orbital. Other notations used here have usual meanings. For the chemical potential term we have adopted the free energy derived from the hyper–netted chain (HNC) closure relation by Singer and Chandler [13],

1 1 ρ dr cαs − h2αs + hαs cαs ∆µ = − β α,s 2 2 ρ 1 = − dr e[−βuαs (r)+tαs (r)] − 1 − tαs (r) − hαs (r)tαs (r) + h2αs (r) β α,s 2 1 ˆ αs (k) cˆαs (k)ρh dk − − 3 (2π) β αs ⎫ ⎬ 1 cˆαs (k)ˆ cγs (k)ˆ ωαγ (k)χ ˆss (k) , + (2.A.20) ⎭ 2 α,γ,s,s

95


where the Greek subscripts refer to the interaction sites of solute molecules, and the Roman to the sites on the solvent. cαs and hαs are the direct and total correlation functions, respectively. χss is the pure solvent site density pair correlation functions and ωαγ the intramolecular correlation function containing the structural information of the solute. The solute–solvent interaction potentials uαs (r) are given as the sum of Coulombic and Lennard-Johns terms; σαs 12 σαs 6 qα qs uαs (r) = − + 4εαs . (2.A.21) r r r Note that the effective charges qα assigned on the solute sites depend on the solute electronic wave function in the present case; γij φi |bα |φj (2.A.22) qα (C, v, R) = ij

where bα is the population operator generates the charge qα on the site α. Although there can be several possibilities to define the operator bλ , one of the promising ways is the least squares fitting procedure to the electrostatic potential originated by the solute electron distribution described in the previous section. The quantity A can be regarded as a functional of the correlation functions hαs , cαs and tαs , as well as the MO coefficient vri and the CI coefficient CI . Imposing the constrains to the orthnormality of configuration state functions and one particle orbital, we have defined the following Lagrangian, 2 CI − 1 − εim (Sim − δim ). (2.A.23) L = A[c, h, t, v, C] − E i

I

m

Variations with respect to the functions yield + * ρ δL = − dr e[−βuαs (r)+tαs (r)] − 1 − hαs (r) δtαs (r) β α,s + {−tαs (r) + hαs (r) − cαs (r)} δhαs (r) ⎫ ⎧ ⎬ ⎨ −1 ωαγ ∗ cγs ∗ χss (r) δcαs (r) + −hαs (r) + ρ ⎭ ⎩ γ,s ! +2 HIJ CJ − ECI δCI I

+2

J

δφi |γij h +

i,j

− γij

λ

−εij |φj ,

∂ bλ ∂qλ

kl

Γijkl gkl

ρ dre[−βuαs (r)+tαs (r)] − β α,s

(2.A.24)

where all the quantities were taken to be real. The second curly bracket with the first one gives the HNC closure relation and the third one is referred to the RISM Ornstein–Zernike (OZ) equation. Now, we introduced the column vector V. The λth element of this vector (V)λ is the averaged

96


electrostatic potential acting on the solute site λ which comes from all the solvent charges; gλs (r) ρ ∂ [−βuαs (r)+tαs (r)] dr ≡ (V)λ . (2.A.25) dr = ρ qs e − ∂qλ β α,s r s The εim is the Lagrange mutiplier related to the MCSCF Fock operator Fij , εim = φi |γmj h + Γmjkl gkl − γmj Vt · b|φj j

=

kl

φi |Fmj |φj

(2.A.26)

j

and the CI matrix element is written as IJ 1 IJ γij φi |h − Vt · b|φj + Γijkl φi |gkl |φj , HIJ = 2 ij

(2.A.27)

ijkl

where b is the column vector with the component bλ . Thus we find that the variational condition for the free energy functional leads to the HNC approximation, the RISM equations and the modified MCSCF equations involving the solvent effect. Equation (2.A.23) is easily reduced to the case of CI and HF wavefunctions. If the variation with respect to the one particle orbital φi is dropped, the equations for the CI method are given. By omitting the term for the CI coefficients, the equation is reduced to the HF approximation. In this case the same solvated Fock operator introduced in the original RISM-SCF method is obtained.

2.2

Energy gradient

The first derivative of the free energy with respect to the nuclear coordinate of the solute molecule Ra is written as ∂Enuc ρ dhαs dtαs ∂A = − + {definition of t} dr[{HNC equation} ∂Ra ∂Ra β α,s dRa dRa dcαs +{RISM equation} ] dRa 1 ∂ω ˆ αγ (k) − cγs (k) χ ˆss (k) dkˆ cαs (k)ˆ 2(2π)3 β ∂Ra α,γ,s,s

1 ∂ φi |γij h + Γijkl gkl − γij Vt · b|φj . + ∂Ra ij 2 kl

(2.A.28)

Using the RISM-MCSCF variational conditions, Eq. (2.A.23), it is easy to show that the free energy gradient is reduced to ∂Enuc 1 ∂ω ˆ αγ (k) ∂A = − cγs (k) χ ˆss (k) dkˆ cαs (k)ˆ ∂Ra ∂Ra 2(2π)3 β ∂Ra α,γ,s,s

1 + + Γijkl (φi φj |φk φl )a 2 i,j i,j,k,l t a a −V q − εij Sij ,

γij haij

i

j

(2.A.29)

97

REFERENCES

where the second term of the right hand side corresponds to the change of solute– solvent distribution function due to the modification of intramolecular correlation. a are calculated by transforming the derivatives The MO integrals haij , (ij|kl)a and Sij of corresponding atomic orbital integrals. The derivatives of effective charges qα are defined as qa =

γij φi |

ij

∂b |φj . ∂Ra

(2.A.30)

With the least squares fitting procedure to the solute electrostatic potential, the electronic contribution q(e) to the solute effective charges can be written as q(e) = −a−1 tr(PB) − λe a−1 1,

(2.A.31)

where the matrices a and B and the Lagrange multiplier λe are defined in [4]. P is the density matrix with the element; IJ Prs = CI CJ γij vri vsj . (2.A.32) I,J

i,j

The fifth term appeared in Eq.(2.A.28) can be rewritten in the following form; t −1 ∂B ∂a−1 Va 1 t t 1 a−1 tr(P − V )− tr(PB) V t qa = t −1 1a 1 ∂Ra ∂Ra

t −1 1 a tr(PB) − tr(PS) ∂a−1 1 − ∂Ra 1t a−1 1 −

Vt a−1 1 ∂S ). tr(P 1t a−1 1 ∂Ra

(2.A.33)

The implementation of the RISM-MCSCF gradient method is rather simple. We only need to add the second and fifth terms to ordinary gradient packages in the electronic structure calculations.

References [1] ACS symposium series, 712 Combined Quantum Mechanical and Molecular Mechanical Methods; J. Gao and M. A. Thompson, Eds.; ACS, Washington DC, 1998 ; M. Orozco and F. J. Luque, Chem. Rev., 100, 4187 (2000). [2] J. Tomasi and M. Persico, Chem. Rev., 94, 2027 (1994); C. J. Cramer and D. G. Truhlar, Chem. Rev., 99, 2161 (1999). [3] J. Gao, Acc. Chem. Res., 29, 298 (1996). [4] F. Hirata, H. Sato, S. Ten-no and S. Kato, in [1] ; F. Hirata, H. Sato, S. Ten-no and S. Kato, ‘The RISM-SCF/MCSCF Approach for the Chemical Processes in Solutions’, in Computational Biochemistry and Biophysics; Oren M. Becker, Alexander D. MacKerell, Jr., ˆ Roux, Masakatsu Watanabe, Eds.; Marcel Dekker Inc., New Benoit York, 2001 .

98


[5] for example, see A. Szabo and N. S. Ostlund, ‘Modern Quantum Chemistry,’ Macmillan Publishing Co. Inc., 1982. [6] M. W. Wong, M. J. Frish and K. B. Wiberg, J. Am. Chem. Soc., 113 (4776,1991). [7] H. L. Friedman, Mol. Phys., 29, 1533 (1974). [8] M. Andersson and G. Karlstr¨ om, J. Phys. Chem., 89, 4957 (1985). [9] H. Sato and S. Kato, J. Mol. Struct.(THEOCHEM), 310, 67 (1994). [10] A. Klamt and G. Sch¨ uu ¨rmann, J. Chem. Soc., Perkin Trans. II, , 799 (1993). [11] S. Ten-no, F. Hirata and S. Kato, Chem. Phys. Lett., 214, 391 (1993); S. Ten-no, F. Hirata and S. Kato, J. Chem. Phys., 100, 7443 (1994). [12] H. Sato, F. Hirata and S. Kato, J. Chem. Phys., 105, 1546 (1996). [13] S. J. Singer and D. Chandler, Mol. Phys., 55, 621 (1985). [14] M Kawata, S Ten-no, S Kato and F Hirata, J. Am. Chem. Soc., 117, 1638 (1995); M Kawata, S Ten-no, S Kato and F Hirata, Chem. Phys., 203, 53 (1996). [15] H. Sato and F. Hirata, J. Am. Chem. Soc., 121, 3460 (1999). [16] H. Sato and F. Hirata, J. Phys. Chem. A, 102, 2603 (1998); H. Sato and F. Hirata, J. Phys. Chem. B, 103, 6596 (1999). [17] H. Sato and F. Hirata, J. Mol. Struct.(THEOCHEM), 461-462, 113 (1999). [18] T. Ishida, F. Hirata and S. Kato, J. Chem. Phys., 110, 3938 (1999). [19] K. Naka, H. Sato, A. Morita, F. Hirata and S. Kato, Theor. Chem. Acc., 102, 165 (1999). [20] Y. Harano, H. Sato and F. Hirata, J. Am. Chem. Soc., 122, 2289 (2000); Y. Harano, H. Sato and F. Hirata, Chem. Phys., 258, 151 (2000). [21] S.-H. Chong, S. Miura, G. Basu, F. Hirata, J. Phys. Chem., 99, 10526 (1995); S.-H. Chong, F. Hirata, Mol. Simulation, 16, 3 (1996); S.-H. Chong, F. Hirata, J. Chem. Phys., 106, 5225 (1997); S.-H. Chong, F. Hirata, Chem. Phys. Lett., 293, 119 (1998); R. Akiyama, M. Kinoshita, F. Hirata, Chem. Phys. Lett., 305, 251 (1999). [22] H. Sato and F. Hirata, J. Phys. Chem. A, 106, 2300 (2002). [23] T. Yamazaki, H. Sato and F. Hirata, Chem. Phys., 325, 668 (2000); T. Yamazaki, H. Sato and F. Hirata, J. Chem. Phys., 115, 8949 (2001). [24] M. Nakahara and C. Wakai,Chem. Lett.,809 (1992); M. Nakahara, DENKI KAGAKU (in Japanese), 62, 108 (1994).

REFERENCES

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[25] H. Sato, A. Kovalenko and F. Hirata, J. Chem. Phys., 112, 9463 (2000).

Chapter 3 CONFORMATIONAL STABILITY OF BIOMOLECULES IN SOLUTION Masahiro Kinoshita Institute of Advanced Energy Kyoto University [email protected]

1. 1.1

Combined RISM-MC approach for predicting peptide conformations Introduction

Biomolecules such as proteins play a variety of essential roles in biological systems. Those noble functions of proteins, however, are closely related to their tertiary structures (conformations). Proteins under their native physiological conditions spontaneously fold into unique conformations in the time scale of milliseconds to minutes. Although the conformations of proteins appear to be dependent on various environmental factors within the cell, it was shown by experiments invitro that unfolded proteins can refold back into their native conformations once the denaturants are removed, suggesting that the conformation of a protein may be determined solely by its amino-acid sequence [1, 2]. To elucidate the functions of natural proteins and, for instance, to design artificial proteins with improved or new functions, the relation between the amino-acid sequence and the conformation must fully be understood. The ultimate goal to be achieved is to develop a reliable approach that allows us to predict conformations of proteins from their amino-acid sequences. A theoretical method of investigating the proteinfolding problem may fall into three major categories. One of them is based on very simplified models (the so-called minimal models) of protein systems [3, 4, 5]. The second one utilizes the prediction from alignment of conformations of X-ray crystallographic data [6]. The third one employs

101

102


all-atom models of protein molecules and tries to make the prediction without relying on empirical factors, that is, from the first principle. One of the most striking facts is that the conformation is stabilized in solvent and effects due to the solvent are enormous and complicated. For example, as will be explained at later sections, a change of the solvent specifications (e.g., addition of salts, addition of alcohol) often causes a drastic change in the conformation stabilized. Although the theoretical method in the first category mentioned above is useful to understand the physical mechanism common in the folding of various proteins, it is not capable of capturing chemical details such as the solvent effects and the specificity of the protein conformation. The method in the second one may happen to give a successful result, but no physical or chemical aspects of the protein folding can be elucidated. We believe that the first-principle prediction in the third category is strongly desired for these reasons. There are, however, two severe difficulties in achieving the goal. First, a protein molecule comprises many atoms and the number of possible conformations is astronomically large. Second, the solvent effects must be incorporated to a full extent. To overcome the first difficulty, powerful conformational sampling methods are being developed. Currently, the Monte Carlo (MC) simulated annealing [7] and the generalized-ensemble algorithms [8] such as the multicanonical method and the replica-exchange method are known as very effective ones (see 1.2). The usefulness of these methods was demonstrated for problems of polypeptide conformation prediction in vacuum. As for the second difficulty, we propose the full reference interaction site model (RISM) theory. In the full RISM theory, the superposition approximation [9, 10] , in which the free energy of a peptide molecule is obtained via the sum of the potential of mean forces between pairs of atoms, is not employed. Moreover, the theory we adopt is the dielectrically consistent version developed by Perkyns and Pettitt [11, 12] . We developed a robust algorithm for solving the RISM equations[13, 14], which is over two orders of magnitude faster than the conventional one. We intend to overcome the second difficulty by the RISM theory with our algorithm. In this section, we briefly describe our hybrid approach for predicting conformations of biomolecules stabilized in solvent [15, 16, 17]. In this approach, the RISM theory is combined with the MC simulated annealing or one of the generalized-ensemble algorithms. Our approach falls into the third category, and both the biomolecule and the solvent are treated on the atomic level. The hybrid approach is implemented for two small peptides as the first step of the research. The solvent effects on the conformational stability of the peptides are discussed and the

Conformational stability of biomolecules in solution

103

important roles played by water are emphasized. The problems to be solved in further studies are also referred to at the end.

1.2

MC simulated annealing and generalized-ensemble algorithms [7, 8]

Despite the great advancement of computer technology in the past decades, simulations of complex systems such as biomolecules are still hampered by the multiple-minima problem. There are a huge number of states giving the energy function local minimum values. Moreover, the barrier to overcome to move from a local-minimum-energy state to an adjacent one is often significantly high. For this reason, the conventional simulation based on the canonical ensemble is no more useful: when the simulation is performed at a low temperature, it often gets trapped in one of the local-minimum-energy states; at a high temperature, on the other hand, it tends to sample only high-energy states having the high state density. The simplest method to avoid such a problem is the simulated annealing. In this method the simulation is started at a high temperature and then the temperature is gradually lowered. It has been shown that in finding out the global-minimum-energy state the simulated annealing is much more powerful than the conventional simulation performed at a constant temperature. Another way to overcome the problem mentioned above is to perform a simulation in a generalized ensemble. One of the most wellknown generalized-ensemble methods is the multicanonical algorithm where each state is weighted by a non-Boltzmann probability weight factor so that a random walk in the energy space may be realized. The random walk allows the simulation to overcome any energy barrier and to sample much wider phase space than by the conventional method. In the replica-exchange method, another generalized-ensemble method, sufficiently many noninteracting copies (or replicas) of the original system at different temperatures are simulated independently and simultaneously by the conventional method. Every few steps, pairs of replicas are exchanged with a specified transition probability. With this treatment a random walk in the energy space is nearly realized. By these generalized-ensemble algorithms one aims at obtaining not only the global-minimum-energy state but also the canonical-ensemble average of a physical quantity at an arbitrary temperature.

1.3

Definition of a new energy function

Let us consider a system comprising a biomolecule with m atoms and N solvent molecules. Each solvent molecule has n atoms, and nN solvent

104


atoms form the solvent. The absolute temperature and the volume of the system are T and V , respectively. In the usual computer simulations the energy function is the potential energy of the whole system. Here, however, we define a new energy function, the solvent-modified potential of the biomolecule. The configuration integral of the system Z can be written as (3.1) Z= exp {−β (EC + ECV + EV )} drV drC , where EC is the conformational energy of the biomolecule (the interaction energy of m atoms plus the sum of torsion energies), ECV the biomolecule-solvent interaction energy, EV the solvent-solvent interaction energy, and β is kB T (kB is the Boltzmann constant). The integration is performed over positions of the m biomolecule atoms and the nN solvent atoms (rV represents all the position vectors of nN solvent atoms and rC represents those of m biomolecule atoms). For an improbable configuration of the m biomolecule atoms EC is infinitely high. Likewise, an improbable configuration of the nN solvent atoms gives rise to an infinitely high value of EV . The ensemble-average of a physical quantity X(rV , rC ) (this represents the instant value for a system configuration) is given by 1 X (rV , rC ) exp {−β (EC + ECV + EV )} drV drC . (3.2) Xav = Z The configuration integral of the pure solvent of nN solvent atoms ZV is written as (3.3) ZV = exp (−βEV ) drV . Using Eq.(3.3), we rewrite Eq.(3.2) as Xav = I1 =

ZV I1 , Z

X (rV , rC ) exp {−β (EC + ECV + EV )} drV drC . exp (−βEV ) drV

(3.4) (3.5)

If X is determined solely by rC like the conformational energy of the biomolecule, the distance between the nitrogen atom at the N-terminus and the oxygen atom at the C-terminus, etc. (since we are mainly concerned with the biomolecule conformation in solvent, this is not a significant limitation), I1 can be arranged as X (rC ) exp (−βEC ) [ exp {−β (ECV + EV )} drV ]drC . (3.6) I1 = exp (−βEV ) drV

105


Notice that ECV is dependent on both rC , rV . Here, we consider I2 given by (3.7) I2 = exp {−β (EC + ECV + EV )} drV . I2 is the integration over positions of the nN solvent atoms with fixing positions of the m biomolecule atoms (i.e., with fixing the conformation of the biomolecule) and related to Fex by I2 = V nN exp (−βFex ) ,

(3.8)

where Fex is the excess free energy of the system comprising the biomolecule in a fixed conformation and the solvent. Fex should be distinguished from Aex , the usual definition of the excess free energy of the system where the conformation of the biomolecule is not fixed:

exp {−β (EC + ECV + EV )} drV drC =

I2 drC

=Z = V nN exp (−βAex ) .

(3.9)

Equating the right hand sides of Eqs. (3.7) and (3.8) gives exp {−β (ECV + EV )} drV = V nN exp {−β (Fex − EC )} . The denominator of Eq.(3.6) is related to Fex0 by exp (−βEV ) drV = V nN exp (−βFex0 ) ,

(3.10)

(3.11)

where Fex0 denotes the excess free energy of the pure solvent of nN atoms. It is convenient to define the solvation free energy of the biomolecule in a fixed conformation ∆µS that is given by ∆µS = Fex − EC − Fex0 .

(3.12)

It should be emphasized that ∆µS is dependent on rC and Eq.(3.12) differs from the usual definition of the solvation free energy where the conformation of the biomolecule is not fixed. Substituting Eqs. (3.10) and (3.11) into Eq.(3.6) and using Eq.(3.12) yield (3.13) I1 = X (rC ) exp {−β (EC + ∆µS )} drC . In a similar fashion, we obtain

106


1 ZV = . Z exp {−β (EC + ∆µS )} drC Substituting Eqs. (3.13) and (3.14) into Eq.(3.4) yields X (rC ) exp {−β (EC + ∆µS )} drC . Xav = exp {−β (EC + ∆µS )} drC

(3.14)

(3.15)

The exact equation Eq. (3.15) implies that the ensemble average of X (rC ) can be calculated using the integration over the conformation of the biomolecule alone when the energy function in the Boltzmann factor is taken to be the solvent-modified potential ET defined by ET = EC + ∆µS .

(3.16)

Hereafter, ET is referred to as the total energy (it is a free energy) of the biomolecule. For a biomolecule in a fixed conformation, we can consider the total energy defined as the sum of the conformational energy EC and the solvation free energy ∆µS . The three quantities vary from conformation to conformation. ET is equivalent to [Fex − Fex0 ] where Fex is the excess free energy of the system comprising the biomolecule in the fixed conformation and the solvent (the solvent is in equilibrium with the peptide molecule) and Fex0 is the excess free energy of pure solvent. The energy function in our approach, the combination of the RISM theory and the MC simulated annealing or a generalized ensemble algorithm, is ET while that in the usual simulations is EC +ECV +EV .

1.4

Combination of RISM and MC simulated annealing [15, 16]

In general, the behavior of a biomolecule in solvent is quite different from that in vacuum. To lower the conformational energy, the biomolecule itself tends to take a more ordered, compact conformation. From the standpoint of the conformational entropy, however, the biomolecule favors a more disordered conformation. The solvent, on the other hand, tries to force the biomolecule to take a conformation whose solvation free energy is as low as possible. The conformation stabilized in the solvent is determined from complicated competition of these factors. Hereafter, we limit our discussions to a peptide molecule that is more amenable to theoretical treatment. In the usual simulations, a peptide molecule and sufficiently many, surrounding solvent molecules are treated simultaneously. However, the number of the solvent molecules required is very large. Moreover, it increases in proportion to m3 (m is the number of the peptide atoms)


107

and becomes unacceptably large for a large peptide or a protein. If the solvent molecules are omitted, the number of degrees of freedom involved in the simulation is drastically reduced. This can be justified by accounting for the solvent effects in a theoretical manner. An important point is that the conformational stability is greatly sensitive to microscopic solvent environment, and any naive treatment of the solvent such as the continuum model will end in failure [18] (see also 2.1 and 3.1). Therefore, we employ a statistical-mechanical theory, the RISM theory, in which the solvent is also treated on the atomic level. In a preliminary prediction of a peptide conformation, the conformational entropy of the peptide is often neglected, and we follow this custom. Our aim is to determine the most stable conformation of a peptide molecule in solvent, and in this case the MC simulated annealing is a powerful tool. To obtain information on the average conformation by accounting for the conformational entropy, one needs to employ a generalized-ensemble algorithm. The combination of the RISM theory and the multi-canonical algorithm is considered in 1.8. For a peptide molecule in a fixed conformation, we consider the total energy ET defined in 1.3 as the sum of the conformational energy EC and the solvation free energy. The important quantities are not the absolute values of EC , ∆µS , and ET but the relative values among different conformations of the peptide molecule. The MC simulated annealing is applied only to the peptide molecule, and for a fixed conformation ∆µS is calculated using the RISM theory (the dielectrically consistent version [11, 12] ) with the hypernetted-chain (HNC) closures. The conformation that gives the lowest value of ET is chosen from among a number of conformations sampled. Our hybrid approach can be summarized as follows. (1) Prepare an initial conformation and calculate EC , ∆µS , and ET . (2) Sample a trial conformation and calculate the three quantities. (3) Reject or accept the trial conformation in accordance with the Metropolis criterion in which the energy function in the Boltzmann factor is ET . (4) Follow steps 2 and 3 and sample a number of conformations. (5) Choose the conformation with the lowest value of ET . Even if EC is higher for the trial conformation, it is accepted if the decrease in ∆µS dominates and ET is lower. We note that the RISM equations, a large set of stiff, nonlinear simultaneous equations, must be solved for all the conformations sampled. However, we employ our robust algorithm [13] that is over two orders of magnitude faster than the conventional one. As a result, the ensembleaveraged structure of the solvent, which is in equilibrium with the pep-

108


tide molecule in each conformation sampled, is efficiently calculated using the RISM-HNC theory to obtain ∆µS . As mentioned above, an MC scheme based on the simulated annealing technique or a generalizedensemble algorithm seems to be the most efficient in sampling the conformational space of the peptide molecule. In such a scheme, the peptide conformation often changes drastically. However, even a slight change is often unacceptable in the usual simulations performed for the whole system comprising a peptide molecule and many solvent molecules, because overlapping of the peptide molecule with many of the solvent molecules is unavoidable. (For this reason, in the usual simulations the MC scheme must be replaced by a molecular dynamics (MD) method where the state of the whole system is gradually changed.) With our approach we can mimic a peptide molecule surrounded by an infinite number of solvent molecules, and it is free from the problem of insufficient number of solvent molecules, which often arises in the usual simulations. Moreover, thanks to the RISM theory the long-range Coulomb potentials can be handled without any problem [13]. In the usual simulations, in which the instant value of the potential energy is the energy function, the absolute value of EV is much larger than that of EC or ECV because the number of solvent atoms is far larger than that of peptide atoms. EV varies much more with changes in the system configuration and the variation of the energy function is governed by that of EV with the result that the energy function is rather insensitive to a conformational change of the peptide molecule. This is particularly true for a large peptide or a protein requiring a great many solvent molecules. In our hybrid approach, on the other hand, both EC and ∆µS are strongly dependent on the peptide conformation. As mentioned in 1.7, when the solvent is water there is a set of conformations strongly rejected by water in which many of hydrophilic atoms are buried and many of hydrophobic atoms are well exposed. In the simulation based on the hybrid approach, the conformation rapidly changes in an initial stage toward the conformations rather acceptable by water (in which many of hydrophobic atoms are buried and many of the hydrophilic atoms are well exposed). For these conformations, the surface of ET is significantly more flattened than that of EC , and in water the barriers in the energy-function can be overcome far more readily than in vacuum. These advantages arise from the feature that the solvent is always in equilibrium with the peptide molecule in a fixed conformation, which is not the case in the usual computer simulations. In the latter a huge number of configurations of the solvent must be sampled to calculate the ensemble-averaged structure of water and ∆µS .


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The most important advantage of our hybrid approach is, however, the readiness of including ions in the solvent and elucidating the salt effects. The effects are enormous for peptides having many ”charged” residues. By the hybrid approach the structure of the ions near a peptide molecule can be calculated for a prescribed, finite salt concentration in the bulk (see Section 3 for more details). In contrast, the usual simulations are not capable of elucidating the salt effects because of the very limited number of the water molecules.

1.5

Conformational stability of Met-enkephalin [15, 18]

(a) Model and theory Let us consider Met-enkephalin (Tyr-Gly-Gly-Phe-Met) with 75 atomic sites in three different solvents including water. The model of a water molecule is the extended simple point charge (SPC/E) model [19]. It is assumed that the peptide is immersed in the solvent at infinite dilution. The potential-energy functions and parameters are adopted from KONF90 [20] that is based on ECEPP/2 [21]. The peptide and solvent molecules comprise the interaction sites (atoms). The interaction between the peptide molecule and a water molecule is expressed as the sum of the site-site pair interactions that have the form uab (r) = qa qb /r + 4εab (σab /r)12 − (σab /r)6 , a = 1, ..., m; b = H, O, (3.17) where subscripts a and b denote the interaction sites in the peptide molecule and those in a solvent molecule (for water b is H or O), respectively, m is the total number of interaction sites of the peptide molecule, qa and qb are the partial charges on site a and site b, respectively. The standard combination rule εab = (εa εb )1/2 , σab = (σa + σb )/2

(3.18)

is employed for calculating the Lennard-Jones (L-J) interaction parameters, σab and εab . In Eq.(3.18), εa and σa are the LJ parameters assigned to site a, and εb and σb are those assigned to site b. One of the other two solvents we consider is a simple, repulsive-potential system (a monoatomic fluid) interacting through ubb (r) = 4εb (σb /r)12 ,

(3.19)

where σb and εb are taken to be the diameter of a water molecule (0.28nm) and the ε-value for oxygen of the SPC/E water, respectively. The interaction between the solvent particle and an atomic site of the

110


peptide is also expressed as the form of Eq.(3.19) (i.e., only the repulsive part of the L-J potential is considered) with the combination rule of Eq.(3.18). The last solvent we consider is an L-J potential system where σb and εb are those for oxygen of the SPC/E water. The solvent particle interacts with an interaction site in the peptide molecule through the L-J potential. The dimensionless number density of the bulk solvent is taken to be that of water at the standard temperature (298K) and pressure (1atm). The conformational energy comprises Coulomb and L-J intramolecular energies and torsion energy. The site-site intermolecular correlation functions are calculated by solving the RISM-HNC equations and the solvation free energy of the peptide molecule in a fixed conformation ∆µS is calculated from the Singer-Chandler formula [22] given by ∆µS =

m

∆µSa ,

(3.20)

a=1

∆µSa = 4πkB T

n b=1

ρb

∞

r 2 {hab (r)}2 /2 − cab (r) − hab (r)cab (r)/2 dr,

0

(3.21) where ρb is the number density of atom b, hab (r) the intermolecular sitesite total correlation function, cab (r) the intermoleculer site-site direct correlation function, and n the total number of the atoms in a solvent molecule. We refer to ∆µSa (the contribution to ∆µS from atom a ) simply as the solvation free energy of atom a. We remark that ∆µSa is dependent not only on the solvophilicity or the solvophobicity of atom a but also on the microscopic environment where atom a is situated. For instance, ∆µSa becomes higher as atom a is less exposed. The basic equations and our algorithm for numerically solving them, a hybrid of the Newton-Raphson method and the Picard iteration strategy, are described in the Appendix. (b) Solvent effects First, we test four different cases to study the solvent effects. In case 1 the solvent is the simple, repulsive-potential system described above. In case 2 the solvent is the L-J potential system. The solvents treated in cases 1 and 2 can be simple models of non-polar solvents. In case 3 the solvent is the SPC/E water but all the site-charges of Met-enkephalin are set at zero to shut off the electrostatic interaction between Metenkephalin and the solvent. In case 4 the solvent is the SPC/E water and the full values are assigned to the site-charges. It is assumed that Met-enkephalin is un-ionized even in case 4, because we have found


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that the effects due to the ionization are not large. When it is ionized and present in the zwitterionic form in case 4, the ∆µS -values themselves change much but the relative values among different conformations, which are much more important, do not significantly change. As for the salt effects, we have found the following: Although the ∆µS values increase slightly with salt addition to water, the relative values among different conformations exhibit no appreciable change, and the qualitative aspects of our conclusions are not altered by salt addition. We note, however, that these results are not always applicable to other peptides (see 1.6(c)). We checked the ∆µS -values of many different, probable conformations of the peptide molecule, which were obtained during the conformational sampling explained in (c). Our finding is as follows. In simple fluids of cases 1 and 2 ∆µS does not vary as much as the conformational energy EC against conformational changes, which implies that the conformational stability is governed by EC . In water of cases 3 and 4, however, ∆µS varies as much as EC or even more, which suggests that ∆µS is very important in stabilizing peptide conformations. This is particularly true in case 4. When atoms with comparatively large site-charges are more exposed to water, for example, ∆µS becomes lower. Thus, it is easy to understand the largest variation in ∆µS in case 4. However, the variation is significantly large even in case 3 where there is no electrostatic interaction between the peptide and the solvent. Unlike non-polar solvents, the structure of water is ordered and orientational due to the hydrogen-bonding network. Therefore, presumably, when a peptide molecule intrudes into water, details of the disturbance caused are considerably dependent on the peptide conformation. In the scaled particle theory [23] for incorporating the solvent effects, water molecules are simply treated as sufficiently small hard spheres that are similar to the simple, repulsive-potential particles considered in our study. This theory assumes that the characteristic differences between a non-polar solvent and water are not due to the water structure but due to the comparatively small size of water molecules, and has been successful for nearly spherical solutes. However, our result is suggestive that such an assumption is not applicable to complicated molecules such as peptides that can take a variety of conformations. (c) Implementation of hybrid approach Our approach, a combination of the RISM theory and the MC simulated annealing, is applied to Met-enkephalin. The scheme for the MC simulated annealing is the same as that employed in vacuum cases except that EC is replaced by the total energy ET . The initial and final tem-

112


peratures for the conformational sampling are set at 500K and 300K, respectively. The temperature is decreased exponentially with the MC sweeps. (Notice that ∆µS is always calculated at 298K.) We perform 8 runs starting from 5 different initial conformations (including both compact and extended ones) of Met-enkephalin in vacuum, case 1, and case 4. In each run, 100-4000 MC sweeps are tested for vacuum and case 1, while only 100-400 MC sweeps are employed for case 4. One MC sweep updates 19 torsion angles in Met-enkephalin and 19 different conformations are sampled per MC sweep. In each run, a conformation is obtained as the one with the lowest value of ET among the conformations sampled. (Notice that it is not always the true lowest-total energy conformation.) That is, 8 different conformations are obtained in each of the three cases. The results for case 1 can be summarized as follows. Since EC is more important than ∆µS in stabilizing the peptide conformations, the observations in case 1 are similar to those in vacuum. It was already shown that in vacuum about 100000 MC sweeps were required to find the lowest-energy conformation. Probably, this is also true in case 1. The lowest-energy conformation in vacuum is still one of the most stable conformations. The 8 conformations obtained are significantly different from one another. The maximum difference in their total energies is 11.8kcal/mol. Thus, the numbers of the total MC sweeps employed are too small to obtain well-converged, low-total energy conformations. The result for case 4 is totally different from that for case 1. The numbers of the total MC sweeps, 100 to 200, are sufficiently large as long as a conformation without bad van der Waals contacts are chosen as the initial one: The maximum difference in the total energy among the conformations is only 1.4kcal/mol. A set of well-converged, low-total energy conformations can be found in far smaller MC sweeps than in vacuum and in simple fluids. We superpose the 8 conformations obtained. The superposition is performed so that backbone structures are best fit in terms of the rootmean-square distance (RMSD). The results in vacuum, case 1, and case 4 are compared in Fig. 3-1. The nuclear magnetic resonance (NMR) experimental result [24] is also included in the figure. It is observed that the conformations in water are much better converged than in vacuum and in simple fluids. In water there are a set of different conformations of Met-enkephalin having almost the same total energies, nearly the lowest values. These conformations exhibit characteristics of almost fully extended backbone structure with significantly large fluctuations in side-chain structure. The RMSD for the 8 conformations obtained in water is 0.11nm-0.24nm and that for the NMR conformations is 0.12nm-


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0.22nm. It can be concluded that our result obtained in water is in good agreement with the NMR experimental observation.

Figure 3.1. Superposition of the eight conformations of Met-enkephalin obtained as the lowest-energy conformations in vacuum (a), in case 1 (b), and in case 4 (c). The NMR result is shown in (d).

As for our algorithm for solving the RISM equations, it was found that the Jacobian matrix calculated for a single conformation and used as part of the input data could be adopted throughout the conformational sampling in all the runs tested, leading to considerable saving of computation time. (We note that the construction and inversion of the matrix forms the most time consuming part.) The RISM equations had to be solved a number of times, but this was performed with moderate computational effort on a workstation. The algorithm never failed to give convergence.

1.6

Conformational stability of C-peptide [16]

(a) Model and theory We then extend our study to the C-peptide fragment (Lys-Glu-ThrAla-Ala-Ala-Lys-Phe-Leu-Arg-Gln-His-Met) of ribonuclease A with 221 atomic sites. The most important feature which is not inherent in Metenkephalin is that there are four positively charged groups in the side chains of lysine-1, lysine-7, arginine-10, and histidine-12, and there is one negatively charged group in the side chain of glutamic acid-2. We refer to these residues as Lys-1+ , Lys-7+ . Arg-10+ , His-12+ , and Glu-2− , respectively. The potential-energy functions and parameters are adopted from KONF90 [20] that is based on ECEPP/2 [21] . The peptide is immersed in the SPC/E water at infinite dilution. The interaction between the peptide molecule and a water molecule is expressed as Eq.(3.17), coupled with Eq.(3.18). Other details are described in 1.5(a). (b) Water effects We decompose the solvation free energy ∆µS into two terms. One of them is ∆µS0 , the solvation free energy in the case where all the sitecharges of the peptide are set at zero, which can be regarded as the con-

114


tribution from the hydrophobic hydration. The other is ∆µSE defined as [∆µS − ∆µS0 ] that can be regarded as the contribution from the electrostatic interaction with water (∆µS = ∆µS0 +∆µSE ). The conformational energy EC can be decomposed into two terms, EC0 , the intramolecular van der Waals energy plus torsion energy, and ECE , the intramolecular Coulombic (electrostatic) energy (EC = EC0 + ECE ). Hereafter, we refer to ∆µSE and ECE as the electrostatic terms, and ∆µS0 and EC0 the non-electrostatic terms. We examined ∆µS and EC for a number of different, probable conformations of Met-enkephalin and C-peptide and observed the following. The variation in the non-electrostatic terms among different conformations for C-peptide is as large as that for Met-enkephalin, but the variation in the electrostatic terms for C-peptide is far larger. For C-peptide, the variation in the electrostatic terms is much larger than that in the non-electrostatic terms, while the opposite is true for Met-enkephalin. The conformational stability is governed by the electrostatic terms for C-peptide and by the non-electrostatic terms for Met-enkephalin. A significant finding (particularly for C-peptide) is that the electrostatic term of the solvation free energy ∆µSE and the electrostatic term of the conformational energy ECE are opposing. Suppose well-exposed, likecharged atoms get closer together, for example. Due to the increased local charge density, these atoms form stronger electrostatic bonding with water molecules and ∆µSE becomes much lower. However, ECE becomes much higher due to the intramolecular Coulombic repulsion among the like-charged atoms. Thus, when ∆µSE becomes lower, ECE tends to become higher. When well-exposed, unlike-charged atoms get closer together, on the other hand, ∆µSE becomes higher with a decrease in ECE . (c) Implementation of hybrid approach Next, we briefly summarize the result of applying our approach, in which the RISM theory is combined with the MC simulated annealing, to Cpeptide in water. The initial conformation is an almost fully extended one (details of this conformation are described in 2.3). In the initial stage, the conformation changes in the direction that the charged groups in the side chains are more exposed to water and the positively charged groups are closer together, and the decrease in ∆µSE dominates. In the later stage, however, further decrease in ∆µSE causes even larger increase in the intramolecular Coulombic repulsion energy, and the conformational change is greatly decelerated. The conformations are thus stabilized quite rapidly, and 100-200 MC sweeps are sufficiently many to attain the stabilization. (One MC sweep updates all of the 64 torsion


115

angles.) The stabilized conformation is shown in Fig. 3-2. The two positively charged groups in side chains of Lys-1+ and Lys-7+ are close together, the two positively charged groups in side chains of Arg-10+ and His-12+ are also close together, and the negatively charged group in side chain of Glu-2− is exposed to water very well. The total energy of this stabilized conformation is 65kcal/mol lower than that of the initial conformation. However, the non-electrostatic terms are about the same, indicating that the stabilization occurs due to the change in the electrostatic terms.

Figure 3.2.

Conformation of C-peptide stabilized in water.

We believe that addition of salts to water will alter the peptide behavior to a large extent, because the charged groups in the side chains are screened by counterions. Behavior of the zwitterionic peptide could be significantly different from that of the un-ionized one, since the charged N- and C-termini can interact with the charged groups of the side chains. We emphasize that these are not true in the case of Met-enkephalin.

1.7

Roles of water [15, 16, 25]

Despite the difference between Met-enkephalin and C-peptide in the conformational stabilization process, both of the two peptides are sta-

116


bilized far more rapidly (i.e., in much fewer MC sweeps) in water than in vacuum. In general, for peptides and proteins there is a set of conformations with high values of ∆µS , in which many of hydrophilic atoms (atoms with high values of site-charges) are buried, many of hydrophobic atoms are well exposed, and the excluded volume is relatively larger. Such conformations are strongly rejected by water and the number of probable conformations is drastically reduced. Moreover, water plays another important role. Here, let us consider the probable conformations that are rather acceptable by water. In these conformations many of hydrophobic atoms are buried, many of hydrophilic atoms are well exposed. Some of these conformations have relatively high values of EC and others have relatively low values. In conformations with relatively high values of EC , like-charged atoms are often close together, there are very few intramolecular hydrogen bonds (contacts of unlike-charged atoms), and so on. In such cases, charged atoms form electrostatic bonding with the water hydrogen atoms or oxygen atoms, and ∆µS is relatively low. In conformations with relatively low values of EC , on the other hand, like-charged atoms are often far apart, there are intramolecular hydrogen bonds, and so on, which leads to relatively high values of ∆µS . Thus, there is a general trend that ∆µS is high when EC is low and ∆µS is low when EC is high. (This is particularly true for C-peptide.) As a result, water flattens the total energy surface, decreasing the total energy difference between conformations that differed by large energies on the vacuum surface and lowering the barriers separating those conformations. This leads to the rapid stabilization of the peptide conformations in water. This is good news, because the number of conformations to be sampled in our hybrid approach is drastically reduced in water, which would remove the computational bottlenecks expected for larger polypeptides and proteins. In the usual computer simulations, the trend that ECV +EV is high when EC is low and ECV +EV is low when EC is high, is no more present (notice that these quantities are not the ensemble-averaged values but the instant ones). However, during sufficiently many steps, the peptide molecule can change its conformation more readily in water than in vacuum. The same is true in the real systems, and peptide conformations are stabilized quite rapidly due to the presence of water.

1.8

Combination of RISM and multicanonical algorithm [17]

In usual simulations, the energy function in the Boltzmann factor is the potential energy of the system comprising a peptide molecule and


117

surrounding solvent molecules, that is, the sum of the conformational energy of the peptide molecule EC , the peptide-solvent interaction energy ECV , and the solvent-solvent interaction energy. The ensemble-averaged value of a physical quantity is obtained after sampling the configurational space of the system. In our approach, on the other hand, the energy function is the total energy ET (ET = EC + ∆µS ) that is determined as a function of the peptide conformation only. The average of a physical quantity is taken over conformations of the peptide molecule sampled. As shown in 1.3 the two averaged values of the physical quantity obtained via the two different routes are equivalent as long as the instant value of the quantity pertains to the peptide conformation alone (like the conformational energy and the end-to-end distance defined below) and the sampling is sufficiently broad. Notice that effects due to the conformational entropy are incorporated in these treatments. In general, with the canonical algorithm at a low temperature, the sampling is often trapped in a local minimum of the energy-function surface because of difficulties in overcoming a barrier. When employed at a high temperature, however, the algorithm only samples conformations with high values of the energy function. In contrast, the multicanonical algorithm is a powerful tool in realizing a random walk in the energyfunction space, leading to broad sampling of the peptide conformations. Moreover, the averaged conformations are calculated at various temperatures from a single set of sampling. (In our hybrid approach, since the temperature for calculating ∆µS is fixed at 298K, the results for the other temperatures are only approximate.) It is worthwhile to recall that with the hybrid approach the barriers in the energy function are much easier to overcome in water than in vacuum. Hence, a sampling algorithm developed through simulations in vacuum will certainly be powerful enough when it is applied to simulations in water. In fact, for a small peptide like Met-enkephalin, even a conventional canonicalensemble simulation performed at 298K gives successful results (we have actually verified this). The approach, the combination of the multicanonical algorithm and the RISM theory, was implemented for Met-enkephalin. It was demonstrated that the multi-canonical simulation is capable of exploring a much broader configurational space than the canonical simulation. We chose 8 representative conformations from those in which ET is lower than 174kcal/mol and the RMSD of the backbone atoms from Gly-2 to Phe-4 are larger than 0.04nm. In Fig. 3-3, they are compared with the 8 conformations obtained from the combination of the RISM theory and the simulated annealing [15] . The lowest-energy conformation in vacuum [26] is also included in the figure. The results for the two com-

118


bined approaches are quite similar to each other. The total energy of the lowest-total energy conformation found in the multicanonical simulation is 172kcal/mol. This is even lower than that (176kcal/mol) of the lowest-total energy conformation obtained in the simulated annealing. We remark that our algorithm for solving the RISM equations never failed to give convergence with a constant Jacobian matrix for about 200000 consecutive evaluations of ∆µS .

Figure 3.3. The lowest-energy conformation of Met-enkephalin obtained in vacuum (the top left). Superposition of the eight lowest-energy conformations obtained from the combination of the RISM theory and the simulated annealing (the top right) and from the multicanonical algorithm combined with the RISM theory (the bottom).

The lowest-energy conformation determined in vacuum has two intramolecular hydrogen bonds and is rather compact. The end-to-end distance (the distance between the nitrogen atom at the N-terminus and


119

the oxygen atom at the C-terminus) of this conformation is 0.48nm. For the lowest-total energy conformation obtained in water, on the other hand, it is 1.17nm in the multi-canonical simulation and 1.36nm in the simulated annealing. They are quite extended due to hydrogen bonds between the peptide and water molecules. The end-to-end distance of the peptide molecule in vacuum exhibits a coil-globule-like transition with increasing temperature. The average value of the distance at 800K in vacuum is ∼1.08nm that is still shorter than that at 298K in water, ∼1.23nm.

1.9

Three-dimensional RISM theory [27, 28, 29]

In principle, the three-dimensional RISM (3D-RISM) theory described in Chapter 4 is significantly more accurate than the RISM theory employed so far which can be distinguished from the 3D-RISM theory by calling it the one-dimensional RISM (1D-RISM) theory. This is because the 3D-RISM theory, in contrast to the 1D-RISM theory, takes orientational average of the molecular Ornstein-Zernike (OZ) equation for solvent molecules only, keeping full description of the shape and orientation of the solute molecule. In reality, a solvent site cannot access to a completely buried atom in the solute molecule. Even if uab (r) is attractive at separation r, site b cannot approach site a if the repulsive core of site a overlaps that of another site in the solute molecule. Though these effects are taken into consideration in the 3D-RISM/HNC (the 3D-RISM theory with the HNC closures), they are not sufficiently incorporated in the 1D-RISM theory with the site-site HNC closures containing no cross-terms (i.e., the 1D-RISM/HNC). As a result, the 1D-RISM/HNC underestimates the excluded volume, particularly for large solute molecules and those with compact shapes comprising many interaction sites with overlapping repulsive cores. This underestimation results in too small values of the partial molar volume of a solute. (The partial molar volume is discussed in Section 4.) Another problem of the 1D-RISM/HNC is that it overestimates the solvophobicity of a solute. When the solvent is water, for example, water molecules are readily oriented with hydrogen atoms towards the hydrophobic solute molecule, which unreasonably enhances hydrogen bonding and causes too strong ordering in the hydration structure. The ordering gives rise to substantial overestimation of the entropic component in the solvation free energy. This overestimation predominates over the underestimation of the excluded volume for a hydrophobic solute, and the 1D-RISM/HNC gives too high a value of the solvation free energy. The 3D-RISM theory with the repulsive bridge cor-

120


rection (3D-RISM/HNC+RBC) accounting for the solvent shape and orientation, gives a significantly more accurate value of the solvation free energy. Even with the 1D-RISM/HNC, however, the relative values calculated among different conformations of the same solute molecule are reasonably accurate. The ∆µS -values of Met-enkephalin in five different conformations calculated by the 1D-RISM/HNC and by the 3DRISM/HNC+RBC are compared in Table 3-1. The values relative to ∆µS of conformation 1 are also given in the table, which indicates that the qualitative aspects of the conclusions are not altered by employment of the 3D-RISM/HNC+RBC. It should be noted that the conformational stability of a peptide molecule is governed not by the absolute values of ∆µS but by the relative values. We have verified that this is also true for C-peptide: ∆µS for the extended conformation is significantly lower than for the conformation with the α-helix structure.

Table 3.1. Total Energies (kcal/ml) of Five Conformations of Met-enkephalin in Ambient Water, Following from the 1D- and 3D-RISM/HNC Theories and from those with the RBC(Repulsive Bridge Correction)-TPT(Thermodynamic Perturbation Theory) Treatment

Conf. (1) (2) (3) (4) (5)

Econf -12.0 12.0 -2.5 0.8 9.7

1D-RISM/ HNC

3D-RISM/ HNC

µsolv 196.6 178.0 202.7 176.8 164.0

µsolv 84.0 72.5 79.0 80.8 71.3

Etot 184.8 190.2 200.2 177.6 173.7

Etot 72.0 84.5 76.5 81.6 81.0

1D-RISM/ 3D-RISM/ HNC+RBC-TPT HNC+RBC-TPT µsolv 15.7 -2.0 24.0 -5.9 -22.7

Etot 3.7 10.0 21.5 -5.1 -13.0

µsolv 40.6 26.4 36.6 30.7 18.8

Etot 28.6 38.4 34.1 31.5 28.5

Unfortunately, our experience has shown that the conformational stability predicted by the 1D-RISM/HNC is no more reliable for larger peptides and proteins with compact shapes. An important point is that the 3D-RISM theory is far more computationally demanding than the 1D-RISM theory and difficult to be applied to very large proteins. In the approach where the RISM theory is combined with the generalized ensemble technique, it is almost impossible to employ the 3D version even for small peptides. Therefore, it is of vital importance to improve the 1D-RISM/HNC theory. Works in these directions are in progress in our group and it is becoming clearer that the modification of the site-site closure equations, the inclusion of appropriate repulsive bridge functions in them, can provide remarkable improvement over the 1D-RISM/HNC (see the last paragraph of 4.3).


1.10

121

Concluding remarks

We have described our hybrid approach, a combination of the RISM theory and a generalized-ensemble technique, for predicting conformations of a peptide molecule stabilized in solvent. It has several advantages that are not inherent in the usual simulations treating the peptide molecule and surrounding solvent molecules simultaneously. Its powerfulness has been demonstrated for Met-enkephalin with 5 residues and C-peptide with 13 residues. Water is clearly distinguished from other solvents in that the solvation free energy of a peptide molecule in water varies to a remarkable extent from conformation to conformation. The conformations with high values of the solvation free energy are strongly rejected by water and the number of probable conformations is drastically reduced in water. Moreover, for conformations acceptable by water (in which many of hydrophobic atoms are buried and many of hydrophilic atoms are well exposed), the surface of the total energy defined by Eq. (3.16) is significantly flattened by water, lowering the energy barriers between local-minimum-energy conformations. In simulations with the hybrid approach, stabilized conformations can be reached in much fewer MC sweeps in water than in vacuum. Also in real systems, water should play essential roles in stabilizing the particular conformations of peptides quite rapidly. In the next stage, we intend to combine the RISM theory with the replica-exchange method [8] that seems to be best suited to the hybrid approach and to apply it to larger peptides and proteins. Thanks to the flattened surface of the energy function, the highest temperature for the replica-exchange method can be set at a sufficiently low value. Furthermore, our recent studies have shown that the temperature difference between two adjacent replicas can be larger in water than in vacuum and that the number of replicas required is considerably reduced. This is in marked contrast to the usual computer simulations where the number increases to a remarkable extent when the simulation is performed in water. The most important task is probably the improvement of the RISM theory: it is to be improved so that it can give reliable results even for much larger peptides and proteins with compact shapes. In the second stage, we intend to incorporate the salt effects in the hybrid approach.

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2. 2.1


Alcohol effects on peptide conformations Introduction

Conformational transitions in protein molecules, especially those of the secondary structures, are very important aspects in protein folding. An example of great interest is the conversion into non-native β-sheet structures in proteins that cause amyloid diseases [30, 31, 32] . Another example is the formation of α-helix structures in the early stage of folding of β-lactoglobulin, the native structure of which is mostly in the β-sheet [33, 34, 35] . Conformations of protein molecules are greatly influenced by the solvent environments. In Section 1, the solvent effects were analyzed by treating small peptide molecules in water. A significant finding is that in water the solvation free energy ∆µS for a peptide molecule varies largely from conformation to conformation and remarkably affects the conformational stability. In fact, the peptide conformations stabilized in water are quite different from those in gas phase. Moreover, addition of salts (e.g., NaCl) to water may alter the conformations to a large extent. Effects of alcohol on peptide and protein conformations [36, 37, 38, 39, 40, 41, 42, 43] are very interesting from the standpoints of both the conformational transitions and the solvent effects mentioned above. Melittin and some fragments of β-lactoglobulin, for instance, take extended (unfolded) conformations in aqueous environments, but when alcohol is added, they turn into α-helix structures. Even when alcohol is further added, the α-helix structure remains until the solvent becomes almost pure alcohol. Thus, alcohol induces peptides and proteins to form α-helix structures significantly more than water. Though the αhelix structure formed is independent of the alcohol species, degree of the induction increases as bulkiness of the hydrocarbon group in an alcohol molecule increases. However, the mechanism of these alcohol effects has long been unknown. The solvent is often treated as a continuum with a particular dielectric constant. This continuum model is, however, not capable of describing the solvent effects. For example, acetonitril and methanol share about the same dielectric constant, but they exhibit fairly different solvation properties for many solutes. When the continuum model is adopted, the solvent has no effects except that the Coulombic interaction between atoms in a solute molecule is simply divided by the dielectric constant. Notice, however, that the division is valid only when the distance between the atoms is far larger than the atomic scale. The solvent effects are sensitively dependent on the conformation of the solvent molecule as well as that of the solute molecule and on the density and ori-


123

entational structure of the solvent near the solute molecule. One might think the following way based on the continuum model of the solvent. Since the dielectric constant of alcohol is lower than that of water, the Coulombic attraction between the amido hydrogen and the carbonyl oxygen in a peptide molecule is stronger in alcohol, with the result that the intramolecular hydrogen bonding is promoted and formation of the α-helix structure is induced. We remark, however, the alcohol effects are not this simple. For example, the dielectric constant of acetonitril is significantly larger than that of ethanol, but the former induces a peptide molecule more to form the α-helix structure. What is more remarkable is that the dielectric constant of trifluoroethanol (TFE) greatly inducing a peptide molecule to form the α-helix structure is only slightly larger than that of ethanol. In this section, we analyze peptide conformations in methanol and ethanol using the RISM-HNC theory. Met-enkephalin and C-peptide, which were considered in Section 1, are chosen in the analyses. Molecular models are employed for methanol and ethanol. The solvent structures near peptide molecules in different conformations and the ∆µS -values are calculated, and the results obtained are compared with those previously obtained for the peptides in water, to elucidate the microscopic origin of the interesting alcohol effects.

2.2

Conformational stability of Met-enkephalin in water, methanol, and ethanol [44]

We consider four different conformations, conformations 1 through 4, for Met-enkephalin. Conformation 1 is the lowest-energy conformation in vacuum and has intramolecular hydrogen bonds. Conformation 4 is one of the conformations stabilized in water and almost fully extended. It has no intramolecular hydrogen bonds. Conformations 1 and 4 are shown in Fig. 3-4. For methanol and ethanol molecules, we employ the optimized potentials for liquid simulations (OPLS) proposed by Jorgensen [45] . The OPLS parameters are compared with the parameters for the SPC/E water in Table 3-2. The temperature is set at the standard temperature. The dimensionless number densities ρV d3 (d = 0.28nm) of water, methanol, and ethanol are 0.7317, 0.3246, and 0.2265, respectively. ”CH3 ” and ”CH2 ” are regarded as single atomic sites, and n = 3 and n = 4 (n is the number of the atomic sites) for methanol and ethanol molecules, respectively. It is assumed that all the ethanol molecules take the trans conformations. An important point is that the number density of water is 2.3 times higher than that of methanol (the number density of hydrogen atoms in water is 4.6 times higher than in methanol), and that

124


of methanol is 1.4 times higher than that of ethanol. Alcohol molecules are larger than water molecules, and this trend is enhanced as bulkiness of the hydrocarbon group in an alcohol molecule increases. Table 3.2. atom water H O methanol H O CH3 ethanol H O CH2 CH3

Potential Parameters Employed σ(nm) 0.0400 0.3160 0.0400 0.3070 0.3775 0.0400 0.3070 0.3905 0.3905

(kcal/mol) 0.046 0.156 0.055 0.170 0.207 0.055 0.170 0.118 0.175

q(-) 0.4238 -0.8476 0.4350 -0.7000 0.2650 0.4350 -0.7000 0.2650 0.0000

Table 3.3. Solvation Free Energies (kcal/mol) for Conformations 1 through 4 of Metenkephalin in Water and Methanol conformation water water-0a methanol methanol-0a 1 197 216 73 87 2 178 209 58 82 3 203 229 75 93 4 177 201 59 77 a ”-0” implies that all the site charges of Met-enkephalin are set to zero.

Table 3-3 gives the ∆µS -values for conformations 1 through 4 of Metenkephalin in water and methanol. The values of the conformational energy EC of the four conformations are -12, 12, -3, and 1kcal/mol, respectively. ”Dash zero” implies that all the site-charges of Met-enkephalin are set at zero to shut off the electrostatic interaction between the peptide and the solvent (i.e., to make the peptide molecule completely hydrophobic). The absolute values of ∆µS in methanol are much smaller than in water. Even when all the site-charges of the peptide are removed, the increase of ∆µS in methanol is significantly less than in water. These trends are enhanced when methanol is replaced by ethanol. For example, the ∆µS -values for conformation 4 of Met-enkephalin in ethanol are 51kcal/mol (in the case of full site-charges) and 67kcal/mol (in the case of zero site-charges). The most important feature observed from the table is that in methanol the ∆µS -values vary considerably less against conformational changes than in water. The maximum differences among the four conformations in ∆µS for the cases of ”Water”, ”Water-0”, ”Methanol”, and ”Methanol-0” are 26, 28, 17, and 16kcal/mol, respect-


125

Figure 3.4. Two conformations of Met-enkephalin considered, conformation 1 (the top) and conformation 4 (the bottom).

126


ively. We emphasize that the conformational stability of a peptide molecule in solvent is governed not by the absolute values of ∆µS but by the relative values among different conformations. Table 3-4 gives the solvation free energies ∆µSa for some individual atoms (the sum of ∆µSa (a = 1, ..., m) constitutes ∆µS of the peptide molecule; see Eq.(3.21)) of Met-enkephalin in water, methanol, and ethanol. Met-enkephalin is in conformation 4 shown in Fig. 3-4. The values calculated with the site-charges set at zero are also included. For the typical hydrophilic atoms with large, negative site-charges in the backbone, ”23 N” in Gly-2 and ”56 O” in Phe-4, the values are negative in water, but they become higher in methanol, and even higher in ethanol. For ”23 N” that is less exposed than ”56 O”, the values in alcohol are positive. For the typical hydrophobic atoms in the side chains, ”43 CD1” and ”47 CZ” in Phe-4, the values are positive in water, but they become lower in methanol, and even lower in ethanol. (The partial charges of ”23 N”, ”56 O”, ”43 CD1”, and ”47 CZ” are -0.356, -0.384, -0.015, and 0.015, respectively.) A similar feature is observed for the hydrophobic atoms with zero site-charges except ”23 N”. For ”23 N” with zero site-charge, the value in ethanol is higher than in methanol. Table 3.4. Solvation Free Energies (kcal/mol) for Some Individual Atomsa of Metenkephalin in Conformation 4: Values in Water, Methanol, and Ethanol atom water methanol ethanol water-0b methanol-0b 23 N Gly-2 -1.62 0.96 1.41 3.41 2.22 56 O Phe-4 -5.98 -3.56 -3.21 2.12 -0.24 43 CD1 Phe-4 1.91 -0.23 -0.76 4.68 2.03 47 CZ Phe-4 2.21 -0.35 -1.00 1.71 -0.60 a For the definition of the solvation free energy for an atom, see Eq.(3.21). b ”-0” implies that all the site charges of Met-enkephalin are set to zero.

ethanol-0b 2.48 -0.72 1.93 -1.14

The pair distribution functions gab (r)(b = H, O) have apparent, main peaks when atom a has a large, negative charge, representing the formation of hydrogen bonding between the peptide atom and the solvent oxygen. The functions for a=”23 N” and ”56 O” and b is H and O in the solvent molecule are shown in Figs. 3-5 and 3-6. The main peak is much higher and sharper for the more exposed atom ”56 O”. Useful information is obtained from calculation of the coordination number Nb defined by rmin

Nb = 4πρb

r 2 gab (r)dr,

(3.22)

0

where ρb is the number density of atom b and rmin is the position of the first minimum of gab (r). Nb for b=H gives the average number of


127

the solvent hydrogen forming electrostatic bonding with atom a of the peptide. The values of Nb calculated are given in Table 3-5. In the last column, the values divided by the value for methanol (i.e., ratios) are given. The Nb -values and ratios for methanol are smaller than for water, and those for ethanol are smaller than for methanol. This is because the number density of hydrogen atoms in methanol is much smaller than in water and that in ethanol is even smaller. The differences among the water, methanol, and ethanol cases in the Nb -values and ratios are larger for ”23 N” than for ”56 O” (e.g., in the case of b=H of ethanol the ratio for a=”23 N” is smaller than for a=”56 O”). The reason for this result is the following: ”23 N” is less exposed than ”56 O”, and due to the steric hindrance by the hydrocarbon group in an alcohol molecule, it becomes more difficult for the alcohol oxygen to form hydrogen bonding with ”23 N”. Since the hydrocarbon group in an ethanol molecule is bulkier than in methanol molecule, the steric hindrance effect for ethanol is larger. These results are well reflected on the ∆µSa -values for ”23 N” and ”56 O” given in Table 3-4. The formation of hydrogen bonding between an atom with a large, negative site-charge of the peptide and the solvent oxygen leads to a large decrease in ∆µSa , but such formation in alcohol becomes more difficult to achieve than in water. This is particularly true for ethanol and for a less exposed atom like ”23 N”.

Figure 3.5. Representative pair distribution functions in water (a), in methanol (b), and in ethanol (c).

128


Figure 3.6. Representative pair distribution functions in water (a), in methanol (b), and in ethanol (c).

Table 3.5. Coordination Numbers of Solvent Hydrogens around Atom a of Metenkephalin in Conformation 4 a 23 N Gly-2

56 O Phe-4

H H H H H H

b of of of of of of

water methanol ethanol water methanol ethanol

Nb 0.196 0.089 0.072 0.883 0.538 0.463

Nb (ratio) 2.20 1.00 0.81 1.64 1.00 0.86

We now discuss the ∆µSa -values for the hydrophobic atoms of the peptide given in Table 3-4. Alcohol molecules are larger than water molecules and the number density of alcohol is lower than that of water. As a result, in alcohol the work required for the cavity formation is less than in water, giving rise to lower values of ∆µSa . This is particularly true for ethanol. There is, however, another significant reason for the result given in the table. The pair distribution functions gab (r) for a=”47 CZ” are shown in Figure 3-7 (these functions remain almost unchanged even when the site-charge of ”47 CZ” is set at zero). The curves for b=CH3 of methanol and for b=CH2 and CH3 of ethanol have relatively high first peaks. We have calculated the coordination numbers Nb for a=”47 CZ” and b=CH3 of methanol and b=CH2 and CH3 of ethanol. Though the number density of ethanol is lower than that of methanol, the result obtained is the following: Nb (b=CH3 of methanol) CH3 COO− > Cl− > Br− > NO3 − > − > CNS− ], and that for cations is [(CH ) N+ > NH+ > ClO− 3 4 4 > I 4 + Rb , K+ , Na+ , Cs+ > Li+ > Mg2+ > Ca2+ > Ba2+ ]. In each of these series, the species to the left decrease the solubility of proteins and stabilize their native structures. The species to the right, on the contrary, increase the solubility and cause destabilization of the native structures. Though the Hofmeister series is not valid for acidic and basic proteins [50, 51] , it is generally applicable to neutral proteins. The series, except for divalent cations, is also applicable to the other neutral substances such as benzene [52] . That is, the effects of monovalent ions on the solubility of various neutral substances follow the Hofmeister series. The microscopic mechanisms of these experimentally known properties, however, have not been elucidated yet. The Hofmeister series is related to change in the water structure arising from the perturbation by ions. The continuum model for salt solutions cannot give any information concerning the structural change of water: They must be treated on the atomic level. In computer simulations, it is very difficult to include ions in water at finite concentrations, because sufficiently many ions must be included and the corresponding number of water molecules becomes unacceptably large. The RISM theory has potential capability of elucidating the salt effects and providing molecular pictures of the structural change of water near a solute molecule and associated change in the solvation free energy. Nevertheless, the RISM theory has never been applied to analyses on the salt effects, because severe numerical instability was often encountered when one tried to solve the RISM equations for a system including ions. We recently developed a robust and very efficient algorithm [14] for solving the RISM equations for salt solutions and made detailed analyses on the salt effects possible. With the algorithm, we can consider a solute

136


molecule with many atomic sites in water including various monovalent ions. In this section, we discuss the effects of salt addition to water on the solvation properties of a peptide molecule such as the structural change of water near the molecule and the solvation free energy, shedding light on elucidation of the Hofmeister series.

3.2

Model and theory

Once ions are included in water, it is crucial to employ the dielectrically consistent version of the RISM theory [11, 12] . The basic equations and the algorithm for numerically solving them are described in the Appendix. In the analysis, we choose acetylglycine ethyl ester (AGE) CH3 CONHCH2 COOCH2 CH3 because the salting-out coefficients of this peptide in various salt solutions are experimentally available [53] . Besides, such a peptide serves as one of the basic models of proteins. The conformation is fixed at the all-trans form in our calculations. The SPC/E model is employed for water. The following two sets of salt solutions are considered to examine the effects of cations and anions on the solvation properties of AGE: LiCl, NaCl, and KCl (set 1) for the cation effects, and NaCl, NaBr, and NaI (set 2) for the anion effects. The temperature and the salt concentration are fixed at 298K and 1M, respectively. The number density and the dielectric constant of each salt solution, which are used as part of the input data in the dielectrically consistent version of the RISM theory, are taken from the experimental data. We adopt the Coulomb plus L-J potential functions for all the interactions between water-solute and ion-solute atomic pairs. That is, the site-site pair potential uab (r) is expressed as Eq.(3.17) (for the NaCl-solution, for example, b=H,O,Na+ ,Cl− ). The AMBER-type potential parameters are employed for the peptide. The ion-ion and ion-water potentials are also expressed as Eq.(3.17). A problem is that there are no reliable values of the L-J potential parameters, σb and εb , for ions. We determine these parameters so that they can give, within the framework of the RISM theory, a good fit to experimentally measured values both of the activity coefficients of the salt solutions and of the salting-out coefficients of argon (a noble gas) in the solutions. Since the εb -values are calculable from the σb -values, ion polarizability, and total number of electrons of the ion using the Mavroyannis-Stephan theory [54, 55] , we only have to determine the σb -values. It has turned out that the σb -values for the ion-ion potentials must be different from those for the ion-water potentials to attain a good fit. The parameter set so determined is used to analyze the solvation properties of AGE (the σb -values for the ion-solute potentials


137

are identical to those for the ion-water potentials). The use of the Coulomb plus L-J potential functions may be oversimplified, because the ion-induced dipole and dipole-induced dipole interactions are not taken into consideration. However, such interactions could be buried in the potential parameters to some extent. For a system consisting of a solute with m sites, water molecules (V), cations (C), and anions (A), the solvation free energy of the solute molecule ∆µS is calculated from ∆µS = IV + IA + IC ,

IX = 4πkB T

m a=1

∞

ρX

(3.23)

r 2 {haX (r)}2 /2 − caX (r) − haX (r)caX (r)/2 dr,

0

X = V, A, C,

(3.24)

where IV is the sum over X=H and O of water. Though IV , IA , and IC are interdependent, they can be regarded as contributions from structures of water molecules, anions, and cations near the solute molecule, respectively [56] . The salting-out coefficient kS is defined by kS = log(S0 /S)/C,

(3.25)

where S0 and S are the solubility of the solute in pure water and that in salt solution of molarity C, respectively. S0 /S is approximately given by , S0 /S = (ρV + ρA + ρC ) exp {∆µS /(kB T )} / ρ0V exp ∆µ0S /(kB T ) , (3.26) where ρ0V is the number density of pure water, and ∆µ0S denotes the solvation free energy in the case of pure water. It is convenient to define ∆∆µS and χ by ∆∆µS = ∆µS − ∆µ0S = χ + IA + IC , χ = IV − ∆µ0S .

(3.27) (3.28)

Notice that χ represents the contribution to the solvation free energy change from the structural change of water near the solute molecule, which is caused by addition of ions. Namely, χ is a measure of a rather indirect effect due to ions through the structural change of water, and [IA + IC ] provides information concerning a more direct effect. It is worthwhile to decompose the salt effect into the hydrophobic and charge

138


effects. For this purpose, we calculate the solvation free energy of an imaginary peptide in which all the site charges are removed. Let the superscript (0) represent a value for the imaginary peptide. For any quantity X we define the charge effect X (q) (i.e., the effect due to the electrostatic interactions) by X (q) = X − X (0) ,

(3.29)

where X (0) denotes the hydrophobic effect. As explained in 1.9, the RISM theory with the site-site HNC closures overestimates hydrogen bonding in the hydration structure near a hydrophobic portion and tends to give too high a value of the solvation free energy. However, the factor overestimated is cancelled by taking, for example, the difference [∆µS -∆µ0S ]. Such cancellation [56] occurs in Eqs.(3.26), (3.27), and (3.28): kS , ∆∆µS , χ, and [IA + IC ] are not affected by the drawback of the RISM-HNC theory, and the conclusions drawn in this section are not vitiated.]

3.3

Solvation free energy of AGE [57]

The solvation free energies of AGE and the salting-out coefficients are summarized in Table 3-12 along with the corresponding experimental data. It is observed that the kS -values follow the order of the Hofmeister series, [Na+ , K+ > Li+ ] and [Cl− > Br− > I− ], that is consistent with the experimental result. The solvation free energies of the imaginary peptide and their components X (0) , which represent the hydrophobic effect, are given in Table 3-13, and the X (q) -values are summarized in (0) (q) Table 3-14. Comparing ∆∆µS with ∆∆µS in the tables, we observe that the hydrophobic effect dominates. Due to the cancellation among (q) (q) (q) (q) χ(q) , IC , and IA (IC + IA ∼ −χ(q) ) the charge effect is very minor. This implies that the overall salt effect does not depend much on the electrostatic characteristics of a solute, namely, whether the solute has partial charges or not, and how the partial charges are distributed to the atoms. As mentioned in the introduction, the Hofmeister series applies to a rather wide class of neutral solutes. The dominant effect due to the hydrophobic hydration can be the reason why the series is so generally applicable.

139


Table 3.12. Solvation Free Energies (kcal mol−1 ) and Salting-out Coefficients of AGE in Various Salt Solutions LiCl NaCl KCl

IV 32.92 32.92 32.06

χ 0.73 0.73 −0.13

IC −2.34 −1.96 −1.27

IA 1.81 1.81 1.77

∆µS 32.39 32.77 32.56

∆∆µS 0.20 0.58 0.37

kS 0.16 0.43 0.28

kS [expt]1 0.10 0.16 0.15

NaCl NaBr NaI

32.92 32.69 32.34

0.73 0.50 0.15

−1.96 −1.98 −1.99

1.81 1.79 1.89

32.77 32.49 32.24

0.58 0.30 0.05

0.43 0.23 0.03

0.16 0.11 0.04

1

Ref. 53

Table 3.13. Solvation Free Energies of Imaginary Peptide in which All the Site Charges are Removed (Unit in kcal mol−1 ) χ(0) −0.60 −0.37 −0.97

IC −0.55 −0.41 −0.08

(0)

LiCl NaCl KCl

IV 50.74 50.97 50.36

(0)

(0)

IA 1.33 1.33 1.31

∆µS 51.53 51.88 51.60

(0)

∆∆µS 0.19 0.55 0.26

(0)

NaCl NaBr NaI

50.97 50.66 50.23

−0.37 −0.67 −1.11

−0.41 −0.42 −0.41

1.33 1.37 1.55

51.88 51.62 51.36

0.55 0.28 0.03

Table 3.14. Effect of Charges on Solvation Free Energies of Peptide in Various Salt Solutions (Unit in kcal mol−1 ) (q)

(q)

(q)

LiCl NaCl KCl

χ(q) 1.33 1.10 0.84

IC −1.79 −1.55 −1.19

IA 0.48 0.48 0.46

∆∆µS 0.01 0.03 0.11

NaCl NaBr NaI

1.10 1.17 1.26

−1.55 −1.56 −1.58

0.48 0.42 0.34

0.03 0.02 0.02

The cancellation occurring among the electrostatic components implies that the decrease in the solvation free energy arising from the ap(q) pearance of the electrostatic interactions in salt solution ∆µS does not 0(q) differ significantly from that in pure water ∆µS . An explanation for this could be given as follows. Suppose that a large, negative site-charge

140


is given to an atom in the peptide molecule. In pure water, the sitecharge is screened solely by polarization and structure reorganization of water molecules. The water hydrogen atoms form electrostatic bonding 0(q) with the peptide atom, leading to a large, negative value of ∆µS . In salt solution, cations are strongly attracted to the atom whereas anions are repelled from it and cations form electrostatic bonding with the (q) (q) atom. IC and IA take an extremely large, negative and positive values, (q) (q) respectively, and the sum [IC + IA ] takes a negative (not large) value. The formation of electrostatic bonding by the water hydrogen atoms with the peptide atom is somewhat disturbed by the structure formation (q) 0(q) of ions, and |IV | becomes correspondingly smaller than |∆µS | : χ(q) (q) (q) takes a positive (not large) value and |χ(q) | ∼ |IC + IA |. Simply, the site-charge is screened either by water molecules (in pure water) or by cooperation of ions and water molecules (in salt solution). When the structure formed by ions and water molecules responding to the sitecharge is integrated over an infinitely long distance, the screening effect is almost the same in pure water and in salt solution. We have seen above that the Hofmeister series is ascribed mainly to (0) (0) the hydrophobic effect ∆∆µS . We next plot the components, χ(0) , IC , (0) and IA , against the salt species in Fig. 3-9. For the series of anions, the solvent reorganization effect χ(0) decreases with the anion size, while the (0) anion-structure effect IA shows the opposite trend. The overall effect is determined essentially by the solvent reorganization effect, because the rate of decrease in χ(0) is significantly greater than that of increase (0) in IA . For the series of cations, χ(0) shows the order Na+ > Li+ > (0) K+ , while the cation-structure effect IC increases as the cation size (0) becomes larger. Although the overall effect ∆∆µS reflects the order of (0) χ(0) , the order between Li+ and K+ reverses due to IC . In summary, the Hofmeister series is mainly determined by the effects due to the structural change of water caused by the presence of ions. For cations, however, the correlation between the solute molecule and cations also makes a significant contribution.


141

Figure 3.9. Contributions to the hydrophobic effect from the solvent reorganization and from the cation and anion structures.

3.4

Structural change of water near AGE caused by salt addition [57]

To discuss changes in the solvation structure around an atom in the peptide molecule, which is caused by the salt addition, we define ∆ga (r) by ∆ga (r) = [gaO (r)in salt solution] − [gaO (r)in pure water],

(3.30)

where gaO (r) is the pair distribution function between atom a in the peptide molecule and the oxygen atom in a water molecule. In Fig. 310, we plot ∆ga (r) for a =”H in the NH group” for the salts of set 1 (A) and those of set 2 (B). The partial charge of this hydrogen atom is 0.252 that is positive. The NH group is sterically protected from the access of water molecules and ions by the surrounding atoms. In Fig. 3-11, ∆ga (r) for a=”O in the C=O group” is shown for the salts of set 1 (A) and those of set 2 (B). This oxygen atom is negatively, highly charged and its partial charge is -0.504. Water molecules and ions are more accessible to the oxygen atom that is more exposed. It is convenient to classify the salt effects on ∆ga (r) into three categories. The first category is related to the ions with smaller sizes, Li+ and Na+ , which bind water molecules rather firmly. These positively hydrated ions cause negative ∆ga (r) when atom a is positively charged, that is, the partial charges of atom a and the ion have the same sign.

142


This is because the ion is repelled from the peptide atom and takes the associated water molecules away from it. We refer to this effect as the positive hydration effect. The effect due to the ions with larger sizes, K+ , Cl− , Br− , and I− , falls into the second category. These negatively hydrated ions give rise to positive ∆ga (r) because the ice-like, open-packed structure is disrupted by the negative hydration and the water configuration around the peptide atom becomes more packed and denser. We refer to this as the negative hydration effect. The last effect in the third category is the replacement of a water molecule in contact with a peptide atom by the ion itself. This occurs when the charges of the peptide atom and the ion have opposite sign and the charge of the peptide atom is significantly high, high enough to remove the hydrated ion from water and attract it. This makes ∆ga (r) become negative for the obvious reason. We refer to this as the ion-exchange effect. Figure 3-10(A) focuses basically on illustration of the cation effect, but it is necessary to account for the effect due to Cl− as a common background. We may expect the negative hydration effect of Cl− , because the ion is known as a negatively hydrated ion. Since the negative hydration effect gives rise to a positive contribution to ∆ga (r), we expect some positive change in gaO (r) as a background. Li+ and Na+ are classified as positively hydrated ions in the order [Li+ > Na+ ] for the strength of binding water molecules. Since the cations are electrostatically repelled from the hydrogen atom in the NH group, those ions are expected to cause a negative contribution to ∆ga (r), which is in competition to the contribution from Cl− . The figure indicates that the positive hydration effect of Li+ is dominant for LiCl, while the negative hydration effect due to Cl− surpasses the positive hydration effect of Na+ for NaCl. K+ is classified as a negatively hydrated ion, and it gives the same effect as Cl− giving rise to a large positive contribution to ∆ga (r). Figure 3-10(B) shows illustration of the anion effect. For anions, the negative hydration effect is dominant as described above. The effect is expected to become stronger with the increase in the ion size and follow the order [Cl− < Br− < I− ]. In the case shown in Fig. 3-11(A), the common anion Cl− makes ∆ga (r) become positive by the negative hydration effect. We have found that the cations, Li+ , Na+ , and K+ , come preferably in contact with the oxygen atom rather than to the solvent-separated position. As a result, the cations cause a negative contribution to ∆ga (r) due to the ion-exchange effect, in the order [Li+ > Na+ > K+ ]. Figure 3-11(A) indicates that the negative hydration effect of Cl− dominates in the case of KCl, while the ion-exchange effect of cations surpasses for LiCl and NaCl. Figure 3-11(B) can be interpreted similarly. Due to the ion-

143


0.015 (A) LiCl NaCl KCl

∆g(r)

0.01 0.005 0 -0.005

0

2

4

r /Å

6

8

10

0.015 (B) NaCl NaBr NaI

∆g(r)

0.01 0.005 0 -0.005

0

2

4

r /Å

6

8

10

Figure 3.10. Change in the water structure around the hydrogen atom in the peptide NH group due to salt effects. (A) solid line, H(peptide)-O(water) in LiCl solution; dotdashed line, H(peptide)-O(water) in NaCl solution; dotted line, H(peptide)-O(water) in KCL solution. (B) solid line, H(peptide)-O(water) in NaCl solution; dot-dashed line, H(peptide)-O(water) in NaBr solution; dotted line, H(peptide)-O(water) in NaI solution.

144


0.04 (A) LiCl NaCl KCl

∆g(r)

0.02 0 -0.02 -0.04

0

2

4

r /Å

6

8

10

0.04 (B) NaCl NaBr NaI

∆g(r)

0.02 0 -0.02 -0.04

0

2

4

r /Å

6

8

10

Figure 3.11. Change in the water structure around the oxygen atom in the peptide C=O group due to salt effects. (A) solid line, O(peptide)-H(water) in LiCl solution; dot-dashed line, O(peptide)-H(water) in NaCl solution; dotted line, O(peptide)H(water) in KCL solution. (B) solid line, O(peptide)-H(water) in NaCl solution; dotdashed line, O(peptide)-H(water) in NaBr solution; dotted line, O(peptide)-H(water) in NaI solution.


145

exchange effect, Na+ decreases the population of water molecules around the oxygen atom in the C=O group as a common background. The anions give a positive contribution to ∆ga (r) by the negative hydration effect, in the order [Cl− < Br− < I− ]. The negative hydration effect exceeds in the case of NaI, while the ion-exchange effect dominates for NaCl. The two effects are balanced in the case of NaBr.

3.5

Concluding remarks

The salts have effects both on the hydrophobic hydration and on the electrostatic interaction. However, effects on the electrostatic interaction largely cancel out, and the salt effects on the hydrophobic hydration become dominant. We believe that this is the reason why the Hofmeister series applies rather generally to the salt effects on a wide variety of solutes including proteins. The series is mainly determined by the effects due to the structural change of water caused by the presence of ions. The salt effects on solvation structure of a peptide molecule have been analyzed in terms of changes in the peptide-water pair correlation functions ascribed to the perturbation by the ions. We have proposed heuristic interpretations for salt effects on the pair distribution functions between peptide and water atoms based on the well-regarded experimental concepts of ion hydration: the positive hydration, negative hydration, and ion-exchange effects. The complicated effects involving interactions among the peptide molecule, water molecules, and ions can be interpreted in a consistent manner using the heuristic model. There is an important question that we have not answered yet: how is the salt effect on the solubility of proteins [58] related to that on their conformational stability? Let us assume that the Hofmeister series is applicable both to the native and denaturated structures of a protein. If the solubility of the protein with the denaturated structure is more sensitive to the salt effects than that with the native one, we can draw a qualitative illustration of Fig. 3-12. The ordinate is for the total energy (the conformational energy plus the solvation free energy) representing the stability of the protein with two structures. The abscissa corresponds to the Hofmeister series. The slope for the denaturated structure is larger than for the native one because of the higher sensitivity assumed above. In pure water, the native structure is more stable. However, moving to the left in the series leads to the higher solubility of the protein and enhanced stabilization of the native structure. Moving to the right, on the other hand, results in the lower solubility and destabilization of the native structure. This could be a reasonable answer to the question mentioned above, which is to be investigated in further studies.

146


Figure 3.12. Qualitative illustration of the salt effect on the solubility of proteins. ”N” and ”D” represent the native and denaturated structures, respectively.

It is very important to point out that the results of analyses on the salt effects are considerably sensitive to the potential parameters (the L-J parameters) employed for ions [59] . The use of a different set of parameters can even lead to qualitatively different conclusions. Therefore, it is strongly desired that the parameters be examined and improved in future studies. Last, we refer to an interesting singular behavior of the RISM theory, which can be encountered when it is applied to a small, zwitterionic peptide in salt (NaCl) solution. When the salt concentration is gradually decreased from a high value (e.g., 5M) for a prescribed conformation of the peptide molecule, the ionic concentration around the molecule continues to increase. Near a threshold value of the salt concentration the increase becomes sudden and the RISM theory eventually looses its solution. A trend of aggregation of the peptide molecules is also observed. This phenomenon can be interpreted as a signal of the metastability limit beyond which the system unavoidably separates into two phases, an almost pure water phase and a salt solution phase where the peptide is dissolved. When a different set of potential parameters is used for the ions, however, no such singular behavior is exhibited, providing a good example of the high sensitivity of the results to the


147

potential parameters. In any case, the singular behavior must further be investigated.

4. 4.1

Partial molar volume of amino acids and pressure effects Introduction

The partial molar volume is a thermodynamic quantity that plays an essential role in the analysis of pressure effects on chemical reactions, reaction rate as well as chemical equilibrium in solution. In the field of biophysics, the pressure-induced denaturation of protein molecules has continuously been investigated since an egg white gel was observed under the pressure of 7000 atmospheres [60] . The partial molar volume is a key quantity in analyzing such pressure effects on protein conformations: When the pressure in increased, a change of the protein conformation is promoted in the direction that the partial molar volume reduces. A considerable amount of experimental work has been devoted to measuring the partial molar volume of a variety of solutes in many different solvents. However, analysis and interpretation of the experimental data are in many cases based on drastically simplified models of solution or on speculations without physical ground, even for the simplest solutes such as alkali-halide ions in aqueous solution. Matters become more serious when protein molecules featuring complicated conformations are considered. Compared to the effort devoted to experimental work, theoretical studies of the partial molar volume have been very limited [61, 62]. The computer simulations for the partial molar volume were started a few years ago by several researchers, but attempts are still limited. As usual, our goal is to develop a statistical-mechanical theory for calculating the partial molar volume of peptides and proteins. The Kirkwood-Buff (KB) theory [63] provides a general framework for evaluating thermodynamic quantities of a liquid mixture, including the partial molar volume, in term of the density pair correlation functions, or equivalently, the direct correlation functions. The RISM theory is the most reliable tool for calculating these correlation functions when the solute molecule comprises many atoms and has a complicated conformation. In this section, we combine the K-B theory with the RISM theory and derive an equation that allows us to calculate the partial molar volume of a polyatomic solute immersed in molecular solvent. The equation is then applied to calculations for the series of 20 amino acids occurring naturally in living systems. These amino acids are chosen because they are of great biological interests as ingredients of proteins and they have

148


extensively been studied in experiments with respect to the partial molar volume. The calculated values of the partial molar volume are compared with the experimental values, and the physical origin of the discrepancy is discussed.

4.2

Model and theory

The K-B theory expresses the thermodynamic quantities such as the partial molar volume and the isothermal compressibility for a solution in terms of the pair correlation functions of constituent molecular species. The expressions in terms of the site-site pair correlation functions are obtained by coupling the K-B theory with the RISM theory. When we consider a two-component system (a solute with m-sites and a solvent with n-sites) and the infinite dilution limit, the partial molar volume of the solute VM and the isothermal compressibility κT can be expressed as, respectively, m VMa , (3.31) VM = κT kB T − VMa = κT kB T ρV . κT kB T = 1/ ρV

1 − ρV

a=1 n

Cab (0),

b=1 n n

(3.32) !/

Cbb (0)

,

(3.33)

b=1 b =1

where a denotes a site in the solute molecule, b or b a site in the solvent molecule, and Cab (0) the Fourier transform of the site-site direct correlation function cab (r) at zero wave number [64] . When the ideal gas is dissolved into a solvent at constant pressure, the solvent volume increases by κT kB T . This volume increase, which has nothing to do with the molecular interactions, originates from the physical cause that the ideal gas makes the solvent volume increase in order to gain the entropy (at a constant pressure). The original K-B theory naturally includes this contribution in the form κT kB T . This is true for solutes of monoatomic molecules. However, it is not so obvious if it applies to polyatomic solutes as well. If atoms in the solute molecule could move freely in the solvent, the ideal contribution would be N κT kB T (N is the number of atoms constituting the solute molecule). In reality, the atoms in a molecule cannot move freely due to the constraints by bonds and bond angles, but they fluctuate in a molecule and this fluctuation becomes more significant as N increases. In all likelihood, the fluctuation adds some extra contribution to the partial molar volume, which increases as N becomes larger. We refer to this ideal contribution due


149

to the internal degrees of freedom as the ideal fluctuation volume. This concept is not explicit in the original K-B theory, because the theory regards a molecule as a single particle without the internal degrees of freedom, although the framework of the theory is general. As is clear from the derivation and resulting Eq.(3.31), the ideal contribution to the partial molar volume is κT kB T , which accounts for the center of mass of the solute molecule, despite that a solute consists of multiple interaction sites or atoms. The result apparently originates from the rigid constraints imposed to the site-site distances for the intramolecular correlation in the RISM theory. The SPC/E model [19] is used for water and the AMBER parameters are employed for amino acids. The interaction between site a in the solute molecule and site b in a water molecule is expressed by Eq.(3.17). The number density and the dielectric constant of water, which are used as part of the input data in the dielectrically consistent version of the RISM theory [11, 12], are taken from the experimental data at the standard temperature (298K) and pressure. The solute-solvent correlation functions are calculated by solving the basic equations for a solute molecule that is immersed in water at infinite dilution (see the Appendix).

4.3

Calculation of partial molar volume of 20 amino acids [64]

(a) Effect of charges on the partial molar volume The VM -values of the 20 amino acids in the zwitterionic form calculated using Eqs. (3.31) - (3.33) are compiled in Table 3-15 (the fifth column) along with the corresponding experimental data [65, 66, 67, 68] (the last column). Also shown in the table are calculated values for the amino acids with un-ionized N- and C- terminus groups (the fourth column), and those for imaginary compounds made up from the amino acids by removing all the site-charges (the third column). The numbers in the parenthesis are differences between the values in the two adjacent columns. The 20 amino acids are classified into five categories in the conventional manner: aliphatic, nonpolar, aromatic, polar, and charged. The VM -values in the third column should be compared with those in the fourth column. It is observed that except for the amino acids with charged residues the effect of the partial charges on VM is not large and in many cases they give a minor, positive contribution to VM with a few exceptions. This suggests that exposure of the polar residues in solvent upon conformational change of a protein molecule does not cause the so-called electrostriction, but rather leads to slightly positive change in the volume. The situation is entirely different for the amino acids with

150


Table 3.15. Partial Molar Volumes (cm3 /mol) of the Twenty Amino Acids in Aqueous Solutions (The values in parentheses represent the difference between the values in the previous and following columns. The values in square brackets represent the partial molar volumes of amino acids with non-dissociated residues.) MWa “aliphatic” Ala 89.10 Val 117.15 Leu 131.17 Ile 131.17 “nonpolar” Gly 75.07 Pro 115.13 Cys 121.16 Met 149.21 “aromatic” His 155.16 Phe 165.25 Tyr 181.24 Trp 204.27 “polar” Asn 132.12 Gln 146.15 Ser 105.10 Thr 119.12 “charged” Lys 147.20 Arg 175.23 Asp 132.10 Glu 146.12 a b

“no-charge”

“unionized”

“zwitterion”

expt.

58.2 68.8 78.5 76.8

( ( ( (

0.3) 0.7) 0.2) 0.4)

58.5 69.5 78.7 77.2

(−8.2) (−5.7) (−5.0) (−5.2)

50.3 63.8 73.7 72.0

60.5b 90.8b 107.8b 105.7b

49.1 71.7 66.1 87.8

(−0.3) (−0.5) ( 0.6) ( 0.3)

48.8 71.2 66.7 88.1

(−8.3) (−9.7) (−7.5) (−5.4)

40.4 61.5 59.2 82.7

43.2b 82.5b 73.4c 104.8b

77.8 88.5 88.9 91.8

(−0.4) ( 0.8) ( 0.7) ( 1.9)

77.4 89.3 89.6 93.7

(−6.2) (−5.2) (−4.9) (−3.2)

71.1 84.0 84.7 90.5

98.9d 121.5b 124.3b 143.4b

65.6 75.1 59.5 67.1

( 0.1) (−0.3) ( 1.3) ( 0.9)

65.7 74.8 60.8 68.0

(−7.5) (−5.6) (−8.2) (−8.4)

58.2 69.2 52.6 59.7

77.3d 93.6d 60.6e 76.9e

88.2 91.1 65.5 74.8

(−5.6) (−3.8) (−2.7) (−4.1)

82.6 87.3 62.8 70.7

(−5.2) (−5.5) (−7.2) (−6.6)

77.4 81.8 55.6 64.1

molecular weight (g/mol) Ref. 66. c Ref. 65. d Ref. 68. e Ref. 67.

[83.4] [84.8] [58.9] [68.1]

108.5c 127.3c 73.8c 85.9c


151

charged residues. The partial charges give a significantly large, negative contribution to VM , ranging from -2.7cm3 /mol to -5.6cm3 /mol. The reduction of the volume is caused by dissociation of the residues. In Table 3-15, the VM -values of the amino acids with non-dissociated residues in the zwitterionic form are represented by the numbers in square brackets. Reduction of the volume due to the dissociation of residues are -6.0, -3.0, -3.3, and -4.0cm3 /mol, respectively, for lysine, arginine, aspartate, and glutamate. Therefore, exposure of such a residue of a protein molecule should cause volume contraction. As a corollary, the residues are in favor of the dissociated form under high-pressure conditions. Ionization of the N- and C-terminus groups also gives a significantly large, negative contribution to the volume, which ranges from -3.2 to -9.7 cm3 /mol, as one can see by comparing the numbers in the fourth and fifth columns. The volume reduction due to dissociation of the residues and ionization of the termini is a manifestation of the electrostriction that is caused by the solute-solvent electrostatic interaction. If the electrostriction occurs, the average distance between the charged group and a solvent molecule should reduce. The first peak position of a solute-solvent pair correlation function gives good information on the average distance. The pair correlation functions between the carboxyl oxygen in the C-terminus group of glycine and the water atoms (O, H) are plotted in Fig. 3-13(A) for the two cases where the carboxyl group is un-ionized and ionized, respectively. We focus on the first peak position for the water oxygen. It is clear from the figure that the first-peak position or the average O-O distance reduces when the carboxyl group is ionized. A similar reduction of the average distance is observed in the pair correlation function between the nitrogen atom in the N-terminus group and the water oxygen, which is shown in Fig. 3-13(B). (b) Group contributions A considerable amount of effort has been devoted to estimation of the partial molar volume of amino acids using a superposition of contributions from the functional groups [65, 69, 70] . Since our theory gives the partial molar volume by the atomic-level treatment, it can shed light on checking the validity of the superposition approximation. We emphasize that the contribution from a functional group, say CH2 , depends very much on its microscopic environment. In Figure 3-14, depicted is the histogram of the contributions from CH2 appearing in the 20 amino acids. This figure strongly suggests that the volume contribution from CH2 is dependent very much on its microscopic environment where the group is situated. The naive way of estimating the partial molar volume of a molecule by means of the superposition of contributions from functional

152


8 (A) O1

6

C

C 2

O H

g(r)

O2-Owater O2-Hwater O3-Owater O3-Hwater

O3 O3'

4 2 0

0

2

4

r (Å)

6

8

10

2.5 (B)

N1-Owater N1-Hwater N2-Owater N2-Hwater

N1H2

2

N2H3

g(r)

1.5 1 0.5 0

0

2

4

r (Å)

6

8

10

Figure 3.13. Pair correlation functions between the oxygen atom in the C-terminus group and water atoms (A) and between the nitrogen atom in the N-terminus group and water atoms (B).

153


groups makes no physical sense and possibly leads to a wrong conclusion. (c) Comparison with experimental data We plot ∆VM = VM,exp − VM (VM,exp ) is the experimentally measured value and VM the theoretically calculated value) against the number of atoms in the amino acids in Fig.3-14. It is obvious that the discrepancy ∆VM increases as the number of atoms N becomes larger. There can be two major reasons for this. The first one is the ideal fluctuation volume discussed in 4.2. The partial molar volume can be decomposed into two terms, the ideal contribution VMid and the excess quantity VMex . The former can further be decomposed into the ideal-gas contribution and the fluctuation volume VMfluc : VM = VMid + VMex ,

(3.34)

VMid = κT kB T + VMfluc .

(3.35)

7 6 Frequency

5 4 3 2 1 0

Figure 3.14. acids.

-4

0 4 8 12 partial molar volume (cm3/mol)

16

Histogram of the contributions from CH2 appearing in the 20 amino

The internal fluctuation of atoms in a solute molecule is entirely frozen by the constraints expressed by the Dirac delta-functions in the RISM theory. If all the constraints were removed, the atoms would move freely, giving the ideal contribution N κT kB T to the partial molar volume. Therefore, a reasonable conjecture is 0 < VMfluc < (N − 1)κT kB T and VMfluc becomes larger with increasing N . The second reason is, as already described in 1.9, in a drawback of the 1D-RISM/HNC. It underestimates the excluded volume of a solute

154


molecule and the degree of the underestimation increases as the solute size becomes larger. In fact, the VM -values from the 3D-RISM/HNC are significantly larger than from the 1D-RISM/HNC and in much better agreement with the experimental data [71] (the 3D-RISM/HNC+RBC values are too large) as shown in Fig. 3-15. Still, even in the 3DRISM/HNC results, the VM -values for amino acids with experimental values exceeding 120cm3 /mol are too small, possibly due to the ideal fluctuation volume. We emphasize, however, that inclusion of repulsive bridge functions in a prudent way leads to remarkable improvement over the 1D-RISM/HNC results [72]. In fact, the improved values are in close agreement with the 3D-RISM/HNC results, which is also illustrated in Fig. 3-15. 1D-RISM/HNC 3D-RISM/HNC 1D-RISM/HNC-BF0

140

(an improved version of the 1D-RISM theory)

Vtheor (cm3/mol)

120

100

80

60

40 40

60

80

100

120

140

3

Vexpt (cm /mol)

Figure 3.15. Partial molar volumes of the 20 amino acids calculated by the three different versions of the RISM theory. V theor and V expt denote the values theoretically calculated and experimentally measured, respectively.

4.4

Concluding remarks

By combining the K-B theory with the RISM theory, we have derived the equation for calculating the partial molar volume VM of a polyatomic solute in solvent. We have calculated the VM -values of the 20 amino acids, constituents of natural proteins. The calculated values are always smaller than the corresponding experimental values. Moreover, the discrepancy becomes larger as the number of the atoms in the amino-


155

acid molecule increases. This result is due to the drawback of the 1DRISM/HNC theory that it underestimates the excluded volume effects and the degree of the underestimation becomes more serious with the increase in the solute size. Another factor responsible for the discrepancy is the internal fluctuation of atoms in the molecule, which is entirely frozen by the constraints expressed by the Dirac delta-functions in the RISM theory but should contribute to an increase in the partial molar volume. The 3D-RISM/HNC is free from the drawback mentioned above and gives more accurate values of the partial molar volume than the 1DRISM/HNC. However, the former is far more computationally demanding and hardly applicable to very large proteins. To make detailed analyses of the pressure effects on protein conformations possible, improvement of the 1D-RISM/HNC is imperative. Based on our very recent studies [72], we believe that it is not difficult to achieve such improvement.

Appendix 1.

Algorithms for solving RISM equations

1.1

Bulk water and salt solution [73, 74]

(a) Basic equations We emphasize that our algorithm is quite general and can be applied to a variety of liquids and liquid mixtures. However, here we describe it for pure water and salt solution (water including one species of salt). The drawback of the original version of the RISM theory, the dielectric inconsistency, becomes serious when salts are included in water, and the theory gives unrealistic structural predictions at finite salt concentration. Therefore, it is crucial to adopt the new version often called the DRISM theory [75, 76]. Hereafter, we refer to the new version simply as the RISM theory. The subscripts ”V”, ”C”, and ”A” denote ”water”, ”cation”, and ”anion”, respectively. It is assumed that a water molecule comprises three interaction sites (H, O, and H) and the ions are monovalent. The site-site Ornstein-Zernike (SSOZ) equation is expressed as ˜˜ ˜ c˜HNC x ˜ ρ−1 + ρ−1 χρ ˜ −1 c˜HNC , ˜ HNC = ρ−1 (I − x cHNC )−1 x η ˜ = ρw ˜ + χ, ˜ x ˜ HNC η

˜ − c˜HNC , =h

(3.A.1) (3.A.2) (3.A.3)

where ρ is the matrix of number densities, I the identity matrix, h the matrix of site-site intermolecular total correlation functions, cHNC the matrix of site-site intermolecular direct correlation functions defined as the solution of the usual hypernettedchain (HNC) equation, w the intramolecular correlation matrix, and ”∼ ” represents Fourier transforms. The matrix χ is introduced to assure the dielectric consistency and given by ˜ ˜ = ρDρ χ (3.A.4)

156


The elements of D are given by ˜ c (k), ˜ 13 = (D) ˜ 31 = (D) ˜ 33 = [j0 (klHH /2)]2 [j1 (kdHz )]2 h ˜ 11 = (D) (D)

(3.A.5)

˜ 12 = (D) ˜ 21 = (D) ˜ 23 = (D) ˜ 32 = −j0 (klHH /2) [j1 (kdHz )] ˜ (D) hc (k), 2˜ ˜ (D)22 = [j1 (kdHz ))] hc (k),

(3.A.6)

2

− dHz = ˜ c (k) = ˜ hc (0) exp(−ak ), a = 0.1, h ˜ c (0) = [( − 1)/y − 3] /(ρT xV ), h 2 (lOH

2 lHH /4)1/2 /2. 2

(3.A.7) (3.A.8) (3.A.9) (3.A.10)

where (D)IJ denotes the (I, J)-element of D, jn (x) is the nth order spherical Bessel function, k the wave number, laa the distance between sites a and a in a single water molecule, ε the dielectric constant of the salt solution, ρT = ρV + ρC + ρA (ρC = ρA ), ρV , ρC , and ρA are number densities of water molecules, cations, and anions, respectively, and xV = ρV /ρT . Refer to Refs. [75] and [76] for the definition of y. The other 16 elements are all zero. The parameters, T (the system temperature), ε, µV , ρV , ρC , and ρA are treated as part of the input data. The matrices of the site-site intermolecular correlation functions are determined from

as

˜ ˜ c=˜ cHNC + b,

(3.A.11)

˜=η ˜ HNC − ˜ b, η

−1 ˜ ˜ ˜=w ˜ + ρh ˜ −1 D + ρ˜ cHNC . w b

(3.A.12)

The matrices are all (5,5) matrices. For 0 0 ρV 0 0 0 0 ρV 0 ρ=0 0 0 0 0 0 0 0 0 0 0

(3.A.13)

example, ρ is a diagonal matrix expressed 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ρV 0 0 ρC 0 0 0 0 0 ρA 0

where ρO and ρH are number densities of oxygen atoms and hydrogen atoms of water, respectively, and ρO = ρH /2 = ρV . The matrix w is expressed as 0 0 0 1 w ˜HO w ˜HH 0 0 0 0 0 0 w 1 w ÕH 0 0 0 0 ÕH 0 ˜ =0 w ˜HO 1 0 0 0 ˜HH w 0 w 0, 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 w ÕH = w ˜HO = sin(klOH )/(klOH ),

(3.A.14)

w ˜HH = sin(klHH )/(klHH ).

(3.A.15)

The matrix h is expressed as 0 ˜ 0 hHH 0 ˜ 0 h 0 OH ˜ =0 h ˜ h 0 HH 0 h 0 ˜ CH 0 h ˜ AH

˜ HO h ˜ hOO ˜ HO h ˜ CO h ˜ AO h

˜ HH h ˜ hOH ˜ HH h ˜ hCH ˜ AH h

˜ hHC ˜ hOC ˜ hHC ˜ CC h ˜ AC h

˜ HA h ˜ OA h ˜ HA h ˜ CA h ˜ AA h

0 0 0 0 0 0. 0 0 0 0


157

When pure water is treated, ρC = ρA = 0 and the matrices become (3,3) matrices. Further, the basic equations can readily be extended to molecular solvents other than water such as alcohol. The closure equation employed is of the HNC type written as cHNCab (r) = exp [−uab (r)/(kB T ) + ηHNCab (r)] − ηHNCab (r) − 1, a, b = H, O, C, A, (3.A.16) where uab is the pair potential between sites a and b belonging to different molecules and kB the Boltzmann constant. Because of the presence of long-range Coulomb potentials, care must be taken in forward and back Fourier transforms of the sitesite correlation functions [77]. It is convenient to define the short-range parts of the functions by (3.A.17) cSR ab (r) = cab (r) + fab (r), a = 1, ..., m, b = H, O, SR ηab (r) = ηab (r) − fab (r),

(3.A.18)

fab (r) = qa qb erf(αr)/(kB T r),

(3.A.19)

where qa and qb are charges on sites a and b, respectively, erf(x) is the error function, and α a constant appropriately chosen. The finally obtained results are independent of α, but the convergence properties are largely influenced by α (more details are described in (b)). The Fourier transform of the above equations yields ãb (k) + fãb (k), c˜SR ab (k) = c

(3.A.20)

SR (k) = ηãb (k) − fãb (k), ηãb fãb (k) = 4πqa qb exp −k2 /(4α2 ) /(kB T k2 ).

(3.A.21) (3.A.22)

We note that fab (k) diverges as k → 0. The numerical forward and back transSR ˜HNCab (k), respectively, and the divergent term forms are applied to cSR HNCab (r) and η SR SR (k) decay very rapidly (exponenis handled analytically. Since cHNCab (r) and η˜HNCab tially) as r → ∞ and k → ∞, they can be accurately transformed numerically with a sufficiently small integration range. The special treatment of Coulomb potentials mentioned so far is simpler and faster than the so-called renormalization technique. A sufficiently long range rL is divided into N grid points (ri = iδr, i = 0, 1, ..., N − 1; δr = rL /N ) and all the functions are represented by their values on these points. Typical values of N and δr in the pure water case are 512 and 0.02d (d=0.28nm). In salt solution cases, N must be set at a larger value: When the salt concentration is 1M, for example, N is 1024 and N must be even larger as the salt concentration becomes lower due to the longer-range ion-ion correlation functions. The simplest iteration strategy is a successive substitution often called the Picard iteration. It can be summarized as follows. (1) Choose initial guesses of ηHNCab (ri ) (a, b=H,O,C,A; i = 0, 1, ..., N − 1). (2) From ηHNCab (ri ) (the input values of the iteration variables), calculate cHNCab (ri ) using the HNC closure equation. (3) Obtain the Fourier transform c˜HNCab (kj )(kj = jδk, j = 0, 1, ..., N − 1; δrδk = π/N ). (4) Calculate η˜HNCab (kj ) using the SSOZ equation. (5) Obtain ηHNCab (ri ) via the back Fourier transformation. These are the output

158


values of the iteration variables and denoted by ηHNCab (ri ). (new) (6) Determine the new input values ηHNCab (ri )from

, (new) ηHNCab (ri ) = ηHNCab (ri ) + ξ ηHNCab (ri ) − ηHNCab ](ri ) , 0 < ξ ≤ 1,

(3.A.23)

where ξ is the mixing parameter. (7) Repeat steps (2)-(6) until the input and output values of the iteration variables become identical within convergence tolerance. If no approximate values of the iteration variables are available, the initial guesses [87] are prepared by ηHNCab (ri ) = qa qb erf(αri )/(kB T ri ).

(3.A.24)

Even with good initial guesses, an extremely small value of ξ is often necessary in the Picard method, requiring a prohibitively large number of iterations. (b) Hybrid algorithm (ri ) by ηti and ηti (t = 1, ..., iP ; i = Hereafter, we denote ηHNCab (ri ) and ηHNCab 0, 1, ..., N − 1), respectively, where iP is the number of distinct pairs (a, b) (for pure water iP =3 and for salt solution iP =10). The solution of the RISM equations can be attributed to the solution of the following set of nonlinear simultaneous equations where ηti form the independent variables: = 0, t = 1, ..., iP ; i = 0, 1, ..., N − 1. Fti = ηti − ηti

(3.A.25)

The number of the independent variables, which equals the number of the functions to be zeroed, is N iP . In solving a set of nonlinear simultaneous equations, the NewtonRaphson (N-R) method can be the most powerful tool. However, direct application of the N-R method to solving Eq.(3.A.25), which is often referred to as the global N-R method, is unacceptable because the dimensionality of the Jacobian matrix is very large (N iP ). In general, the construction and inversion of the matrix must be repeated in the N-R method. We note that the time required for inverting a matrix becomes longer in proportion to n3D (nD is the dimensionality of the matrix). In summary, the N-R method is potentially capable of achieving convergence in only several iterations, but the large matrix must be handled with the result of a large amount of computer storage requirements and very long computing time. The Picard method, on the other hand, is very simple and requires no handling of such a large matrix, but it is often unstable and extremely slowly convergent. One of the most judicious algorithms is a hybrid of the Picard method and the N-R method, which utilizes only the advantageous aspects of the two methods. The hybrid algorithm was pioneered by Gillan [78] for one-component, bulk simple fluids and further developed by Kinoshita et al. [73, 74, 77, 79 - 86] in extending it to multi-component, molecular fluids both in the bulk and at surfaces. In the iteration cycle (2)-(6) described in (a), a change in an input variable ηti with for all i. a small value of i leads to significant changes of all the output variables, ηti In contrast, a change in an input variable ηti with a large value of i causes essentially no changes of the output variables. We refer to ηti with small i and those with large i as the coarse variables and the fine variables, respectively. The numerical instability encountered in the Picard method is ascribed to the coarse variables. Therefore, the iterative loop is decomposed into two sub-loops, the inner and outer loops. In the inner loop the coarse variables are made converged using the N-R method while in the outer loop the fine variables are successively updated in accordance with the Picard

159


iteration strategy. ηti with i ≤ iD behave as the coarse variables, and iD δr is in the range from 1.1σab to 1.2σab where σab is the repulsive core size (the sigma-value in the Lennard-Jones potential) for pair (a, b). The number of the coarse variables specially treated in the inner N-R loop, namely, the number of the independent variables is iD iP . However, the number can further be decreased in the following manner (this decrease is crucial in 1.2). ηti with i≤ iD are decomposed into the coarse parts atz and the fine parts ∆ηti using the projective representation ηti =

κt

atz Ptzi + ∆ηti

(3.A.26)

z=1

where Ptzi is the discretized zth basis function for pair t at r = iδr and κt is the number of basis functions for pair t. We use the roof basis functions for Ptzi with unit height and width 2ibt δr. We note that κt and ibt may differ for different pairs t. Both the fine parts of the coarse variables (∆ηti ) and the fine variables (ηti with i > iD ) are converged in the outer Picard loop. For a given approximation of the fine parts and the fine variables, the coarse parts are converged in the inner loop using the N-R method. The fine parts and the fine variables are then updated using the Picard method and the coarse parts are again converged in the inner loop. This cycle is repeated until the fine parts and the fine variables become unchanged. One cycle of the inner N-R calculation can be summarized as follows: (1) Starting with the input values of the coarse parts atz , construct ηti from Eq.(3.A.26) and obtain ηHNCab (ri ). (ri ) via steps (2)-(5) described in (a) and obtain ηti . (2) Calculate ηHNCab from (3) Evaluate the output values of the coarse parts atz by decomposing ηti atz =

κt ibt −1

ηti Qtzi ,

(3.A.27)

i=0

where Qtzi are the discretized conjugate basis functions determined from Ptzi . The N-R iteration solves for the roots of Φtz = atz − atz , t = 1, ..., iP ; z = 1, ..., κt .

(3.A.28)

Notice that the independent variables in the N-R iteration strategy are atz . The dimensionality of the Jacobian matrix J is reduced to iP κt (κt ∼ 10). An analytical expression of the matrix can be derived and arranged in a compact form so that the matrix can be constructed quite efficiently using the Fast Fourier Transform (FFT). If the initial guesses are not sufficiently good, the simple application of the N-R method may cause divergence in the first inner loop. Even in such cases, however, replacement of J by [J +I /ζ] (ζ > 0) leads to quite successful convergence (ζ must be smaller against more numerical instability). In the inner loop, the calculation is considered converged once Ein =

iP κt

|Φtz | /(iP κt ) < εin

(3.A.29)

t=1 z=1

is satisfied. There is no need to calculate and invert the matrix at every N-R iterative step. For instance, the calculation and inversion should be skipped if Ein is smaller

160


than, for example, 10−1 . The outer-loop iterations are finished once

Eout =

Nf (new) (old) (new) − γJ /γJ γJ /Nf < εout

(3.A.30)

J =1

is satisfied. Here, γJ denote all the iteration variables treated in the outer loop, superscripts ”(old)” and ”(new)” represent the old and new values, respectively, and Nf is the total number of the variables. The recommended values of εin and εout are 10−2 and 10−5 , respectively. However, the independent variables in the inner N-R loop are updated at least once regardless of the εin -value. With this treatment, εin is also sufficiently small (εin < 10−5 ) when the outer-loop iterations are finished. (c) Numerical examples We give a brief idea of the powerfulness and high speed of the hybrid algorithm. First, let us consider the pure water case. With the initial guesses expressed by Eq.(3.A.24) and α=3.0, converged values of the iteration variables are obtained in only 8 outerloop Picard iterations with ξ=1.0. Too large or too small a value of α gives rise to divergence in the first N-R loop. With α=3.0, just for the first 6 iterations in the first inner N-R loop, the replacement of J by [ J + I/ζ] (ζ=7.5) is required. The Jacobian matrix (nD =36) calculated at the last of the first inner loop is used for all the succeeding inner loops. This leads to considerable saving of the computation time. The number of the total N-R iterations is 16. With the conventional algorithm of the Picard iteration strategy, convergence under a specified condition must be achieved in a far more arduous manner. In the first step, very small values are assigned to the site-charges. In the next step, the site-charges are gradually increased until the full values are reached. Hundreds of iterations are required to obtain convergence in each new calculation using the last solution as the initial guess. In contrast, the hybrid algorithm allows us to use the full site charges in the first calculation. In general, convergence becomes more difficult to achieve once ions are included. Take 1M-NaCl solution as an example in the salt-solution case. First, ρC (= ρA ) is set at a very small value (e.g., 10−4 times the true value). For the three pairs, (H,H), (H,O), and (O,O), the correlation functions converged in the pure water case serve as sufficiently good initial guesses. For the other 7 pairs, the initial guesses are prepared from Eq.(3.A.24) with α=2.0. Converged values of the iteration variables are obtained in 6 outer-loop Picard iterations and the total of 11 inner-loop N-R iterations. Next, ρC is increased to, for example, 10−1 times the true value. Convergence is achieved in 5 Picard iterations and the total of 6 N-R iterations. With the true value of ρC , 8 Picard iterations and the total of 11 N-R iterations finish the calculation for the 1M-NaCl solution. The replacement of J by [ J + I /ζ] is not required at all and ξ is always fixed at 1.0. The construction and inversion of the Jacobian matrix (nD =112) was carried out only 9 times in the entire calculation comprising the three steps with different values of ρC .

1.2

Solute immersed in water and salt solution [77, 86]

(a) Basic equations We emphasize that our algorithm is quite general and can be applied to a solute immersed in a variety of liquids and liquid mixtures. However, here we describe it

161


for a solute immersed in pure water and salt solution (water including one species of salt). The subscripts ”V”, ”C”, ”A”, ”E”, and ”S” denote ”water”, ”cation”, ”anion”, ”salt solution”, and ”solute”, respectively. It is assumed that the solute molecules are present at infinite dilution. The calculation process is then split into two steps where bulk salt solution (step 1) and salt solution near a solute molecule (step 2) are treated, respectively. The site-site intermolecular total correlation functions calculated in step 1 are used as input variables for step 2. The calculation in step 1 is performed using the basic equations and the hybrid algorithm described in 1.1. We consider step 2 hereafter. It is assumed that the solute molecule, a water molecule, a cation, and an anion have m, 3, 1, and 1 interaction sites, respectively. The SSOZ equation is expressed as ˜ EE − ˜ ˜ SS c˜SE H ˜ SE = w cSE , (3.A.31) η ˜ SE − ˜ ˜ =h η cSE , (3.A.32) SE

˜ EE = w ˜ EE , ˜ EE + ρE h H

(3.A.33)

where H EE , η SE , and w SS are (5, 5), (m, 5), and (m, m) matrices (in the pure water case, they are (3, 3), (m, 3), and (m, m) matrices), respectively. ρE is the matrix of number densities of water molecules, cations, and anions in the bulk, h is the matrix of site-site intermolecular total correlation functions, c is the matrix of sitesite intermolecular direct correlation functions, w is the intramolecular correlation matrix, and ” ” represents Fourier transforms. H EE depends on properties of the bulk salt solution alone and is part of the input data for step 2. w SS , w EE C ρE , hEE , and H EE are symmetrical matrices. The expressions of some of the matrices are 0 0 0 ρV 0 0 0 0 0 0 0 0 0 ρV 0 0 0 0 0 0 ρE = 0 0 0 0 0 ρV 0 0 0, 0 0 0 0 0 0 ρ C 0 0 0 0 0 0 0 ρA 0 0 0 c˜1H c˜1O c˜1H c˜1C c˜1A 0 0 . .. . . .. . . . c˜SE = 0 . . . . . 0 0 c˜mH c˜mO c˜mH c˜mC c˜mA 0 0 0 1 w ˜12 · · · w ˜1m 0 0 0 0 w 1 ··· w ˜2m 0 0 ˜21 0 ˜ SS = 0 . w .. .. 0 , .. 0 .. . . . 0 0 0 0 w ˜m2 · · · 1 0 ˜m1 w w ãa = w ã a = sin(klaa )/(klaa ),

0 0 0 0 0, 0 0

(3.A.34)

where k is the wave number and laa is the distance between sites a and a in the solute molecule. The closure equation employed is of the HNC type given by cab (r) = exp {−uab (r)/(kB T ) + ηab (r)} − ηab (r) − 1, a = 1, ..., m; b = H, O, C, A, (3.A.35) where uab is the pair potential between sites a and b (a and b denote sites in the solute molecule and in the salt solution, respectively). The special care described in 1.1(a)

162


is taken in handling the long-range Coulomb potentials. In the pure water case we have tested another set of fab (r) and fab (k): fab (r) = {1 − exp(−r)} qa qb /(kB T r), fãb (k) = 4πqa qb / k2 (k2 + 1)kB T ,

(3.A.36) (3.A.37)

and found that rL with δr = 0.02d (d=0.28nm) and N =512 is too short: The charge neutrality is poorly satisfied and the solvation free energy is quite erroneous (N =1024 or N =2048 is required). The renormalization technique suffers from a similar shortcoming. (b) Hybrid algorithm As in cases of bulk water and salt solution, all the functions are discretized. The range rL must be longer when ions are included in water. Typical values of N and δr in the pure water case are 512 and 0.02d (d=0.28nm). When the salt concentration is 1M, for example, N is 1024 and N must be even larger as the salt concentration becomes lower. We iterate on ηab (ri ) (there are 2m and 4m distinct pairs in the pure water and salt-solution cases, respectively). The coarse variables, ηab (ri ) for ri ≤ Dab (Dab Dab are treated as the fine variables. The coarse parts are converged in the inner N-R loop and the fine parts and the fine variables are successively updated in accordance with the Picard method in the outer loop. The values of ibt and κt (t = 1, ..., 2m or 4m)can all be different. A problem is that for a biomolecule m is very large and the dimensionality of the Jacobian matrix is correspondingly large, and the construction and inversion of the matrix becomes considerably time-consuming. To overcome this problem, we choose Dab such that exp[−uab (ri )/(kB T ) + ηab (ri )] in the closure equation is sufficiently smaller than 1.0 for ri ≤ Dab . Namely, Dab chosen is smaller than but close to σab . As a result, the Jacobian matrix becomes almost independent of the solute moleculewater and -ion correlation functions. The matrix can then be treated as part of the input data: It is constant against changes in all the iteration variables. In other words, the construction of the matrix is required only once. At each N-R iterative step, the linear set of equations written as Jx = b

(3.A.38)

must be solved for x (J is the Jacobian matrix, x the column vector the components of which are unknown corrections of the independent variables and b the column vector comprising known numbers). This is solved using Crout’s method for the LU (L and U represent lower triangular and upper triangular, respectively) decomposition followed by the forward and back substitutions. The linear set of equations is subsequently solved with the same J but a different right-hand side b for the N-R iterations. This can be done efficiently without calculating the inverse of J . Further, since J is diagonally dominant (this is also a great advantage of our algorithm), pivoting is not required at all. With the deliberate considerations described in the last paragraph, construction of the Jacobian matrix is required only once and its inversion is unnecessary, keeping the computational load at the minimum amount. We remark that the LU decomposition can be accomplished much more rapidly than the matrix inversion by a factor


163

of 3. As for the computer storage requirements, in the case where the solvent is pure water, the computer memory to be used for the Jacobian matrix is comparable with that for the intramolecular correlation matrix. In the case of salt solution, the former is significantly larger than the latter, but no serious problem is posed with current computer-system capabilities unless m is extremely large. Since the matrix is determined only from the bulk properties and the intramolecular correlation functions of the solute molecule, it is completely independent of the initial guesses of the iteration variables. Therefore, the matrix is never pathological and the calculation never becomes unstable even with crude initial guesses, convergence in the inner loop is achieved in only 2 iterations during the first few outer loops and in only a single iteration afterwards. (The initial guesses written as Eq.(3.A.24), ηab (r) = qa qb erf(αr)/(kB T r), are very good.) The replacement of J by [J + I / ζ] is not required at all. This is in marked contrast with the usual N-R method where the matrix can be pathological and give rise to severe instability when only crude initial guesses can be prepared. With the conventional algorithm of the Picard iteration strategy, convergence under a specified condition must be achieved in a far more arduous manner. In the first step, the basic equations are solved with no intramolecular constraints and with zero site-charges for the solute molecule. In the next step, the constraints that are weighted by some factors ranging from 0 to 1 are imposed on the distance between atomic pairs, and the factors are gradually increased to 1. In the final step, the sitecharges are added to the atoms step-by-step with a small increment. Hundreds of iterations are required to obtain convergence in each new calculation using the last solution as the initial guess. In contrast, the hybrid algorithm allows us to use the full constraints and site-charges in the first calculation. We once compared our algorithm with the Picard method for acrolein (a solute molecule with m=8) in pure water and found that the results from the two sources were the same but our algorithm was over 100 times faster. It is worthwhile to add that the difference between the conventional algorithm and our hybrid one in terms of the speed becomes even larger if the convergence criterion adopted becomes more severe. Another great advantage of our algorithm is that the Jacobian matrix calculated for a particular conformation of the solute molecule can also be used for other different conformations if the corrections are damped: The new values of the independent (new) (old) (I = 1, ..., M ) is determined from the old values xI from variables xI (new)

xI

(old)

= xI

+ β∆xI , 0 < β < 1, I = 1, ..., M,

(3.A.39)

where ∆xI denote the corrections, the components of x in Eq.(3.A.38), β is a damping factor, and M the total number of the independent variables. This robustness is very useful when the RISM theory is combined with the MC simulated annealing technique or a generalized-ensemble algorithm as is done in 1.4 and 1.8. In such a combination, the RISM equations must be solved for a number of different conformations of a peptide molecule. Moreover, the peptide conformation often changes drastically during the conformational sampling, and the solution for the last conformation is not always a sufficiently good initial guess for the new one. Nevertheless, we have verified that only a single Jacobian matrix calculated for a particular conformation can be employed for the whole set of conformations sampled. (c) Numerical examples To give a brief idea of the powerfulness and high speed of our algorithm, let us consider Met-enkephalin (m=75) in conformation 4 immersed in pure water. With the initial

164


guesses expressed by Eq.(3.A.24) with α=2.5, converged values of the iteration variables are obtained in 49 outer-loop Picard iterations with ξ=1.0. The total number of the N-R iterations is 51. With the Jacobian matrix (nD =1438) calculated for conformation 4, Met-enkephalin in conformation 1 is treated. The correlation functions converged in the case of conformation 4 are used as the initial guesses. Convergence is achieved in only 27 Picard iterations with ξ=1.0 and the total of 29 N-R iterations with β=0.8. We note that conformation 1 is quite different from conformation 4. When the new conformation is not significantly different from the last one, the converged result is obtained in several Picard iterations. We next consider a dipeptide (m=23) immersed in 1M-NaCl solution. Again, convergence becomes more difficult to achieve once ions are included. With the initial guesses expressed by Eq.(3.A.24) with α=2.5, converged values of the iteration variables are obtained in 77 outer-loop Picard iterations with ξ=0.6 and the total of 78 N-R iterations (the dimensionality of the Jacobian matrix nD is 956).

1.3

Concluding remarks of the Appendix

We have developed a robust and very efficient algorithm for solving the full RISM equations for a molecular solute with many atomic sites (interaction sites) immersed in water or salt solution. It is a hybrid of the Newton-Raphson (N-R) and Picard methods. The Jacobian matrix is part of the input data and made completely independent of the correlation functions to be calculated. Its inversion, which would be remarkably time-consuming, is avoided. The long-range Coulomb potentials are handled in a special manner so that the number of grid points N can be minimized. In the combined RISM-MC approach [88, 89, 90] proposed in 1.6 and 1.8, the RISM equations must be solved for a number of different conformations of a peptide molecule. Further, the peptide conformation often changes drastically during the conformational sampling, and the solution for the last conformation is not always a sufficiently good initial guess for the new one. Nevertheless, only a single Jacobian matrix calculated for a particular conformation can be employed for the whole set of conformations sampled. Thus, the time required for the construction and the LU decomposition of the matrix takes only a negligibly small percentage of the total time consumed in the combined approach. In terms of the high speed as well as the insensitivity to the initial guesses, our hybrid algorithm predominates over any other algorithm such as the Picard method, the global N-R method, and the conjugategradient type methods [91]. For the three-dimensional RISM equations, however, the hybrid algorithm becomes problematic and the modified method of direct inversion in the iterative subspace (the modified DIIS method [87] ), which is described in the Appendix of Chapter 4, is significantly more useful. Last, we comment on the computer storage requirements in the hybrid algorithm. When a very large peptide or a protein in salt solution is treated, the size of the Jacobian matrix will become huge. We believe, however, that even such a problem can be overcome by modifying how to construct the hybrid algorithm: There is no need to apply the N-R method to all the solute-water and -ion correlation functions. Or if necessary, we might consider a special technique of avoiding direct manipulation of the huge matrix, in which there is no need to actually generate it and store it [92] . If such a technique is employed, the robustness and speed will be somewhat deteriorated, but the resultant algorithm is still far superior to the Picard method.

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Chapter 4 THREE-DIMENSIONAL RISM THEORY FOR MOLECULAR LIQUIDS AND SOLID-LIQUID INTERFACES Andriy Kovalenko Institute for Molecular Science, Okazaki National Research Institutes Myodaiji, Okazaki 444-8585, Japan [email protected]

Present address: National Institute for Nanotechnology, National Research Council of Canada ECERF Bldg., W6-025, University of Alberta, 9107-116 Str., Edmonton, AB T6G 2V4 Canada [email protected]

1.

Introduction

For a liquid of molecular species modelled by the sum of spherically symmetric potentials centered on molecular interaction sites, it is natural to base the consideration on site-site radial correlation functions averaged over orientations of molecules. This level of description is provided within the conventional, site-site RISM approach which yields full access to the thermodynamics of molecular liquids. Owing to the orientational reduction of the intermolecular correlations, it possesses a significant advantage of numerical simplicity and small computational load. In the case of a large solute of arbitrary form immersed in a molecular solvent, however, it is highly desirable to have a more detailed picture of the solvation shell around the solute. A typical example is an interface of a molecular liquid to a corrugated solid surface. The face of a crystalline lattice can be roughly modelled as an assembly of atoms interacting with solvent interaction sites by spherically symmetric potentials, for instance, comprising the electrostatic and Lennard-Jones (LJ) terms [1, 2, 3]. Even this simplified system constitutes considerable difficulties

169

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for numerical solution of the site-site RISM integral equations. With no possibility to apply periodic boundary conditions, a large cluster of solute atoms is used to adequately represent the infinite surface [4]. The spatial distributions of liquid over the solid surface can be estimated from the radial distributions between atoms of the solid and liquid in the superposition approximation. Moreover, an essential shortcoming of this approach is that it cannot be applied to surface models including potentials other than central symmetric. Advanced classical potentials fitting the results of ab initio molecular orbital calculations for electrolyte species at a metal cluster are essentially anisotropic. Spohr and Heinzinger [5] designed the potentials of water, and alkali and halide ions near Pt(100) and mercury surfaces as a set of exponential functions parametrized in terms of atom-atom distances and their projections on the surface plane. Based on the Spohr-Heinzinger form, Berkowitz and co-workers [6, 7, 8] modified the water-surface potentials so as to incorporate the lattice symmetry and corrugation explicitly. Zhu and Philpott [9, 10] proposed the potentials comprising an adjustable inverse power term, an electrostatic image, and an anisotropic LJ term to represent respectively chemical bonding, surface polarization, and effective many-body effects. Furthermore, ab initio methods for electronic structure produce electronic densities and potentials specified on a functional basis (typically Gaussian) or on a grid. Therefore, it is highly desirable to have an integral equation theory operating with three-dimensional (3D) potentials and distributions of the liquid at the surface. For similar reasons a 3D theory is of demand also for single atomic as well as molecular solutes. Moreover, it corresponds to the higher level of description of many-body effects for solvent at a solute of arbitrary shape. For liquids in contact with a face of a periodic crystal, the correlation functions can be represented as Steele’s expansion into sums of Fourier components periodic on the surface lattice [11, 12]. The OrnsteinZernike (OZ) integral equation then reduces to a linear matrix equation for the expansion coefficients dependent on the distance to the surface [13, 14, 15]. This approach, however, is not very convenient since the surface symmetry entirely determines a particular form of the periodic functional basis. Much more flexible is to describe the solute-solvent system within the supercell technique employing a set of 3D plane waves. The convolution in the OZ equation can be then approximated by means of the 3D fast Fourier transform (3D-FFT) procedure. For the first time, this approach was elaborated by Beglov and Roux [16] for numerical solution of the 3D-OZ integral equation with the hypernetted chain (HNC) closure for the 3D distribution of a simple LJ solvent around non-polar solutes

Three-dimensional RISM theory

171

of irregular shape, and by Ikeguchi and Doi [17] for the 3D-OZ equation with the Percus-Yevick (PY) closure for the sphericalized water of Pratt and Chandler [18] around a hydrocarbon molecule. To account for electrostatic interactions, Beglov and Roux [19] formulated the 3D-OZ integral equations with the mean spherical approximation (MSA) closure for the distribution of a liquid of spherical molecules with an embedded dipole around a polar solute, and solved them by using Blum’s method of rotationally invariant expansions [20, 21]. However, this approach is difficult in representation of such chemical specificities as hydrogen bonding and in description of strongly aspherical solvent molecules (for instance, alcohols). An alternative description of a molecular solvent in contact with a solute of arbitrary shape is provided by the 3D generalization of the RISM theory (3D-RISM) which yields the 3D correlation functions of interaction sites of solvent molecules near the solute. It was first proposed in a general form by Chandler, McCoy, and Singer [22] and recently developed by several authors for various systems: by Cortis, Rossky, and Friesner [23] for a one-component dipolar molecular liquid, by Beglov and Roux [24, 25] for water and a number of organic molecules in water, and by Hirata and co-workers for water [26, 27], metal-water [26, 28] and metal oxide-water [31] interfaces, orientationally dependent potentials of mean force between molecular ions in a polar molecular solvent [29], ion pairs in aqueous electrolyte [30], and hydration of hydrophobic and hydrophilic solutes: alkanes [32], polar molecule of carbon monoxide [33], simple ions [34], protein [35], amino acids and polypeptides [36, 37]. It should be noted that accurate calculation of the solvation thermodynamics for ionic and polar solutes in a polar molecular liquid requires special corrections to the 3D-RISM equations to eliminate the electrostatic artifacts of the supercell treatment employed in the 3D-RISM approach [30, 34]. The effective interparticle potentials change drastically as the system turns from a gaseous into liquid phase. Therefore, the most consistent description of a liquid-solid interface is achieved in a Car-Parrinello (CP) type approach [38, 39], when the classical potentials are determined from the first principles simultaneously with evaluation of molecular dynamics (MD) by solving the many-electron, all-particle problem using the Kohn-Sham formulation of the density functional theory (KS-DFT). Its big advantage consists in taking proper account of the mutual influence of the solid and liquid including many-body quantum effects, and moreover, of nonlinear coupling between the electron and classical subsystems. Price and Halley [40] have developed such an approach combining the KS-DFT in the local density approximation (LDA) for

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metal valence electrons and MD for liquid molecules for ambient water in contact with the Cu(100) surface. Another self-consistent combination of the KS-DFT/LDA for the jellium model of metal with Monte Carlo (MC) classical simulation for liquid was elaborated by Shelley et al to describe a mercury-water interface [41]. They examined also the effect of surface roughness introduced as regular corrugations corresponding to the (100) and (111) planes of a face centered cubic (FCC) lattice. Along with all its merits, the approach of CP type is limited in respect of the size of the system under consideration even to a greater extent than classical MD simulations. It quickly becomes problematic in description of processes that take place on a long time scale or that involve a great number of solvent particles. For instance, to describe the copper-water interface, Price and Halley [40] applied several substantial simplifications. The water molecule sites and the metal slab atoms were represented by local pseudopotentials with small Fourier series cutoffs. The latter were constructed for a singly valent element roughly modelled after copper. Outside the metal slab, the electron subsystem was contracted to the region near the metal surface. Finally, the total time used for collection of statistics was relatively small by MD standards, however, sufficient to obtain meaningful results. In view of the above limitations of the CP technique it is of much interest to couple the KS-DFT for valence electrons with, instead of MD, integral equation theory of liquids. Halley, Price and co-workers allowed for the response of the metal on the pseudopotential of solvent, however, neglecting the effect of the metal field on the solvent profiles taken as the stepwise function [42, 43] or the distribution of hard spheres in the mean spherical approximation (MSA) [44]. Patey and co-workers developed a self-consistent theory of a metal represented by the jellium model in contact with a liquid of hard spheres with embedded point dipole [45], and mixture of charged hard spheres and solvent of hard spheres carrying point dipoles [46] or dipole, quadrupole and octupole moments characteristic of a water molecule in the liquid phase [47]. They used the one-dimensional implementation of the KS-DFT/LDA for a jellium metal slab of a finite thickness [48, 49], and the OZ equation in the reference hypernetted chain (RHNC) approximation for liquid [21]. The orientational dependence of the density profiles of liquid near the surface was treated by using the method of rotational invariant expansions [20]. The mutual influence of the metal and liquid was accounted for with their mean electrostatic potentials averaged over the plane parallel to the surface. Numerical solution of both the KS-DFT/LDA and OZ/RHNC equations was based on the algorithms adapting the Newton-Raphson method for solution of the liquid structure integral


173

equations [168, 169, 170, 171]. A failing of the jellium model is that it predicts a negative surface energy for metals with a small Wigner-Seitz radius, for instance, transition elements [49]. As has been demonstrated, this is mainly owed to the absence of the discrete lattice structure [50]. Yamamoto and Kinoshita [51] considered a much more realistic, lattice model of the Pt(111) surface described with norm-conserving pseudopotentials in contact with a dipolar liquid by using the coupled KS DFT/LDA and OZ/RHNC equations. The calculation was performed in the cell with the two-dimensional periodicity of the (111) surface of an FCC lattice. Similarly to the previous works [45, 46, 47, 41], the quantum potential of liquid acting on metal valence electrons as well as the long-range part of the classical potential of metal acting on liquid particles were represented with their mean electrostatic potentials. Notice that a more consistent description is gained by allowing for the coupling between metal electrons and liquid molecules by means of pseudopotentials of molecules averaged over the classical distributions of the liquid [28]. The treatment of a molecular solvent by using the rotational invariants expansion becomes increasingly cumbersome with asphericity of solvent molecules because of the slow convergence of multipole expansions. A description of a molecular liquid may be achieved by applying a reference HNC integral equation theory (RHNC) to a central force (CF) model of liquid, which considers formation of molecules of liquid from constituent atomic species interacting via pairwise CF classical potentials. However, such potentials are usually strongly non-additive and very specific to a particular system. An example is the BJH flexible water model of Bopp, Jancso, and Heinzinger [52] constructed so as to reproduce in MD simulations the structural, thermodynamic, and intramolecular vibration properties of bulk ambient water. Ichiye and Haymet [53] showed that the integral equation theory for BJH water becomes successful on adding bridge diagrams in the RHNC approximation. Forstmann and co-workers [54, 55] described the CF water at a planar wall with the 9-3 LJ potential and electrostatic images by using the Lovett-Mou-Buff-Wertheim (LMBW) equation [56, 57, 58]. It requires at input the spatially inhomogeneous direct correlation functions of two atomic species at the surface, which are not easily obtainable and have been approximated by the direct correlation functions of the bulk liquid. However, this level of description seems to be inadequate for the CF water model because formation of water molecules stems from subtle cancellation of the strongly non-additive interatomic interactions. Consistent description of the CF water at an interface requires solution of the spatially inhomogeneous OZ integral equations [21] in the RHNC

174


approximation. For a lateral surface structure taken into account this becomes unfeasible at present due to computation time and memory limitations. The CF approach becomes even more hopeless for complex polyatomic solvents, such as alcohol. On the other hand, the RISM method which bypasses formation of solvent molecules yields the structure and thermodynamics for realistic models of a variety of molecular liquids in good qualitative agreement with experiment. The site-site RISM/HNC theory has been coupled with the ab initio molecular orbital (MO) theory in a self-consistent field (SCF) calculation of the electronic and solvation structure of a solute molecule immersed in molecular solvent, referred to as the RISM-SCF method [59, 60, 61]. Since the site-site treatment of the solute-solvent correlations involves the approximation of radial averaging, it constitutes a bottleneck of the RISM-SCF method. Although this approach yields reasonable results for the thermodynamics of solvation for many solute species and solvents [62], it lacks a 3D picture of the solvation structure for complex solutes and oversimplifies the contribution to the solvation properties from highly directed electron orbitals of the solute molecule. The 3D-RISM integral equation approach for molecular solvent is best suited to be coupled with the electronic structure methods using the planewave basis set. Kovalenko and Hirata [28] developed a selfconsistent combination of the KS-DFT and 3D-RISM methods for the electronic and classical properties of a metal-water interface. As another hybrid self-consistent approach, Sato, Kovalenko, and Hirata [33] combined the 3D-RISM theory with the ab initio multi-configurational selfconsistent field method (3D-RISM-MCSCF) to describe the hydration of a carbon monoxide molecule in ambient water. It avoids the effective charge fitting procedure that is required in the original RISM-SCF approach and treats the solute electronic distribution directly. These methods are much more suitable to represent non-classical, directed electronic distributions, such as those arising from π or δ-orbitals. The computed solvent distributions at the interface or around the solute molecule are comparable to those produced by the electronic methods combined with molecular simulations. Since the 3D-RISM approach requires much less computational time than simulation methods, it is possible to combine it with more sophisticated ab initio electronic structure methods, and to consider large systems and processes occurring at long time scale. In conclusion to these introductory notes, it is necessary to emphasize that the 3D-RISM theory of ion-molecular solutions provides an advanced description as compared to the reaction field continuum models of solvent. Treating the 3D-RISM integral equations is not much more computationally expensive than the electrostatic 3D boundary value prob-


175

lem. Besides the solvent electrostatics, the 3D-RISM theory evaluates the effects of solvent chemical specificities, such as hydrogen bonding, on the solvation properties and electronic structure of the solute. Unlike the dielectric continuum approach, it does not require empirical parameters, such as the dielectric permittivity and the size and shape of the cavity accommodating the solute in the dielectric continuum. Although they can be adjusted to match experimental data for solvation thermodynamics, this is very peculiar to a particular chemical system and hence not much transferable to new objects. For instance, it is very problematic to distinguish within the dielectric continuum solvent method between the structures of hydrogen bonding in water and alcohol. No doubt, molecular simulations constitute the ultimate tool in modelling. However, they are very demanding computationally, and thus not feasible for large systems and slow processes. Although less precise, the 3D-RISM method is by far more cost-efficient. It provides good qualitative description of the thermodynamics as well as detailed structure of molecular solutions and interfaces. Several examples are considered below to illustrate the 3D-RISM predictive capability for various chemical and biological species in molecular solution.

2.

3D-RISM integral equation

The solute-solvent 3D-RISM equation complemented with the 3DHNC closure was first obtained by Beglov and Roux [24] within the density functional method by reduction of the generalized closure of Chandler, McCoy, and Singer for nonuniform polyatomic systems [22]. In an alternative way, the 3D-RISM integral equation was derived by Kovalenko and Hirata [28] from the six-dimensional (6D), molecular OZ equation for a solute-solvent mixture of rigid molecules at infinite dilution. The latter is written as [21] (4.1) huv (r12 , Ω1 , Ω2 ) = cuv (r12 , Ω1 , Ω2 ) 1 dr 3 dΩ3 cuv (r13 , Ω1 , Ω3 ) ρv hvv (r32 , Ω3 , Ω2 ), + Ω where the superscripts u and v denote solute and solvent species, respectively, huv (r12 , Ω1 , Ω2 ) are the total correlation functions (TCFs), and cuv (r12 , Ω1 , Ω2 ) the direct correlation functions (DCFs) dependent as separation r12 of molecules on the orientations Ω1 and Ω2 as well of the corresponding species, Ω = dΩ is the normalization factor of the orientational integral, and ρv is the solvent number density. In the case of multi-component solvent the values of the solvent superscripts v vary over the solvent species, and the summation convention applies in addition to the integration dr 3 dΩ3 .

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The standard, site-site RISM theory performs full orientational reduction of the molecular OZ equation (4.1). As distinct, the 3D-RISM approach averages out just the solvent molecular orientations but keeps the orientational description of the solute molecule. This partial averaging reduces by definition the solute-solvent 6D-TCF huv (r12 , Ω1 , Ω2 ) to the 3D-TCFs of solvent interaction sites γ around the solute molecule, 1 uv uv h γ (r 1γ ) ≡ h γ (r1γ , Ω1 ) = dΩ2 huv (|r 1γ − r 2γ |, Ω1 , Ω2 ), (4.2) Ω where r 1γ = r γ −r1 is the intermolecular vector from the solute molecule origin 1 to solvent site γ, and r 2γ = r γ − r 2 is the intramolecular vector in a solvent molecule from its origin 2 to site γ with solvent molecule orientation Ω2 to be averaged over. Accordingly, the Fourier transforms of the 3D site TCFs are obtained by orientational reduction of the Fourier transform of the 6D-TCF, 1 uv dΩ2 dr 1γ eik·r 1γ huv (|r 1γ − r 2γ |, Ω1 , Ω2 ) h γ (k) = Ω 1 dΩ2 huvγ (r 1γ ) eik·r 2γ , (4.3) = Ω with the phase factors eik·r2γ arising due to the spatial displacements of the interaction sites γ from the molecular origin 2. To evaluate analytically the orientational integration over the orientation of the “third particle” Ω3 in (4.1), one has to assume that the solute-solvent 6D-DCF cuv (r12 , Ω1 , Ω2 ) can be decomposed into partial contributions of sites α of solvent molecule 2, cuvα (r 1α ). (4.4) cuv (r12 , Ω1 , Ω2 ) = α

This decomposition constitutes the basic assumption of the 3D-RISM theory. It is a reasonable approximation because the long-range asymptotics of the 6D-DCF reduces to the interaction potential between the solute and solvent molecules, cuv (r12 , Ω1 , Ω2 ) ∼ −uuv (r12 , Ω1 , Ω2 )/(kB T ) a classical model is assumed to be additive, for r12 → ∞, which in uuv (r uuv (r12 , Ω1 , Ω2 ) = α 1α ). Written in the reciprocal space it takes α

the form cuv (k, Ω1 , Ω2 ) =

dr 12 eik·r 12 cuvα (r 1α )

α

=

α

cuvα (k) e−ik·r 2α .

(4.5)

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Next, the molecular OZ equation (4.1) is transformed into the reciprocal space and averaged over solvent orientation Ω2 . Substitution of (4.3) and (4.5) into (4.1) gives huv γ (k) =

α

+

1 Ω

cuvα (k)

1 Ω2 α

dΩ2 eik·(r 2γ −r 2α )

(4.6)

−ik ·r 3α dΩ2 dΩ3 cuv α (k) e

× ρv hvv (k, Ω3 , Ω2 ) eik·r 2γ .

(4.7)

The orientational average of the phase shift exponent in the first term is the familiar intramolecular correlation matrix 1 Ω

dΩ2 eik·(r 2γ −r 2α ) =

v ) sin(klαγ vv ≡ ωαγ (k) v klαγ

(4.8)

of a rigid molecule with site separations lαγ (implying lαα ≡ 0). In the convolution term, the dependence on solvent orientation Ω 3 is now factored out of the 3D-DCFs cuvα into the phase factor e−ik·r 3α . This allows one to carry out the integration over the orientation of the “third” particle, which gives the site-site radial TCFs of solvent, hvv αγ . Finally, one obtains the 3D-RISM equation written in the reciprocal space as vv (k) + cuvα (k)ρv hvv huvγ (k) = cuvα (k)ωαγ αγ (k),

(4.9)

where the summation convention applies for repeating site indices (and solvent species in the case of multi-component solvent). The 3D site uv total and direct correlation functions huv γ (r) and c γ (r) are specified on a 3D uniform grid in a rectangular supercell to enable calculation of their forward and backward Fourier transforms by using the 3D fast Fourier transform (3D-FFT) technique [63]. The full orientational averaging applied along similar lines to the molecular OZ equation for pure solvent leads to the site-site OZ equation for the solvent correlations, which can be written in the direct space as vv vv vv vv vv v vv hvv αγ (r) = ωαµ (r) ∗ cµν (r) ∗ ωνγ (r) + ωαµ (r) ∗ cµν (r) ∗ ρ hνγ (r), (4.10)

where ∗ means convolution in the direct space and summation over repeating site (and species) indices.

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3.


Closures for the 3D-RISM theory

The closure relations to the 3D-RISM equation are constructed in analogue to those of the site-site RISM theory. The hypernetted chain (HNC) closure to the RISM equations for the radial site-site correlation functions appeared to be most successful and adequate for the description of the structure and thermodynamics of polar liquids [64, 65, 66, 67]. The molecular HNC closure is written as [21] ( guv (r12 , Ω1 , Ω2 ) = exp −βuuv (r12 , Ω1 , Ω2 ) (4.11) ) uv uv + h (r12 , Ω1 , Ω2 ) − c (r12 , Ω1 , Ω2 ) , where guv (r12 , Ω1 , Ω2 ) = huv (r12 , Ω1 , Ω2 )+1 is the solute-solvent molecular distribution function, and β = 1/(kB T ) in the inverse temperature that scales energies. In the extended atom limit the molecular TCF huv (r12 , Ω1 , Ω2 ) can be decomposed into the partial site contributions similarly to the 3D-DCFs and the interaction potential. This yields the 3D-HNC closure as ( ) uv uv uv (4.12) guv γ (r) = exp −βu γ (r) + h γ (r) − c γ (r) , uv with the 3D site distributions guv γ (r) = h γ (r) + 1. For molecular systems with strong site-site attraction the 3D-HNC approximation (4.12) can cause divergency. The HNC closure is known to give pathological predictions in the theory of associating liquids for mixtures of species with highly asymmetric size ratio [68], and presents difficulties in convergence of the OZ/HNC equations for associating liquidsolid distributions at a crystalline surface with a strong adsorption potential [15]. The 3D-RISM/HNC equations for a metal-water interface with the metal electron distribution forming a surface dipole has a similar problem with convergence [28]. To obviate this obstacle Kovalenko and Hirata elaborated a partial linearization of the HNC closure [28, 29, 30] which will be referred to as the KH approximation. The 3D-KH closure is written as ( ) exp duv for duvγ (r) ≤ 0, uv γ (r) (4.13) g γ (r) = uv 1 + d γ (r γ ) for duvγ (r) > 0, uv uv duvγ (r) = − βuuv γ (r) + h γ (r) − c γ (r).

It combines the HNC approximation [21] in the spatial regions of density depletion where guv γ (r) < 1, and the linearization of the entire exponent in the enrichment regions of guv γ (r) > 1. The latter enforces uv (r) > 1, which thus reduces to the mean (r) when g cuvγ (r) = −βuuv γ γ


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spherical approximation (MSA) [21] applied to the spatial regions of density enrichment. The distribution function and its first derivative are continuous at the junction points, duvγ (r) = 0, by construction. Notice that this way of mixing the two approximations is distinct from the modified HNC closure with cross-over of Foiles, Ashcroft, and Reatto [69, 70] that switches continuously from hard sphere behavior at small r to mean-spherical run at larger r. The MSA is known to be appropriate for the description of critical fluids, and their phase equilibria and critical lines [71, 72], whereas the HNC solutions pathologically diverge close to the coexistence curve [70, 73]. On the other hand, the MSA and its extensions produce unphysical negative regions of the distribution functions in the cases of strong interaction potentials and high liquid density. The KH approximation efficiently combines the advantages of these two closures. High peaks of short-range structure as well as longrange enrichment tails of the correlations h(r) ∼ r −1 exp(−r/ζ) with length ζ → ∞ in the critical regime are treated within the MSA, while its HNC counterpart in the depletion regions ensures that the physical condition of g > 0 is satisfied. The KH approximation shows further advantage in the context of RISM theory. The RISM/HNC equations strongly overestimate clustering and become divergent for fluids comprising charged species in a vapor phase and near critical lines, as association of fluid molecules increasingly strengthens. For instance, Lue and Blankschtein [74] described the liquid-vapor phase diagram of water by using the RISM as well as the Chandler-Silbey-Ladanyi (CSL) integral equation [21] with a corrected closure comprising the HNC part and the first bridge diagrams that contribute at zeroth order in the density. Because of problems with the bridge diagrams for the strong hydrogen bonding attraction, they treated the vapor branch of water by means of the second virial coefficient. In contrast, the RISM/KH theory performs well over a wide range of fluid density from gas to liquid. The KH approximation has been shown to be adequate for treatment of phase transitions of strongly associating molecular fluids, in particular, liquid-vapor coexistence of water [76], methanol, and hydrogen fluoride [77]. The RISM/KH theory provides appropriate description for the association structure in a mixture of water with tert-butyl alcohol possessing strong amphiphilic character, which substantially changes with the molar fraction of the constituents [78]. At the same time, the partial linearization (4.13) does not affect the structure of ambient water significantly as compared to the common HNC closure [28]. The first peak of the oxygen-oxygen distribution, for instance, becomes somewhat lower but wider, and so both the KH and HNC approximations yield very similar results for the coordination

180


numbers of ambient water. It is worth noting also that the KH approximation improves the RISM predictions for the water compressibility. For ambient water, the RISM/KH theory yields χT = 5.2 × 10−5 bar−1 in considerably better agreement with experiment [75] giving 4.5 × 10−5 bar−1 , as compared to the value 7.3 × 10−5 bar−1 following from the RISM/HNC approach. The chemical potential of solvation following from the 3D-RISM approach (4.9) in the 3D-HNC approximation (4.12) is derived similarly to the closed analytical expression obtained within the OZ/HNC theory by Morita and Hiroike [79] and generalized to the RISM/HNC equations by Singer and Chandler [80]. It is readily extended to the 3D-RISM/HNC equations as [28] 1 uv 1 ( uv )2 HNC v uv uv h (r) − c γ (r) − h γ (r) c γ (r) . dr ∆µsolv = ρ kB T 2 γ 2 γ (4.14) For the KH approximation (4.13), the solvation chemical potential keeps an analytical form too: 1 ( uv )2 ( uv ) KH v h (r) Θ −h γ (r) dr ∆µsolv = ρ kB T 2 γ γ 1 uv uv h (r) − (r) c (r) , (4.15) −cuv γ γ 2 γ where Θ(x) is the Heaviside step function which puts the term h2 in effect only in the regions of density depletion. While expression (4.14) is simply a 3D generalization of Singer and Chandler’s formula, the existence of the closed analytical form (4.15) in the case of the KH closure (4.13) is a non-trivial result. An exact analytical differential d∆µsolv independent on the path of the thermodynamic integration over the solute-solvent interaction does not exist for the MSA or PY closures. This is related essentially to the contribution from the repulsive core edge spreading in the process of “switching on” the interaction between fluid particles. Notice, however, that an analytical expression for the chemical potential can indeed be obtained for the PY closure by assuming a particular thermodynamic integration path with a linear functional dependence between the interaction potential and distribution functions [81]. In the KH closure the repulsive core region is obtained in the HNC approximation possessing the exact differential, whereas the MSA treatment regions are smoothly joint. The partial linearization (4.13) in the enrichment regions is applied to the whole exponent exp(−βu + h − c), including the entire interaction potential. This allows one to identify

181


the exact differential and the analytical form (4.15) for the chemical potential (see Appendix). Since the site-site RISM equation (4.10) is derived by full orientational reduction of the molecular OZ integral equation, the site-site HNC closure can be seen as an approximation to the corresponding orientational average of the molecular HNC closure. The site-site HNC closure, however, treats different sites of a molecule as entirely independent, non-bonded species, since it does not contain cross-terms. Nevertheless, it provides the right asymptotics of the site-site direct correlation functions at long range, whereas the influence of other sites through ab the intramolecular bonds represented by the intramolecular matrix wαγ yields a qualitatively correct short-range solvation structure. On the other hand, the interrelation between the site distributions due to the steric constraints of the intramolecular bonds is imperfect in this approximation. This results in the well documented inconsistency of the RISM/HNC theory in treatment of so called auxiliary sites [82, 83]. It becomes especially noticeable for hydration of large compact molecules including neutral or weakly charged atoms buried inside the solute and inaccessible to solvent. Moreover, the separation between the hydrogen and oxygen peaks of water around an ion in aqueous solution is yielded by the RISM/HNC approach to be somewhat larger than the OH intramolecular bond length of a water molecule [84, 85, 67]. A similar shortcoming arises in the 3D-RISM description for a pair of anions in water, discussed below in Section 6. For a strongly attractive potential between the Cl− -Cl− ion pair solute and the hydrogen site of a water molecule, the corresponding peaks of the water oxygen and hydrogen site distributions can be more apart than it follows from the length of the intramolecular OH bond [30]. Such an unphysical peak has been observed also for the water hydrogen density distribution near the negatively charged carbonyl oxygen of the N -methylacetamide solute [24]. Lue and Blankschtein [86] studied the structural and thermodynamic properties of hydrocarbons in water at infinite dilution following from the site-site RISM integral equations complemented with the closures other than the HNC approximation. They found that the quantitative predictive capabilities of the RISM theory are substantially improved by adapting the Martynov-Sarkisov (MS) closure [87] or its generalization, the Ballone-Pastore-Galli-Gazzillo (BPGG) closure [88] proposed originally for simple liquids. The 3D-BPGG closure can be similarly constructed as ( ) uv (r) = exp −βu (r) + 1 + s huvγ (r) − cuvγ (r) guv γ γ

1/s

−1 .

(4.16)

182


For s = 2 it becomes the 3D-MS closure, and for s = 1 reduces to the 3D-HNC approximation. An essential weakness of both the sitesite BPGG and 3D-BPGG closures is that they can be used only for non-polar solute species. In the case of a solute including charged sites, the solute-water site direct correlation function acquires the asymptotics of the 3D solute-solvent site electrostatic potential, cuvγ (r) ∼ −βuuv γ (r), usually very strong at intermediate separations r. They cancel out in the HNC closure (4.12). On the contrary, the BPGG approximation (4.16) mistreats this electrostatic asymptotics, which results in divergence. The 3D-RISM/HNC equations (4.9) and (4.9) appropriately treat the excluded volume of the solute regarded as a whole. However, the imperfectness persists for solvent molecules since their orientation is reduced. Cortis, Rossky, and Friesner [23] proposed to perform explicit orientational averaging of the Boltzmann factor for a short-range repulsive part of the initial orientationally dependent interaction potential. The 3D-HNC closure thus modified to include multi-site correlations between solvent sites around the solute is written as 1

2 uv uv(r) βu α (r α ) g γ (r γ ) = exp − Ω2

α

× exp −βuuv(a) (r γ ) + huvγ (r γ ) − cuvγ (r γ ) , γ

(4.17)

where the site potential is broken up into a short-range core repulsion and a long-range attraction, uv(r) uv(a) (r γ ). (4.18) uuv γ (r γ ) = u γ (r γ ) + u γ 4 3 ) ( accounts for the orientationally averaged The factor exp −βuuv(r) Ω repulsion between the entire molecular cores. This considerably improved the structure predictions for polar as well as non-polar models of one-component liquids of N2 and HCl-like molecules [23]. With the appropriate choice of the subdivision (4.18), such a repulsive bridge correction yields qualitatively correct, improved results for both the structure and thermodynamics of hydrophobic hydration of polyatomic solutes [23, 25, 32]. Another version of the 3D-HNC closure is possible in analogy with the so called RISM-2 theory [89]: 1

2 ( ) uv uv βu α (r α ) exp huvγ (r γ ) − cuvγ (r γ ) . (4.19) g γ (r γ ) = exp − α

Ω2

The RISM-2 approach yields a significantly improved description of the dielectric properties of a polar liquid [90, 91]. However, it works well in

183


the low density region but fails for a higher density and strong polarity of liquid [91]. The orientational average of the entire core repulsion can be decoupled into those of the partial site repulsive potentials [32], 1

2

2 1 uv(r) uv(r) exp −βu α (r α ) u α (r α ) , (4.20) exp −β Ωvγ

α

Ωvγ

α

where the averages of both the entire and site repulsions depend on the position r γ of site γ. This relation is exact for diatomics, but for the larger number of sites it treats different site pairs of the solvent molecule independently. Nevertheless, it is reasonable to use the approximation (4.20) since the hydrogen-hydrogen correlation is not crucial in the case of water solvent. The orientational averages of the site contributions in (4.20) are obtained simply by convolving them with the intramolecular vv . With the approximation (4.20), the closure (4.17) takes the matrix ωαγ form

uv uv uv uv(r) (r) = exp −βu (r) + h (r) − c (r) + b (r) (4.21) guv γ γ γ γ γ uv(r)

with the 3D repulsive bridge correction (3D-RBC) b γ (r) specified as

*

+ vv uv(r) (r) = (r) ∗ exp −βu (r) . (4.22) ω exp buv(r) γ αγ α α=γ

In the approximation (4.17) or (4.21)–(4.22) the solvation chemical potential is not available in a closed analytical form. It can be obtained by the familiar Kirkwood recipe of numerical thermodynamic integration over the coupling parameter of the solute-solvent interaction. In the case of the solute-solvent potential comprising the LJ and Coulomb terms, it is calculated by integration over the LJ diameter and charges [25, 32]. This requires repeated solution of the time consuming 3D-RISM integral equations at each step of integration over the “switching” parameter, usually 10 to 25 increments to provide good convergence [25]), which drastically increases computational load. To reduce computational burden, the excess chemical potential following from the repulsive bridge correction of form (4.22) can be approximated by means of thermodynamic perturbation theory as [32] ∆µsolv = ∆µHNC solv + ρv kB T

γ

Vcell

(4.23)

dr guv(HNC) (r) exp buv(r) γ γ (r) − 1 ,

184


where the contribution ∆µHNC solv is obtained by the HNC expression (4.14) from the correlation functions obtained in the absence of the bridge uv(r) uv(HNC) (r) is the distribution function in correction b γ (r) = 0, and g γ the absence of the bridge correction. Calculation of the excess chemical potential (4.23) thus requires to solve just the 3D-RISM/HNC equations as the repulsive bridge correction (4.22) does not involve the correlation functions. The integration in the expressions (4.14), (4.15), and (4.23) is performed over the supercell volume, which should be taken sufficiently large to cover two to three solvation shells. In the case of a solute containing charged groups, however, the 3D site direct and total correlation functions cuvγ (r) and huvγ (r) calculated from the 3D-RISM equations must be corrected for the artifacts of the supercell treatment, as explained in Section 6.

4.

Hydrophobic hydration

The RISM/HNC approximation, unlike other closures, is well suited for the description of liquid of molecular species carrying site charges. However, its predictions are rather poor for the thermodynamics of hydrophobic hydration of such neutral solutes as hydrocarbons [86] or rare gases [92]. It systematically overestimates the hydrophobicity, gives a wrong positive dependence of the excess chemical potential on hydrophobic solute size, and the incorrect sign of the excess hydration enthalpy. As discussed above, these drawbacks originate essentially from the fact that the RISM/HNC approximation underestimates multi-site correlations of water sites near the solute, which results in too close approach of water hydrogens to the hydrophobic solute and thus in too strong ordering in the hydration structure. This leads to substantial overestimation of the entropic component in the free energy of hydrophobic hydration. Lue and Blankschtein [86] have obtained substantial improvement for the hydration chemical potential of alkanes in ambient water predicted by the solute-solvent site-site RISM equation complemented with the MS or BPGG closure. However, these closures fail for solutes comprising both hydrophobic and charged hydrophilic groups. Du, Beglov, and Roux [25] have proposed and parametrized an empirical choice of the bridge functions in the form of the sum of exponents with parameters adjusted to fit experimental data for the hydration chemical potential, and parametrized them for a number of polar as well as non-polar organic molecules in water. A simple bridge correction appropriate for the hydrophobic hydration can be constructed from the orientationally averaged Boltzmann factor

185


20

8

3 D -R IS M /H N C -R B C -T P T 3 D -R IS M /H N C R IS M /H N C -R B C -T P T R IS M /H N C -R B C R IS M /H N C M D sim u latio n E x p erim en t

15

R IS M /H N C -R B C R IS M /H N C -R B C -T P T R IS M /H N C M D sim u la tio n E x p e rim e n t

4

∆µ (k c a l/m o l)

∆µ (k c a l/m o l)

6

10

5 2 0 0

Ne

Ar

Kr

Xe

Figure 4.1. Excess chemical potential of hydration of the rare gases in ambient water.

M e th an e E th a n e P ro p a n e B u ta n e

Figure 4.2. Excess chemical potential of hydration of the alkanes in ambient water.

for the solute core repulsion, in particular in the RBC form (4.22). Reasonable results are obtained for the short-range core repulsion specified as the sum of the r −12 terms of the 12-6 LJ site-site potentials,

uv 12 σαγ uv(r) uv(r) uαγ (rαγ ) = 4 αγ . (4.24) u γ (r γ ) = rαγ α∈u α∈u uv(r)

The orientational average (4.22) of the oxygen core repulsion u O (r γ ) prevents an approach of water hydrogens too close to the solute core. A significant part of the RBC effect providing the correct dependence of the hydration chemical potential on the solute size comes from the tail of the r −12 potential (4.24). On the orientational averaging in (4.22), it yields a soft repulsive tail in the bridge corrections for both oxygen and uv(r) uv(r) hydrogen sites b O (r γ ) and b H (r γ ). Figure 4.1 illustrates the RISM approach for the chemical potential of hydration of rare gas atoms in ambient water. As expected, the RISM/HNC equations overestimate the hydrophobicity of the rare gases, and give a positive dependence of the hydration chemical potential on

186


2

g A r-O

Ar - O

1

0

0

2

4

1 .5

r (Å )

6

8

g A r-H

1 .0

0 .5

Ar - H 0 .0

0

2

4

r (Å )

6

8

Figure 4.3. Radial argon-water oxygen and hydrogen distribution functions following from the RISM/HNC and RISM/HNC-RBC approaches.

the atom size, which is in contradiction with the tendency observed in the MD simulations for the same solute potential model [93], and with the experiment [94]. Notice, however, that it has the right sign and remains less than 10 kcal/mol, which is much smaller in magnitude than typical values for ionic solutes [84, 67], and can be considered as a first estimate for the hydration chemical potential. Similar problems arise also in description of solvation of rare gases in salt aqueous solutions [92]. The HNC-RBC closure (4.22) essentially improves the predictions of the RISM theory. The hydration chemical potential calculated by using the thermodynamic integration over the LJ diameter of the solute fits the simulation results up to 0.5 kB T . Although somewhat less accurate, the TPT treatment (4.23) gives the hydration chemical potential in quite good agreement with the simulation results for both magnitude and the solute size dependency. The difference corresponds to the second order effect of the TPT. Decomposition of the hydration chemical potential into the energetic and entropic components reveals that the RISM/HNC approximation strongly overestimates the latter [32]. With the RBC applied, the agreement with the simulations is drastically improved for the entropic term and somewhat refined for the average solute-water poten-


187

tial energy. Figure 4.3 shows the modification introduced by the RBC to the argon-water distribution functions. With the RBC the ascending slopes of the first peaks of both the oxygen and hydrogen distributions fit the simulation results much better. The water density in this contact region gives significant contribution to the free energy dependence with respect to the solute size. The positions of the first and second maxima also get closer to the simulation ones. The lower height of the peaks is offset by the shallower first maxima, and so the agreement with the simulations is much better for the coordination numbers of the first hydration shell [32]. Calculation of the excess chemical potential of hydrophobic hydration of polyatomic solutes is illustrated on the hydration of hydrocarbons. Figure 4.2 exhibits the hydration chemical potentials for the molecules of several alkanes modelled with the united-atom OPLS force field of Jorgensen and co-workers [95, 96]. For simplicity, butane is considered only in its trans isomer. Much as for the rare gas solutes, the site-site RISM/HNC equations without the bridge correction overestimate the hydrophobicity of hydrocarbons and give rapid increase of the hydration chemical potential with the hydrocarbon chain length. The 3DRISM/HNC approach exhibits somewhat smaller but still too big discrepancy with the simulation results [96]. With the RBC applied, both the site-site and 3D-RISM theories show drastic improvement. They correctly predicts the order of increase of the hydration chemical potential for the alkane series found in the simulations, ∆µethane < ∆µmethane < ∆µpropane < ∆µbutane , with the ethane molecule most water-soluble. The difference between the hydration chemical potential found in experiment [137] is smaller, and the order for methane and propane is reversed, ∆µpropane < ∆µmethane . The experimental order is reproduced when the all-atom model of the hydrocarbon groups is used, as has been shown in the simulations [96]. As expected, the 3D-RISM/HNC-RBC approach fits the simulation values better because it treats the shapes and orientation of polyatomic solutes explicitly. In the case of the site-site RISM approach, the bridge correction to the HNC closure is constructed similarly to (4.21) as [32]

uv uv uv uv(r) (r) = exp −βuuv gαγ αγ (r) + hαγ (r) − cαγ (r) + bαγ (r)

(4.25)

uv(r)

with the site-site RBC bαγ (r) specified as

*

+ vv uv(r) (r) = (r) ∗ exp −βu (r) . ω exp buv(r) αγ µγ αµ µ=γ

(4.26)

188


In the site-site HNC-RBC closure (4.25) the averaging is performed only over the orientation of solvent molecules. If the averaging over the solute orientation is added, the exclusion effect of the polyatomic solute on the distributions of water sites is significantly overestimated. Similarly to (4.23), the excess chemical potential can be obtained in the TPT approach as ∆µsolv = ∆µHNC solv + ρkB T

αγ

∞

4π

(4.27)

(HNC) r 2 dr gαγ (r) exp buv(r) αγ (r) − 1 ,

0

Notice that if the convolution in Eq. (4.26) is evaluated by means of the nonlinear fast Fourier transform a cutoff should be applied to the convouv(r) lution results so as to eliminate the effect of numerical noise in bαγ (r) at large r on the integral value in Eq. (4.27). Although less accurate than the 3D-RISM/HNC-RBC theory, the site-site RISM/HNC-RBC approximation is much more fast and cost efficient. It yields the hydration chemical potential quite close to the simulation data and correctly predicts the order of ∆µ with the hydrocarbon chain length (Figure 4.2). The use of the TPT approximation (4.27) introduces a comparably small error, and holds ∆µ in qualitative agreement with the simulations. In application to proteins the 3D-RISM theory was used to estimate conformational stability of Met-enkephalin in ambient water [35]. The hydration free energy was calculated for its several representative conformations [97, 98]: (1) conformation corresponding to the lowest energy in the gas phase as determined by the multicanonical MC algorithm [99]; (2) one of the trial folded conformations obtained in the course of multicanonical MC search; (3) shrunk conformation corresponding to the backbone dihedral angles given in [100] which are determined from NMR experiments for Met-enkephalin in aqueous solution with 50 mM sodium dodecyl sulfate as buffer; (4) fully extended one as implied by the NMR results [100]; and (5) conformation corresponding to the lowest total energy of Met-enkephalin in pure water found by means of the hybrid algorithm of the MC simulated annealing and the 1D-RISM approach [98]. The total energy of a conformation is defined as the sum of the potential energy of the solute conformation and the hydration free energy, Atot = Econf + µsolv . The former component is minimal for conformation 1 by its construction. In all the five conformations the hydration chemical potential yielded by the 1D- and 3D-RISM/HNC theories is essentially positive, corresponding to hydrophobic hydration. However, both the 1D- and 3D-HNC approximations overestimate the hydrophobicity, the latter giving better results. Table 4.1 makes a com-

189


Table 4.1. Total free energy (kcal/mol) of five conformations of Met-enkephalin in ambient water, following from the 1D- and 3D-RISM integral equation approaches in the HNC-RBC-TPT as well as HNC approximations.

conf.

Econf

1D-RISM/ HNC µsolv Atot

(1) (2) (3) (4) (5)

−12.0 12.0 −2.5 0.8 9.7

196.6 178.0 202.7 176.8 164.0

184.8 190.2 200.2 177.6 173.7

3D-RISM/ 1D-RISM/ 3D-RISM/ HNC HNC+RBC-TPT HNC+RBC-TPT µsolv Atot µsolv Atot µsolv Atot 84.0 72.5 79.0 80.8 71.3

72.0 84.5 76.5 81.6 81.0

15.7 −2.0 24.0 −5.9 −22.7

3.7 10.0 21.5 −5.1 −13.0

40.6 26.4 36.6 30.7 18.8

28.6 38.4 34.1 31.5 28.5

parison of the energy for these conformations, following from the 1D- as well as 3D-RISM approaches. In all the approximations the potential energy of the solute in the intermediate extended conformation 2 strongly increases and overruns the significant decrease in the hydration chemical potential. The 1D-RISM/HNC approach yields the maximal decrease of µsolv in conformation 5. It also gives some increase of µsolv in conformation 3, although the latter is characterized by the dihedral angles found in experiment. This is reasonable since these angles are obtained in the presence of the buffer which, presumably, alters the stable conformation of Met-enkephalin in solution significantly. Conformation 5 thus appears to be the most stable as predicted by the 1D-RISM/HNC theory. The 1D-RBC-TPT treatment (4.25)–(4.27) significantly decreases the hydration chemical potential, bringing the total energy of the most favorable conformations 4 and 5 to negative values. This is similar to the tendency observed for the 1D-RISM/HNC+RBC-TPT theory for alkanes which essentially improves the calculated values of the hydration chemical potential, however underestimates it to some extent. However, the order of Atot in different conformations is not changed, and the difference between them is even emphasized. The 3D-RISM approach yields a detailed picture of the hydration shell around the functional groups of Met-enkephalin [35]. The 3DRISM/HNC theory gives a decrease of µsolv for all conformations 3, 4, and 5 as compared to µsolv for conformation 1. However, none of them is sufficient to overrun the increase in Econf . In conformation 5 the 3D-RBC-TPT correction (4.21)–(4.23) deepens the gain in µsolv which appears to be slightly more than the loss in Econf . This result is expected to be further improved by optimization of the short-range potential (4.24) in the 3D-RBC correction (4.22). It is, however, too early to make

190


any conclusive statement regarding performance of the 3D-RISM/HNCRBC-TPT theory. In principle, the total free energy of a conformation, Atot , should include also a chain entropy contribution. The latter is essentially zero for the global-minimum conformation 1, but amounts to a considerable negative value for extended conformations. A comparison to experiment requires careful parametrization of the interaction potentials for the peptide sites. Unfortunately, both these issues cannot be conclusively addressed in the absence of reliable molecular simulations of protein with explicit water.

5.

Potential of mean force between molecular species in solution

The solvent mediated potential of mean force (PMF) between molecular solutes is defined as

uu uu uu (4.28) w (r12 , Ω1 , Ω2 ) ≡ w (r 12 , Ω12 ) = −kB T ln g (r 12 , Ω12 ) For two solute molecules immersed in solvent at infinite dilution, the 6D distribution function guu (r 12 , Ω12 ) between the solutes, dependent on their relative orientation Ω12 can be obtained from the 3D solute-solvent site correlations [29]. The molecular OZ equations for the solute-solute correlations has the form (4.29) huu (r12 , Ω1 , Ω2 ) = cuu (r12 , Ω1 , Ω2 ) 1 dr 3 dΩ3 cuv (r13 , Ω1 , Ω3 ) ρv hvu (r32 , Ω3 , Ω2 ), + Ω where the solute index u denotes the first and second solute species. Reduction of solvent orientations in Eq. (4.29) leads to the set of coupled integral equations: RISM equation (4.10) for the site-site correlations of pure solvent, 3D-RISM equation (4.9) for the 3D solute-solvent correlations, and 6D-RISM equation for the 6D solute-solute correlations, 5 12 huv (r), (4.30) huu (r, Ω12 ) = cuu (r, Ω12 ) + cuv (r) ∗ ρv Ω γ

γ

5 12 rotates the 3D site correlation functions huv (r) of the second where Ω γ solute through the angle of the relative orientation of the solutes. Complemented with the site-site HNC closure, Eq. (4.10) produces the sitesite correlations of bulk solvent. They are then input to the 3D-RISM equations (4.9) with the 3D-HNC closure (4.12) giving the 3D correlations of solvent sites around each of the solutes. Employing the molecular HNC closure to the solute-solute equation (4.30),

guu (r 12 , Ω12 ) = exp −βuuu (r 12 , Ω12 ) + huu (r 12 , Ω12 ) − cuu (r 12 , Ω12 ) , (4.31)


191

immediately yields their PMF as v5 uv wuu (r, Ω12 ) = uuu (r, Ω12 ) − kB T cuv γ (r) ∗ ρ Ω12 h γ (r),

(4.32)

uuu (r

where 12 , Ω12 ) is the interaction potential between the two solutes. Here the effect of solvent correlations on the PMF between two solutes reduces to the convolution of the 3D solute-solvent site correlations of the solutes. It is calculated by applying the 3D-FFT to the 3D site correlation functions, first rotated according to the relative arrangement of the solutes. Notice that instead of the HNC closures, the KH approximation can be employed for each of the integral equations (4.10), (4.9), and (4.30). The use of the 3D-FFT technique introduces the supercell periodicity artifact into the electrostatic component of the PMF, which requires special correction. The energy of the electrostatic interaction between solutes comprising a set of molecular sites is written as 1 u u 4π k2 η2

(r , Ω ) = Q Q exp ikr − uuu 12 αγ α γ 2 (C) 12 Vcell k 4 k=0 αγ /η) erfc(rαγ , (4.33) + rαγ αγ 5 12 r γ − r α is the vector between molecular sites α of where r αγ = r 12 + Ω solute 1 and γ of rotated solute 2, Quγ and Quα are the site charges of the solutes, Vcell is the supercell volume, the summation over wavevector k is carried out over its 3D grid values by means of the 3D-FFT, and η is the broadening length parameter to be adjusted to optimize convergence. Since η is much smaller than the supercell size, direct-space summation of the compensating charge potential over the periodic images of the solutes is replaced in (4.33) by its tabulation on the supercell grid, with determined subject to the minimum the site separation distance rαγ image convention [101]. In (4.33), the positions of the molecular sites 5 12 r γ , are those r γ used to synthesize the soluteof the second solute, Ω solvent site potential uuvγ (r) but turned through angle Ω12 about the “origin” of the second solute. The dielectric constant of a polar liquid can be obtained via the low-k behavior of the site-site correlation functions. The RISM approach is known to yield essentially trivial results for the dielectric constant of polar liquid [102, 103]:

RISM = 1 + 4πβρv µ2 /3,

(4.34)

where µ is the dipole moment of a solvent molecule. At infinite dilution, a simple way to correct the PMF between two ionic solutes for the

192


phenomenological static dielectric constant of solvent expt is to replace the theoretical electrostatic potential with the phenomenological one. This yields good results for pairs of atomic ions in a polar molecular solvent [66] and for alkali halides in water [84]. The correction to the orientationally dependent PMF (4.32) is w ˜uu (r, Ω12 ) = wuu (r, Ω12 ) −

uuu (C) (r, Ω12 )

RISM

+

u ˜uu (C) (r, Ω12 )

expt

,

(4.35)

u u where u ˜uu αγ Qα Qγ /rαγ is the electrostatic interaction po(C) (r, Ω12 ) = tential between the solutes not distorted by the supercell periodicity. The replacement (4.35) simultaneously eliminates the periodicity artifact of the supercell treatment affecting the electrostatic asymptotics of the PMF wuu (r, Ω12 ) as well as of the interaction potential uuu (r, Ω12 ). The 3D-RISM/KH-HNC approach was applied to calculate the orientationally dependent PMF between the N,N-dimethylaniline cation (DMA+ ) and the anthracene anion (AN− ) in the acetonitrile solvent (CH3 CN) [29]. The interaction potentials of the ions and solvent comprise the Coulomb and 12-6 LJ terms, .

ab 6 / 12 ab Qaα Qbγ σαγ σαγ ab ab + 4εαγ − uαγ (r) = r r r ab = (σ a + σ b )/2 and with the Lorentz-Berthelot combination rules σαγ α γ a εb )1/2 . Figure 4.4 depicts the molecular models of the solutes = (ε εab αγ α γ and solvent. The site charge Qaα , LJ size and energy parameters σαa and εaα , and the geometry of the solutes are obtained from the fit to ab initio electronic structure calculations, and those of the solvent are taken from the optimized potential for liquid simulations (OPLS) force field. The 3D site distributions of ambient acetonitrile solvent around each of the DMA+ and AN− solutes following from the 3D-RISM/KH integral equations are illustrated in Figures 4.5 and 4.6 showing their sections in the molecular planes of the solutes. Analysis of the 3D site distribution profiles reveals that, in general, the orientations of acetonitrile solvent molecules are determined by the electrostatic interaction with the partial charges of the ionic solutes DMA+ and AN− . Cartoons in Figure 4.6 schematically show the most probable arrangements of solvent molecules associated to the solutes. In the first solvation shell, acetonitrile molecules are located over and under the DMA+ molecular plane with the MeCN nitrogen bound at the dimethylamino group and the MeCN methyl group bound at the phenyl ring (positions 1 and 1 ). Otherwise, MeCN molecules are oriented with nitrogen N towards the DMA+ solute

193

Three-dimensional RISM theory + 0 .1 5

C H 3C N

+ 0 .2 8

C

Me

-0 .4 3

z

N

+ 0 .1 2 6 6

z y

+ 0 .2 4 9 2

M e1

+ 0 .3 1 5 8

+ 0 .1 3 6 6

H 1 -0 .6 5 9 7 + 0 .1 6 6 4

+ 0 .0 2 1 5

H3

N

C2

C1

+ 0 .0 2 1 5

+ 0 .1 2 5 8

H5

C3

C6

C5

+ 0 .0 2 1 5

+ 0 .2 4 9 2

Ha

-0 .4 5 2 4

H2

+ 0 .1 2 6 6

Cd

Cc + 0 .1 2 6 6

+ 0 .1 5 8 4

Ca

C d' Cb C c'

C b'

M e2 -0 .4 5 2 4

+ 0 .3 1 5 8

H d'

-0 .2 2 4 5 -0 .2 7 4 2

H c' + 0 .1 2 6 6

C a' H a' + 0 .1 2 6 6

-0 .6 5 9 7

H4

+ 0 .1 6 6 4

-0 .2 2 4 5 H c -0 .2 7 4 2

+ 0 .1 5 8 4

+ 0 .5 5 3 9

C4

-0 .0 0 2 7

x

x

Hd

+ 0 .1 2 6 6

y

DM A

+

AN

-

Figure 4.4. Interaction site models of the N,N-dimethylaniline cation DMA + and anthracene anion AN− solutes, and acetonitrile solvent molecules CH 3 CN.

(positions 2, 3, 3 , 4). Acetonitrile molecules are either strongly associated with the methyl group at the plane of the AN− rings (positions 1, 1 , 2, 2 , 2a, 2a ), or aligned around the AN− solute periphery (positions 3, 3a to 6, 6a). The associated molecules are depicted schematically with small or no tilt with respect to the solute plane. The 3D site correlation functions of acetonitrile solvent around the DMA+ and AN− solutes obtained from the 3D-RISM/KH theory have been used to calculate the PMF between the solutes by using the HNC approximation (4.32) with the correction (4.35). The PMF profiles were calculated at the six relative orientations of the DMA+ and AN− molecules corresponding to the parallel arrangements of their symmetry planes. They are denoted by four angles between the molecular planes PDMA and PAN , and molecular axes xDMA and xAN : PAN , xDMA xAN , xDMA PAN , PDMA xAN ). (PDMA Figure 4.8 illustrates characteristic positions of DMA+ over AN− corresponding to each of these relative orientations: the planes of the molecules are parallel (parts a and b) or perpendicular (c to f), and the DMA+ symmetry axis lies in parallel to the AN− molecular plane (a to d) or in perpendicular to it (e and f), denoted respectively as Ω12 = ( ), ( ⊥ ), (⊥ ), (⊥ ⊥ ⊥), (⊥ ⊥ ⊥ ), and (⊥ ⊥ ⊥ ⊥). Figure 4.9 presents the sections of the PMF by the symmetry planes of the AN− molecule at relative orientation of the solutes Ω12 = ( ). The planes of the two solutes are parallel, with the DMA+ molecular

194


M e-C -N

M e-C -N

M e-C -N

XOY

YOZ

XOZ

Figure 4.5. Sections of the 3D site distributions of acetonitrile solvent MeCN around the dimethylaniline cation DMA+ , obtained by the 3D-RISM/KH theory.

195


M e-C -N

M e-C -N

M e-C -N

XOY

YOZ

XOZ

Figure 4.6. Sections of the 3D site distributions of acetonitrile solvent molecules MeCN around the anthracene anion AN− , obtained by the 3D-RISM/KH theory.

196


z z

y

2

1 1 N

Me

C

3

3a

4a

x

5a

2 4

Me

2a

y

x

N C

6 6a

3'

5 4

3

1'

2' 1'

DMA+

AN -

2a'

Figure 4.7. Preferred arrangements of acetonitrile solvent molecules MeCN in the first solvation shells of the DMA + and AN− solutes, according to the 3D site distributions following from the 3D-RISM/KH theory.

z

(a)

y

x

z

z

Ω12=( || || || || )

Ω12=( || ⊥ || || )

x

Figure 4.8.

x

z

y

Ω12=(⊥ || || || )

y

(b )

z

(c)

Ω12=(⊥ ⊥ ⊥ || )

y

x

z

(d ) Ω12=(⊥ ⊥ || ⊥)

y

x

(e)

(f) Ω12=(⊥ ⊥ ⊥ ⊥)

y

x

Representative relative orientations Ω 12 of the DMA+ and AN− solutes.


197

axis oriented along the AN− longer axis. The coordinate system of the AN− solute is taken as a reference, and the DMA+ molecule is shifted with respect to it so that its carbon atom C1 moves in the XOY , Y OZ, and XOZ coordinate planes, keeping the relative orientation of the solutes. The cartoons in the right hand column illustrate the relative arrangements of the solutes situated in the corresponding planes of the left hand column plots. To make the PMF minima more readable on 3D plots, the PMF is depicted with opposite sign, −w ˜ uu (r 12 , Ω12 ), so that PMF wells become peaks and vice versa. The strongest attraction well of depth 18.3 kcal/mol between the solutes is obtained for the contact ion pair (CIP) when the DMA+ dimethylamino group is situated over the the A while the DMA+ phenyl first ring of the AN− at a separation of 3.3 ˚ − ring is extended beyond the AN molecule (peaks at x = −4.5, y = 0, z = ±3.3 ˚ A on the plot marked XOZ. Owing to the repulsion of the + DMA phenyl ring from the AN− solute, the attraction well decreases to 10.2 kcal/mol as DMA+ shifts to the AN− center (peaks at x = 0, y = 0, z = ±3.6 ˚ A on the XOZ and Y OZ plots). Further shift moves the DMA+ dimethylamino group away from the AN− , and the repulsion of the DMA+ phenyl ring dominates the PMF. Accordingly, the two CIP “ridges” at z = ±3.5 − 4 ˚ A on the XOZ plot get lower. At x = 4.5 ˚ A the DMA+ phenyl ring is in face to face contact with the AN− side ring, whereas the dimethylamino group is located well beyond the contact. For the solutes approaching along the z-direction, the PMF at this arrangement has a barrier of height 4.3 kcal/mol and then a local minimum at the level of 2.9 kcal/mol above zero (points of the crests at x = 4.5, y = 0, z = ±3.85 ˚ A on the XOZ plot). For the molecules situated in one plane and DMA+ in contact with AN− by its dimethylamino group, edge to edge, the attraction well depth amounts to 8.9 kcal/mol (peak at x = −9.2, y = 0, z = 0 ˚ A on the XOY plot in Figure 4.9). However, when DMA+ contacts AN− by its phenyl ring the local minimum of PMF is very shallow (about 0.05 kcal/mol) and is again at a level of 1.3 kcal/mol above zero (small peak at x = 10.3, y = 0, z = 0 ˚ A on the XOY and XOZ plots. Thus the CIP configurations of the DMA+ and AN− solutes contacting by their rings are energetically unfavorable. Figure 4.10 shows the coordinate planes sections of the PMF (left hand column plots) for the parallel planes of the solute molecules but the DMA+ molecular axis oriented across the longer axis of the AN− solute, orientation Ω12 = ( ⊥ ). The attraction well depth amounts to about 19.4 kcal/mol for the DMA+ nitrogen atom N located at 3.25 ˚ A on A over the AN− central ring (peaks at x = 0, y = 2.5, z = ±3.25 ˚ the left hand Y OZ plot). This PMF well is also close to the deepest

198


z

XOY

y

x x

YOZ

z

y

XOZ

z

y x

Figure 4.9. Sections of the potential of mean force (of opposite sign) −w ˜ uu (r 12 , Ω12 ) + − between the DMA and AN solutes in acetonitrile solvent MeCN at their relative orientation Ω12 = ( ) as in Figure 4.8a. The corresponding right hand cartoons show arrangements of DMA+ being shifted around AN− in the section planes XOY , Y OZ, and XOY . Result of the 3D-RISM/KH-HNC theory.


199

among all the relative arrangements of the solutes. The similar arrangement but with the DMA+ nitrogen site N located over the center of the AN− central ring corresponds to the strongest attraction between the DMA+ and AN− molecules in the dielectric continuum corresponding to acetonitrile at 298 K [104]. In that position the positively charged dimethylamino group of DMA+ is closest to the AN− carbon atoms Ca,a carrying the maximal negative charge, whereas the DMA+ carbons C2,3 charged negatively are attracted by the positive charge of the AN− carbon Cb and hydrogens Ha,c . It should be emphasized that the significant difference in the effective potential of DMA+ contacting AN− by the dimethylamino group and by the phenyl ring is mainly a solvent effect. The short-range solvent contribution to the PMF brings about effective interactions between the groups of the solute molecular sites that differ substantially from the direct interaction potential between the solutes. The DMA+ dimethylamino group charged positively is attracted to the AN− solute, whereas the DMA+ phenyl ring experiences repulsion, much stronger than the electrostatic interaction in the dielectric continuum. For comparison the right hand column of Figure 4.10 depicts the PMF obtained within the primitive solvent model representing ambient acetonitrile as dielectric continuum with the dielectric constant MeCN expt = 37.3. It scales the (r, Ω ) of the interaction potential between the electrostatic term u ˜uu 12 (C) solutes. The section planes and the solute orientations are same as on the corresponding left hand column plots. The shapes of the potential are determined largely by the interplay of the Coulomb interaction between the extra charges of the solutes screened by the dielectric continuum and the short-range LJ repulsion between the cores of the solutes. This yields the attraction well of depth varying from 1.4 kcal/mol for the solutes being in contact by their edges (right hand XOY plot), to 2.5 kcal/mol for those contacting by the planes of their molecules when the extra charges get closer to each other (right hand Y OZ and XOZ plots). The electrostatic potential of the solute multipole charges screened by the dielectric continuum brings about only a small difference into the attraction well depth at different relative arrangements of the solutes. Figure 4.11 exhibits the PMF obtained for the DMA+ symmetry axis directed in perpendicular to the AN− molecular plane, with relative orientations Ω12 = (⊥ ⊥ ⊥ ) and (⊥ ⊥ ⊥ ⊥). For the solutes in contact by their rings at this relative orientation, the first minimum of the PMF is at the level of 2.5 kcal/mol above zero, making such an arrangement much less stable. The PMF attraction well is deepest when the DMA+ dimethylamino group contacts the AN− anywhere over its plane (peaks at x = 0, y = 0, z = 5.4 ˚ A on the Y OZ and XOZ plots). The attraction

200


XOY

YOZ

XOZ

-~ wuu

p rim itiv e so lv en t m od el

Figure 4.10. Sections of the potential of mean force −w ˜ uu (r 12 , Ω12 ) (left hand column plots) at the relative orientation of the DMA+ and AN− solutes Ω12 = ( ⊥ ), as in Figure 4.8b. Result of the 3D-RISM/KH-HNC theory (left column plots), and of the primitive dielectric continuum model for the solvent (right hand column plots).

201


XOY

YOZ

XOZ

(a)

(b )

Figure 4.11. Sections of the potential of mean force at the relative orientation of the solutes as in Figure 4.8e and 4.8f, Ω12 = (⊥ ⊥ ⊥ ) and (⊥ ⊥ ⊥ ⊥) (plots in columns a and b, respectively).

varies rather smoothly over the AN− molecule, and so the PMF well is wide along the AN− molecular plane. It reaches the maximum of 20.1 kcal/mol over the AN− center, for the DMA+ molecular plane parallel to the AN− x-axis.

202


For each CIP arrangement of the solutes, the minimum of the PMF is observed when the nitrogen atom N of the DMA+ dimethylamino group is located over the AN− central ring. The relative orientations of the solutes are grouped in order of decreasing stability of the CIP as follows: Ω12 = (⊥ ⊥ ⊥ ), (⊥ ⊥ ⊥ ⊥), ( ⊥ ) ( ), (⊥ ), and (⊥ ⊥ ⊥). A substantial result is that acetonitrile solvent changes the most stable relative arrangement of the DMA+ and AN− molecules as compared to gas phase. The deepest minimum of the PMF among all the orientations considered is that of Figure 4.8e, whereas the interaction potential has the strongest attraction at the orientation of Figure 4.8b.

6.

Solvation chemical potential of an ionic cluster in electrolyte solution

The potential of mean force between two particles of a liquid mixture of ionic and molecular species breaks up into their direct interaction and the solvent contribution which can be presented as a difference in the excess free energy of their solvation in a given relative arrangement and at infinite separation, wab (r12 , Ω1 , Ω2 ) = uab (r12 , Ω1 , Ω2 ) + µab (r12 , Ω1 , Ω2 ) − µa − µb , (4.36) where uab (r12 , Ω1 , Ω2 ) is the interaction potential of particles of species a, b, r12 = |r 2 − r 1 | is their separation, and Ω1 , Ω2 are their orientations in the case of molecular species. To obtain the solvation free energy of a pair of particles at a fixed relative position, they are regarded as a single “solute” immersed in a “solvent” of all other particles of the solution and treated by using the 3D-RISM integral equations. This approach can be readily extended to formation of a multi-particle cluster in solution described by a multi-dimensional solvation free energy. For solvent comprising several species, the 3D solute-solvent RISM equation is written as

ab (r) + ρa hab (r) , (4.37) hubγ (r) = cuaα (r) ∗ ωαγ αγ where hubγ (r) and cubγ (r) are the 3D total and direct correlation functions for site γ of solvent species b around the whole composite solute denoted ab (r) = δ δ(r − la )/(4π(la )2 ) is the intramolecular matrix for as u, ωαγ ab αγ αγ a , and ρa the mixture of species a with intermolecular site separations lαγ is the number density of solvent species a. The site-site total correlation functions hab αγ (r) between site α of species a and site γ of species b of solvent are obtained from the site-site RISM equations for an ionmolecular mixture. According to either 3D-HNC closure approximation


203

(4.12) or 3D-KH one (4.13) used to complement the 3D-RISM integral equation (4.37), the chemical potential of solvation of the composite solute is obtained from the expression (4.14) or (4.15). As has been discussed, the RISM/HNC approach leads to the trivial result (4.34) for the static dielectric constant of polar liquid. To force the phenomenological value expt of the dielectric constant of polar solvent, Cummings and Stell [146] proposed to scale the long-range Coulomb asymptotics of the site-site DCFs. This can be achieved this by multiplying the Coulomb terms of all the site-site interaction potentials in the HNC closure with a constant close to unity. Such a correction to the RISM/HNC equations yields sensible results at infinite dilution [66, 84], but fails at finite ionic concentrations because of an inherent inconsistency between the ionic and dipolar routes to the solution dielectric constant [105, 85]. It was shown that the the form of a dielectric bridge correction proper to an ion-molecular solution is more complex [106, 111]. Based on that form, Perkyns and Pettitt [105] introduced the dielectrically consistent RISM theory (or DRISM) yielding the dielectric constant independent of the Debye screening effects. The bridge correction is applied only to the correlations between polar species, whereas the corrections to the ion-ion correlations result from indirect coupling with the corrected dipole-dipole correlations. In a similar way, inserted in the 3D-RISM equation (4.37), the site-site correlations hab αγ (r) following from the DRISM approach ensure the dielectric consistency of the solvation thermodynamics of the composite solute. The consistent dielectric correction modifies the site-site HNC closure by an additional term similar to a bridge function of molecular theories,

ab ab ab ab (r) = exp −βuab (r) + h (r) − c (r) + b (r) , gαγ αγ αγ αγ αγ

(4.38)

ab (r) = hab (r) + 1 are the site-site radial distribution functions, where gαγ αγ and uab αγ (r) are the site-site interaction potentials. The bridge correction bab αγ (r) makes up for a part of the diagrams proper to dielectric screening [106, 111] that are missed in the site-site RISM/HNC approximation as compared to the orientationally dependent, molecular OZ/HNC equations. On the other hand, the correction provides phenomenological improvement over the dielectric constant following from the molecular OZ/HNC theory [107, 108]. It is convenient to renormalize [109] the DRISM equations with respect to the correction bab αγ (r). This transfers the correction from the closure (4.38) to the DRISM integral equation

204


which takes the form ab ac a cd db db hab αγ − ζαγ = (ωαµ + ζαµ ) ∗ cµν ∗ (ωνγ + ζνγ ) ac ac d db db + ζαµ ) ∗ ccd + (ωαµ µν ∗ ρ (hνγ − ζνγ ),

(4.39)

ab (r) is the renormalized correction defined as where ζαγ ab ac db ac cd d db (r) = ωαµ ∗ bcd ζαγ µν ∗ ωνγ + ωαµ ∗ bµν ∗ ρ ζνγ .

(4.40)

the intramolecular matrix of a multi-component mixture has the form ab (k) = δ sin(kla )/(kla ), and the closure returns to the HNC form ωαγ ab αγ αγ

ab ab ab (r) = exp −βuab (4.41) gαγ αγ (r) + hαγ (r) − cαγ (r) . Perkyns and Pettitt [105] demonstrated that the ionic and dipolar routes to the dielectric constant of solution are consistent both with and without Debye screening, provided the screened correction is specified in the reciprocal space as ab ζαγ (k) = j0 (kxaα ) j0 (kyαa ) j1 (kzαa ) hc (k) j0 (kxbγ ) j0 (kyγb ) j1 (kzγb ), (4.42)

where j0 and j1 are the spherical Bessel functions of the zeroth and first order, daα = (xaα , yαa , zαa ) is the position of site α with respect to the origin of the molecule with its dipole moment µa = α Qaα daα oriented along z-direction (µax = 0, µay = 0). Notice that ion-dipolar as well as ab are zero. The coefficient at k 2 in the ion-ion species elements in ζαγ low-k expansion of hab αγ (k) determining the dielectric constant complies with that for pure polar molecular liquid proposed by Høye and Stell [106]. The correction function hc (k) in (4.42) is assumed in a smooth, non-oscillatory form that does not change the qualitative short-range structure obtained from the RISM/HNC theory. It goes to zero at large k and has the value hc (k = 0) providing the consistency between the ionic and dipolar routes to the dielectric constant of the mixture:

( )

expt − 1 1 − 3 exp −s2 k2 /4 , (4.43) hc (k) = ρdip y where ρdip = a ρadip is the total number density of dipolar species, and s is the parameter controlling the length on which the dielectric correction is “switched on”. Its particular value does not affect the results crucially, and is suggested [105] to be chosen s = 0.5 ˚ A. Worth noting is that instead of the HNC closure, the renormalized DRISM integral equation (4.39) can be complemented with the KH ap-

205


proximation written for the site-site correlations as [78] ( ) exp dab for dab αγ (r) αγ (r) ≤ 0, ab gαγ (r) = ab for dab 1 + dαγ (r) αγ (r) > 0,

(4.44)

ab ab ab dab αγ (r) = −uαγ (r)/(kB T ) + hαγ (r) − cαγ (r).

It was shown for the RISM/HNC theory [65, 66] that the dielectric constant which is related to the coefficients of k2 in the low-k expansion of the site-site total correlation functions hab αγ is determined solely ab by their renormalized long-range part ζαγ . For the 3D-RISM equation (4.37), the renormalized long-range part of the 3D site total correlation functions hubγ (r) takes the form

ua ab a ab (4.45) ζ ub γ (r) = −βφ α (r) ∗ ωαγ (r) + ρ hαγ (r) ,

where

a φua α (r) = Qα

dr

q u (r ) |r − r |

a is the electrostatic potential φua α (r) between the solvent site charge Qα and the solute charge density distribution q u (r). Eq. (4.45) is linear and yields ζ ub γ (r) comprising a superposition of the solvent reactions on each solute charge element dQu = q u (r)dr. Using at input the solvent site-site total correlation functions hab αγ obtained from the DRISM equation (4.39) ensures the dielectric consistency of these elementary reactions and hence of the 3D site correlations following from the 3D-RISM equation (4.37). Thereby it is not necessary to include an explicit bridge function to the 3D-HNC or 3D-KH closures (4.12) or 3D-KH) since the consistent dielectric correction is introduced indirectly through the modification of hab αγ , much as in the DRISM approach for ionic species at infinite dilution. It should be noted this treatment makes up for a medium with a uniform dielectric constant expt surrounding the solute. The microscopic, shortrange behavior of the dielectric function (k) is represented by the O(k4 ) terms of the distribution functions and is described at the HNC level. The electrostatic part φubγ (r) of the interaction potential between site γ of solvent species b and the solute comprising several ions i is calculated within the supercell technique by means of the Ewald summation [101]. It is synthesized on a 3D supercell grid as

φubγ (r) =

Qbγ 4π k2 η2

Qi 2 exp ik · (r − r i ) − Vcell k 4 k=0 i erfc(∆r /η) i , + ∆ri i

(4.46)

206


where r i is the position of solute ion i and Qi is its charge, ∆ri is the distance to ion i determined subject to the minimum image convention [101], and the summation over wavevector k is calculated by the 3D-FFT. The remaining part of the solute-solvent site potential is also tabulated by using the minimum image convention, since it is typically represented by the 12-6 LJ potential and falls off at the supercell length. Notice that the 3D site direct correlation functions of the periodic solute, cubγ (r), are defined up to a constant in the solute electrostatic potential (4.46). It cancels out in the solvation chemical potential (4.14) or (4.15) as well as in the 3D-RISM equation (4.37), as follows from the local electroneutrality of solvent and the compressibility theorem. The 3D-RISM integral equations involve convolution of the 3D solutesolvent site correlation functions which is performed by means of the 3D-FFT within the supercell technique [16, 17, 19, 23-37]. It was shown by Hu ¨nenberger and McCammon[110]thatin computer simulations,the artificial periodicity imposed by the use of the Ewald or related methods produces significant errors in the solvation free energy andthe potentials of meanforce of ions in dielectric continuum. The effect can be qualitatively explained as follows [110]. The solvent in the central unit cell is perturbed by the periodic copies of the ionic solute and thus less available to solvate the central ion. Therefore, it is more difficult to solvate the set of solute periodic copies than the non-periodic solute. A similar problem arises in the 3D-RISM integral equation approach. Calculation of the solvation thermodynamics for a charged solute immersed in a polar solvent requires special measures to eliminate the supercell treatment artifacts which can become huge [30]. The correction to cubγ (r) distorted by the supercell treatment replaces their electrostatic asymptotics for the periodic solute by that for a single solute. The corrected 3D site direct correlation functions are written as Qu α . (4.47) c˜ubγ (r) = cubγ (r) + βφubγ (r) − βQbγ |r − r α | α The 3D site total correlation functions hubγ (r) have to be corrected for the small but detrimental shift caused by the supercell background charge, ˜ ub (r) = hub (r) − ∆ζ ub . h γ γ γ

(4.48)

For a uniform background charge peculiar to the Ewald method of handling the electrostatics, the shift is given by the expression

Qα 4πncell ab a ab lim (k) + ρ h (k) , (4.49) ω ∆ζγub = αγ αγ kB T k→0 aα k2


207

where ncell = Qu /Vcell is the background chargedensity for the supercell volume Vcell and the solute net charge Qu = α Quα . For the reaction field approach, the supercell background charge can be regarded as a 3D distribution ncell (k) [30]. The limit in (4.49) should be taken after the summation to provide cancellation of the singularities at k = 0 due to the local electroneutrality of solvent. The supercell periodicity does not affect the long-range asymptotics of the 3D site total correlation functions. They decay at least as [102, 111] hubγ (r) ∼ r −6 , and the periodic images of the solvation structure are well separated in space from each other for the supercell including two or three solvation shells. The 3D site direct correlation functions can be corrected by (4.47) after solving the 3D-RISM equations. However, the correction (4.48) to the 3D site total correlation functions cannot be applied in this way since it would violate the equality of g˜ubγ (r) = 0 inside the solute repulsive core. The shift ∆ζ ub γ (r) strongly differs for molecular and ionic species of solvent. Its magnitude corresponds to the mean density of an “excess cloud” of solvent species b screening the charge Qu . Naturally, the mean density of screening charge is higher for Debye screening with an exponential decrease of the excess density than for dielectric screening with a power decay. Therefore the correction ∆ζ ub γ appears to be of −3 the order of 10 for molecular species of solvent; however, it ranges up to 10−1 for ionic species. To enforce the distribution function to vanish inside the solute repulsive core, the closure should be applied to the cor˜ ub (r) = 0. For instance, the rected 3D site total correlation functions, h γ 3D-HNC closure relation (4.50) modified with account for the correction (4.48) takes the form

(4.50) g˜ubγ (r) = exp −βuubγ (r) + hubγ (r) − cubγ (r) − ∆ζ ub γ , where g˜ubγ (r) = gubγ (r) − ∆ζ ub γ are the corrected 3D site distribution functions. The 3D-KH closure (4.13) is modified in a similar way as ( ) exp dubγ (r) for dubγ (r) ≤ 0, ub (4.51) g˜ γ (r) = for dubγ (r) > 0, 1 + dubγ (r) dubγ (r) = −uubγ (r)/(kB T ) + hubγ (r) − cubγ (r) − ∆ζ ub γ. After solving the 3D-RISM equations (4.37) with one of the modified ˜ ub (r) and closures (4.50) or (4.51), the corrected correlation functions h γ ub c˜ γ (r) can be inserted into the expression for the solvation chemical potential, Eq. (4.14) or (4.15), according to the closure used. The effect ˜ 2 /2 − h2 /2 = h∆ζ + (∆ζ)2 /2, of the correction (4.48) on the first term, h is relatively small. Owing to the local electroneutrality of solvent, the

208


periodicity correction (4.47) cancels out in their second term −˜ cubγ (r) on summation over sites and species. The periodicity corrections improve mainly the last cross-term. For the composite solute comprising two ˜ c = − 1 (h(sr) c(sr) + 1 h(sr) β(φ˜1 + φ˜2 ) − ions, it can be decomposed as − 12 h˜ 2 2 ) 1 ˜ ˜2 )c(sr) + 1 β(ζ˜1 + ζ˜2 )(φ˜1 + φ˜2 ) , where “(sr)” means a short-range ( ζ + ζ 1 2 2 part of the correlations, and a tilde denotes the long-range part for a non-periodic solute. The last term accounts for the mutual electrostatic interactions between the ions constituting the composite solute and the charges of their solvation shells, and obviously is most sensitive to the periodicity artifact. Notice that even for the non-periodic solute, the integration in (4.14) or (4.15) can be carried out just over the supercell volume, because the long-range Coulomb asymptotics cancels out in the ˜ c/2. Worth noting second term −˜ c and is cut off in the third term −h˜ also is that it makes no sense to position the composite solute ions apart by more than half the supercell size. Although the electrostatic part of the solvation chemical potential is forced to be that for the non-periodic solute, the short-range components in such an arrangement originate improperly from the interactions with the solute images in the adjacent periodic supercells. The calculation of interionic PMFs from the relation (4.36) by using the 3D-RISM approach is illustrated below on the example of aqueous solution of sodium chloride. For alkali and halide ions in water a point of significant importance is the choice of the interionic and ion-water potentials. They are modelled by the sum of the Coulomb and 12-6 LJ terms, with the LJ parameters of the ions elaborated by Smith and Dang [115] for their simulation of this system. The cross-terms of the LJ potentials are obtained from the standard Lorentz-Berthelot mixing rules. More discussion on this potential choice can be found in [30]. Water is represented by the extended simple point charge (SPC/E) model [127]. By direct comparison of the properties of NaCl in water represented with a polarizable model and in non-polarizable SPC/E water, Smith and Dang [115] also demonstrated that polarization fluctuations play a minor role in this system. They recommended the SPC/E water model to be used for aqueous solutions, since it includes self-polarization energy corrections. The results of the 3D-RISM/HNC theory for the PMFs between sodium chloride ions in aqueous solution are presented in Figure 4.12. In addition, Figure 4.13 shows the PMFs between sodium chloride ions at infinite dilution calculated without the periodicity corrections (4.47) and (4.48). The periodicity artifact manifests in the PMF as a very large shift of approximately 35 kcal/mol negative for unlike and positive for like ions, which arises because of the electrostatic energy of the solvent charge distribution in the compensating background charge


209

of the supercell. Moreover, the Ewald summation distorts the Coulomb asymptotics of the PMF between the ions by an additional slope of about 5 kcal/mol on half the supercell length. It should be stressed that the supercell artifact arises in the PMFs of not only like but also unlike ion pairs. Although the solvation chemical potential of the unlike ion pair µ+− (r+− ) is distorted much less because of the zero net charge of the pair, the solvent contribution to the PMF obtained from Eq. (4.36) involves the chemical potentials of individual ions µ+ and µ− which are strongly distorted. The shifts in both µ+ and µ− are positive, and thus add up to the big error in the calculated PMF. For a like ion pair the net charge of the pair and so the background charge are doubled, and the shift in the solvation chemical potential, say µ++ (r++ ), becomes approximately four times higher than that in µ+ . This results in the error in the PMF for like ions which has roughly the same magnitude but the sign opposite to that for unlike ions. Scaled by the supercell size, the magnitudes of all these shifts agree well with those observed in the electrostatic calculations [110] and simulations for simple ions in water [112]. The electrostatic corrections (4.47) and (4.48) provide cancellation of the Ewald summation artifacts in the PMFs with an accuracy of 0.05 kcal/mol or better. The precision of the cancellation is seen well for the PMF of the Na+ –Cl− ion pair tending with distance to its asymptotics of the dielectrically screened Coulomb interaction between the ions. It is also evident from all the sodium chloride PMFs at a finite salt concentration which do fall off quickly in the presence of Debye screening. Consider the PMF for a single pair of Na+ and Cl− ions in water. Figure 4.14 makes a comparison of the PMF obtained from Eq. (4.36) in the 3D-RISM treatment with the results of the site-site DRISM approach, the primitive model of dielectric continuum solvent, and the MD simulation data for the same potential models [115]. The 3D-RISM/HNC theory yields the contact ion pair (CIP) minimum with the close position and the same depth as that of the simulation curve. Although the DRISM/HNC theory gives the CIP distance in even better agreement with the MD simulation, it overpredicts the CIP minimum depth by a factor of 2.4 because of the underestimation of the steric exclusion effect for the ions in pair. The 3D-RISM approach treats the latter explicitly and thus produces additional repulsion of water molecules. The position of the solvent-separated ion pair (SSIP) minimum of the PMF is predicted correctly by both the 3D-RISM/HNC and the site-site DRISM/HNC approaches, although it is shallower than in the MD result. The use of the 3D-KH approximation rather than the 3D-HNC closure does not affect the PMF behavior substantially. This rises the CIP

210


2

in f in ite d ilu tio n

w ( k cal/m o l )

C l-- C l1

N a +- N a + 0

N a +- C l-

-1

3

4

5

N a +- N a +

6

7

8

C l- - C l-

1

w ( k cal/m o l )

r (Å )

0

N a +- C l-

C = 1 .0 6 9 m o l / l

-1 3

4

5

r (Å )

6

7

8

Figure 4.12. Potentials of mean force between sodium chloride ions in ambient aqueous solution, following from the 3D-RISM/HNC theory.

211


w ( k cal/m o l )

-3 3 -3 4 -3 5

N a +- C l-

-3 6 -3 7 3

4

5

w ( k cal/m o l)

40

6

7

8

6

7

8

C l-- C l-

38

36

r (Å )

N a +- N a +

34 3

4

5

r (Å )

Figure 4.13. Potentials of mean force between sodium chloride ions at infinite dilution in ambient aqueous solution, following from the 3D-RISM/HNC theory without the corrections eliminating the supercell periodicity artifacts.

minimum by about 0.4 kcal/mol, and produces the dissociative barrier between the CIP and SSIP arrangements much closer to the simulation. The position of the SSIP minimum obtained in both the 3D-HNC and 3D-KH approximations is in qualitative agreement with the simulation, although its depth is underestimated. The CIP minimum obtained from both the site-site and 3D-RISM treatment is located at smaller distances as compared to the primitive model of continuum solvent. This shift is attributed to the entropy gain associated with the pairing of unlike charges being enhanced in a molecular solvent [84]. Figure 4.15 illustrates the thermodynamic decomposition of the PMF obtained from the 3D-RISM and site-site DRISM approaches for the Na+ –Cl− ion pair, w(r) = E(r) − T S(r), (4.52) ( ) where S(r) = − ∂w(r)/∂T N,V . The improved agreement of the former with the simulation can be traced back to the energetic component of the PMF. The 3D-RISM/HNC description better represents the steric effects in the ion pair, but little affects the entropic part. The 3D-KH approximation further increases the repulsive energetic component of the PMF as well as the height of the dissociation barrier as compared to the

212


N a + - C l-

w ( k cal/m o l)

1

0

-1

-2

-3 2

3

4

5

6

7

8

r (Å )

Figure 4.14. Potential of mean force between the Na+ and Cl− ions in ambient water at infinite dilution. Results following from the 3D-RISM/HNC and 3D-RISM/KH approach (solid and short-dashed lines, respectively), and from the ion-ion radial distribution function obtained by the site-site DRISM/HNC theory (dash dotted line). Molecular simulation data [115] (open squares). Prediction of the primitive continuum model of solvent (long dashed line).

3D-HNC closure, and slightly enhances ordering of the solute hydration structure. Figure 4.16 presents the results of the 3D-RISM/HNC theory for the 3D site distributions of water oxygen and hydrogen around the Na+ –Cl− ion pair in three arrangements: the CIP and SSIP, and that corresponding to the PMF barrier between them. The cartoons below the plots illustrate possible orientations of water molecules in the first hydration shell. In all the arrangements water molecules are oriented in general with the oxygen and hydrogen sites towards the corresponding unlike charges of the Na+ –Cl− ion pair. In the first hydration shell around Na+ , water hydrogen sites are located in one shell separated from that of oxygen sites by a distance of about 0.6 ˚ A. The next maximum of A outwards and hydrogen sites around Na+ is located by more than 2 ˚ belongs to the second hydration shell. This suggests water molecules around the Na+ ion are oriented with their dipoles directed outwards and both hydrogens tilted equally (orientations 1 and 1 in each arrangement of the ion pair). However, the orientation of water around the Cl− ion is somewhat different from what might be expected for just dipoles in the electrostatic field of the ion. There are two concentric hydrogen maxima in the first hydration shell around Cl− . The first,

213

Three-dimensional RISM theory 3

N a + - C l-

w ( k cal/m o l)

2 1

E 0

w -1 -2

-T S

-3 2

3

4

5

6

7

8

r (Å )

Figure 4.15. Decomposition of the potential of mean force between Na+ and Cl− ions pair at infinite dilution in water, w(r), into the energetic and entropic components, E(r) and −T S(r). Results of the 3D-RISM/HNC and 3D-RISM/KH approaches versus the site-site DRISM/HNC theory (solid, short dashed, and dash dotted lines, respectively).

sharp peak of hydrogen sites is located inside the shell oxygen sites at the separation close to the length of the intramolecular OH bond. The other, much smoother but well pronounced maximum of hydrogens in the first hydration shell is located at some distance outside the first shell of oxygen sites. This is indicative of the orientation of water molecules with one hydrogen directed to the Cl− ion and the other one somewhat tilted outward (orientations 2 and 2 ). The distributions are almost uniform over the “half-sphere” covering Cl− , whereas the oxygen profile in the first shell around Na+ has a rather high maximum in the vicinity of the “pole” over Na+ (on the interionic axis). The second hydration shell is much less pronounced for both oxygen and hydrogen sites. In any arrangement of the Na+ and Cl− ions, water oxygen sites are strongly localized in the region of contact between the ions (closer to the former), seen as two high peaks on the oxygen site profile. The corresponding hydrogen sites are concentrated in the wider and lower peak shifted by about 0.6 ˚ A outward the interionic axis OX, which is stretched in the x-direction up to 1.6 ˚ A but twice as narrow in the direction perpendicular to it. This indicates that such localized water molecules are oriented with their dipole moments almost in normal to the interionic axis (orientations 3 and 3 ). Next to the high oxygen peak in the contact region are a rather wide void, and then a much lower

214


o xy g en

h y d rog en

C IP a rran g em en t

b a rrier arra n g em en t

S S IP arra n gem en t

N a + (-1.58 ,0,0)

N a + (-1.93 5,0 ,0 )

N a + (-2.45 5,0 ,0 )

C l - (1.58,0,0 )

4

N a+

C l-

1'

2'

3' 4'

3

1

1'

i ty

N a+

4 2

ca v

3 1

C l - (1.93 5,0 ,0)

4 2

C l-

4'

2

3 1

2'

3'

C l - (2.45 5,0 ,0)

N a+

C l3'

2'

1' 4'

Figure 4.16. Distributions of water around the Na+ –Cl− ion pair, following from the 3D-RISM/HNC approach. Interionic separations corresponding to the CIP minimum, barrier maximum, and SSIP minimum of the Na+ –Cl− PMF (plots in the left, middle, and right columns, respectively). Distributions of water oxygen and hydrogen sites (plots in the upper and middle row, respectively). Arrangements of water molecules in the first hydration shell around the ion pair (cartoons in the lower row). The dashed line marks hydrogen bonding between water molecules of the hydration shell.


215

oxygen peak located closer to Cl− which is more pronounced in the CIP arrangement. The orientation of water molecules in the latter peak is similar to that around the Cl− ion: one hydrogen is directed to the ion and the other is tilted outward, which gives respectively a higher peak at the end of the inner “crown” and a smooth maximum on the outer part of the first shell of hydrogen sites around Cl− . Two such orientations in the contact region are depicted in the cartoons as molecules 4 and 4 . Notice that in the CIP position, the water molecule at orientation 3, strongly bound at Na+ is capable of forming a hydrogen bond with molecule 3 attracted by Cl− (dashed line in the CIP cartoon). It seems the energy gain coming from such hydrogen bonds contributes noticeably to the stability of the CIP arrangement of Na+ –Cl− . As the distance in the Na+ –Cl− ion pair increases, a part of the oxygen-localized water molecules between the ions turn with one hydrogen toward Cl− (orientation 3 at the barrier arrangement in Figure 4.16). Because of to the additional attraction of the hydrogen site to the Cl− ion, this localized position is energetically more stable and the oxygen as well as hydrogen site peaks get higher. However, the PMF acquires a maximum top due to the work done against the ionic attraction and the energy of creation of a cavity between the ions. The oxygen maximum near the “pole” over Na+ shifts toward Cl− to the position schematically represented in the barrier cartoon by molecules 1 and 1 . With the ions moving farther apart the cavity is filled with water molecules forming the intermediate water layer and the hydration free energy overrides the work necessary for separating the ions in the SSIP arrangement. Notice, however, that water molecules are not distributed uniformly throughout the SSIP gap. The steric constraint of the two ions attracting each other orders water molecules in the gap into a ring with the diameter of the oxygen repulsive core, providing their close packing. Owing to the varying orientation of the localized water molecules (orientations 3 versus 3 in the SSIP cartoon), however, water hydrogen sites are distributed along the SSIP gap much more smoothly and complete the hydrogen shell surrounding the Cl− ion. The high oxygen peak is depleted as compared to the barrier arrangement and returns to the value somewhat higher than that for the CIP. The oxygen shell around Na+ becomes more uniform. Nevertheless, it has the maxima at its “pole” over Na+ and closer to Cl− , marked in the SSIP cartoon as the positions of water molecules 1 and 1 , respectively. Figure 4.17 depicts the PMFs for Na+ –Na+ and Cl− –Cl− obtained from the 3D description versus the site-site treatment. Compared to the latter approach, the former does not alter the PMF between two Na+ ions significantly. The first minimum rises and shifts to some extent,

216


w ( k c a l/m o l )

2

N a+ - N a+

1

0

3

4

5

r (Å )

6

w ( k c a l/m o l )

2

7

8

C l- - C l-

1

0

3

4

5

r (Å )

6

7

8

Figure 4.17. Potentials of mean force for the Na+ –Na+ and Cl− –Cl− ion pairs at infinite dilution in ambient water, following from the 3D-RISM/HNC and site-site DRISM/HNC approaches (solid and dash dotted lines, respectively).

and the small oscillation at r = 4.3 ˚ A disappears, whereas the second minimum just shifts slightly. However, the change to the 3D-RISM does strongly affect the PMF for the Cl− –Cl− ion pair. The CIP minimum becomes significantly higher as compared to infinite separation and the dissociation barrier diminishes by a factor of 2.8. Although there is a second oscillation of the PMF, it is insufficient to stabilize a SSIP arrangement of two Cl− ions. Again, the modification in the PMF with the change to the 3D approach is accounted mainly for the energetic rather than entropic component, as is evident from its thermodynamic decomposition (4.52) shown in Figure 4.15. Notice that in contrast to the Na+ –Cl− ion pair, the energetic component of the PMF for Cl− –Cl− in the 3D-RISM approach is close to zero up to the repulsive core of the interionic potential, so the extrema of the PMF are formed mainly owing to its entropic part.


217

Figure 4.18 exhibits the 3D distributions of water oxygen and hydrogen around the Na+ –Na+ ion pair at the interionic distances corresponding to the PMF extrema in Figure 4.12. Water molecules in the two hydration shells are oriented with the dipole moments along the electrostatic field of the ion pair, outwards from the ions. The first hydration shell has a sharp maximum of water oxygen sites, which is surrounded by a smoother and lower shell of hydrogens at a distance of about 0.6 ˚ A. Such orientations are represented in the cartoons by molecules 1, 2 and 2 . The first minimum of the PMF for two Na+ occurs at the A, substantially larger than their LJ diainterionic distance rmin = 4.1 ˚ A. In this arrangement there is a high maximum of meter, σNa+ = 2.35 ˚ water oxygen sites localized in the region between the ions on a ring of diameter 2.4 ˚ A (molecules 3 and 3 in the first-minimum drawing), even A. Unlike the Na+ –Cl− smaller than the oxygen LJ size, σO = 3.166 ˚ ion pair where the water molecules in the SSIP gap are ordered by the pressure of the ions attracting each other, those localized between the two Na+ ions are ordered by the mutual repulsion of the ions. It creates the force pushing the bridging water molecules to the center of the gap and reducing the diameter of the localization ring. The hydrogen sites of bridging molecules are located in the peak wider and shallower than that of the oxygens. Such molecules are oriented with one hydrogen outwards, in normal to the interionic axis (orientation 3), or with the dipole moment perpendicular to the axis (orientation 3 ). The hydrogen peak is narrower along the axis than perpendicular to it, which indicates the preferential orientation of the bridging molecules with the molecular plane perpendicular to the interionic axis. The localized water molecules fill the gap between the ions, and the first minimum of the PMF for Na+ –Na+ corresponds rather to a SSIP arrangement with one separating water layer. Also it is seen that at the first minimum of the PMF the first hydration shell is more dense around the “poles” over each of Na+ , on the interionic axis (molecule 1 in the left column drawing). In the arrangement at the top of the PMF barrier the gap between the ions is widened, and so the ring-like peak of bridging water molecules merges into a single maximum between the ions, wide along the interionic gap. The maximum of the hydrogen site distribution corresponds to the bridging molecules with both hydrogens situated in the plane perpendicular to the interionic axis (molecule 3 in the middle column cartoon). The distribution in the first hydration shell now has a minimum at the “poles” (molecule 1) and two sequential maxima (molecules 2 and 2 ) with going to the region between the ions.

218


oxy gen

h y d rog en

1st-m in im u m arran g em en t

b a rrier a rran g em en t

2n d -m in im u m a rra n gem en t

N a + (-2.05 ,0,0)

N a + (-2.49 ,0,0)

N a + (-3.07 ,0,0)

N a + (2.05 ,0,0)

N a + (2.49 ,0,0)

3

3' 1

Na+ 2

N a + (3.07 ,0,0)

1

Na+ 3

Figure 4.18.

2'

Na+ 2

3'

Na+ 2'

1

Na+

3'

2

Same as in Figure 4.16, but for the Na+ –Na+ ion pair.

Na+ 2'

219


ox yg en

h y d rog en

C IP a rra n gem en t

b a rrier a rran g em en t

S S IP arran g em en t

C l - (-2.105 ,0,0)

C l - (-2.525 ,0,0)

C l - (-3.455 ,0,0)

C l - (2.10 5,0,0)

3

C l - (2.52 5,0,0)

C l - (3.45 5,0,0)

3 C l-

1

C l-

ca v

C l-

i ty

3 1

C l-

1

C l-

C l3'

3' 2

3' 2'

Figure 4.19.

2

2'

2

Same as in Figure 4.16, but for the Cl− –Cl− ion pair.

2'

220


The interionic distance at the second minimum of the PMF is wide enough to allow formation of two water layers separating the Na+ ions. Each of the ions is thus “wrapped up” in its own, closed first hydration shell. The distribution of water oxygen sites in each of the shells is somewhat higher around the “poles” over the Na+ ions (molecule 1 in the right column cartoon) as well as between the ions (molecules 3 and 3 ). The latter peak has a small local minimum at the interionic axis, so there is some ordering water molecules between the ions into a ring of diameter about 1.2 ˚ A, represented by the positions of molecules 3 and 3 . Water molecules in the shell around each of Na+ are oriented with the dipole moments outward the ion (orientations 1, 2 and 2 ). Water molecules in the two layers separating the ions are oriented with both hydrogens toward the symmetry plane between the ions (orientation 3 ) or with one hydrogen to the center and other away from the ion pair (orientation 3). These hydrogens form a single layer in the symmetry plane with a rather high maximum in the center. Figure 4.19 shows the hydration structure around the Cl− –Cl− ion pair, corresponding to the PMF extrema in Figure 4.12. For each of the ions it is similar to that around Cl− in the Na+ –Cl− ion pair. The oxygen peak of the first hydration shell is sharp and varies only slightly over the shell, except for the region between the ions. Water molecules are bonded to Cl− by one of their hydrogen sites, with the other site tilted outwards (orientations 1, 2 and 2 in all cartoons of Figure 4.19). As a consequence, the first hydration shell has two layers of hydrogen sites, the inner high and sharp and the outer shallow and smooth. The second hydration shell is scarcely visible. As distinct from the two Na+ ions, the first minimum of the PMF for the Cl− –Cl− ion pair arises in a CIP arrangement, at the interionic separation close to the size of the repulsive core of the interionic potential. The repulsion of the ions is balanced out by bridging water molecules bonded with the hydrogens to both ions and located in close contact with them in a narrow peak (molecules 3 and 3 in the CIP cartoon). Two peaks in the inner shell of hydrogen sites arise at the contact of bridging molecules with each of the ions. The hydrogen distribution in the outer shell has a shallow but wellpronounced peak over over bridging water molecules. These hydrogens possessed by water molecules bonded to just one ion can form hydrogen bonds with bridging water molecules (dashed line in the CIP cartoon). There is an unphysical peak of hydrogens in the CIP center, where the ionic attraction acting on hydrogen sites is strongest, but where they cannot penetrate because of the steric constraint. It is related to the well documented inconsistency of the RISM/HNC theory in treatment of so called auxiliary sites [82, 83] which manifests in the interrelation between


221

the oxygen and hydrogen site profiles owed to the steric constraints of the intramolecular bonds being imperfect. For a strongly attractive solutewater site potential the corresponding peaks of the water oxygen and hydrogen site distributions can be more apart than it follows from the length of the intramolecular OH bond. A similar unphysical peak has been observed for the water hydrogen density distribution near the negatively charged carbonyl oxygen of the N -methylacetamide solute by Beglov and Roux [24]. This drawback in the hydration structure can be treated by introducing the bridge corrections (4.21) taking account for the orientationally average of the Boltzmann factor for a core repulsive part of the intermolecular potential [23, 24, 32]. The hydration structure of Cl− –Cl− at the interionic separation corresponding to the PMF barrier is very similar to that in the CIP arrangement. There arises in addition a cavity between the ions, and bridging water molecules get closer to the gap center (molecules 3 and 3 in the barrier drawing). Local minima appear near the “poles” of the inner hydrogen shell. Since the distribution near the “poles” of the oxygen shell remains smooth, this can be interpreted as additional turn of such water molecules so that the hydrogen bonded to the ion is tilted to the interionic axis and the other is directed in perpendicular to it (orientation 1 in the barrier cartoon). Figure 4.19 also displays the Cl− –Cl− ion pair at the interionic separation corresponding to the second minimum of the PMF following from the site-site DRISM/HNC approach, although it vanishes on the 3D-RISM curve. In this arrangement, a complete, almost uniform hydration shells is formed around each of the Cl− ions. The two shells of oxygen sites merge into a water monolayer separating the ions. The oxygen peak of double height is uniform along the gap between the ions and as sharp as each of the shells. The orientations of separating water molecules are now similar to those in the other part of the shell. Their non-bonded hydrogens are located in the symmetry plane between the ions or slightly tilted to it (orientations 3 and 3 in the SSIP cartoon, respectively). The shallow peak of hydrogen sites at the junction of the outer hydrogen shells between the ions becomes hardly visible. It seems to be formed by water molecules located both inside and outside the gap (orientations 2 and 3 in the SSIP drawing). To obtain the PMF between ions in solution at a finite salt concentration, the 3D-RISM description is applied to two labelled ions considered as a composite solute immersed in solvent of all other ions and molecular species constituting the solution. The ion-ion as well as all other sitesite radial distributions input to the 3D-RISM equation are follow from the site-site DRISM theory. The PMFs following from Eq. (4.36) would be same as those corresponding to the distribution functions yielded by

222


the site-site DRISM equations if the closures to the integral equations were exact. The PMFs calculated as a difference between the solvation chemical potential of the ion pair and individual ions is more accurate. Much as for the case of infinite dilution, the 3D-RISM approach yields better results for the energy part of the PMFs by taking explicit account of the steric effects for solvent molecules between the ions in pair, and thus improves the PMF minima depth [30]. The geometry of an ionic cluster in solution can be obtained more precisely by applying the 3DRISM treatment to the whole cluster. Such a multi-dimensional PMF dependent on the relative positions of the ions is rather complicated in view of numerous local minima appearing owing to interference of the first and second minima of the pair PMFs. Figure 4.12 draws a comparison between the 3D-RISM/HNC results for the PMFs of sodium chloride in water at infinite dilution and at a finite concentration of c = 1.069 mol/l. The curves for Na+ –Cl− and Na+ –Na+ at the finite concentration are qualitatively similar to those obtained by Perkyns and Pettitt in the site-site DRISM approach [105]. A prominent feature of the PMFs for sodium chloride at concentration c = 1.069 mol/l is that their short-range structure is very similar to that at infinite dilution. In fact, the short-range behavior of the PMFs is almost same, except for the long-range Coulomb component decaying fast. A reasonable interpretation is that it is screened out by solution ions since the Debye length at the 1M salt concentration decreases to ˚ κ−1 D = 2.9 A. The ionic screening decreases the stability of ion pairs for attracting unlike charges and increases it for repelling like charges. The CIP minimum for Na+ –Cl− becomes significantly shallower with respect to the zero level of infinite separation, whereas those of Na+ –Na+ and Cl− –Cl− ion pairs are now situated below zero. Besides, the CIP and SSIP separations somewhat increase for unlike ions and decrease for like ions. Thus, the short-range parts of the ion-ion PMFs in aqueous solution change little with salt concentration. This justifies the use of the PMFs calculated at infinite dilution for the description of aqueous solutions at the McMillan-Mayer level [117, 122, 131]. The 3D-RISM approach yields in 3D picture of the solvation structure around the pairs of sodium chloride ions in the aqueous solution at 1M concentration. It is presented in detail in [30]. The 3D site distributions of water around the ion pairs resemble those at infinite dilution. Their peaks get slightly lower because of the decrease of the partial density of water at the finite salt concentration. Figures 4.20–4.22 show the 3D-RISM/HNC results for the 3D distributions of solution ions around the labelled ion pairs Na+ –Cl− , Na+ –Na+ , and Cl− –Cl− , respectively. The first solvation shell peaks are significantly higher for solution ions

223


ch lo rid e

so d iu m

C IP a rra n g em en t

b a rrier a rra n g em e n t

S S IP a rra n g em en t

N a + (-1.585,0,0 )

N a + (-1.925,0,0 )

N a + (-2.485,0,0 )

C l - (1.58 5,0,0)

C l - (1.92 5,0,0)

C l - (2.48 5,0,0)

Figure 4.20. Distributions of Na+ and Cl− ions around the labelled Na+ –Cl− ion pair in aqueous solution of sodium chloride at concentration c = 1.069 mol/l. Result of the 3D-RISM/HNC approach. Interionic separations corresponding to the CIP minimum, barrier maximum, and SSIP minimum of the Na + –Cl− PMF (plots in the left, middle, and right columns, respectively). Distributions of chloride and sodium ions (plots in the rows from top to bottom, respectively).

224


ch lo rid e

so d iu m

1 st-m in im u m a rr a n g em en t

b a rrier a rra n g em en t

2 n d -m in im u m a r ra n g em en t

N a + (-1.99 ,0,0)

N a + (-2.5,0,0)

N a + (-2.81 5,0,0)

Figure 4.21.

N a + (1.99,0,0)

N a + (2.5 ,0,0)

N a + (2.8 15 ,0,0)

Same as in Figure 4.20, but for the labelled Na+ –Na+ ion pair.

ch lo rid e

so d iu m

C IP a rra n g em en t

b a rrie r a rra n g em e n t

S S IP a rra n g e m en t

C l - (-2.1 2,0,0)

C l - (-2.5 64,0,0)

C l - (-3.3 34,0,0)

Figure 4.22.

C l - (2.12,0,0)

C l - (2.5 64,0,0)

C l - (3.3 34,0,0)

Same as in Figure 4.20, but for the labelled Cl− –Cl− ion pair.

225


C l-

4 N a+

C l-

2 1

N a+ N a+

3 C l-

Figure 4.23. Possible cluster arrangement of ions and water molecules in aqueous solution of sodium chloride at concentration c = 1.069 mol/l.

in contact with the labelled ion of opposite charge. This gives the “halfcrowns” of the ion distributions around the labelled Na+ –Cl− pair. A salient feature of the first ionic shells around the ion pairs is the huge narrow peaks corresponding to localization of solution ions of opposite charge at a contact with both ions of the pair. They appear in the 3D distributions of solution ions around both unlike and like ion pairs, and are indicative of ionic ordering arising in the NaCl aqueous solution at 1M concentration. These peaks correspond to ionic bridges which stabilize the contact arrangement of like ions. Simultaneously, the bridging ions somewhat decrease the stability of unlike ions in contact as compared to the pair of unlike ions at infinite dilution. Notice that the most probable separations between like ions, 4 to 4.2 ˚ A, and those between unlike ions, ˚ 2.8 to 3 A, are somewhat bigger but close to the interionic distances in the NaCl FCC lattice, 3.99 ˚ A and 2.82 ˚ A, respectively. This suggests formation of ionic clusters in the 1M solution. All the high peaks at the contact between the ions of both unlike and like ion pairs, Na+ and Cl− at Na+ –Cl− , Na+ at Cl− –Cl− , and Cl− at Na+ –Na+ , correspond to a multi-ion cluster comprising equal number of unlike ions. The cartoon in Figure 4.23 depicts such an ionic cluster together with associated water molecules in its first solvation shell. A similar ionic cluster structure has been deduced in experimental investigation of the three molal aqueous solution of erbium bromide [132]. The distance of the first solvation shell of Na+ ions around the Na+ –Na+ pair from each of the ions is very close

226


to the first minimum of the PMF for Na+ –Na+ (plots in the bottom row in Figure 4.21), which also confirms the consistency of the description. The first shell of Na+ around the Na+ –Na+ pair is separated from the pair by a shell of water molecules (molecule 1 in Figure 4.23), similarly to the Na+ –Na+ arrangement at the first minimum of the PMF. This is also clear from Figure 4.21. In the region between the ions the first shell has a rather high peak, split into three well pronounced subpeaks. They indicate favorite locations of Na+ ions near the Na+ –Na+ pair, in the proximity of the high peak of Cl− ions. Similar effects are observed for Cl− around Cl− –Cl− (plots in the top row in Figure 4.21). However, now the Cl− ions are connected by hydrogen-bonded, bridging water molecules (molecules 2 to 4 in Figure 4.23).

7.

Self-consistent 3D-RISM approach

The site-site RISM approach suffers from imperfect treatment of excluded volume effects. Local closures, such as the site-site HNC approximation regard interaction sites of the molecule as independent species, whereas coupling of the site correlations through the intramolecular vv in the RISM integral integral equation cannot provide the matrix ωαγ intramolecular steric constraints be accurately satisfied in the repulsive core region. Such a structural inconsistency manifests especially strongly for ions in polar solvents. It leads to incorrect size dependence of the thermodynamics of hydrophobic hydration. As has been discussed above, this shortcoming can be alleviated by introducing bridge corrections so constructed and parametrized as to effectively account for the core repulsion averaged over molecular orientation. Another way of improving the solvation structure and thermodynamics consists in the self-consistent (SC) 3D-RISM approach which has been applied to water [27] and simple ions in water [27, 34]. For a simple ion immersed in a polar molecular liquid, the description simplifies to 3D correlations of the ion around a solvent molecule regarded as a second solute. The interaction between the ion and the labelled molecule is mediated by the solvent of density ρs . At infinite dilution the molecular OZ equation for the solvent-ion correlations has the form 1 si si dr dΩ css (|r − r |, Ω, Ω )ρs hsi (r , Ω ), (4.53) h (r, Ω) = c (r, Ω) + Ω where the superscripts i and s denote the ion and solvent. The total and direct correlation functions between ion and the solute molecule dependent on their separation r and the solute molecule orientation Ω are equivalent on the 3D correlations of the ion around the solute molecule, hsi (r, Ω) ≡ hsi (r) and csi (r, Ω) ≡ csi (r). The orientational dependence of

227


the 6D argument of css (r, Ω, Ω ) ≡ css (r, Ω ) can be reduced in part by integrating out the orientational degrees of freedom in (4.53). Similarly to the additivity assumption (4.4) of the 3D-RISM theory, the solventsolvent 6D-DCF is decomposed into the partial 3D site contributions css α (r). This is written in the reciprocal space as css (k, Ω ) = css (4.54) α (k) exp(−ik · r sα ), α

where the dependence on solvent orientation Ω goes into the phase factor exp(−ik · r sα ). Substituting the relation (4.54) into the MOZ equation (4.53) transformed to the reciprocal space allows one to carry out the integration over the orientation Ω . The orientational integration reduces the 3D solvent-ion TCF to the radial TCFs between the ion and solvent site α, 1 si dΩk hsi (k) exp(−ik · r sα ), (4.55) hα (k) = Ω according to the definition of the orientational average, 1 si dΩr hsi (r sα + r). (4.56) hα (r) = Ω This leads to the 3D-RISM equation for the 3D correlations hsi (r) and csi (r) of the ion around the solvent molecule, s si hsi (r) = csi (r) + css α (r) ∗ ρ hα (r).

(4.57)

To treat the problem self-consistently the ion-solvent site radial TCFs hsi α (r) are calculated by numerical orientational averaging of the 3D solvent-ion TCFs hsi . Numerical integration in the direct space (4.56) constitutes a non-trivial problem [133, 134] and a considerable computational burden. Instead, it is substantially easier to apply the averaging procedure (4.55) to the Fourier expansion of the 3D-TCF hsi (r) [27, 34]. This simply reduces the expansion in 3D plane waves to that in spherical waves, 1 dΩr hsi (k) exp(−ik · r sα ) (r) = hsi α Ω k sin(kr) exp(−ik · r sα )hsi (k), (4.58) = kr k |k|=k where the phase factor exp(−ik · rsα ) arises owing to the shift of the averaging center from the origin to site α. The orientational averaging

228


procedure thus consists in performing the 3D-FFT, summing the expansion coefficients that have equal absolute values of wavevector k, and then synthesizing hsi α (r) of spherical waves on a linear one-dimensional radial grid. Finally, the radial distribution functions are transformed by the linear 1D-FFT, and then splined to obtain hsi α (k) at the absolute values of wavevector k necessary for the 3D convolution in (4.57). The parameters of the one-dimensional grid are chosen in such a way as to override the resolution of the 3D grid in the direct and reciprocal space. The functions hsi α (r) are calculated by Eq. (4.58) inside a sphere of size equal to the largest dimension of the supercell, r < 12 max (Lx , Ly , Lz ), and are padded with zeros outside it. As compared to numerical orientational integration in the direct space by such orientational quadrature schemes as Lebedev’s quadrature [23], the orientational averaging by resummation of the Fourier harmonics is substantially faster. It should be noted that the procedure (4.58) distorts the averaging radial correlation function by an artifact of small oscillations with the frequency equal to the cutoff of the plane wave set used [27]. However, this is of no importance since such oscillations are smoothened out in the convolution procedure in (4.57) and thus introduce a negligible error. On having converged the SC-3D-RISM equations the ion-solvent site radial distributions hsi α (r) can be finally calculated by numerical orientational averaging of the 3D solvent-ion distribution functions hsi (r) in the direct space according to the definition (4.56). Accurate numerical integration over the sphere is provided, for instance, by employing the 700-point set of the Repulsion scheme within the Spherical Harmonic Reduction or Elimination by a Weighted Distribution (SHREWD) quadrature method [134]. For a system with electrostatic interactions, it is appropriate to complement Eq. (4.57) by the molecular HNC closure approximation, which relates the 3D correlations between the ion and the entire solvent molecule. This provides consistency of the solvent-ion correlations and, in particular, ensures the steric constraints for the ion-solvent site distributions. The solvent-ion SC-3D-RISM/HNC equations for the 3D solvent-ion TCFs hsi (r) can be cast in the form ( ) s si (4.59) hsi (r) + 1 = exp −βusi (r) + css α (r) ∗ ρ hα (r) , where hsi α are calculated by using the procedure (4.58). Notice that the KH approximation can be used as well. It is straightforward to extend the description to a finite concentration of ions by including ionic species in the solvent surrounding the labelled ion and molecule. The solution at a finite ionic concentration includes at least two ionic species owed to the electroneutrality condi-

229


tion, and the SC-3D-RISM/HNC equations (4.59) should be written for the correlations of each ionic species around the solvent molecule. The 3D solvent site DCFs css α (r) to be input into Eq. (4.59) are determined from the SC-3D-RISM equations for a one-component molecular liquid [27], ( ss ) ss s ss (4.60) hss α (r) = cα (r) ∗ ωαγ (r) + ρ hαγ (r) , where the site-site radial TCFs of solvent hss αγ (r) are calculated by the numerical orientational averaging of the Fourier transform as hss αγ (r) =

sin(kr) k

kr

exp(−ik · r sα )hss γ (k).

(4.61)

|k |=k

The closure to Eq. (4.60) can be chosen in the form of the 3D-HNC approximation, ss ss ss hss α (r) + 1 = exp (−βuα (r) + hα (r) − cα (r)) ,

(4.62)

where uss α (r) is the interaction potential between a molecule and site α of solvent. Unlike Eq. (4.59) for the 3D solvent molecule-ion correlations, the 3D-HNC closure (4.62) for the 3D solvent molecule-site correlations suffers in part from the imperfectness discussed above in treatment of the excluded volume of auxiliary sites. However, this is not crucial since such an effect in the correlations between polar molecules is much smaller than in those between ions and polar molecules. It can be alleviated by adding bridge corrections to the 3D-HNC closure (4.62). An alternative formulation of the solvent-ion 3D-RISM equation can be obtained by the orientational reduction of the MOZ equation (4.53) with the term representing interaction over the “third particle” written as d3 hss (13)csi (32). In this case the orientational averaging is applied to the ion-solvent 3D-DCF csi (r) to self-consistently calculate the ionsolvent site radial DCFs csi α (r). The orientational average 1 si dΩr csi (r sα + r) cα (r) = (4.63) Ω comprises partial sites contributions, and should be decomposed into csi α (r) by using the relation ss si csi α (r) = ωαγ (r) ∗ cγ (r),

(4.64)

where the summation convention applies, as before. In a similar manner the 3D site DCFs of pure solvent can determined alternatively from the

230


molecular site Ornstein-Zernike integral equation (MSOZ) proposed by Cortis, Rossky, and Friesner [23], ss ss s ss ss hss α (r) = cα (r) ∗ ωαγ (k) + ρ hα (r) ∗ cαγ (r),

(4.65)

where the site-site radial DCFs css αγ (r) are obtained from the 3D solvent (r) by numerical orientational integration site DCFs css α 1 dΩr css css (4.66) αγ (r) = γ (r sα + r) Ω and decomposition of the averaged DCFs css αγ into the partial site contributions according to ss ss css αγ (r) = ωαµ (r) ∗ cµγ (r).

(4.67)

It should be noted that the solvent SC-3D-RISM equations (4.60) and (4.61) are not equivalent to the MSOZ equations (4.65)–(4.67). The additive approximation (4.67) is exact only in the limit of the long-range asymptotics and thus decreases the level of accuracy of the solvent-ion correlations. The decomposition procedure (4.67) can be unstable and even divergent in the case of large partial charges of molecular interaction sites. Furthermore, the numerical orientational averaging (4.66) is problematic because of the long-range electrostatic asymptotics the 3D site DCFs css γ (r) which in addition is distorted by the artifact of supercell periodicity. Such difficulties with convergence have been observed by Cortis, Rossky, and Friesner [23] in the MSOZ results for polar molecular fluids. Same drawback is peculiar to the decomposition (4.67) for the solvent site-ion DCFs. The closed analytical expressions (4.14) or (4.15) for the excess chemical potential of solvation are no more valid in the SC-3D-RISM approach, since the orientational averaging (4.56) breaks the symmetry of the 3D-RISM integral equation with respect to the solvent indices. Nevertheless, the solvation chemical potential obtained from the SC-3DRISM/HNC equation (4.59) does take the HNC form (4.14) within the additive approximation (4.64), (see Appendix). The use of the 3D-KH closure leads with account of (4.64) to the solvation chemical potential (4.15). The excess energy of solvation at infinite dilution can be obtained by taking the isochoric temperature derivative [135, 136, 129],

∂(β∆µ) = ∆µ + T ∆sV , (4.68) ∆ε = ∂β V where ∆sV is the excess entropy at constant volume. The alternative decomposition of the excess chemical potential of solvation into the excess enthalpy ∆h and the excess entropy ∆sP at constant pressure can


231

be achieved by the use of the isobaric temperature derivative [137],

∂(β∆µ) = ∆µ + T ∆sP . (4.69) ∆h = ∂β P The difference between the two entropic terms is given by [129]

s ∂∆µ , (4.70) T (∆sP − ∆sV ) = ∆h − ∆ε = T αP ρ ∂ρs T

where αP = ∂(log V )/∂T is the isobaric thermal expansion coeffiP cient of the solvent. Applying Eq. (4.68) to the solvation chemical potential (4.A.17) yields the solvation energy in the form s ∆ε = ρ kB T dr βusi (r)gsi (r) 1 1 + csi (r)δT hsi (r) − hsi (r)δT csi (r) , 2 2

(4.71)

where δT f (r) = T (∂f (r)/∂T ) denotes a temperature derivative of the correlation function f (r). Although such derivatives can be obtained by differentiation of the RISM integral equations [136, 129, 67], it is much simpler to calculate them numerically by using the central finite difference approximation, δT f (r; ρ, T ) ≈ T

f (r; ρ, T + ∆T ) − f (r; ρ, T − ∆T ) . 2∆T

(4.72)

The first term of expression (4.71) is the average solute-ion interaction energy, and the rest are the solvent reorganization energy. The solvation enthalpy obtained from Eqs. (4.70) and (4.71) is written as ρs kB T dr hsi (r)δρ csi (r) ∆h = ∆ε + T αP ∆µ + 2

(4.73) −csi (r)δρ hsi (r) , where δρ f (r) = ρs (∂f (r)/∂ρs ) is the density derivative which is calculated by the central finite difference similarly to Eq. (4.72). Since the SC-3D-RISM approach employs the 3D-FFT technique to calculate the 3D-space convolution in the integral equations (4.57) and (4.60), it requires corrections to the electrostatic asymptotics of the correlation functions to alleviate the periodic supercell artifacts. For the

232


correlations of an ion around a polar molecule, the 3D solvent-ion DCF has the asymptotics of the ion-dipole rather than Coulomb potential, c˜si(lr) (r) = −βQi

α

Qα ds · r ∼ −βQi 2 |r − r α | r

(4.74)

where Qi is the ion charge, and ds = α Qα r α is the dipole moment of the solvent molecule with site charges Qα . The potential of mean force between the ion and the labelled solvent molecule and hence the 3D solvent-ion TCF hsi (r) have the same ion-dipole asymptotics but scaled by the dielectric constant of the solvent. The solvent SC-3DRISM/HNC equations (4.60)–(4.62) are partially averaged over solvent orientations. Without any dielectric correction to the 3D-HNC closure (4.62), they yield the trivial dielectric constant of solvent RISM given by (4.34), which leads to the asymptotics of the 3D solvent-ion TCFs ˜ si(lr) (r) ∼ − βQi ds · r . h

RISM r 2

(4.75)

A dielectrically consistent correction to the SC-3D-RISM description can be introduced similarly to the site-site DRISM theory of Perkyns and Pettitt [105]. Within the supercell technique, the electrostatic part φsi(p) (r) of the interaction potential usi(p) (r) between the labelled solvent molecule and the ion is calculated by using the Ewald summation method [101]. On solving the SC-3D-RISM/HNC equations, the distorted periodic asymptotics of the 3D solvent-ion DCF is replaced with the theoretical expression (4.74), c˜si (r) = csi (r) + βφsi (r) − βQi

α

Qα . |r − r α |

(4.76)

The asymptotics (4.75) of the 3D solvent-ion TCF is restored in a somewhat different way. The asymptotic correction is significant at a large separation between the solvent molecule and the ion, and should vanish at the repulsive core. However, a small inaccuracy in the cancellation of the periodic and non-periodic asymptotics of gsi (r) inside the core multiplied by the large value of the repulsive potential leads to a big error in the internal energy obtained by the integration (4.71). This can be avoided by adding the correction to the potential of mean force rather than to the distribution function, Qα βQ i . (4.77) g˜si (r) = gsi (r) exp βφsi (r) −

RISM α |r − r α |

233


With the ion-dipole asymptotics (4.74) and (4.75), calculation of the solvation chemical potential from expression (4.A.17) requires analytical treatment of the long-range contributions. They appear in the terms ( si )2 ˜ si (r)˜ ˜ (r) and h csi (r) but cancel out in c˜si (r) owing to the elech troneutrality of the solvent molecule. On separating out the electrostatic terms, the integration can be performed merely over the supercell volume, 1 (˜ si )2 1 ˜ si h (r) − h (r)˜ csi (r) − c˜si (r) ∆µHNC = ∆µ(es) + ρs kB T dr 2 2 Vcell !

RISM − 1 ( si(b) )2 (r) βφ , (4.78) + 2 2RISM where the long-range asymptotics is defined as the electrostatic potential of the site charges broadened by half-width δ, φsi(b) (r) = Qi

Qα erf(|r − rα |/δ) α

|r − r α |

,

(4.79)

rather than (4.74) for point charges. This provides convergence of the electrostatic contribution to the chemical potential which takes the form ( )2

RISM − 1 s

− 1 (es) (es) =− ρ kB T dr βφsi(b) (r) = − RISM I . ∆µ 2 2 2 RISM 2 RISM (4.80) With allowance for the electroneutrality of the solvent molecule it is expressed through the one-dimensional integral

∞ exp(−k2 δ2 /2) sin(klαγ ) (es) s 2 = 8ρ βQi Qα Qγ dk − 1 (4.81) I k2 klαγ 0 αγ which can be easily evaluated by numerical integration. In a similar way, the ion solvation energy (4.71) is calculated with account for the supercell periodicity as ∆ε = ∆ε(es)

+ ρs kB T Vcell

(4.82)

1 ˜ si (r) dr β u ˜si (r)˜ gsi (r) + c˜si (r)δT h 2 !

2

+ 1 1 ˜ si βφsi(b) (r) . − h (r)δT c˜si (r) + RISM 2 2 2RISM

234


˜ si (r) are the temperature derivatives calculated where δT c˜si (r) and δT h by using the finite differences (4.72). The electrostatic term obtained on separating out the ion-dipole asymptotics (4.79) from the 3D correlation functions and their derivatives entering (4.82) has the form ∆ε(es) = −

RISM + 1 (es) I , 2 2RISM

(4.83)

where the temperature derivative of the dielectric constant RISM given by (4.34) is taken into account. Finally, the ion solvation enthalpy (4.73) is calculated as ∆h = ∆ε

+ ρs kB T 2 αP Vcell

dr

(4.84)

1 ( si )2 1 si h (r) − h (r)csi (r) − csi (r) 2 2 ! 1 si 1 si s si s si + h (r)δρ c (r) − c (r)δρ h (r) , 2 2

where no electrostatic correction is necessary for the latter integral since its electrostatic components cancel out. Application of the SC-3D-RISM/HNC theory to ambient water described with the SPC model yields a detailed 3D structure of the water hydrogen bonding [27]. It is typically visualized as four hydrogen bonds at the vertices of a tetrahedron around the central water molecule, with two of them being extensions of the intramolecular bonds. Figure 4.24 exhibits the profiles of the 3D distributions of oxygen and hydrogen sites around the central molecule, in the molecular plane (parts a, b), and in the plane perpendicular to the molecule axis and passing through its oxygen site (parts c, d). The narrow high peaks of the oxygen and hydrogen profiles in parts a, b, and d correspond to hydrogen bonding between water molecules. The hydrogen bonds are strongly localized in front of the hydrogen sites. Oxygens of water molecules bound to the central molecule are situated in two narrow spots against its hydrogen sites. On the other hand, hydrogens bound to the oxygen site of the central molecule at its opposite flank are arranged more loosely in an “arc” between the third and fourth vertices of the tetrahedron. This 3D picture agrees well with the results of molecular simulations [138]. Notice that the high narrow peaks of the hydrogen bonding turn on orientational averaging into the wider and much shallower first peak of the oxygen-hydrogen radial distribution. The bonding positions in part d correspond to the tetrahedral configuration characteristic of hydrogen


235

bonding of water molecules. The peaks of the hydrogen profile split out from the first maximum and shift towards the oxygen site of the central molecule. At the same time those of the oxygen profile remain within the first solvation shell since the hydrogens are located inside the oxygen LJ core. The other peaks of the solvation shell of each site are located in the places of the maximal repulsion. The first and second peaks are higher and the oscillations are deeper in front of each of the sites. The four slight maxima of the first oxygen solvation shell in part b result from the neighboring hydrogens in the tetrahedral configuration of water hydrogen bonding. The 3D site DCFs of water, css αγ (r), were further used within the SC3D-RISM approach to calculate the structure and thermodynamics of hydration of Na+ and Cl− ions [34], and of such hydrophobic solutes as the rare gas atoms [32]. The orientationally dependent distribution of solvent molecules around the ion can be equally seen seen as a 3D spatial distribution function of the ion around the labelled solvent molecule. Plots in the left column of Figure 4.25 show sections of the 3D distribution of the Cl− ion around the labelled molecule of ambient water, passing through its oxygen. Water oxygens form the first and second solvation shells that are of small height about 1.2 to 1.6 almost (max1) everywhere, except for the two narrow peaks of height gW−Cl− = 73.7 at the arrangements of the Cl− ion located in front of one of the water hydrogens. The latter are attributed to the formation of water molecules hydrogen-bonded to the Cl− ion owing to the asymmetry of the electrostatic field of a water molecule, well known from the 1DRISM/HNC [84, 67] and 3D-RISM/HNC treatment [30] as well as simulations [112, 119, 139]. The 3D distribution of the Na+ ion around the labelled water molecule is depicted in the right column of Figure 4.25. It has a single (max1) large maximum gW−Na+ = 50.3 corresponding to the arrangement with the Na+ ion facing the water oxygen. Notice that the first peak is narrow in the molecular plane where it is concentrated mainly near the OX axis, whereas it becomes a wide “arc” stretched over the oxygen in perpendicular to the molecule plane. Such arrangements of the Na+ ion on the “arc” around the oxygen correspond to the directions usually occupied by hydrogen bonds around a water molecule in ambient water. Unlike the hydrogen bonding to Cl− , water molecules show a simple dipole-like orientation around the Na+ ion. As compared to the 1D-RISM/HNC theory, the SC-3D-RISM/HNC treatment substantially improves the prediction for the positions of the first and second hydration shell peaks and minima around the Cl− and

236


R[\JHQ

K\GURJHQ

Figure 4.24. Three-dimensional distributions of oxygens and hydrogens in ambient water around the labelled molecule, following from the SC-3D-RISM/HNC theory. Sections by the molecular plane XOY (upper plots), and by the plane XOZ perpendicular to the water dipole moment and passing through the oxygen site (lower A, and the plots). The oxygen site is situated at the coordinate origin r O = (0, 0, 0) ˚ A in the XY plane. The hydrogen hydrogen sites are at r H = (±0.8165, 0.5773, 0) ˚ bonding peaks are marked with bold lines.

237


7 3 .4

5 0 .3

5 0 .3

w a ter - C l -

w a ter - N a +

Figure 4.25. Three-dimensional water-ion distributions of Cl − and Na+ at infinite dilution in ambient water (left and right hand columns, respectively), following from the SC-3D-RISM/HNC theory. Sections by the coordinate planes XOY , XOZ, and Y OZ passing through the oxygen site (upper, middle, and lower plots, respectively). The labelled water molecule is situated in the plane XOY , with its dipole moment A along the axis OX and the oxygen and hydrogen sites located at r O = (0, 0, 0) ˚ A. The numbers at the distribution peaks show their and r H = (0.5774, ±0.8165, 0) ˚ height.

238


O -C l g O -N a +

g O -C l -

2

4 2

1 0

O -N a +

6

3

1

2

3

4

5

6

0

7

3

r (Å )

1

2

3

4

5

6

7

r (Å )

4

g H -C l -

g H -N a +

H -C l -

3 2

H -N a +

2

1 1 0

1

2

3

4

5

r (Å )

6

7

0

1

2

3

4

5

6

7

r (Å )

Figure 4.26. Radial distribution functions between water oxygen and hydrogen sites and the Cl− and Na+ ions in ambient aqueous solution at infinite dilution. Orientational averages of the 3D water-ion distributions obtained from the SC-3DRISM/HNC theory (solid lines), results following from the site-site RISM/HNC approach (dash dotted lines), and molecular simulation data [112] (short dashed lines).

Na+ ions (Figure 4.26). On orientational averaging, the narrow high peaks turn into much lower first peaks of the chlorine-water oxygen and (max1) (max2) hydrogen radial distributions of height gO−Cl− = 3.0 and gH−Cl− = 2.3. Notice that the unphysical penetration of water hydrogen sites towards the negatively charged Cl− ion, typical for the former is completely eliminated. The extrema of gH−Cl− (r) now fit the simulation results well. The first peak position of gO−Cl− (r) is improved too. As a shortcoming, the amplitude of the solvation shell oscillations is underestimated. For the water-Na+ radial distribution functions, the SC-3D-RISM/HNC treatment provides some improvement in the first peak height, whereas the oxygen-sodium peak height is worsened. Notice, however, that the oxygen peak in the SC-3D-RISM approach is noticeably wider. This yields the oxygen running coordination number of the first solvation (3D−RISM) (RISM) = 4.9, which is higher than NO−Cl− = 4.4 folshell to be NO−Cl− lowing from the 1D-RISM approach and is closer to the simulation value (MD) NO−Cl− = 5.8. Both integral equation theory [136, 129, 84, 67] and molecular simulations [119, 139, 140, 113, 114, 120, 126, 141, 142, 115, 143, 112, 144, 145] for the hydration of simple ions indicate that its thermodynamics as

239


Table 4.2.

Thermodynamics of solvation of the Cl− and Na+ ions in ambient water.

Approach

∆µ kcal/mol

∆ε kcal/mol

SC-3D-RISM/HNC

−79.5 −48.5a −83.9 −86.0 −80.5c −73.9d

−93.5 −53.7a −100.1

−80.1 −48.5a −77.3 −82.9 −89.6c −96.4d

−90.2 −53.7a −88.8

∆h kcal/mol

∆sV cal/mol

∆sP cal/mol

−98.9

−47.0

−64.8

−105.7

−54.3 −16. −20.1e

−73.0

Cl−

1D-RISM/HNC MD b experiment

−88.2c −81.8d

−25.8c −26.5d

Na+ SC-3D-RISM/HNC 1D-RISM/HNC MD b experiment

−95.6

−34.0

−52.0

−94.1

−38.6 −14. −23.8e

−56.4

−99.9c −106.6d

−34.5c −34.2d

a Electrostatic contributions (4.80) and (4.83) to the excess chemical potential ∆µ and energy ∆ε of hydration. b From [112]. c From [183]. d From [184]. e From [185].

well as structure are very sensitive to the details of the ion potentials, such as softness of the ion repulsive core. With the appropriate choice of the parameters for the site-site interactions, Yu, Roux, and Karplus [129] achieved a good agreement of the 1D-RISM/HNC theory for hydration of simple ions with experiment. An important check, however, is provided by comparison against simulation with the same potential model. Table 4.2 shows the hydration thermodynamics of the Cl− and Na+ ions in ambient SPC/E water. The ions are described with the potential parameters elaborated by Dang and co-workers [126, 141, 142, 115, 143], and latter used by Rasaiah and co-workers in molecular simulations [112, 144, 145]. The predictions of both the 1D-RISM/HNC and the SC-3D-RISM/HNC approaches are qualitative agreement with the simulations and experiment. The SC-3D-RISM treatment improves the excess internal energy of hydration, ∆ε, more sensitive to the behavior of the solute-solvent distributions in the repulsive core region. This in turn results in a noticeable improvement for both the hydration entropies at constant volume and pressure, ∆sV and ∆sP , prone to errors when calculated as a difference between close values [135, 136, 129], T ∆sV = ∆ε − ∆µ, and T ∆sP = ∆h − ∆µ. Further improvement can be obtained by introducing bridge corrections into the solvent-ion as well as solvent-solvent closures to refine the short-range structure and to en-

240


sure the dielectric consistency. The latter can be achieved by adapting the consistent dielectric corrections of Perkyns and Pettitt [105] for the 3D-HNC closure (4.62).

8.

Combined Kohn-Sham DFT and 3D-RISM approach for a metal-liquid interface

It is very attractive to couple the 3D-RISM method with the KS-DFT for the electronic structure to self-consistently obtain both classical and electronic properties of solutions and interfaces. The 3D-RISM approach using the 3D-FFT technique naturally combines with the KS-DFT in the planewave implementation. The planewave basis set is convenient for the simple representation of the kinetic and potential energy operators, and is frequently employed for large systems. The hybrid KS-DFT/3D-RISM method is illustrated below by the example of a metal slab immersed in aqueous solvent [28]. In a self-consistent field (SCF) loop the electronic structure of the metal solute in contact with molecular solvent is obtained from the KS-DFT equations modified for the presence of the solvent. The electron subsystem of the interface is assumed to be at the zeroth temperature, whereas its classical counterpart to have temperature T . The energy parameter of the KS-DFT is replaced by the Helmholtz free energy defined as , , , A ne (r), {ργ (r)} = Emet ne (r) + ∆µsolv ne (r), {ργ (r)} ,

(4.85)

where Emet is the electron energy of the metal, ∆µsolv is the excess chemical potential coming from the metal-solvent interaction, ne (r) is the electron density distribution, and ργ (r) are the classical density distributions of molecular sites γ of solvent. The metal energy is determined by the standard KS-DFT expression written in atomic units as Emet

,

, 1 ne (r)ne (r ) drdr ne (r) = Ts ne (r) + drne (r)vi (r) + 2 |r − r | , (4.86) +Exc ne (r) ,

, gas where Ts ne (r) is the kinetic energy of a non-interacting, electron in its ground state with density distribution ne (r), Exc ne (r) is the exchange-correlation energy, and vi (r) is the potential of metal ionic cores. From the minimal property of the free energy functional (4.85), , δA ne (r), {ργ (r)} = 0, δne (r)

(4.87)

241


subject to the normalization condition for Ne metal valence electrons, (4.88) drne (r) = Ne , one obtains the self-consistent KS equation modified owed to the presence of solvent,

1 − ∆ + vi (r) + vH (r) + vxc (r) + vsolv (r) ψj (r) = εj ψj (r), (4.89) 2 where the Hartree potential is vH (r) =

dr

ne (r ) , |r − r |

(4.90)

the electron density distribution is determined by summation over the Ne lowest occupied eigenstates with allowance for their double occupancy by electrons with opposed spins, ne (r) =

Ne

|ψj (r)|2 ,

(4.91)

j=1

the exchange-correlation potential is the functional derivative , δExc ne (r) , vxc (r) = δne (r) and the solvent potential is defined as , δ∆µsolv ne (r), {ργ (r)} . vsolv (r) = δne (r)

(4.92)

(4.93)

The total free energy is obtained as Atot

Ne

, ne (r)ne (r ) drdr + E = (r) − n xc e |r − r | j=1 , + ∆µsolv ne (r), {ργ (r)} − drvsolv (r)ne (r). 1 εj − 2

drvxc (r)ne (r) (4.94)

As has been widely accepted in Car-Parrinello (CP) simulations [38, 39], the exchange-correlation energy and potential are described in the local density approximation (LDA) [49]. A reasonable level of accuracy is achieved with the LDA including the correlation part by Ceperley and Alder [148] as parametrized by Perdew and Zunger [149], applied to

242


the case of the unpolarized gas. Further refinement can be achieved by employing the generalized gradient corrections (GGA), for example the GGA functional of Perdew, Burke, and Ernzerhof [150]. It is convenient to split up the classical effective potential acting on site γ of a solvent molecule near a metal surface into the contribution of metal ionic cores and that of metal valence electrons, iv ev uuv γ (r γ ) = u γ (r γ ) + u γ (r γ ).

(4.95)

The former term comprises the effective Coulomb interaction between the metal lattice ions and the solvent site, and the exchange repulsion between their core electrons which can be represented semiempirically by the 12-6 LJ potential, Qi Qγ σiγ 12 σiγ 6 iv + 4 iγ − , (4.96) u γ (r γ ) = riγ riγ riγ i

where Qi and Qγ are the effective charges of a metal ionic core and solvent site γ, respectively, iγ and σiγ are the energy and diameter parameters of the LJ interaction, and riγ = |r γ − r i |. The effective potential ueγ (r γ ) of metal valence electrons acting on a single solvent site near the surface is obtained in the linear response approximation. The total energy change δE due to the external pseudocharge variation δqext is given to second order by the expression [49]

1 drδvext (r) δne (r) δE = − dr δqext (r) vi (r) + vH (r) + 2

1 (4.97) + εF + δεF δNe , 2 where δqext (r ) (4.98) δvext (r) = − dr |r − r | is the variation of the electrostatic pseudopotential owed to the external pseudocharge density δqext (r) “felt” by an electron, δne (r) is the induced electron density, δNe = dr δne (r) is the variation of the total number of electrons, and εF is the Fermi level. The first term in Eq. (4.97) accounts for the electrostatic energy of interaction between the perturbing charge and the unperturbed, unpolarized metal. The second term represents interaction of the perturbing charge with the screening electron distribution, that is, with its electrostatic image on the metal surface. The last term vanishes because of the constant number of electrons in the system under consideration: δNe = 0. The other contributions to (4.97), in particular the exchangecorrelation energy, cancel out [49]. Shelley et al [41] strongly argue for

243


neglecting the surface polarization effects altogether in the case of a dense liquid near a metal surface. Although the image potential of a single water molecule can be quite large, several simulations for pure water near a metal surface have revealed a very large degree of cancellation in the interaction between all of the water molecules and all of their images [151, 152, 41]. Similar results have been found for the metal-dipolar fluid interface [153]. Therefore the image term, quadratic in the perturbing charge, can be neglected in the linear response reaction (4.97). Moreover, it should be noted that for a single molecule of an ionic or polar adsorbate located at the interface and disturbing its (translational) lateral symmetry, the image potential is effectively allowed for as the corresponding change in the density distributions of metal valence electrons and solvent charged sites, obtained in solving the SCF loop for the effective potential and the charge distributions. Variational differentiation of the energy (4.97) with respect to the external pseudocharge density yields the classical potential acting on solvent sites. It consists of the contributions from the effective charges of metal lattice ions and from metal valence electrons, Qi Qγ δE = + ueγ (r γ ). (4.99) δqext (r γ ) |r γ − r i | i

The external potential of a solvent site acting on metal valence electrons can be presented with a solvent site pseudopotential vγps (r). The contribution of metal valence electrons to the classical site potential thus reduces to the energy of the metal valence electron density in the site pseudopotential, e ps uγ (r γ ) = − dr qγ (|r − r γ |) vH (r) = dr vγps (r) ne (r − r γ ). (4.100) The effective potential of solvent acting on metal valence electrons is obtained according to Eq. (4.93) by functional differentiation of the excess chemical potential (4.15) with respect to the electron density distribution. This gives v (4.101) dr hγ (r ) vγps (|r − r |). vsolv (r) = ρ γ

Derivation of this expression in the case of the solvation chemical potential specified in the 3D-KH approximation is given in Appendix. Notice that the mean field potential (4.101) follows essentially from the use of the solvation chemical potential in either form (4.14) or (4.15). The self-consistent KS-DFT/3D-RISM equations are solved by means of the dynamical relaxation (DR) method [26, 27, 28] described in the

244


Appendix. First, the Kohn-Sham equations are converged for the metal slab without the solvent. Thereafter the DR procedure is applied simultaneously to the KS-DFT equations for the electron wave functions, and to the 3D-RISM/KH equations for the solvent site correlation functions. On each DR step for ψj (k), the new wave functions of M states are orthonormalized by the Gram-Schmidt process. Then the exchangecorrelation potential vxc (r) is recomputed, the metal Hartree potential vH (r) is obtained from the Poisson equation by using the 3D-FFT, and the metal-solvent site potential uuv γ (r) is calculated. Next, several DR steps are performed for the classical distributions, tuv γ (r), and the solvent effective potential vsolv (r) is calculated evaluating the convolution (4.101) by using the 3D-FFT. The iterational loop is completed with recalculating the total effective potential of the KS equation, veff (r). Iterations are continued until the given tolerance for the root mean square residuals of the electronic and classical distributions ψj (k) and tuv γ (r), as well as for the convergence of the KS effective potential veff (r). It should be emphasized that in contrast to the CP simulation technique in which each molecular dynamics step is performed only after self-consistency of the electron degrees of freedom is reached, the combined KS-DFT/3D-RISM equations are being converged simultaneously and just once. However, the classical distributions have to “respond” quickly to variations of the electron subsystem in the process of relaxation. The 3D solvent site total correlation functions huv γ (r) result from subtle cancellation of the strong attractive metal-solvent site potential uuvγ (r) and the solvent screening effective field given by the term uv uv tuv γ (r) = h γ (r) − c γ (r) in the closure (4.13) to the 3D-RISM equation (4.9). Therefore the effective potential of solvent acting on metal valence electrons, vsolv (r), determined by huvγ (r) is very sensitive to the degree of convergence of the 3D-RISM equations. Nevertheless, there is no need to reach the full convergence of the classical distributions at each relaxation step for the electron wave functions. To ensure robust fictitious motion of the system to the solution, it is sufficient to keep the root mean square residual of the 3D-RISM equations an order of magnitude lower than that of the KS-DFT, Eqs. (4.A.41) and (4.A.43) in the Appendix, by performing 10 to 30 relaxation steps for the former per one relaxation step for the latter. This adds a little to the computation time since the most time consuming is the KS-DFT part (the Gram-Schmidt orthonormalization requiring on the order of M 2 N operations, with N plane waves kept in the eigenstate expansions). The whole procedure converges in mere several hundreds relaxation steps for the KS equation, and thus compares favorably with the CP simulation in computational time.


245

The self-consistent KS-DFT/3D-RISM-KH theory has been applied to the copper-water interface [28]. For the sake of comparison, the potential models of the metal as well as water molecules were taken from the CP simulation available for this system [40]. The five-layer slab of 4 × 4 metal atoms per layer, arranged in an FCC lattice with (100) surfaces and lattice constant a = 2.55 ˚ A is positioned across a tetragonal supercell ˚ of size 10.2 × 10.2 × 40.8 A (Figure 4.27). The rest of the supercell is filled with ambient water at temperature T = 298 K and number density ρ = 0.03334 ˚ A−3 . The local pseudopotential of the metal ionic cores is constructed to represent a singly valent element roughly modelled after copper, and adjusted to recapitulate the experimental value of the Cu slab work function. Water molecules are taken to be closed-shell systems with the local pseudopotential comprising those centered on the oxygen and hydrogen sites with the corresponding site charges of the TIPS water model [130]. It is fitted to the all-electron KS-DFT calculation for a single water molecule and the ab initio results for the minima of the metal slab-water molecule interaction energy. Bulk ambient water is described by the simple point charge (SPC) model [155]. Figures 4.28 to 4.32 present the results of the self-consistent KS-DFT/3D-RISM-KH method applied to this model of the copper-water interface. Figure 4.28 shows the total effective potential veff and its components, and the metal valence electron density ρe , averaged in the plane parallel to the slab surface as a function of the z-coordinate along the larger supercell side. The work function for the present model of the metal slab amounts to 4.85 eV. Owing to the water solvent the work function decreases by 0.50 eV. This decrease is close to the value of 0.6 eV obtained for the similar system in the CP simulation by Price and Halley [40], and agrees with the order of magnitude typically found in experiment [156, 157, 158]. Substantial oscillations set up in the solvent effective potential vsolv , and hence in the total effective potential veff owed to the hydration layers with alternating excess of oxygen and hydrogen sites of water molecules. The water effect is most noticeable in the first hydration layer, close to the metal surface. Inside the metal the solvent contribution vsolv flattens out at the level of 0.63 eV. The metal electrostatic potential ves and the total effective potential veff oscillate in accord with the periodicity of the metal slab lattice. However, the change in veff owed to the solvent is smooth inside the slab and manifests in the shift by 0.5 eV causing the decrease of the work function. The change in ves is simply an increase in its surface drop. The metal surface dipole rises in response to the enhancement of electron escape from the metal owed to the water surface polarization. Therefore the change in ves is opposite in sign to the water surface dipole and some-

246


Figure 4.27. Arrangement of the metal slab atoms in the tetragonal supercell with periodicity in three dimensions. The slab faces are (100) surfaces of a FCC lattice. The number of atoms in each layer of the slab is 4 × 4 per supercell. Sections of the supercell by the planes perpendicular to the slab face and going through its diagonal AC (part a) and side AB (part b), and by the plane of the slab surface layer (part c). On-top, bridge, and hollow site adsorption positions are marked as T, B, H. Solid and dashed circles denote metal atoms located in the section plane and in the next lattice layer, respectively.


Figure 4.28. Transverse average of the total effective potential of the Kohn-Sham equation veff and its metal electrostatic and solvent effective terms ves and vsolv versus the z-coordinate of the supercell (part a). Transverse average of the metal valence electron density ρe (part b). The metal slab in contact with water (solid lines) and in vacuum (dashed lines).

247

Figure 4.29. Transverse average of the 3D site distributions of water oxygen and hydrogen gO and gH (solid and dashed lines, respectively) as a function of the z-coordinate of the supercell (part a), and their portion as a function of the distance ∆z = z − zisurf from the surface layer of metal slab A atoms positioned at zisurf = ±3.6 ˚ (part b).

what reduces the resulting decrease in the work function. This is unlike the CP simulation [40], where the change in the metal electrostatic potential has the same sign as the potential of the water surface dipole and produces about half the decrease in the work function. This can be attributed to the effect of a few of chemisorbed water molecules, which adsorb and desorb at particular on-top positions during the CP simulation. These tightly bound molecules strongly deplete the metal valence electron density around them in the region of the electron density tail, close to the metal surface layer. This gives a considerable reduction of the metal surface dipole and decreases the work function in addition to the effect of the water surface polarization. The behavior of the metal valence electron density ρe and the positions of its peaks are similar to those found in the CP simulation. On the metal side there arise oscillations of ρe with alternating depletion and

248


enrichment regions extending deeply into the metal slab. The water solvent induced change in ρe is quite small as compared to the whole density of the metal slab. Outside the metal it has just one positive peak at z = ±6.6 ˚ A, in the region of the water surface dipole and then exponentially decays into the water bulk. Figure 4.29 depicts the water site distribution profiles averaged along the metal slab surface, as a function of the z-coordinate across the supercell. A layering of water molecules near the metal surface is clearly seen. The distance ∆z = z − zisurf from the first surface layer of metal atoms zisurf to the oxygen and hydrogen peaks of the first hydration layer max1 = 2.47 ˚ max1 = 2.52 ˚ A and ∆zH A, practically coincide in position, ∆zO respectively. The minima marking the boundary of the first hydration min1 = 4.07 ˚ min1 = 3.98 ˚ A and ∆zH A. The layer are close as well, ∆zO occupation numbers of a surface adsorption site defined as the number of water sites γ in the first hydration layer per surface unit cell area a2 , Nγ(1) (1)

ρv = 2 a

zγmin1

dr guv γ (r),

zi (1)

amount to NO = 0.800 for oxygen and NH = 0.794 for hydrogen, showing that there is no excess of either of water sites in the first hydration layer and water molecules are oriented predominantly in parallel to the surface. However, the first peak of gH is lower and widened asymmetrically outwards the surface, and its first minimum is shallower than that of gO . The outer hydrogen sites correspond to a considerable part of water molecules tilted with hydrogens outwards the metal, which induces a surface dipole decreasing the work function. At the same time there is a small excess of hydrogen sites close to the surface (∆z = 1.5 to 2.0 ˚ A) coming from some water molecules tilted with hydrogens towards the metal, and resulting in a small decrease in the solvent potential vsolv before it flattens out in the slab. Such orientations at the potential of zero charge conform with experiment [159]. Worth noting is also that the surface occupation numbers smaller than one are consistent with the first maximum separation of the bulk water distribution gOO (r) being larger than the lattice constant a = 2.55 ˚ A of the Cu(100) metal slab; water molecules are bigger and cannot occupy adsorption positions at all surface metal atoms simultaneously. The subsequent oscillations with the maxima at ∆z = 5.36; 8.13; 10.84 ˚ A for the oxygen site profile and 5.61; 8.48; 11.19 ˚ A for the hydrogen one fall off in the water bulk. The shift between the corresponding gO and gH maxima increases with distance from the surface, showing the decay of water orientational ordering far from the metal surface.


249

The behavior of the transverse average of the water site distributions at the metal surface discussed above is in agreement with both the classical molecular dynamics results [160, 161, 162] and the CP simulation [40]. However, the lateral distributions obtained from the KS-DFT/3DRISM-KH theory contrast with the classical MD simulations, but agree well with the CP simulation. Figure 4.30 displays the 3D oxygen and hydrogen site distributions of water in contact with the metal, gO (r) and gH (r), following from the KS-DFT/3D-RISM-KH theory. Shown are the profiles in the plane passing through the on-top and hollow-site positions in normal to the metal surface. Figure 4.31 depicts the profiles in the plane over the surface at the separation close to the first maxima of the distributions. The strongest adsorption is observed at bridge and hollow site positions, in agreement with the results of the CP simulation. The classical MD simulations employ the classical metal-water potential derived from extended H¨ uckel molecular orbital calculations for a metal cluster with a single water molecule [163], which has the biggest energy of adsorption of a single molecule at the on-top and smallest at hollow site location. Hence the simulations using this potential input yield the water site distributions with the high maxima of gO (r) over metal surface atoms. In contrast to the classical MD simulations the KS-DFT/3D-RISM approach allows for modification of the metal-water potentials in dense water owed to the many-body quantum effects. In dense liquid the electronic structure of adsorption of a given molecule is strongly disturbed by the neighbors, and so the substantial part of the adsorption potential of the molecule is represented in the first approximation by its effective average potential. As has been discussed above, the latter is determined in the linear response approximation by the metal electrostatic potential ves . Figure 4.32 exhibits the sections of the electrostatic potential ves (r) and valence electron density ρe (r) of the metal slab in the same plane as the water distribution profile in Figure 4.30. It is seen that the metal electrostatic potential is strongest at the hollow site and weakest at on-top adsorption positions. The attraction of oxygen sites to the surface follows these shapes. On the other hand, the electrostatic potential for hydrogens has the opposite sign and is repulsive. The repulsion is weakest when hydrogens are located at the on-top positions. Owing to the geometry of water molecules, this pushes oxygen sites away from the on-top locations as well. The shortrange repulsion between ionic cores of metal atoms and water sites is shaped as a (100) surface array of nearly hard spheres, which deepens the attractive wells at the hollow positions. The repulsion owed to the overlap of water oxygen cores with the cloud of metal valence electrons is somewhat weaker over metal atoms than at the hollow and bridge

250


ox y g en

oxygen

h yd rogen

h yd rogen

Figure 4.30. Three-dimensional distributions of oxygen and hydrogen sites of ambient water in contact with the (100) FCC metal slab. Section by the plane passing through the ontop and hollow site adsorption positions of the (100) surface. Metal ionic cores of the slab are located as in Figure 4.27a, and are schematically shown with humps in the void gγ (r) = 0.

Figure 4.31. Section of the 3D water oxygen and hydrogen site distributions by the plane situated in parallel to the (100) FCC metal slab at the distance of 2.45 ˚ A from the first surface layer. The section passes through the peaks of the distributions in the first hydration shell. The positions of metal ionic cores are same as in Figure 4.27c.

site positions. However, this lateral corrugation becomes considerable only inside the metal at the first surface layer, but not in water at the distance of the adsorption potential minima. It should be pointed out that the above distributions of water over the metal surface as well as the results of the CP simulation of Price and Halley [40] are obtained for the model taking no account of directionality of d-electrons. For such a simple model metal the valence electron density and the attractive electrostatic potential possess maxima at hollow site positions, whereas the full-electron calculations for a copper slab yield the electron density with the maxima at on-top adsorption positions [49]. The explicit account for d-electrons of a transition metal might drastically change the above discussed distribution of water over the surface, making the distribution more biased towards the on-top position. In any case, the effective interaction potential between the metal surface and liquid water differs from that of a single adsorbed water

251

Three-dimensional RISM theory m eta l electro sta tic p o ten tia l

m eta l electro n d en sity

Figure 4.32. Section of the metal electrostatic potential ves (r) and the metal valence electron density ρe (r) of the (100) FCC metal slab in contact with ambient water by the plane passing through the on-top and hollow site adsorption positions. The positions of metal ionic cores are same as in Figure 4.30.

molecule, and should allow for many-particle quantum effects of dense liquid. A more consistent treatment of the metal-water interface would be achieved by treating a particular adsorbed water molecule besides the mean field effect of dense water. Within the 3D-RISM approach other ionic and molecular species adsorbed from the liquid onto the metal surface can be readily considered, including charge transfer between the electrode and adsorbate. The description can be advanced to a more realistic model of metal atoms by employing the Vanderbilt ultrasoft pseudopotentials [164], widely used in CP simulations [165, 166]. This would allow to explicitly consider d-electrons forming highly directional chemical bonds at transition metal surfaces.

9.

Hybrid 3D-RISM-SCF and ab initio MO method for solvated molecules

In the RISM-SCF procedure coupled with the multi-configurational self-consistent field (MCSCF) approach [59, 60, 61], the solvent effective potential modifying the electronic structure of the solute molecule is (0) incorporated by adding a solvent term to the Fock operator Fi of an

252


isolated solute, (solv)

Fi

(0)

= Fi

− fi

Vs(solv) bs ,

(4.102)

s

where fi is the occupation number of orbital φi , bi is the population (solv) is the solvent mean field assumed operator of solute site s, and Vs to be the electrostatic potential of a solvent molecule averaged with the radial site-site total correlation functions hsγ (r) between solute site s and solvent site γ, ∞ hsγ (r) Vs(solv) = 4πρv Qs , (4.103) r 2 dr r 0 γ where the solute-solvent site-site correlations are obtained from the RISM/HNC integral equations. Notice that the Fock operator (4.102) can be derived from the variational principle for the solute free energy comprising the electron part determined by the ab initio MO part and the solvation chemical potential given in the site-site HNC and mean field approximations, similarly to Eq. (4.85) for the hybrid KS-DFT/3DRISM approach. The SCF loop is completed by the “back” effect of the solute electronic structure on the solute-solvent classical distributions owed to the modification of the gross population of partial charges Qs on solute sites s. To be incorporated into the site-site RISM technique, it has to be fitted to a Coulomb potential of point effective charges centered on the solute sites. This approximation of radial averaging constitutes a bottleneck of the site-site RISM-SCF method, since it oversimplifies the effect of directed orbitals of the solute molecule on its solvation structure. The solute site charge fitting required in the site-site RISM-SCF treatment is eliminated for the ab initio MO method coupled with the 3DRISM approach explicitly treating the solute electron distribution in the SCF loop. The effective potential of solvent acting on the solute electrons, V (solv) (r), is obtained by functional differentiation of the excess chemical potential of solvation with respect to the electron density distribution of the solute, Eq. (4.93). In the 3D-KH as well as 3D-HNC approximations (4.15) and (4.14) this leads to the solvent effective potential in the mean field form (4.101), expressed in terms of the pseudopotential of a solvent molecule acting on an external electron. It comprises partial contributions vγps (r) centered on the interaction sites of the solvent molecule. The classical effective potential energy of the solute acting on solvent site γ,

uv(sr) (n) (e) (r) = u (r) + Q (r) + φ (r) , (4.104) φ uuv γ γ γ

253

Three-dimensional RISM theory uv(sr)

is broken up into the short-range interaction uγ (r) between the solvent site and the whole solute represented by the sum of the 12-6 LJ potentials over the solute sites, and the electrostatic energy of the solvent site effective charge Qγ in the fields of the solute nuclei and electrons, φ(n) (r) and φ(e) (r). Within the supercell treatment used for the 3D-RISM approach, the Coulomb potentials of the solute nuclei φ(n) (r) are summed by means of the Ewald method. The electron electrostatic potential φ(e) (r) is obtained in the reciprocal space from the Poisson equation, (4.105) −k2 φ(e) (k) = 4πρ(k), and then synthesized on the direct space grid by using the 3D-FFT. This requires evaluation of the solute electron density ρ in reciprocal space at k-grid nodes [except for k = 0 determining only the constant term in φ(e) (r)], which are calculated from the usual density matrices Dµν as Dµν χµ (r)χ∗ν (r) exp(ik · r) ρ(k) = drρ(r) exp(ik · r) = dr =

µ,ν

Dµν

µ,ν

drχµ (r)χ∗ν (r) exp(ik · r) =

Dµν Iµν (k), (4.106)

µ,ν

where χµ (r) are the usual basis set functions represented by the sum of Gaussian functions. The Fourier integrals Iµν (k) are obtained analytically, which crucially reduces the computational expenses for transformation of the solute electron density ρ(r) from the Gaussian set representation Dµν to the planewave form ρ(k). Notice that the classical solvent site potential uγ (r) is directly related to the solute electron density coefficients Dµν through Eqs. (4.105) and (4.106), not involving any charge fitting procedure. The 3D-RISM-MCSCF approach has been applied to carbon monoxide (CO) solute in ambient water [33]. Since it is known that the HartreeFock method predicts the electronic structure of CO in wrong character [167], the CASSCF method (2 core, 8 active orbitals, 10 electrons) in the basis sets of double zeta plus polarization (9s5p1d/4s2p1d) augmented with diffuse functions (s- and p-orbitals) was used. Water was described by the SPC/E model [127] and the site-centered local pseudopotential elaborated by Price and Halley for CP simulation [40]. The 3D-RISM/KH integral equations for the water distributions specified on A were solved at each a grid of 643 points in a cubic supercell of size 20 ˚ step of the SCF loop by using the method of modified direct inversion in the iterative subspace (MDIIS) [27, 29] (see Appendix). Figure 4.33 presents the water oxygen and hydrogen distributions in the plane containing the CO molecular axis. The first hydration shell

254

Molecular theory of solvation C O - O (w ater)

C O - H (w ater)

4

4

2

2 z (Å )

6

z (Å )

6

0

0

-2

-2

-4

-4

-6

-6 -4

-2

0

x (Å )

2

4

-4

-2

0

x (Å )

2

4

Figure 4.33. Three-dimensional distributions of water oxygen and hydrogen sites around the hydrated molecule of carbon monoxide.

peak of the water oxygen distribution is rather uniform around the CO molecule. The water oxygen peak at the contact is highest against the (max) = 2.6, whereas in the other parts of the shell it CO carbon, gO amounts to about gO = 2.0. Accordingly, the oscillation of the water oxygen distribution in the second hydration shell is larger next to the first peak at the CO carbon on the molecular axis. The water hydrogen distribution has a high separate peak at the contact with the CO carbon, (max) = 3.2, due to its negative charge. The hydrogen peak is shifted by gH ˚ 1 A with respect to the water oxygen shell towards the CO carbon. Next follows the first shell of hydrogens with the maximum of gH = 1.2 to 1.4, surrounding the whole CO molecule together with the above hydrogen peak and the first shell of oxygens. The hydrogen peak is split apart the first hydrogen shell by a region of hydrogen depletion, gH = 0.7, coinciding with the oxygen shell. The separation between the hydrogen and oxygen shells is about 0.3 to 0.35 ˚ A almost everywhere with outward hydrogens, except for the region over the CO carbon where it amounts to 1 ˚ A. The second hydration shell distributions have shallow and wide maxima shaped barely as two ellipsoids around the CO molecule, with hydrogens shifted with respect to oxygens outwards the solute by about


255

Figure 4.34. Radial distributions of water oxygen and hydrogen sites around the hydrated CO molecule: C(CO)–O(water), O(CO)–O(water), C(CO)–H(water), and O(CO)–H(water), parts (a) to (d), respectively. Results of the 3D-RISM-MCSCF and site-site RISM-MCSCF approaches (solid and dashed lines, respectively).

0.3 ˚ A. This corresponds to the orientation of water molecules similar to that in the uniform part of the first hydration shell. Thus water molecules located in the major part of the first hydration shell closer to the CO oxygen or over it are oriented with their both hydrogens tilted outwards the CO solute, whereas water molecules over the CO carbon near the CO molecular axis are hydrogen-bonded to it by one of the hydrogens and the other hydrogen tilted outwards the CO carbon. The hydrogen bonding of water molecules over the CO molecule to its carbon, evident from the 3D site distributions of water becomes much less evident in the site-site description. Figure 4.34 draws a comparison of the RDFs between the CO and water interaction sites, obtained by numerical orientational averaging of the 3D site distributions resulting from the 3D-RISM-MCSCF treatment, and those following from the site-site RISM-MCSCF approach. The CO carbon-water hydrogen bonding manifests in fact only in small shoulders of the CO-water RDFs.

256


Other features of the RDFs are similar in both the methods: a small shoulder of C(CO)-O(water) around 4.0 ˚ A, the second peak around 6.0 ˚ A in C(CO)-O(water), and rather broad characters in C(CO)-H(water) RDF’s. Whereas the C(CO)-H(water) and O(CO)-H(water) maxima are very similar in the 3D and site-site approaches, the 20 percent decrease of the first peaks height of the C(CO)-O(water) and O(CO)-O(water) distributions is related to the use of the 3D-KH closure (4.13) rather than the 3D-HNC one (4.12). It is, however, counterbalanced by the widening of the peaks, and so the coordination numbers of the first hydration shell are not affected much. Notice that coordinations of water hydrogens to the CO oxygen arise around 2.5 ˚ A in the O(CO)-H(water) RDF following from the 3D-RISM-MCSCF method. This corresponds to the hydrogen bonding discussed above and is not reproduced well by the site-site RISM-MCSCF approach.

257


Appendix 1.

Free energy functions in the KH approximation

Similarly to the expressions found by Singer and Chandler [80] for the RISM/HNC equations, the KH approximation (4.13) allows one to obtain the free energy functions in a closed analytical form avoiding the necessity of numerical coupling parameter integration. The derivation is analogous for both RISM and 3D-RISM/KH equations [28], and is shown here in the context of the 3D approach. The excess part of the solvation chemical potential, in excess over the ideal translational term, can be related to the 3D site correlation functions by the Kirkwood’s “charging” formula ∆µsolv = ρv

γ

1

dλ

dr

0

∂uuv γ (r; λ) uv g γ (r; λ), ∂λ

(4.A.1)

where the coupling parameter λ “switches on” the interaction between the solute and solvent. The KH closure (4.13) then takes the form g uv γ (r; λ) =

( ) exp (duv γ (r; λ) 1 + duv γ (r; λ)

for for

duv γ (r; λ) ≤ 0, duv γ (r; λ) > 0,

uv uv uv duv γ (r; λ) = −λβu γ (r) + h γ (r; λ) − c γ (r; λ).

(4.A.2) (4.A.3)

Differentiating Eq. (4.A.3) with respect to λ gives

⎧ dhuv dcuv γ γ ⎪ ⎨ g uv − −βuuv γ γ + dλ dλ = uv uv ⎪ dλ ⎩ −βuuv + dh γ − dc γ γ dλ dλ

dg uv γ

for

duv γ ≤ 0,

for

duv γ > 0.

(4.A.4)

On simple rearrangement, Eq. (4.A.4) yields

uv βuuv γ g γ

⎧ ( uv )2 dcuv dcuv ⎪ 1 d hγ γ γ ⎪ ⎨ − − huv γ 2 dλ dλ dλ = uv uv ⎪ ⎪ ⎩ − dc γ − huv dc γ γ dλ dλ

for

duv γ ≤ 0,

for

duv γ

(4.A.5)

> 0.

uv Since the joint point duv γ = 0 corresponds to h γ = 0, Eq. (4.A.5) can equally be written as uv βuuv γ g γ =

dcuv dcuv 1 d ( uv )2 ( uv ) γ γ h γ Θ −h γ − − huv , γ 2 dλ dλ dλ

(4.A.6)

where Θ(x) is the Heaviside step function. The last term in Eq. (4.A.6) is to be transformed so as to bring the whole expression into the form of an exact differential d∆µsolv /dλ. Using the 3D-RISM equation (4.9) written in the form huv γ (r) =

vv dr cuv α (r − r ) χαγ (r ),

vv v vv where χvv αγ (r) = ωαγ (r) + ρ hαγ (r), one can see that

(4.A.7)

258 ∂ ∂λ γ


∂ vv uv drdr cuv α (r ; λ) χαγ (|r − r |) c γ (r; λ) ∂λ αγ ∂ uv = 2 dr huv (4.A.8) c γ (r; λ). γ (r; λ) ∂λ γ

uv dr huv γ (r; λ) c γ (r; λ) =

On substituting Eq. (4.A.6) into (4.A.1) with account of (4.A.8) and integrating over λ, one obtains the excess chemical potential of solvation for the 3D-KH closure (4.13) as 1 ( uv )2 ( uv ) 1 uv v uv uv h dr = ρ k T (r) Θ −h (r) − c (r) − (r) c (r) . ∆µKH h B solv γ γ γ γ γ 2 2 γ (4.A.9) Applying similar analysis to the Helmholtz free energy per unit volume, A = Aid +Aex , one gets its excess part as 1 v 2 1 ( uv )2 ( uv ) (4.A.10) dr h γ (r) Θ −h γ (r) − cuv AKH ex = − (ρ ) kB T γ (r) 2 2 γ dk 1 vv uv v vv uv v + ln det 1 − ω (k)c (k)ρ (k)c (k)ρ + kB T Tr ω . 2 (2π)3 The free energy functional (4.A.10) is stationary with respect to variation of the correlation functions, which leads to the integral equation (4.9) and the closure (4.13). Hence, the pressure can be evaluated by differentiating only the explicit density dependence in (4.A.10), 1 ( uv )2 ( uv ) 1 dr h γ (r) Θ −h γ (r) − cuv P KH = ρv kB T + (ρv )2 kB T γ (r) 2 2 γ 1 dk + kB T (4.A.11) ln det 1 − ω vv (k)cuv (k)ρv 3 2 (2π)

−1 vv uv v uv v uv − Tr ω (k)c (k)ρ 1 − ω (k)ρ (k)c (k) . Extension of the KH expressions (4.A.9)–(4.A.11) to a molecular mixture (including the 1D-RISM approach) is obvious. Notice that all they are analogous to the corresponding forms derived by Singer and Chandler [80] from the RISM/HNC equations, except for the difference in the term h2 (r)/2 which is switched off in the spatial regions of density enrichment, h(r) > 0.

2.

Solvation chemical potential in the SC-3D-RISM approach

The excess chemical potential of solvation of an ion in a molecular solvent is obtained by the common procedure of “switching on” the solute-solvent interaction, ∆µ = ρ

usi (r )

dr

δusi (r)g si (r).

(4.A.12)

0

For the 3D solvent-ion correlations obtained from the MOZ equation (4.53) complemented with the HNC closure approximation, the integration over the interaction in

259


(4.A.12) can be performed analytically. The 3D solvent-ion MOZ/HNC equation can be written as

ρ hsi (r, Ω) + 1 = exp −βusi (r, Ω) + dr dΩ css (|r − r |, Ω, Ω )hsi (r , Ω ) . (4.A.13) Ω Functional variation of Eq. (4.A.13) leads after simple rearrangement to βg si (r, Ω)δusi (r, Ω) = − δhsi (r, Ω) ρ dr dΩ css (|r − r |, Ω, Ω )δhsi (r , Ω ) + (4.A.14) Ω ρ dr dΩ css (|r − r |, Ω, Ω )δhsi (r , Ω ). + hsi (r, Ω) Ω On insertion into (4.A.12), the first and second terms of expression (4.A.14) can be immediately integrated over the interaction. Taking into account the MOZ equation (4.53), and adding integration over orientation Ω to symmetrize the third term, one has (4.A.15) β∆µHNC = − ρ drcsi (r) +

ρ2 Ω2

drdr dΩdΩ

usi (r )

hsi (r, Ω) css (|r − r |, Ω, Ω ) δhsi (r , Ω ).

0

The last integral over the interaction in (4.A.15) is taken by using the symmetry of the variation

(4.A.16) δ hsi (r, Ω) css (|r − r |, Ω, Ω ) hsi (r , Ω ) = δhsi (r, Ω) css (|r − r |, Ω, Ω ) hsi (r , Ω ) + hsi (r, Ω) css (|r − r |, Ω, Ω ) δhsi (r , Ω ). Using Eq. (4.53), this results in the expression for the chemical potential of solvation of the ion in the familiar closed analytical form

2 1 1 hsi (r) − hsi (r)csi (r) − csi (r) . dr (4.A.17) ∆µHNC = ρkB T 2 2 Making functional variation of the 3D-RISM/HNC equation (4.59), and inserting the result into (4.A.12) with allowance for (4.57) gives β∆µHNC = − ρ dr csi (r) (4.A.18) + ρ2

drdr

usi (r ) 0

hsi (r)

si css α (r − r ) δhα (r ).

α

The latter term in this expression cannot be integrated analytically over the interaction since the variation

si (4.A.19) δ hsi (r) css α (r − r ) hα (r ) si si si = δhsi (r) css css α (r − r ) hα (r ) + h (r) α (r − r ) δhα (r ) α

α

does not possess the symmetry as in Eq. (4.A.16). Nevertheless, taking into account the decomposition (4.67) employed in the derivation of the 3D-RISM integral equation

260


(4.57), the second term of expression (4.A.18) is equivalent to that in (4.A.15), and the variation (4.A.19) is same as (4.A.16). Therefore within the additive approximation (4.67) the solvation chemical potential obtained from the 3D-RISM/HNC equation (4.59) does take the form in Eq. (4.A.17).

3.

Solvent effective potential coupling the KS-DFT and 3D-RISM equations

Consider the effective potential of solvent acting on metal valence electrons, defined by Eq. (4.93) as the functional derivative of the solvation free energy with respect to the metal valence electron density, ne (r). Its derivation can be done similarly to the above expression for the excess chemical potential of solvation, ∆µsolv , in the 3D-KH approximation. The variation of Eq. (4.A.9) is written as ( uv ) uv uv dr huv δ∆µsolv = kB T ρv γ (r) δh γ (r) Θ −h γ (r) − δc γ (r) γ

) 1 ( uv . − δ huv γ (r) c γ (r) 2

The variation of the 3D-KH closure (4.13), ) uv ( uv uv g γ −βδuuv for huv γ + δh γ − δc γ γ ≤ 0, = δg uv γ uv uv uv for huv −βδu γ + δh γ − δc γ γ > 0, rearranges as ( uv ) uv uv uv uv uv uv g uv γ βδu γ = h γ δh γ Θ −h γ − δc γ − h γ δc γ .

(4.A.20)

(4.A.21)

(4.A.22)

The last term in this expression is evaluated by analogy with Eq. (4.A.8) as

uv uv dr huv dr huv =2 (4.A.23) δ γ (r) c γ (r) γ (r) δc γ (r). γ

γ

The comparison of Eqs. (4.A.22) and (4.A.23) with Eq. (4.A.20) gives uv dr g uv δ∆µsolv = ρv γ (r) δu γ (r).

(4.A.24)

γ

From Eqs. (4.95) and (4.100) one has the relation between the variations of the classical site potential and the metal valence electron density, δuγ (r) = vγps (|r − r |). δne (r )

(4.A.25)

Substituting Eq. (4.A.25) into (4.A.24) and taking the functional derivative yields the effective potential of solvent, δ∆µsolv = ρv (4.A.26) dr gγ (r ) vγps (|r − r |). vsolv (r) ≡ δne γ Using the electroneutrality of solvent it can also be rewritten as dr hγ (r ) vγps (|r − r |). vsolv (r) = ρv

(4.A.27)

γ

Notice that this expression means simply the mean field approximation and is the same for the HNC and KH approximations. In fact, it is predetermined by the solvation free energy (4.A.1) constructed as an average of the interaction over the distribution.


4.

261

Algorithms for solving the RISM equation

In numerically solving the integral equations of liquid state theory the distribution functions are usually discretized on a grid, and the convolution in the OZ equation is performed by employing the fast Fourier transform. This reduces the OZ equation together with the closure relation to a set of nonlinear algebraic equations. Methods for solving the equations can be classified as direct or iterative. Direct methods execute in a predictable number of operations and provide a quadratic convergence in the region close enough to the solution. However, they cannot be readily applied since it requires calculating and inverting the Jacobian matrix, which is prohibitively wasteful for a large number of grid points, especially for a 3D grid. Iterative methods do not share this deficiency and are very simple to implement. At the same time, a great shortcoming is their slow convergence. This becomes of vital importance for time consuming 3D calculations. Rapidly convergent and efficient algorithms have been devised out for solving liquid theory integral equations on a one-dimensional grid. They consist in expanding the solution into a coarse basis of a few functions recapitulating its general behavior, calculating the expansion coefficients by the Newton-Raphson method, and then refining the solution by Picard iterations. For the first time Gillan [168] introduced this method with the coarse basis of piecewise linear functions to solve the OZ integral equation for simple atomic liquids. Labik, Malijevsky, and Vonka [169] proposed to perform the expansion in the reciprocal space at several points of low values of the wavevector k, that is to use the coarse basis comprising several low-k sine functions. This allowed them to analytically obtain the Jacobian for calculation of the coarse expansion by the Newton-Raphson method. Kinoshita and co-workers [170, 171, 172, 45, 46, 47] developed the Newton-Raphson approach to solve the OZ and RISM equations for a mixture of ion-multipole liquids in the bulk and near surfaces. They elaborated highly efficient preconditioning and calculating the Jacobian in Newton-Raphson solution, which provides extremely fast convergence. The Newton-Raphson method is very difficult to use in the 3D case, since an estimated number of coarse functions required is a third power of that necessary for a one-dimensional grid. Another combination of a direct and iterative method, referred to as direct inversion in the iterative subspace (DIIS), has been developed in the context of ab initio molecular orbital calculations [173, 174, 175]. Its essence consists in minimizing the residual in the subspace spanned by the consecutive iterated vectors, which serve as basis functions approximating the solution at the current stage of the iterational process. The basis vectors are then being updated after several conventional self-consistent field (SCF) cycles [173] or by using the approximate Fock matrix [174, 175]. DIIS greatly improves convergence, especially closer to the terminal iterative steps. However, it often wastes time in the early stages of the iterational procedure, outside the region of quadratic convergence. An algorithm similar to the DIIS method with three basis vectors was used earlier for solution of the OZ equation in a charged system [176]. The DIIS procedure updated by Picard iterations was applied to solve the RISM/HNC equations combined with the SCF calculations [177]. It was proposed to update the DIIS vectors by the modified Broyden (MB) method which is a multidimensional secant method [178]. This significantly improves convergence as compared to the pure MB method. The modified DIIS method [27, 29, 30] updates the DIIS basis vectors at every iteration by using the approximate residual obtained in the DIIS extrapolation. It ensures great acceleration of convergence which is especially important and constitutes a challenging

262


task in the case of 3D integral equations. In the quadratic region, when the root mean square residual is typically less than 10 −3 to 10−4 , the convergence rate can reach an order of magnitude per iteration. When the 3D-RISM theory is incorporated in a self-consistent field (SCF) loop together with the Kohn-Sham DFT approach for the electronic structure [28], or coupled with the ab initio molecular orbital method [33], the resulting equations are characterized by a very strong instability and nonlinearity. For the Kohn-Sham DFT method, in particular, this is caused by the well known electrostatic instability with respect to long-range fluctuation of the valence electron charge [154]. Under such circumstances, robust convergence of the SCF integral equations can be provided by using the method of dynamical relaxation [26, 28, 27]. The procedure evaluates fictitious dissipative dynamics of the iterated vector relaxing to the solution of the equations, and is closely related to the methods of simulated annealing and CarParrinello molecular dynamics [38, 39]. The convergence rate appears to be highest at the beginning of the process, and later on slows down, nevertheless remaining significantly faster than plain iterations.

4.1

MDIIS algorithm for acceleration of convergence

The modified direct inversion in the iterative subspace (MDIIS) method combines the simplicity and relatively small memory usage of an iterational approach with the efficiency of a direct method. It comprises two stages: minimization of the residual linearly approximated with last successive iterative vectors used as a current basis, and then update of the basis with the minimized approximate residual by a properly scaled parameter. , Consider the integral equation R f (r) = 0 for a function f (r) defined in the 3D space, discretized on a grid of k points. The solution is thus represented by a vector f (r k ), and the integral equation , - reduces to a set of k nonlinear equations zeroing the values of the residual R f (r) at the points r k , that is zeroing the residual vector , R(r k ) ≡ R f (r k ) = 0.

(4.A.28)

, The integral equation can equally by specified as a functional Φ f (r) to be minimized, and - then the set of nonlinear equations (4.A.28) is obtained as R(r k ) = , δΦ f (r k ) /δr k = 0. The number of equations of the set (4.A.28) is equal to the number of the integral equations (the number of correlations between sites and components) times the number of grid points, which is typically more than 64 3 in the 3D-RISM case. This gives a very big sparse system of algebraic equations, in addition strongly nonlinear. The underlying idea consists in approximating the solution by consecutive iterated vectors f (j) (r k ) which serve as basis functions linearizing it at the current stage of the iterational process. As m vectors have been obtained in course of iteration, a much better approximation to the solution of Eq. (4.A.28) can be constructed in the subspace spanned by the basis functions f (j) (r k ), j = 1, . . . , m as their linear combination,

f (∗) (r k ) =

m j=1

cj f (j) (r k ),

(4.A.29)

263

Three-dimensional RISM theory by requiring that the corresponding linear combination of the residual vectors, R(∗) (r k ) =

m

cj R(j) (r k ),

(4.A.30)

j=1

approximates the zero vector in the mean square sense, subject to the constraint that the approximated residual (4.A.30) lies in the (m − 1)-dimensional plane passing through m iterative points of the basis residual vectors and linearizing the residual R(r), m ∂ 3 (∗) (∗) 4 R |R = 0 and cj = 1. ∂cj j=1 This leads to the system ⎛ S11 ⎜ .. ⎜ . ⎜ ⎝ Sm1 −1

of m + 1 linear equations for the expansion coefficients, ⎞ ⎞ ⎛ ⎞⎛ 0 c1 . . . S1m −1 .. .. ⎟ ⎜ .. ⎟ ⎜ .. ⎟ .. ⎟ ⎟ ⎜ ⎜ . . . ⎟ (4.A.31) ⎟⎜ . ⎟ = ⎜ . ⎟, ⎝ ⎠ ⎠ ⎝ 0 ⎠ . . . Smm −1 cm −1 λ ... −1 0

where Sij is the inner scalar product of the residual vectors, defined on the grid of k points as 4 3 R(i) (r k )R(j) (r k ), Sij = R(i) |R(j) = k

and λ is a Lagrangian multiplier yielding the squared norm of the minimized residual (4.A.30). Referred to as the DIIS, this method has been developed by Pulay in the context of ab initio molecular orbital calculations [173, 174, 175]. On the minimization, further improvement is possible only after updating the iterative subspace of f (j) (r k ) by new vectors introducing new dimensions that cannot be reduced to a linear combination of the previous ones. The basis vectors are then being updated by several conventional self-consistent field iterative loops [173] or through the approximate Fock matrix at the minimum point (4.A.29) [174, 175]. A modification to the DIIS method consists in updating the DIIS basis at every iteration with the vector obtained by moving from the approximated minimum position (4.A.29) in the approximated gradient direction (4.A.30): f (m+1) (r k ) = f (∗) (r k ) + ηR(∗) (r k ),

(4.A.32)

where η is the step parameter to be chosen for the best convergence. The approximated minimal residual (4.A.30) is orthogonal to the DIIS subspace in which the minimization has been done, that is to the (m − 1)-dimensional plane passing through the DIIS basis residuals, and is a reasonable guess of further search. The optimal value of η typically falls in the range 0.1 to 1 and is inversely related to the norm of the Jacobian matrix of the residual function (4.A.28), |δR(r)/δf (r)|, much as in the damped iteration approach. Notice that the next DIIS minimization in the updated subspace, in fact, improves the guess (4.A.32) by the best correction that can be constructed as the linear combination of the DIIS basis residuals R(j) (r), j = 1, . . . , m. The MDIIS procedure starts with one vector, sequentially increments the number of stored vectors up to m and the DIIS matrix size to m × m, and thereafter discards the earlier vectors. A typical number of vectors to be stored for significant acceleration of convergence is m = 10 to 20. A larger m is not efficient for computers with

264


32-digit arithmetics since the DIIS matrix (4.A.31) becomes badly conditioned owing to accumulation of roundoff errors. In the case of 3D integral equations, the number of stored vectors is frequently reduced to m = 5 due to memory limitations, although m = 10 provides considerably faster convergence [27]. Kawata et al [178] proposed to update the DIIS basis by the the modified Broyden (MB) multidimensional secant method [179, 180]. However, it does not seem that in general the MB update could offer an essential advantage over the MDIIS procedure. It was pointed out by Hamilton and Pulay [175] that the DIIS technique is very close to conjugate gradient type methods. Therefore, the successive DIIS minimization by itself improves the update (4.A.32), as indicated above. Outside the quadratic region the root mean square value of the residual, n )2 1/2 ( (j) ¯ (j) = 1 R (r k ) R n k=1

does not necessarily decrease monotonically with iterations. The new residual, even somewhat larger than the previous one, updates and usually improves the representation of the residual function R(r k ) by the DIIS vectors. Therefore such a rise is typically followed by a substantial drop at the next steps. However, if the root mean square value of the residual calculated at the next point (4.A.32) appears to be substantially larger than the smallest one for the DIIS vectors, ¯ (1) , . . . , R ¯ (m) ), ¯ (m+1) > Kr min(R R

(4.A.33)

it is worthwhile to restart the MDIIS procedure from the point with the smallest residual since the old DIIS vectors no longer model the behavior of the residual function R(r) properly. A reasonable value of the restarting threshold (4.A.33) is one order of magnitude: Kr = 10. The procedure should be restarted also if the matrix in Eq.(4.A.31) becomes ill conditioned.

4.2

Dynamical relaxation algorithm for acceleration of convergence

Essential acceleration as compared to Picard iteration is achieved with dynamical relaxation (DR) [26, 27]. Its advantage is very smooth and robust convergence, which is very important if the solutions of the integral equations are unstable [28]. Similarly to the simulated annealing and Car-Parrinello approaches [38, 39], solution of the initial equation (4.A.28) is replaced with evaluation of fictitious dynamics of the system described by the equation of motion for components of the iterated vector, ηf˙(r f , t) + R [f (r f , t)] , µf¨(r k , t) = −˜

(4.A.34)

where t is fictitious time, µ the fictitious mass, and η is the damping parameter. The residual R [f (r k , t)] acts here as a force returning the system to an equilibrium position at which it is zero, satisfying Eq. (4.A.28). The term depending on the fictitious velocity, f˙(r k , t), introduces dissipation ensuring the system evolves to the solution in a robust way. The equation of motion is integrated by using the Verlet algorithm [101], but somewhat modified to ensure stability of the relaxation. Damping is applied in the form of reduction of the change in f (r k , t) after a current step, much as in the iterational scheme, so that the function at the next time step ∆t is , f (r k , t + ∆t) = f (r k , t) + (1 − η˜) f (r k , t) − f (r k , t − ∆t) + R(r k , t)∆t2 , (4.A.35)


265

where the fictitious mass is scaled to m = 1, and the modified damping parameter η˜ is specified in the range 0 < η˜ < 1. Owing to the damping the relaxation slows down as the system approaches the solution. However, the convergence rate can be improved by adjusting the step and damping parameters in the process of relaxation. The values of the parameters typically used to solve the SCF integral equations for the electronic and classical structure are ∆t = 1, η˜ = 0.1 for the Kohn-Sham equations and ∆t = 1, η˜ = 0.5 for the 3D-RISM equations. The step can be increased when a small value of the residual is reached. A more sophisticated finite difference algorithm would in principle allow a longer time step ∆t to be used and hence reduce the total number of steps required to reach a given accuracy. However, a higher-order difference scheme does not increase the maximal stable time step significantly, but requires a large additional amount of memory to store the corresponding accelerations from a larger number of steps [154]. The DR method provides substantial acceleration of convergence of liquid structure calculations as compared to damped iterations, especially at the beginning stage of the iterational process [26, 28, 27]. Notice that the acceleration is brought about by the “coasting” term, f (r k , t) − f (r k , t − ∆t), acting similarly to the corrections introduced into the steepest descent direction in the technique of conjugate gradients [63, 154].

4.3

Residuals and initial vectors

The MDIIS or DR methods of convergence require definition of residuals. Their construction is illustrated below for the 3D-RISM/DRISM integral equations for aqueous solution discussed in Section 6. The residual of the Kohn-Sham DFT equations combined with the 3D-RISM approach in the SCF loop considered in Section 8 is considered as well. The solvent DRISM/HNC equations (4.39) and (4.41) for the bulk solvent distributions (or equally the DRISM/KH equations) are iterated for the function rcvv αγ (r) specified in the direct rather than reciprocal space. The residual is set up as ( ) ab ab ab (4.A.36) R rcab αγ (r) ≡ Rαγ (r) = r g(r)αγ (r) − 1 − rhαγ (r), ab where rhab αγ (r) is first obtained from rcαγ (r) by means of the DRISM integral equation ab (r). In solving (4.39) (4.39), and then inserted into the HNC closure (4.41) to get gαγ the 1D Fourier transforms are performed by using the technique of the fast Fourier transform (FFT) with nonlinear exponentially spaced grid [181, 182]. Notice the factor r at the iterated vector and the residual, which emerges naturally from the use of the 1D-FFT. It gives proper weighting of the residual (4.A.36) throughout the range of r, and is very important for preconditioning of the initial vectors. The DRISM/HNC equations are not renormalized with respect to the long-range Coulomb part of the site-site potential. Instead, the Coulomb long-range asymptotics divergent at k = 0 is separated out and handled analytically when performing the Fourier transformations [92, 27, 28, 29, 30]. This treatment is simpler than the renormalization technique [109], and does not affect the convergence rate in the case of the MDIIS method of solution. The forward Fourier transforms of the site-site DCFs are calculated as r

ab(sr) (4.A.37) (r) − βQaα Qbγ erf − , rcab αγ (r) = rcαγ τ 2 2

4π k τ ab(sr) kcab exp − , (4.A.38) (k) − βQaα Qbγ αγ (k) = kcαγ k 4

266

Molecular theory of solvation ab(sr)

where τ is the broadening length parameter, and the short-range part, rcαγ (r), is transformed numerically by the nonlinear FFT. The ion-ion total correlation functions of an ion-molecular solvent possess the Coulomb singularity 1/k2 at vanishing ionic concentration [66, 111], and are long-range enough to present difficulties, even for the nonlinear FFT, also in the case of weak Debye screening at small ionic concentrations. Therefore the backward Fourier transform of the site-site total correlation function is performed by k2 τ 2

Qaα Qbγ 4πk ab , (4.A.39) exp − khab αγ (k) = kΛαγ (k) − β 2 2 expt k + κD 4 κ2 τ 2

Qaα Qbγ ab rhab exp − D αγ (r) = rΛαγ (r) − β expt 4 κ τ r

1 exp(−κD r) 1 − erf D − × 2 2 τ

κ τ r , (4.A.40) − exp(κD r) 1 − erf D + 2 τ with its short-range part kΛab (k) transformed numerically by the nonlinear FFT, a 2 αγ 1/2 where κD = 4πβ ρ Qa is the inverse length of Debye screening of solvent. a∈ ions

The iterations of either the MDIIS or DR method are started from the initial ab(sr) guess of rcαγ (r) = 0. The long-range part in (4.A.37)-(4.A.38) specified as the Coulomb potential of the broadened site charges represents both the asymptotics of the direct correlation function and its behavior in the core region, where it goes to a constant as r → 0 [176, 27]. This allows us to begin convergence at once for a given temperature and full molecular charge, and to avoid the procedures of gradually decreasing temperature (so called “cooling”) and increasing site charges (“charging”) usually employed to facilitate convergence [65, 66, 23, 178]. The solute-solvent 3D-RISM equation (4.9) complemented with the 3D-HNC or 3D-KH closure (4.12) or (4.13) are solved for the difference of the 3D site total and direct correlation functions, tubγ (r) = hubγ (r)−cubγ (r). This provides -smaller sensitivity , to the initial guess of the iterated vector. The residual R tubγ (r) is specified as a difference between the distribution functions obtained from the integral equation and its closure:

ub ub g ub γ (r) = exp −βu γ (r) + t γ (r) , ub cubγ (r) = g ub α (r) − 1 − t α (r),

ab a ab hubγ (r) = cua α (r) ∗ ωαγ (r) + ρ hαγ (r) , ub Rubγ (r) = g ub γ (r) − 1 − h γ (r).

(4.A.41)

Much as for the 1D solvent-solvent correlations, the renormalization of the 3DRISM/HNC equations is not necessary in respect to convergence. Nor is it required for the 3D fast Fourier transform (3D-FFT) employed to evaluate the convolution in Eq. (4.A.41). For a periodic solute neutralized by a compensating background charge, the Coulomb potential of the solute charge is screened at a supercell length. Therefore the 3D site direct, and hence total correlation functions, are free from the Coulomb singularity at k = 0 and can be transformed directly by the 3D-FFT. Similarly to the 1D case, neither “cooling” nor “charging” procedures are necessary, which reduces the computation time greatly. Iteration is started from the initial guess

267

REFERENCES

possessing the long-range asymptotics of tubγ (r) and fitting its behavior inside the cores of the solute ions. It is specified as the electrostatic potential of the solute but with its charges smeared by the Gaussian distribution to half width δ, (r) = tub(0) γ

βQbγ k2 δ 2

4π . Qi 2 exp ik(r − r i ) − Vcell k 4 k =0 i=1,2

(4.A.42)

ub(0)

The smearing limits t γ (r) as r → r α and is introduced to mimic the run of tubγ (r), bounded inside the cores. The Kohn-Sham DFT equation (4.89) is solved for the electron wave functions ψj (k) of M = Ne /2 eigenstates, specified on the 3D grid in the reciprocal space. Its residual vector is written as Rj (k) = εj ψj (k) −

2 1 (0) veff (k − k ) ψj (k ), k + k ψj (k) − 2 k

(4.A.43)

where k(0) is the special mean value point in the Brillouin zone of the supercell, and veff = vi + vH + vxc + vsolv is the effective potential. For the local pseudopotential of metal ions used, the potential term is evaluated by transforming the wave function to the direct space by the 3D-FFT, evaluating the product veff (r) ψj (r) on the 3D grid and transforming it back to the reciprocal space. The new set of M wave functions obtained from the residual (4.A.43) after a current DR step of Eq. (4.A.35) are not mutually orthogonal and must be orthonormalized, much as in the CP technique [38, 39, 154]. This can be done by means of the iterative orthogonalization used by Car and Parrinello [39], as well as by the GramSchmidt orthogonalization process. As an advantage, the latter ensures convergence of the wave functions to the lowest KS eigenstates and breaks spurious symmetries which may arise in the course of iteration [154]. To accelerate convergence of the KS equations, it is convenient to apply the preconditioning technique [154] in the form elaborated by Price and Halley for the CP simulation [40]. The residual (4.A.43) is multiplied with the mixing factor α = min( 2/kc2 , 2/k2 ) dependent on wavevector magnitude k, with the k-dependence threshold kc = 1.69. This in effect represents a modification of the higher kinetic energy matrix elements of the Hamiltonian, which crucially accelerates the rate of convergence while still giving convergence to very nearly the correct ground state. The relaxation of the electron wave functions ψj (k) is started from a random distribution over states j and wave vector argument k. This also allows one to avoid spurious symmetries which may occur in the choice of initial conditions [154]. The (0) electron wave functions converged for the metal slab without the solvent, ψj (k), are then used to get the initial guess of the classical distributions at the second stage including the solvent,

(r) = β ui(C) (r) + ue(0) (r) , tuv(0) γ γ γ e(0)

(4.A.44)

where uγ (r γ ) is the potential (4.100) of the metal valence electron density corres(0) ponding to ψj (k). As above, the relaxation begins at once for the given solvent temperature and site charges, avoiding the “charging” and “cooling” procedures.

268


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Chapter 5 DYNAMICAL PROCESSES IN SOLUTION Song-Ho Chong Dipartimento di Fisica, INFM Center for Statistical Mechanics and Complexity, Universit` a di Roma ‘La Sapienza’, Piazzale Aldo Moro 2, I-00185, Roma, Italy

1.

Introductory remarks on the theory for dynamics of simple liquids

The study of collective excitations (collective density fluctuations) in molecular liquids has been a subject of intense investigations through experiments [1, 2, 3] and molecular dynamics simulations [4-9] in the past decade. These investigations have established that there are at least two different kinds of collective excitations in molecular fluids: the ‘acoustic’ mode which originates from the center-of-mass motions of constituent molecules and the ‘optical’ mode which is rooted in the rotational (librational) motions. The appearance of the optical mode is a distinctive feature in the dynamics of polyatomic fluids: it does not show up in liquids consisting of spherical particles. Since the rotational motions give rise to the local charge-density fluctuations, the optical mode is also responsible for the dielectric properties of polar liquids. A detailed understanding of these collective excitations at a microscopic level is prerequisite for understanding a variety of timedependent phenomena occurring in polar liquids, such as solvation dynamics and solvent dynamical effects on charge transfer reactions in solutions [10 -17]. In this chapter we review a recently developed microscopic theory to study such collective excitations in molecular liquids based on the interaction-site representation. As an application we also present a molecular theory for dynamics of solvated ion, in which the ionic dynamics is described in terms of the response of the solvent collective excitations to the solute perturbation. But before embarking on the main subject, let us make a brief survey of the historical developments of the theory

277

278


for dynamics of simple liquids since our theory for molecular liquids is a natural extension along these developments. Throughout this chapter we consider a homogeneous and isotropic fluid composed of N rigid molecules (or spherical particles) in a volume V at the inverse temperature β = 1/kB T , and the thermodynamic limit with density ρ = N/V is implied.

1.1

Memory-function formalism

The study of liquid-state dynamics is an important branch of nonequilibrium statistical physics. Over the past 40 years this active field of research has seen new and important developments. These developments, however, are largely limited to a system of spherical particles, and are not directly applicable to the description of dynamics of molecular systems. Nonetheless, ideas and concepts acquired during the growth of a modern theory constitute the bases in constructing molecular-liquid theories, and in the following main features of those developments for simple liquids are briefly surveyed. Let us begin with the description of the general framework which constitutes a convenient starting point of any microscopic theory for liquid dynamics. The time evolution of a dynamical variable (or a set of dynamical variables), which we denote as a row vector A (an example is given in Eq. (5.22)), is in general described by the Liouville equation dA(t) = iLA(t), dt

(5.1)

where L denotes the Liouville operator [18, 19, 20]. This equation can be formally integrated to give A(t) = exp(iLt)A.

(5.2)

(Hereafter the absence of any indication for time t in dynamical variables means that the latter are evaluated at t = 0.) However, for a strongly interacting system, the formal solution (5.2) is useless in practice. An exception is the case in which the system can naturally be characterized by a certain ‘small parameter’ whose presence would suggest a perturbative treatment of the part of L which is effectively small. Unfortunately, this convenient framework does not work in dense (i.e., strongly interacting) systems such as liquids, where no obvious small parameter exists which could be used as an expansion parameter. In such a situation a frequently adopted strategy is to rephrase the problem in a different way, still avoiding any approximation. Although this new description is again formal and no real progress has apparently

279

Dynamical processes in solution

been made, the establishment of the alternative framework is strongly biased by a number of physical arguments, with the ultimate expectation that practical approximation schemes may become available. Since its introduction in the 1960s by Zwanzig and Mori [21, 22, 23], the memory-function formalism based on projection operators has pervaded many theoretical approaches dealing with the dynamics of strongly interacting systems. Indeed, the idea of describing a many-body system by a limited number of relevant variables characterized by a relatively simple dynamics appears to be extremely appealing. The central point of the framework is the introduction of a projection operator P, which projects an arbitrary dynamical variable X onto the subspace spanned by A: P X ≡ A(A, A)−1 (A, X).

(5.3)

Here the inner product of two row vectors A1 and A2 is defined as the canonical ensemble average, (A1 , A2 ) ≡

1 A† A2 , N 1

(5.4)

where A†1 denotes a column vector adjoint to A1 , and the factor of 1/N is purely a matter of convention for later convenience. When applied to Eq. (5.1), the framework yields an equation, called the generalized Langevin equation (GLE) [18, 19, 20] t dA(t) = iΩ A(t) − dτ K(τ ) A(t − τ ) + f (t), (5.5) dt 0 in which iΩ, K(t) and f (t) are respectively referred to as the proper frequency, the memory function, and the fluctuating random force given by (defining Q = 1 − P) ˙ (A, A)−1 , iΩ = (A, A) K(t) = (f , f (t)) (A, A)−1 , ˙ f (t) = exp( iQLt ) QA.

(5.6) (5.7) (5.8)

(Notice that when A is a vector representing a set of dynamical variables, Ω and K(t) become matrices whereas f (t) is a vector. When A is a scalar consisting of only one relevant dynamical variable, the frequency Ω is identically zero since all autocorrelation functions are even functions of time.) An important consequence of the framework is that (A, f (t)) = 0,

(5.9)

280


i.e., the fluctuating force is orthogonal to A at all times. From Eq. (5.5) it is straightforward to obtain the corresponding equation for the time-correlation function C(t) ≡ (A, A(t)). Exploiting the orthogonality of f (t) to A, one obtains t dC(t) = iΩ C(t) − dτ K(τ ) C(t − τ ), dt 0

(5.10)

(5.11)

which is also called GLE or the memory-function equation. It should be noticed that no approximation has been made in deriving Eqs. (5.5) and (5.11), and both equations are still equivalent to Eq. (5.1). One of the most important properties of the memory function is that it seems to lend itself readily to simple approximations. For example, the following equation t dC(t) =− dτ K(τ )C(t − τ ), (5.12) dt 0 which is valid for a single-variable case, tells us that if we assume K(t) decays extremely rapidly, e.g., K(t) ≈ δ(t) (Markov approximation), then the correlation function decays exponentially. Although this form of a correlation function is not rigorously correct, it is well known that such an approximate form leads to classical theories such as Debye theory of dielectric relaxation. Since such a simple form for K(t) leads to a useful expression for the correlation function C(t), one might hope that more detailed assumptions for K(t) would lead to an even better C(t). Indeed, the framework is shown to provide a sound basis for the buildup of a modern theory for dynamics of fluids, which turned out to be quite successful even in the description of the dense-liquid dynamics.

1.2

Basic definitions and equations

A detailed description of the time evolution of spatial correlations in liquids requires the introduction of a time-dependent generalization of the radial distribution function. It is the van Hove correlation function [24] which retains the microscopic nature of the system and yet are tractable within the current development in the statistical mechanical theory of liquids. The relevant dynamical variable which defines the van Hove correlation function is the local number density of particles ρ(r, t) ≡

N i=1

δ(r − ri (t)),

(5.13)

281


where ri (t) denotes the position of particle i at time t. It is often more convenient to work with its space Fourier transform ρ(k, t) =

N

exp(ik · ri (t)).

(5.14)

i=1

Besides the local number density, it is usual to incorporate the associated particle current as well into the set of relevant dynamical variables. Since the current is a vector quantity, it is convenient to separately consider the longitudinal and transversal components of the current, whose microscopic expressions are respectively given by jL (k, t) ≡

N

vi,z (t) exp(ik · ri (t)), jT (k, t) ≡

i=1

N

vi,x (t) exp(ik · ri (t)),

i=1

(5.15) where we have chosen k along the z axis, and vi,z (vi,x ) denotes the z (x) component of the velocity of particle i. The number and the longitudinal current densities satisfy the continuity equation ρ(k, ˙ t) = ik jL (k, t),

(5.16)

where the dot denotes the time derivative and k = |k|. On the other hand, the dynamics of the number density and the transversal current density is effectively uncoupled. The time-correlation functions of these basic dynamical variables are defined by F (k, t) ≡ (ρ(k), ρ(k, t)), (5.17) CL (k, t) ≡ (jL (k), jL (k, t)),

CT (k, t) ≡ (jT (k), jT (k, t)),

(5.18)

in terms of the inner product defined in Eq. (5.4), which are respectively referred to as the intermediate scattering function, the longitudinal current correlation function and the transversal current correlation function. The intermediate scattering function F (k, t) is nothing but the space Fourier transform of the van Hove correlation function. It is also convenient to introduce the time Fourier transform of F (k, t), ∞ dt eiωt F (k, t), (5.19) S(k, ω) ≡ −∞

which is called the dynamic structure factor. This function is closely related to the cross section for coherent scattering of neutrons [18, 19, 20].

282


Similarly, the longitudinal and transversal current spectra are respectively defined by ∞ ∞ iωt dt e CL (k, t), CT (k, ω) ≡ dt eiωt CT (k, t). CL (k, ω) ≡ −∞

−∞

(5.20)

The continuity equation (5.16) implies the following relation: CL (k, ω) =

ω2 S(k, ω). k2

(5.21)

In the following we shall consider the application of the memoryfunction formalism to the calculation of the intermediate scattering function F (k, t) for simple liquids. For this purpose a natural choice for the vector A is a set of conserved variables consisting of the microscopic density and the longitudinal current density:

(5.22) A(k, t) ≡ ρ(k, t) jL (k, t) . (It is sometimes more conventional to incorporate the local temperature fluctuations as well into A, in particular when one wishes to relate the resulting theory with hydrodynamics. However, the essential points of the theory would not be altered with a simplified choice of the dynamical variables.) Substituting Eq. (5.22) into Eq. (5.11) yields the following GLE for F (k, t) in the time domain [18, 19, 20]: t dτ KL (k, t − τ )F˙ (k, τ ) = 0, (5.23) F¨ (k, t) + ωk2 F (k, t) + 0

where kB T , m S(k)

KL (k, t) = βm R(k), exp(iQLt)R(k) , ωk2 = k2

kB T ρ(k). R(k) = j˙ L (k) − ik m S(k)

(5.24) (5.25) (5.26)

Here m is the mass of a particle, S(k) the static structure factor, and ωkn the normalized nth frequency moment of S(k, ω) defined by ∞ 1 n dω ω n S(k, ω)/S(k). (5.27) ωk ≡ 2π −∞ Equation (5.23) has a typical form of that for a damped oscillator; ωk2 and KL (k, t) represent the characteristic frequency and the (non-


283

Markovian) damping term, respectively. For later convenience we introduce the Laplace transform of Eq. (5.23) /−1 . 2 ω k S(k), (5.28) F˜ (k, z) = − z − ˜ L (k, z) z+K where the Laplace transform f˜(z) of a function f (t) is defined by ∞ ˜ dt exp(izt) f (t) (Im z > 0). (5.29) f (z) = i 0

As mentioned in Sec. 1.1, the memory-function formalism which leads to Eq. (5.23) is, in a sense, merely a formal rephrasing of the original Eq. (5.1), shifting the difficulty in determining F (k, t) to that in evaluating the memory function KL (k, t). A priori, the only exact result that we know about the memory function is its initial value [18, 19, 20, 25] 2 − ωk2 ≡ ∆(k) with KL (k, t = 0) = ωk,L

2 ωk,L ≡ ωk4 /ωk2 , (5.30) in terms of the normalized frequency moments of S(k, ω) defined in Eq. (5.27). Thus to solve our real problem one has to separately develop theories for determining the time dependence of KL (k, t). In the following subsections the main features of three such theories developed for simple liquids are briefly summarized. These theories constitute the bases for our theory dealing with molecular liquids.

1.3

Viscoelastic model for memory function

The simplest assumption that we may make about the time dependence of KL (k, t) would be the exponential decay law KL (k, t) = exp[−t/τ (k)] ∆(k),

(5.31)

where we have noticed Eq. (5.30), and τ (k) is a yet unknown wavevectordependent relaxation time. This approximation has the merit of leading to the following simple analytic form for S(k, ω), S(k, ω) =

2∆(k)ωk2 [1/τ (k)] S(k), (5.32) ω 2 [ω 2 − ωk2 − ∆(k)]2 + [1/τ (k)]2 [ω 2 − ωk2 ]2

which has the correct first three nonzero frequency moments regardless of the model for τ (k) [18, 19, 20]. This type of relaxation mechanism was introduced in a somewhat different context by J. C. Maxwell in the 19th century. The time constant τ (k) controls the specific collective behavior

284


of the system [18, 19, 20]: for times t τ (k) Eq. (5.31) describes the solid-like response of the system whereas for times t τ (k) it represents the viscous mechanisms inherent to liquid, and this is why Eq. (5.31) is conventionally referred to as the viscoelastic model. To proceed further on the basis of Eq. (5.31) we must have some approximate scheme to evaluate the relaxation time τ (k). In the absence of any well-based microscopic theory it is wisest to treat the relaxation time τ (k) as an adjustable parameter. Nonetheless, it is tempting to look for some relatively simple prescription for this quantity. For example, Lovesey has used an argument based on a scaling of the memory function to derive the approximate relation [26, 27] 7 (5.33) 1/τ (k) = ξ ∆(k). The proportionality constant ξ can be determined by requiring that the asymptotic form of the resulting S(k, ω) given in Eq. (5.32) satisfies a certain property. Lovesey proposed a particularly simple prescription in which ξ is determined by requiring that the resulting S(k, ω = 0) coincides with√the exact free-particle result in the k → ∞ limit; this implies ξ = 2/ π [26, 27]. The validity of the viscoelastic model (5.32) has been tested against experimental and molecular dynamics simulation results [26, 27, 28]. The detailed comparison has established that the viscoelastic model works remarkably well for wavenumbers k ≈ km , where km denotes the first peak position of the static structure factor S(k). However, it has also been found that the situation is not so satisfactory for smaller wavenumbers, where the viscoelastic model is shown in some circumstances to yield even qualitatively incorrect results. This failure was attributed to the fact that the ‘single relaxation time’ model (5.31) cannot describe both the short-time behavior of the memory function, dominated by the so-called binary collisions, and in particular the intermediate and long-time behavior where in the liquid range additional slow processes play an important role (see the next subsection). It is obvious that these conclusions demand a more rigorous consideration of the memory function, which lead to the development of the modern version of the kinetic theory. Nevertheless, the viscoelastic model provides a rather satisfactory account of the main features of microscopic collective density fluctuations in simple liquids at relatively large wavenumbers, and its value should not be undervalued.

1.4

Generalized kinetic theory

A theory into which a detailed account is taken of the collisional processes underlying the dynamics of fluids is referred to as the kinetic

285


theory. A special feature of any microscopic kinetic theory is the appearance of correlations defined in phase space rather than in configuration space alone as in Eq. (5.13); the retention of the momentum variables (see Eq. (5.34)) is essential if one wishes to take into account the dynamics of collisional processes since momentum variables of colliding particles change appreciably before and after collisions. Up to the early 1970s a kinetic approach to the time-dependent properties of fluids was synonymous with a framework based on the Boltzmann equation and its extension by Enskog, in which a central role is played by those dynamical events referred to as uncorrelated binary collisions [29]. Because of this feature the Boltzmann equation is in general not applicable to dense fluids, where the collisions are so frequent that they are likely to interfere with each other. The uncorrelation ansatz is clearly equivalent to a loss of memory, or to a Markov approximation. As a result, for dense fluids the traditional kinetic approach should be critically revised to allow for the presence of non-Markovian effects. The bases for such a generalized kinetic theory were set during the 1970s and the early 1980s by several independent workers. The final form of the theory is largely owed to the works of Sjögren et al. [30, 31], which are based on the results of the so-called renormalized kinetic theories by Mazenko and co-workers [32]. The fundamental dynamical variable in the kinetic framework is the phase-space density defined by f (rp; t) ≡

N

δ(r − ri (t)) δ(p − pi (t)),

(5.34)

i=1

and its time-correlation function C(rp; r p ; t) ≡ fˆ(r p ; 0)fˆ(rp; t),

(5.35)

where fˆ(rp; t) ≡ f (rp; t) − f (rp; t) = f (rp; t) − ρf0 (p),

(5.36)

with f0 (p) denoting the normalized Maxwell distribution of momenta. It is often more convenient to work in the Fourier space C(k, pp ; t) ≡

1 ˆ f (k, p ; 0)∗ fˆ(k, p; t), V

(5.37)

in which fˆ(k, p; t) denotes the spatial Fourier transform of fˆ(rp; t). Like any other dynamical variable, the phase-space density satisfies the equation of motion of the form f˙(t) = iLf (t), and the memory-

286


function formalism can be applied to yield the following GLE: ˙ C(k, pp ; t) = dp iΩ(k, pp )C(k, p p ; t) t − dτ dp M (k, pp ; τ )C(k, p p ; t − τ ). (5.38) 0

The expression for the proper frequency iΩ(k, pp ) reads [18, 19, 20] i i (k · p) δ(p − p ) − (k · p) ρf0 (p)c(k), (5.39) m m where c(k) denotes the direct correlation function. The first term on the right-hand side of Eq. (5.39) stems from the ‘free-streaming’ portion (i.e., the kinetic energy term) of the Liouville operator, and the second term accounts for the static correlations occurring between different particles due to interatomic forces. The memory function M (k, pp ; t) represents effects from the dynamics of collisional processes. Before embarking on the survey of the results of the generalized kinetic theory, let us see briefly how the basic equations of classical kinetic theories can be recovered by means of the memory-function equation (5.38). For instance, the Vlasov equation can be obtained by completely ignoring the memory term in Eq. (5.38): iΩ(k; pp ) =

i ˙ C(k, pp ; t) − (k · p)C(k, pp ; t) m i = − (k · p)ρ f0 (p)c(k) dp C(k, p p ; t), m

(5.40)

in which only the static correlation, coming from the proper frequency term given in Eq. (5.39), is taken into account in a mean-field way through the direct correlation function c(k). Thus all the dynamical features of collisional processes embodied in the memory function is neglected, and this is why the Vlasov approximation is sometimes called the collisionless approximation. As another example let us see how the Boltzmann equation, which incorporates the effect from dynamics of collisional processes in some definite way, can be derived based on Eq. (5.38). At low densities a Markov approximation for the memory function becomes rigorous in the low wavevector limit [33]. Equation (5.38) in this limiting case can be written as ˙ (5.41) C(k, pp ; t) = dp iΩ(k, pp ) − Γ(pp ) C(k, p p ; t), where

Γ(pp ) ≡ 0

∞

dt lim lim M (k, pp ; t). k→0 ρ→0

(5.42)


287

Within the linearization schemes implying small departures from equilibrium, Eq. (5.41) is found to coincide with the Boltzmann equation [33], with an identification Γ(pp ) = ρ σ0 (pp ),

(5.43)

where σ0 (pp ) involves the typical quantities relevant for an isolated binary collision. The predictions of the Boltzmann equation and its earlier generalizations become increasingly incorrect as the density approaches the values typical of the liquid state, since for dense fluids the collisions cannot be considered any more as uncorrelated, or equivalently since the memory function develops an increasing non-Markovian character. The question of how to generalize the classical Boltzmann equation to make it applicable to dense fluids had been a long-standing problem. The answer provided by the modern version of the kinetic theory is the natural emergence of two basic dynamical mechanisms governing the time decay of the memory function: M (k, pp ; t) = MB (k, pp ; t) + M (k, pp ; t).

(5.44)

The first decay channel, MB , is a rapidly decaying function of time describing the effects of essentially uncorrelated binary collisions, which is well known and is the only relaxation mechanism in the traditional kinetic theories of gases. The second mechanism M , which is often referred to as the non-binary collision term, is the most important outcome of the generalized kinetic theory, and accounts for additional slow processes coming from correlated collisional effects. Now let us see how an approximate form for the memory function for F (k, t), i.e., KL (k, t) in Eq. (5.23), comes about on the basis of the results of the generalized kinetic theory. This can be done by relating the phase-space correlation function to more familiar ones, such as F (k, t), CL (k, t) and CT (k, t). For this purpose it is convenient to switch from a continuous to a discrete matrix representation of the phase-space correlation function by introducing a complete set of orthonormal momentum functions Hl (p); these are generally chosen to be the Hermite polynomials satisfying the orthonormality relations [34] (5.45) dp f0 (p)Hl (p)Hl (p) = δl,l , with the equilibrium distribution f0 (p) playing the role of a weight factor. The first five members (l = 0, 1, · · · , 4) of the set are associated with the conserved variables: Hl (p) = {1, pz /p0 , px /p0 , py /p0 , (1/61/2 )(p2 /p20 − 1)},

(5.46)

288


where p0 = (m/β)1/2 . (Strictly speaking, the last one is conserved only for hard-sphere systems.) The phase-space density can then be expanded as Al (k, t)Hl (p), (5.47) f (k, p; t) = f0 (p) l

where Al (k, t), which is referred to as the mode, is given by Al (k, t) = dp fˆ(k, p; t)Hl (p).

(5.48)

It is readily verified that the first four Al s represent the density and current modes, (5.49) A0 (k, t) = ρ(k, t), A1 (k, t) = jL (k, t)/p0 ,

A2(3) (k, t) = jT (k, t)/p0 .

(5.50)

It should be noticed that the time evolution of these (quasi-conserved) hydrodynamic mode variables becomes very slow as k → 0. The expansion (5.47) for the phase-space density implies a similar decomposition (mode expansion) for the phase-space correlation function Hl (p) Cll (k, t) Hl (p ), (5.51) C(k, pp ; t) = f0 (p)f0 (p ) Cll (k, t) =

dp

l,l

dp Hl (p) C(k, pp ; t) Hl (p ),

(5.52)

and similarly for iΩ(k, pp ) and M (k, pp ; t). In view of Eqs. (5.49) and (5.50) it is straightforward to verify that F (k, t), CL (k, t) and CT (k, t) can be expressed in terms of Cll as follows: F (k, t) = C00 (k, t)/ρ, CL (k, t) = C11 (k, t)/ρβm,

CT (k, t) = C22(33) (k, t)/ρβm.

(5.53) (5.54)

In the discrete space the Laplace transform of the memory-function equation (5.38) reads ˜ ll (k, z) C˜l m (k, z). Ωll (k) + M z C˜lm (k, z) + Clm (k, t = 0) = − l

(5.55) The elements of Ωll (k) can be obtained from Eq. (5.39) and a modeexpansion similar to Eq. (5.52), and read 1 k dp f0 (p) Hl (p) (k · p) Hl (p) − ρ c(k) δl,1 δl ,0 . Ωll (k) = m (βm)1/2 (5.56)


289

For later convenience it is more useful to rewrite Eq. (5.55) in such a way that only a limited number of modes appears explicitly. This task of making explicit the appearance of only the H modes of interest can be accomplished by introducing a suitable projection operator which projects any variable onto this restricted set of modes. After a repeated application of this projection operator on Eq. (5.55), one obtains the exact result [35, 36, 37] z C˜lm (k, z) + Clm (k, t = 0) H−1 ˜ ll (k, z) + m =− ˜ ll (k, z) C˜l m (k, z), (5.57) Ωll (k) + M l =0

in which all the couplings between the H modes of interest and the remainder are transferred into the new memory function m ˜ ll (k, z). Let us now consider F (k, t) = C00 (k, t)/ρ and its memory function KL (k, t) (see Eq. (5.23)). We first notice that in view of Eqs. (5.53) and (5.54) the continuity equation, (5.16) or (5.21), implies that k C˜11 (k, z), (βm)1/2 k2 ˜ C11 (k, z). z z C˜00 (k, z) + ρS(k) = βm z C˜01 (k, z) = −

(5.58) (5.59)

According to the latter equation the memory-function equation for C˜11 (k, z) is required if one wishes to obtain that for F (k, t) = C00 (k, t)/ρ. Let us write the memory-function equation for C˜11 (k, z) according to the new formulation (5.57). If we choose H = 4 (the first four members in Eq. (5.46)), i.e., if we consider only the hydrodynamic modes as relevant modes, one obtains ˜ 11 (k, z) + m ˜ 11 (k, z) C˜11 (k, z), z C˜11 (k, z) + ρ = −Ω10 (k)C˜01 (k, z) − M (5.60) where we have noticed Ω11 (k) = 0 and exploited the orthogonality of the longitudinal (A1 ) and transversal (A2 and A3 ) current modes. Using Eq. (5.58) to eliminate C˜01 (k, z) and Ω10 (k) = [k/S(k)](βm)−1/2 , which follows directly from Eq. (5.56), one finds * +−1 ˜ 11 (k, z) + m ˜ 11 (k, z)] . (5.61) C˜11 (k, z) = −ρ z − [k2 /βmS(k)z] + [M Substituting this result into Eq. (5.59) yields !−1 2 /βmS(k)] [k . C˜00 (k, z)/ρ = −S(k) z − ˜ 11 (k, z) + m z + [M ˜ 11 (k, z)]

(5.62)

290


A comparison of this equation with Eq. (5.28) leads to an identification KL (k, t) = M11 (k, t) + m11 (k, t),

(5.63)

where we have noticed C00 (k, t)/ρ = F (k, t) and Eq. (5.24). At this point it should be noted that M11 (k, t) can further be decomposed into its binary and non-binary contributions as implied by Eq. (5.44): (k, t), (5.64) M11 (k, t) = MB,11 (k, t) + M11 whose decay mechanisms are fast and slow, respectively. On the other hand, m11 (k, t) on the right-hand side of Eq. (5.63) involves couplings to all the non-hydrodynamic modes, and consequently its time evolution (especially in the small-k regime) should retain ‘fast’ characters which involves couplings only to the hydrodynamic compared to M11 modes: taken together, MB,11 (k, t) and m11 (k, t) are responsible for the rapid short-time decay of the memory function KL (k, t). Following these arguments, Eq. (5.63) shall be written as (k, t) KL (k, t) = [MB,11 (k, t) + m11 (k, t)] + M11 ≡ [KL (k, t)]fast + M11 (k, t),

(5.65)

where the first and second terms in the last expression shall be respectively referred to as the fast and slow portions of the memory function. , Rather than going through the derivation of the expression for M11 we only cite the resulting expression obtained by the generalized kinetic theory [31] (an alternative derivation based on the mode-coupling theory will be described in the next subsection): 2 ρkB T dq qz c(q) + pz c(p) M11 (k, t) = 3 (2π) m × F (q, t)F (p, t) − FB (q, t)FB (p, t) , (5.66) where p ≡ k − q. This expression clarifies that the slow portion of the memory function is basically due to the couplings to the density modes. The term involving the ‘binary’ intermediate scattering functions FB has (k, t) very small at short times. This reflects the effect of making M11 the intuitive concept that any couplings to the collective modes of the medium (here, the density modes) requires some finite time to build up. In practice, this feature can be accounted for by a suitable approximation for FB which has a required property FB (k, t) ≈ F (k, t)

up to O(t2 ) for t 1.

(5.67)

291


For example, Sjögren proposed the following approximation [31] FB (k, t) ≡ [F0 (k, t)/Fs (k, t)]F (k, t),

(5.68)

where Fs (k, t) and F0 (k, t) denote the self part and the free-particle expressions of the intermediate scattering function, respectively. In any case, whatever the precise form is adopted for the ‘binary’ correlation FB , its intrinsically rapid decay has the effect of making the last product term FB FB in Eq. (5.66) virtually negligible beyond a microscopically (k, t) short-time interval; in fact one can show that the effects of M11 4 are O(t ) in the short-time regime [31]. Hence, beyond a comparatively (k, t) is ultimately ruled short-time interval, the long-lasting ‘tail’ of M11 by the product of the density correlation functions F F . Turning our attention to the fast portion of the memory function, [KL (k, t)]fast , it is clear that this part is badly defined since it consists of a strictly binary part MB,11 and the term m11 representing couplings to all the ‘fast’ non-hydrodynamic (i.e., H ≥ 4) modes. However, it turns out in practice that a detailed knowledge of the functional forms of MB,11 and m11 is hardly necessary, and one can proceed as follows. As mentioned above, the effects of non-binary slow contributions start as O(t4 ). As a result, up to O(t2 ) in the short-time region, the full and the fast portion of the memory function coincide, and the functional form of the fast portion can be deduced by analyzing the short-time behavior of the full memory function. This idea can be employed to write [KL (k, t)]fast = ∆(k) g(t/τ (k)),

(5.69)

where we have noticed Eq. (5.30), and the ‘shape function’ g(x) should ensure a sufficiently fast decay such that g(x) ≈ 1−x2 for small x. Often adopted are g(x) = exp(−x2 ) and sech2 (x). The functional form (5.69) comes from the following time-expansion of the full memory function t2 d2 KL (k, t) 2 dt2 = [1 − t2 /τ 2 (k) + · · · ] KL (k, 0),

KL (k, t) = KL (k, t = 0) +

t=0

+ ··· (5.70)

where only even powers of time appear due to the even character of KL (k, t), and we have defined 1/τ 2 (k) ≡ −

¨ L (k, 0) K , 2 KL (k, 0)

(5.71)

which can be expressed in terms of the frequency moments of S(k, ω) as in Eq. (5.30) [20].

292


The framework based on Eqs. (5.65), (5.66) and (5.69) was shown to be successful in the cases where density modes rule the long-time behavior of the memory functions, namely, for dense liquids [38]. Numerical results for S(k, ω) at small wavevector regions, where the viscoelastic model of the previous subsection was found to give unsatisfactory results, were considerably improved compared to molecular dynamics calculations and experiments [38]. As the density decreases the situation becomes more complicated because of the intervening effects of other decay channels involving collective current variables. In particular, in dense gases the dominant non-binary contributions are known to stem from couplings involving slow transversal current modes, which are ultimately responsible for the appearance of well-known t−3/2 tails. Actually, these couplings to the current modes can also be accounted for by the generalized kinetic framework with some further approximations [30, 31]. However, in general, the overall quantitative relevance of these channels is small in the liquid range [38], and it is hardly necessary in a first approximation to include those decay modes. In principle, the inclusion of several kinds of decay channels paves a way for a microscopic description of dynamics of a fluid over a quite extended range of states.

1.5

Mode-coupling theory

It is well established that the principal results of the generalized kinetic theory, especially the functional form of the slow portion of the memory function, can be obtained also by a direct mode-coupling approach [18, 19, 20]. The basic idea behind the mode-coupling theory is that the fluctuation of a given dynamical variable decays, at intermediate and long times, predominantly into pairs of hydrodynamic modes associated with quasi-conserved dynamical variables. The possible ‘decay channels’ of a fluctuation are determined by ‘selection rules’ based, for example, on time-reversal symmetry or on physical considerations. It is reasonable to expect that for dense fluids the decay of the memory function at intermediate and long times is dominated by those mode correlations which have the longest relaxation times. The sluggishness of the structural relaxation processes typical of dense liquids suggests that the slow decay of the memory function at long times is basically due to couplings to wavevector-dependent density modes of the form A2 (q, p) ≡ ρ(q)ρ(p).

(5.72)

(Here p refers to a wavevector, not a momentum variable.) The slow portion of the memory function can then be expressed as a sum of products of the wavevector-dependent density modes provided a decoupling (or factorization) approximation is made (see below).


293

The simplest way of extracting the dominant slow portion of the memory function is to introduce another projection operator P2 which projects any variable onto the subspace spanned by A2 (q, p). Translational invariance of the system implies that the only A2 (q, p), whose inner products with a dynamical variable X(k) are non-zero, are of the form A2 (q, k − q) = ρ(q)ρ(k − q), (5.73) for various qs, and we define the second projection operator P2 in terms of A2 as follows: 1 A2 (A2 , A2 )−1 (A2 , X(k)), (5.74) P2 X(k) ≡ 2 q where the factor of 1/2 is to avoid the double counting. (From here on, in order to simplify the notation, A2 (q, k − q) will be simply denoted as A2 , and we shall often adopt the convention of writing the wavevectors as if they were discrete.) The first approximation of the mode-coupling treatment corresponds to replacing the time-evolution operator exp(iQLt) by its projection on the subspace spanned by A2 : exp(iQLt) ≈ P2 exp(iQLt)P2 .

(5.75)

At the same time the anomalous propagator exp(iQLt) on the right-hand side is replaced by the conventional one, the final expression being exp(iQLt) ≈ P2 exp(iLt)P2 .

(5.76)

Under these approximations KL (k, t) in Eq. (5.25) reads KLMCT (k, t) = βm(P2 R(k), exp(iLt)P2 R(k)).

(5.77)

The second approximation usually made in the mode-coupling approach is to assume that the four-variable correlation functions are factorized into products of two-variable correlation functions: ρ∗ (q)ρ∗ (k − q)ρ(q , t)ρ(k − q , t) ≈ N 2 F (q, t)F (|k − q|, t)(δq,q + δq,k−q ).

(5.78)

For example, the denominator of Eq. (5.74) reads (A2 , A2 ) ≈ N S(q)S(|k − q|).

(5.79)

The third approximation conventionally employed is the convolution approximation [39], under which the application of P2 to R(k) given in

294


Eq. (5.26) yields [18, 19, 20] P2 R(k) =

−iρ kB T qz c(q)ρ(q)ρ(k − q). N m q

(5.80)

By substituting Eq. (5.80) into Eq. (5.77) and using the factorization approximation, one eventually obtains the following memory function (p ≡ k − q): 2 ρkB T dq qz c(q) + pz c(p) F (q, t)F (p, t), (5.81) KLMCT (k, t) = 3 (2π) m which is essentially identical to Eq. (5.66) up to the presence of the ‘binary’ intermediate scattering functions FB . This difference comes due to the abandonment of the full phase-space description in the modecoupling approach. Requiring that the slow portion of the memory function should evolve as t4 in the short-time regime, which is one of the principal results of the generalized kinetic theory, leads to Eq. (5.66) up to the choice of the functional form for FB . Since mode-coupling approaches were originally introduced just to account for long-lasting features, the equivalence KLMCT (k, t) and the ‘tail’ of Eq. (5.66) is hardly surprising. Besides accounting for the latter, the generalized kinetic framework of the previous subsection has the practical advantage of dealing also with the short-time features of the memory functions, both in their ‘binary’ portions and in the initial increase of the ‘correlated-collision’ contributions.

1.6

Toward a theory of the dynamics of molecular liquids

There are two quite different ways of describing the equilibrium structure of molecular liquids, one based on the rotational invariant expansions and another based on the interaction-site model [19, 40]. It is clearly advantageous to use the latter in developing theories for dynamics of molecular liquids because it is capable of treating the general class of polyatomic fluids without too much difficulties. This feature is in contrast to other theories based on the rotational invariant expansions [41, 42, 43, 44, 45], in which theories become very complicated when there is no symmetry in a molecule. As we have seen so far in this section, in the theoretical development for the time-correlation functions of liquids at wavelengths and frequencies of a molecular scale, the memory-function formalism based on projection operators has played a key role. By combining this convenient framework and the interaction-site representation of molecules,


295

it is possible to develop theories for dynamics of polyatomic fluids. Along this line, Hirata proposed the site-site Smoluchowski-Vlasov (SSSV) theory [46], and Friedman and co-workers developed a reference memory function approximation (RMFA) [47-50]. The SSSV theory is a generalization of the Smoluchowski-Vlasov theory of Calef and Wolynes [41] based on the interaction-site model, and its application has shown that the theory predicts some of the essential features of van Hove correlation functions of water [46]. However, the SSSV theory in its current form is valid only in the diffusion regime, and the non-Markovian effects in the memory function, which are important in the dense-liquid dynamics, cannot be properly taken into account. The RMFA is generated by setting the normalized lth memory function of the time-correlation function of interest equal to the lth normalized memory function of the time-correlation function of a reference dynamical variable [47, 48]. Specializing this prescription in various ways leads to known approximation schemes, such as the Vineyard and Kerr approximations, and their generalization to molecular liquids. A dielectric form of the RMFA has been applied quite successfully to reproduce the ‘optical’ mode excitation profiles in liquid water [49, 50]. However, this dielectric approach cannot be applied to the description of another important collective excitation in water, the ‘acoustic’ mode, and it is desirable to develop a theory that accounts for all the characteristic features of the collective excitations in molecular liquids. Besides these limitations the most unwelcome feature common to the SSSV and RMFA theories is that they need dynamical information from outside, usually from molecular dynamics simulations. For example, diffusion coefficients of constituent atoms of a molecule are required in the SSSV theory prior to its application, whereas in the RMFA the memory function of a reference dynamical variable has to be determined in advance from molecular dynamics simulations. What is intended in the present contribution is to review a recently developed theory of the dynamics of molecular liquids which overcomes all these unwelcome features of currently available theories: a self-contained theory that requires only the knowledge of parameters for potential functions and molecular geometry (such as bond lengths) as in simulation studies; a theory whose memory function incorporates the nonMarkovian effects; a theory which accounts for all the characteristic features of collective excitations in molecular liquids. This will be accomplished by generalizing successful frameworks described so far in this section to molecular liquids based on the interaction-site representation. The rest of this chapter is organized as follows. Section 5.2 establishes the basic theoretical framework based on the memory-function formalism

296


and on the interaction-site model for molecular liquids. The generalized Langevin equations are derived for collective and single-particle site-site density correlation functions, and a formal definition of memory functions is given in terms of projected random forces. A simple exponential model and a mode-coupling approximation for memory functions are developed by extending the methods described in Secs. 5.1.3 through 5.1.5. Then our theory is applied in Sec. 5.3 to the study of collective excitations in a model diatomic molecular liquid. There we clarify how the acoustic and optical modes originate from the translational and rotational motions of constituent molecules, respectively. In Sec. 5.4 a molecular theory of the dynamics of solvated ion in the model diatomic liquid is presented, in which the ionic frictions are described in terms of the response of the solvent collective excitations to a solute displacement. It is demonstrated that the resulting expression for the ionic friction coefficient can be naturally decomposed into the hydrodynamic, dielectric and their cross terms, and that the acoustic and optical modes of solvent are responsible for the hydrodynamic and dielectric frictions, respectively.The systems analyzed in these Secs. 5.3 and 5.4 are rather simple and suitable for gaining insights into the physics of dynamical processes in molecular liquids. These insights can be employed to the analysis of a more realistic system, and Sec. 5.5 deals with the collective excitations and the dynamics of ions in water. Section 5.6 concludes the chapter.

2.

Interaction-site-model description of molecular-liquid dynamics

In this section the generalized Langevin equation (GLE) for density correlation functions for molecular liquids is derived based on the memory-function formalism and on the interaction-site representation. In contrast to the monatomic liquid case, all functions appearing in the GLE for polyatomic fluids take matrix forms. Approximation schemes are developed for the memory kernel by extending the successful frameworks for simple liquids described in Sec. 5.1.

2.1

Basic dynamical variables

We begin with the definition of two basic dynamical variables, ρ and j. ρ is a row vector whose component is a local density of atoms (sites) α at time t in Fourier k space,

ρα (k, t) ≡

N i=1

α

eik·ri (t) ,

(5.82)

297


where rαi (t) specifies the location of α atom in the ith molecule at time t. Hereafter the Greek subscripts and superscripts refer to the interaction sites of a molecule and the Roman letters label the molecules unless mentioned otherwise. j is a row vector of which each component is a longitudinal current density, jα (k, t) ≡

N

α

α vi,z (t)eik·ri (t) ,

(5.83)

i=1

where the wavevector k is chosen such that it is along the z axis of α (t) denotes z component of the the space-fixed laboratory frame, and vi,z velocity of α atom in the ith molecule at time t. (In this section only the longitudinal component of current is explicitly taken into account, and the subscript L indicating the ‘longitudinal’ will be often omitted from here on for the notational simplicity.) These two vectors satisfy the following continuity equation: ˙ ρ(k, t) = ik j(k, t).

(5.84)

The site-site intermediate scattering function matrix, which is a generalization of the intermediate scattering function for monatomic systems, is defined in terms of ρ and the inner product given in Eq. (5.4) as F(k, t) ≡ (ρ(k), ρ(k, t)).

(5.85)

Similarly we define the matrix of the site-site longitudinal current correlation functions by J(k, t) = (j(k), j(k, t)).

(5.86)

The matrices of the site-site dynamic structure factors and the longitudinal current spectra are respectively defined as the time Fourier transforms of the corresponding time-correlation functions: ∞ dt eiωt F(k, t), (5.87) S(k, ω) ≡ −∞ ∞ dt eiωt J(k, t). (5.88) CL (k, ω) ≡ −∞

The continuity equation (5.84) implies that these two matrices are connected through the relation CL (k, ω) =

ω2 S(k, ω). k2

(5.89)

298


The initial value of F(k, t) is the matrix of the site-site static structure factors F(k, 0) ≡ χ(k) = w(k) + ρh(k), (5.90) where w(k) and h(k) are the intramolecular and intermolecular total correlation function matrices defined by 1 3 −ik·rαi ik·rβ 4 e e i , (5.91) wαβ (k) ≡ N i 1 3 −ik·rα ik·rβj 4 i e e . (5.92) ρhαβ (k) ≡ N i

j=i

The intramolecular function for a rigid molecule takes the form wαβ (k) = j0 (klαβ ),

(5.93)

where j0 (x) is the zeroth-order spherical Bessel function and lαβ denotes the ‘bond’ length between α and β sites. (Notice that for α = β lαβ = 0, and consequently wαα (k) = 1.) The inverse of the matrix χ(k) which is required later can be related to the matrix of the site-site direct correlation functions c(k) based on the RISM equation [51, 52], h(k) = w(k)c(k)w(k) + w(k)c(k)ρh(k),

(5.94)

χ−1 (k) = w−1 (k) − ρ c(k).

(5.95)

and is given by

The initial value of J(k, t) can be discussed as follows. Having assumed the molecule to be rigid, the velocity of the atom α consists of translational and rotational contributions: α C = vi,z + [wi × δrCα vi,z i ]z ,

(5.96)

C represents z component of the center-of-mass velocity of the where vi,z the vector joining the ith molecule, wi its angular velocity, and δrCα i center-of-mass and α atom. Due to the statistical independence of the velocities of different molecules at the same time, the initial value J(k) ≡ J(k, 0) can be written as

Jαβ (k) =

N β 1 α β −ik·(rαi −rβj ) α β −ik·(rα 1 −r1 ) . vi,z vj,z e = v1,z v1,z e N

(5.97)

i,j=1

Furthermore, owing to the statistical independence of translational and rotational velocities at the same time, J(k) has a generic form trans rot (k) + Jαβ (k), Jαβ (k) = Jαβ

(5.98)

299


where β

trans C 2 −ik·(r1 −r1 ) (k) = (v1,z ) e , Jαβ α

(5.99) β −ik·(rα 1 −r1 )

Cβ rot Jαβ (k) = [w1 × δrCα 1 ]z [w1 × δr1 ]z e

. (5.100)

The evaluation of the elements of Jtrans (k) is readily accomplished to evaluate due to the statistical independence of the translational velocities and coordinates: kB T trans wαβ (k), (k) = (5.101) Jαβ M where M denotes the total mass of the molecule. On the other hand, elements of Jrot (k) depend on the molecular model. It is straightforward to obtain elements of Jrot (k) once a definite molecular model is specified, and those for a diatomic molecule, which are required in Sec. 5.3, are given by 2 kB T rot zα zβ [j0 (klαβ ) + j2 (klαβ )], (k) = (5.102) Jαβ 3 I in terms of the moment of inertia I and the coordinates in the bodyfixed frame (cf. Fig. 5.1). The elements of Jrot (k) for a three-site water model to be used in Sec. 5.5 can be found in [53]. The dynamics of single molecule (or tagged molecule) can be discussed in terms of the self-part of the intermediate scattering function matrix Fs (k, t) ≡ ρs (k)† ρs (k, t),

(5.103)

defined with the tagged molecule density ρsα (k, t) = exp(ik·rα1 (t)). (Here the 1st molecule is chosen as the tagged molecule.) Since all the formulations to be presented below for the collective correlation functions F(k, t) can easily be generalized to those for Fs (k, t), the latter will not be repeated for brevity. The interested reader is referred to [54] and [55].

2.2

GLE for density correlation functions

Combining our primary dynamical variables ρ and j to form a new row vector ( ) A(k, t) ≡ ρ(k, t) j(k, t) , (5.104) we introduce the following projection operator P which acts on a row vector X: (5.105) P X ≡ A(A, A)−1 (A, X). The memory-function formalism described in Sec. 5.1.1 leads to the following GLE in the time domain [18, 19, 20, 25]: t 2 ˙ ¨ dτ K(k, t − τ )F(k, τ ) = 0, (5.106) F(k, t) + ω k F(k, t) + 0

300


which can be considered as a matrix generalization of Eq. (5.23). In this equation ω nk denotes the normalized nth frequency moment matrix of S(k, ω) defined by ∞ 1 n ω k ≡ dω ω n S(k, ω) χ−1 (k), (5.107) 2π −∞ which is a generalization of Eq. (5.27). Since (1/2π) dω ω 2 S(k, ω) = k2 J(k) as can be derived from Eqs. (5.88) and (5.89), the second frequency moment matrix is given by ω 2k = k2 J(k)χ−1 (k),

(5.108)

which is a generalization of Eq. (5.24) to molecular liquids. K(k, t) in Eq. (5.106) is the memory-function matrix (or simply called the memory kernel), whose formal expression reads K(k, t) ≡ (R(k), exp(iQLt)R(k))J−1 (k),

(5.109)

where Q ≡ 1 − P and the fluctuating random force vector is defined by ˙ R(k) ≡ j(k) − ikρ(k)χ−1 (k)J(k).

(5.110)

The initial value of K(k, t) can be expressed in terms of the normalized frequency matrices as in Eq. (5.30) [18, 19, 20, 25]: K(k, 0) = ω 2k,L − ω 2k ≡ ∆(k)

with

ω 2k,L ≡ ω 4k ω 2k −1 . (5.111)

To obtain GLE in the frequency domain, let us define the Laplace transform as in Eq. (5.29). Equation (5.106) then takes the form

−1 ˜ z) = − zI − zI + K(k, ˜ ω 2k F(k, z)

−1

χ(k),

(5.112)

where I denotes the unit matrix and we have noticed Eq. (5.90). It follows directly from Eq. (5.29) that ˜ ω + i ) = F (k, ω)/2 + i F (k, ω)/2, lim F(k,

→0

where F (k, ω) = −2 F (k, ω) = 2

0

(5.113)

∞

dt sin(ωt) F(k, t),

(5.114)

0 ∞

dt cos(ωt) F(k, t).

(5.115)

301


Applying these relations to Eq. (5.112) one obtains the following GLE in the frequency domain: −1 ω 2 ω 2k χ(k), K (k, ω) S(k, ω) = Y(k, ω)[K (k, ω)]−1 Y(k, ω) + 4 (5.116) in which Y(k, ω) is defined by Y(k, ω) ≡ ω 2 I − ω 2k +

2.3

ω K (k, ω). 2

(5.117)

Simple exponential model for K(k, t)

Now we have a formally exact equation of motion for F(k, t), Eq. (5.106). To obtain F(k, t), it is required to develop an approximation scheme for the memory kernel K(k, t). Here we propose a simple model for the memory-function matrix which is a direct generalization of the one for monatomic liquids developed by Lovesey [26, 27] (see Sec. 5.1.3). We write K(k, t) in Eq. (5.106) as (5.118) K(k, t) = exp −t τ −1 (k) ∆(k), where we have noticed Eq. (5.111), and τ (k) is a time-constant matrix to be specified below. It should be noted that this form of K(k, t) guarantees that the first three non-zero moment matrices of S(k, ω) are correct regardless of the model for τ (k). Let us introduce a matrix U∆ (k) which diagonalizes ∆(k): (k)∆(k)U (k) = diag ∆λ (k) , (5.119) U−1 ∆ ∆ where diag[∆λ (k)] represents a diagonal matrix whose λth diagonal element is an eigenvalue of the matrix ∆(k). According to Eq. (5.111), the matrix U∆ (k) diagonalizes the initial value of the memory kernel K(k, 0). In extending the method of Lovesey to molecular systems it turned out that one needs to impose an additional restriction on the model for τ (k): we assume that the matrix U∆ (k) diagonalizes the memory kernel K(k, t) for all the times. This implies that the memoryfunction matrix in Eq. (5.118) can be written as

(5.120) K(k, t) = U∆ (k) diag e−t/τλ (k) U−1 ∆ (k)∆(k). In the frequency domain it follows from Eqs. (5.114) and (5.115) that . / ω U−1 K (k, ω) = −2 U∆ (k) diag ∆ (k)∆(k), (5.121) ω 2 + (1/τλ (k))2

302


.

1/τλ (k) K (k, ω) = 2 U∆ (k) diag 2 ω + (1/τλ (k))2

/ U−1 ∆ (k)∆(k).

(5.122)

Now we specify 1/τλ (k) by generalizing the method of Lovesey [26, 27]: we write τλ−1 (k) in the following form: 7 (5.123) τλ−1 (k) = ξλ ∆λ (k). The constant ξλ is determined by requiring that the resultant S(k, ω = 0) coincide with SG (k, ω = 0) in the k → ∞ limit, where SG (k, ω) is defined as the time-Fourier transform of the Gaussian-approximated self part of the intermediate scattering function matrix. The details of this procedure are summarized in [53], and the resulting expressions for ξλ for a diatomic molecule and water can be found in [54] and [53], respectively. An important point is that, based on this prescription, one can calculate ξλ , and hence τλ−1 (k), once a concrete molecular model (such as bond length and interaction potential) is specified.

2.4

Mode-coupling theory

Although it turned out that a number of essential features concerning dynamics of molecular liquids can be well captured by the theory of the previous subsection (see Secs. 3 and 5), an intense investigation through experimental, theoretical, and molecular-dynamics simulation studies for simple liquids has revealed that the microscopic processes underlying various time-dependent phenomena cannot be fully accounted for by a simplified memory-function approach [18, 19, 20]. In particular, the assumption that the decay of memory kernels is ruled by a simple exponential-type relaxation must be significantly revised in view of the results of the kinetic framework developed for dense liquids (see Sec. 5.1.4). This motivated us to further improve the theory for dynamics of polyatomic fluids. As described in Sec. 5.1.4,memory functions generally consist of the fast and slow portions: the fast portion is due to the rapidly decaying ‘binary collision’ contributions, whereas the slow portion stems from correlated collisional effects. Hence, a starting point of our argument here is that the memory kernel K(k, t) for molecular systems also consists of its fast and slow portions: K(k, t) = Kfast (k, t) + Kslow (k, t).

(5.124)

Since, as discussed in Sec. 5.1.5, the functional form for the slow portion of the memory kernel can be obtained also by a direct mode-coupling approach, we shall review here the basic features of the mode-coupling


303

theory for molecular liquids. However, since the memory kernel obtained from the direct mode-coupling approach is valid only in the long-time region, we have to separately treat the fast portion so that the short-time regime can also be adequately described. This will be done exploiting the ‘frequency sum rules’, and the fast and slow portions of the memory kernel will be connected according to the prescription by Sj¨ ogren [31, 38].

2.4.1 Fast portion of the memory kernel. As discussed in Sec. 5.1.4, the initial decay of the full memory kernel is dominated by its fast portion, which can be deduced by analyzing the short-time behavior of the full memory kernel. At sufficiently short times, the full memory kernel K(k, t) can be expanded as t2 d2 K(k, t) = K(k, 0) + K(k, t) +... 2 dt2 t=0 (5.125) = I − t2 τ −2 (k) + . . . K(k, 0), where only even powers of time appear due to the even character of K(k, t), and we have defined a matrix, 1 ¨ K(k, 0)K−1 (k, 0). (5.126) 2 By differentiating Eq. (5.106) with respect to t repeatedly and setting t = 0, it follows that τ −2 (k) ≡ −

K(k, 0) = ω 4k ω 2k −1 − ω 2k , ¨ −K(k, 0) = ω 6k ω 2k −1 − (ω 4k ω 2k −1 )2 ,

(5.127) (5.128)

in terms of normalized frequency moment matrices defined in Eq. (5.107). Equation (5.125) implies the following approximate form for the fast portion of the memory kernel: (5.129) Kfast (k, t) ≈ g tτ −1 (k) K(k, 0), where the shape function g(x) satisfies g(x) ≈ 1 − x2 for small x, and the function g of a matrix is defined in terms of its expansion in a Taylor series. If U(k) is a matrix which diagonalizes τ −2 (k) it follows that (5.130) U−1 (k)τ −2 (k)U(k) = diag τλ−2 (k) , where τλ−2 (k) denotes the λth eigenvalue of the matrix τ −2 (k). Then the fast portion of the memory kernel can be summarized as

(5.131) Kfast (k, t) = U(k) diag g(t/τλ (k)) U−1 (k)K(k, 0).

304


2.4.2 Slow portion of the memory kernel. We next investigate the long-lasting tails of the memory kernel exploiting the modecoupling approach. As discussed in Sec. 5.1.5, the basic idea behind the mode-coupling theory is that the decay of the memory kernel at intermediate and long times is determined by its couplings to the pair density modes. For molecular liquids based on the interaction-site representation, the pair density modes take the form Aλµ (q, p) ≡ ρλ (q)ρµ (p).

(5.132)

The slow portion of the memory kernel can then be expressed as a sum of products of the wavevector-dependent density modes provided a decoupling (or factorization) approximation is made (see below). As described in Sec. 5.1.5, the simplest way to extract the dominant slow portion of the memory kernel is to introduce a projection operator P2 which projects any variable onto the subspace spanned by Aλµ (q, p). Based on Eq. (5.109), the prescription as described in Sec. 5.1.5 leads to the following expression as the starting formula: KMCT (k, t) = (P2 R(k), exp(iLt)P2 R(k))J−1 (k).

(5.133)

The evaluation of the elements of P2 (R(k))α is involved, and the interested reader is referred to original paper [55]: see also the latest version of the derivation which exploits a somewhat different route [56, 57]. Here we only cite the resulting expression, which reads P2 (R(k))α = −

iρ qz cλµ (q)Jαµ (k)ρλ (q)ρµ (k − q). N q

(5.134)

λ,µ

By substituting Eq. (5.134) into Eq. (5.133) and noting that ρ∗λ (q)ρ∗µ (k − q)ρλ (q , t)ρµ (k − q , t) ≈ N 2 Fλλ (q, t)Fµµ (k − q, t)δq,q + N 2 Fλµ (q, t)Fµλ (k − q, t)δq,k−q ,

(5.135)

under the factorization approximation one obtains * ρ dq qz2 cλµ (q)cβν (q)Fµν (q, t)Fλβ (p, t) [KMCT (k, t)]αβ = (2π)3 λ,µ,ν + (5.136) + qz pz cλµ (q)cβν (p)Fµβ (q, t)Fλν (p, t) Jαλ (k), where p = k − q. Equations (5.134) and (5.136) are respectively the extensions of Eqs. (5.80) and (5.81) to molecular liquids.

305


At this point it may be tempting to identify the mode-coupling contribution given by Eq. (5.136) with the slow portion of the full memory kernel. However, as mentioned in Secs. 5.1.4 and 5.1.5, the slow portion should evolve as O(t4 ) in the short-time regime, and we have to manipulate Eq. (5.136) in order to guarantee this feature: this is exactly the extra price we should pay due to the abandonment of the full phasespace description. Following the works of Sj¨ ogren [31, 38] we introduce the following auxiliary function: fαβ (k, t) ≡

0 (k, t) Fαβ s (k, t) Fαβ

,

(5.137)

0 (k, t) denotes an element of the following intermediate scatwhere Fαβ tering function matrix: 1 (5.138) F0 (k, t) ≡ exp − ω 2k,s t2 w(k), 2

which is essentially that for a non-interacting molecule. It is readily verified that F0 (k, t) and Fs (k, t) coincide up to O(t2 ) in the shorttime regime by construction. It is also obvious that the former decays more rapidly than the latter, which also has the long-lasting portion like F(k, t) [55]. Thus, the auxiliary function defined in Eq. (5.137) satisfies the following relations: fαβ (k, t) = 1 up to O(t2 ) for t 1, lim fαβ (k, t) = 0. t→∞

(5.139) (5.140)

0 (k, t) and It should be also noticed that fαβ (k, t) is symmetric since Fαβ s Fαβ (k, t) are symmetric matrices. By inserting this auxiliary function into Eq. (5.136) according to Sj¨ ogren’s prescription, we eventually obtain the slow portion of the memory kernel which evolves as O(t4 ) in the short-time regime: ρ dq [Kslow (k, t)]αβ = (2π)3 λ,µ,ν * 2 × qz cλµ (q)cβν (q)

× 1 − fµν (q, t)fλβ (p, t) Fµν (q, t)Fλβ (p, t)

+ qz pz cλµ (q)cβν (p)

+ × 1 − fµβ (q, t)fλν (p, t) Fµβ (q, t)Fλν (p, t) Jαλ (k), (5.141)

306


where p = k − q. This expression is an analogue of Eq. (5.66) for simple liquids. Numerical results for F(k, t) can be obtained self-consistently by solving Eqs. (5.106), (5.124), (5.131) and (5.141).

3.

Collective excitations in diatomic liquids

In this section the theory described in the previous section is applied to investigate collective excitations in a model diatomic liquid. It is also clarified how these excitations originate from the translational and rotational motions of constituent molecules. As in other studies [5, 6, 7, 9, 49, 50] this will be done by analyzing the longitudinal current spectrum, rather than the dynamic structure factor, since the former always exhibits a peaked structure even when the inelastic peaks in the latter are not well defined [18]. The physical information contained in these two quantities is essentially the same due to Eq. (5.89). Furthermore, the physical meaning of the longitudinal current spectrum is clearer since it can be interpreted as the wavenumber-dependent density of states in the system.

Z

z = d

A

A

A

dA l

AB

O

Y d B

B

z = -d B

B

Figure 5.1. Schematic representation of a diatomic molecule, consisting of atoms A and B, in the body-fixed molecular frame where the origin is taken to be the center of mass and the z axis is along the principal axis of the molecule. zA and zB denote the z coordinates of atoms A and B, respectively, and lAB the bond length.

307


2

A-A

1 0

χ (k)

2

A-B

1 0 2

B-B

1 0 0

1

2

3

4

5

6

-1

k [Å ] Figure 5.2. The site-site static structure factors of the diatomic molecule calculated from the extended version of RISM.

3.1

System description

We consider a liquid consisting of rigid diatomic molecules depicted in Fig. 5.1 with constituent atoms, A and B, of masses mA = 36 g mol−1 and mB = 4 g mol−1 separated from each other by a fixed distance of A. Atoms A and B carry partial charges qA = − 0.25 e and lAB = 2.0 ˚ qB = + 0.25 e, respectively, and the resultant dipole moment is 2.4 D. A and A /kB = 200 K for The Lennard-Jones parameters are σA = 4.0 ˚ A and B /kB = 100 K for atom B. The number atom A, and σB = 2.0 ˚ density is 0.012 molecules ˚ A−3 and the temperature is 250 K. Figure 5.2 exhibits the site-site static structure factors, Eq. (5.90), calculated from the extended version of the RISM integral-equation theA−1 for A-A ory [58, 59]. The peak positions are kmax = 1.69 and 1.65 ˚ and B-B pairs, respectively. Note that in the k → 0 limit all the site-site structure factors coincide [40], and we define χ(0) ≡ χAA (0) = χAB (0) = χBB (0).

(5.142)

We also define the following quantity for later convenience: χ (0) ≡ χAA (0) + χBB (0) − 2χAB (0), where χαβ (0) = limk→0 d2 χαβ (k)/dk 2 .

(5.143)

308

3.2


Eigenmodes of the system

In this subsection we investigate eigenmodes of the system and how each atom in the molecule contributes to the modes. To find the eigenmodes of F(k, t), we neglect the damping term represented by the memory kernel in Eq. (5.106), leading to ¨ t) = −ω2 F(k, t). F(k, k

(5.144)

Elements of ω 2k for the diatomic molecule can be calculated from Eq. (5.108) with Eqs. (5.98), (5.101), and (5.102), and the eigenmodes can be obtained by diagonalizing this matrix. That ω 2k for the diatomic molecule is a two-by-two matrix leads naturally to the possible existence of two modes, which turn out to be acoustic and optical modes as is explained below. In Fig. 5.3 we report the results of the diagonalization of ω 2k . Solid line in Fig. 5.3(a) represents the dispersion curve corresponding to the acoustic branch. It is seen that the dispersion behavior of the acoustic mode is very similar to that of a monatomic system. The k → 0 limit of the eigenfrequency of this mode is given by [5] 2 (k → 0) = ωacou

kB T k2 , M χ(0)

(5.145)

i.e., that of the ordinary sound mode which propagates with the isothermal sound velocity, consistent with the fact that energy fluctuations are not explicit dynamical variables in our treatment here. The contributions from each atom to the acoustic mode can be extracted in the following way. Diagonalizing the matrix ω 2k corresponds to turning the description of the system in terms of the densities of each atom, ρA (k) and ρB (k), to that in terms of their linear combination: xA (k) ρA (k) + xB (k) ρB (k),

(5.146)

where xA (k) and xB (k) are the components of the eigenvector corresponding to the mode. It can be readily found for the acoustic mode, lim [xA (k), xB (k)] ∝ [1, 1].

k→0

(5.147)

This is consistent with the sound mode steming from the center-of-mass (translational) motions of the molecules, i.e., each atom in the molecule equally contributes to this mode. Figure 5.3(b) summarizes xA (k) and xB (k) of the acoustical branch at various wavevectors, normalized such that x2A (k) + x2B (k) = 1. It is seen from the figure that xA (k) ≈ xB (k) holds well in the small-k (up to ≈ 1.0

309

Dynamical processes in solution 50 (a)

ω [ps-1]

40 30 20 10 0

xA

(b)

1 0.5

xB

0

xB

(c)

1 0.5

xA

0 0

1

2

3 k [Å-1]

4

5

6

Figure 5.3. (a) The eigenfrequencies as evaluated by diagonalizing ωk2 . The solid and dashed lines represent the eigenfrequencies of the acoustic and optical modes, respectively. (b) xA (solid line) and xB (dashed line) defined in the text corresponding to the acoustic mode. (c) xA (solid line) and xB (dashed line) corresponding to the optical mode. xA and xB are normalized such that x2A + x2B = 1.

˚ A−1 ) region. In the large-k (i.e., k kmax ) region, however, it is found that xA (k) ≈ 1 with xB (k) ≈ 0, i.e., the dispersion curve of the acoustical branch in this region reflects the self-motion (the single-particle motion) of the heavier atom A. (The collective and single-particle nature of the modes will be discussed in the next subsection.) In the intermediate-k (i.e., k ≈ kmax ) region xA (k) ≈ xB (k) does not hold, and xB (k) becomes even negative (see Fig. 5.3(b)). As will be clarified below, this fact reveals that the rotational motion is also involved in this k-region. We next turn our attention to the dispersion relation of another branch of ω 2k , presented as dashed line in Fig. 5.3(a). This branch is well separated from the acoustic branch at all wavevectors, and it is apparent from the figure that this mode does not vanish in the k → 0 limit. From an analogy with solid state physics, this mode will be re-

310


ferred to as the ‘optical’ mode. The eigenfrequency in the k → 0 limit is found to be given by 2 (k → 0) = ωopti

4kB T 2 l , 3Iχ (0) AB

(5.148)

and the corresponding eigenvector, using the notation in Eq. (5.146), turns out to be (5.149) lim [xA (k), xB (k)] ∝ [zA , zB ]. k→0

Note from Fig. 5.1 that zA and zB are opposite in sign (the liquid model used here gives limk→0 [xA (k), xB (k)] = [−0.11, 0.99] when the norm of the eigenvector is normalized to unity). Thus the atoms A and B contribute to this mode with the out-of-phase fashion in terms of Eq. (5.146). Since the magnitude of zα gauges the efficiency of the atom α for participating in the orientational motion, the optical mode is evidently related to the rotational motion of the molecules. This fact is also obvious by noting that Eq. (5.148) depends on the moment of inertia of the molecule. Figure 5.3(c) presents xA (k) and xB (k) corresponding to the optical mode at various wavevectors. (Note that, since generally ω 2k is not a symmetrical matrix, the eigenvectors corresponding to the acoustic and optical modes are not orthogonal to each other.) It is seen that [xA (k), xB (k)] ∝ [zA , zB ] is well satisfied in the small-k (up to ≈ 1.0 ˚ A−1 ) region. In the larger-k region it is found that the dispersion curve is dominated by the single-particle contribution from the lighter atom B, i.e., xB (k) ≈ 1 with xA (k) ≈ 0.

3.3

Longitudinal current spectra

The resonances determined by ω 2k are shifted and damped by the memory kernel K(k, t). Here we analyze the collective excitations in our model liquid by solving the full GLE which incorporates the damping effect. As a model for the memory kernel we employ a simple exponential model described in Sec. 5.2.3. The longitudinal current spectra are calculated from Eqs. (5.89) and (5.116) with the exponential model for K(k, t) given by Eq. (5.120), and the results for A-A and B-B pairs are presented as solid lines in Figs. 5.4 and 5.5, respectively. Also added as dashed lines in these figures are the single-particle contributions (i.e., contributions from the self-motions) calculated from a similar theory [54] with the aim of elucidating the collective nature of the excitations in the small-k region. A number of observations can be made concerning the theoretical results given in Figs. 5.4 and 5.5. We first note that two peaks are

311

Dynamical processes in solution 1.0

k = 1.0

k = 2.2

k = 0.7

k = 1.9

k = 0.4

k = 1.6

k = 0.1

k = 1.3

0.5 1.0 0.5 1.0 0.5 1.0 0.5 0 0

20

40

20

40

-1

ω [ps ]

Figure 5.4. Solid lines: the longitudinal current spectra, CL,AA (k, ω), of the pair A-A as a function of ω at the indicated k (in ˚ A−1 ) values, in arbitrary units. Dashed lines: the single-particle contributions.

observable in both of the spectra at k = 0.1 ˚ A−1 . However, CL,AA is dominated by the contribution from the lower-frequency acoustic mode, and the higher-frequency peak (≈ 20 ps−1 ) is only slightly seen. The lower-frequency peak also appears in CL,BB because both atoms A and B participate in the center-of-mass density fluctuations as discussed above. The higher-frequency peak corresponding to the optical mode is exaggerated in CL,BB since the lighter atom B is much more responsible for the orientational motion of the molecule. The collective nature of these two excitations in the small-k region is apparent by comparing solid lines with dashed ones in Figs. 5.4 and 5.5. In the large-k region, on the other hand, the spectra are almost identical to those from the singleparticle contributions: in this wavevector region CL,AA is dominated by the contribution from the self-motion of atom A, whilst that of atom B determines the overall shape of CL,BB . The resultant low and high peak frequencies (dispersion relations) of the longitudinal current spectra are shown as solid lines in Figs. 5.6(a) and 5.6(b), respectively. It is seen from Fig. 5.6(a) that the dispersion curve of the low-frequency acoustic mode is very similar to that of monatomic fluids. The dispersion behavior of the optical mode presented

312


k = 1.0

k = 2.2

k = 0.7

k = 1.9

k = 0.4

k = 1.6

k = 0.1

k = 1.3

0.5 1.0 0.5 1.0 0.5 1.0 0.5 0 0

20

40

20

40

-1

ω [ps ] Figure 5.5. Solid lines: the longitudinal current spectra, CL,BB (k, ω), of the pair B-B as a function of ω at the indicated k (in ˚ A−1 ) values, in arbitrary units. Dashed lines: the single-particle contributions.

in Fig. 5.6(b) is qualitatively in accord with that of water reported by Resat et al. [49], who calculated it based on a dielectric theory which takes into account the rotational motion of water. The reasons for this qualitative correspondence of the dispersion behavior of the optical mode between our results and those of Resat et al. [49] (but for water) are two-fold: similarity in the liquid models and that in the way of the description of the mode. First, although our model diatomic liquid seems to be very different from water, they have a similarity in that both of them have quite heterogeneous mass distributions of constituent atoms: the translational motion is almost carried by the atom A in our model and by oxygen atom in water while the rotational motion is dominated by that of B and hydrogen atoms. Second, we have found in the previous subsection (but within the nondamping approximation) that the optical mode in the small-k region can be described in terms of the following linear combination: zA ρA (k) + zB ρB (k).

(5.150)

313


(a) 20

ω [ps-1]

10

0 (b) 40 30 20 10 0 0

1

2

3 k [Å-1]

4

5

6

Figure 5.6. The dispersion relation of the (a) acoustic and (b) optical modes as evaluated from the peak positions in the longitudinal current spectra (solid lines), by 2 (upper dashed diagonalizing ωk2 (lower dashed lines), and by diagonalizing ωk,L lines) [54].

This is ‘nearly’ the same as the basic dynamical variable (the local charge density) in the dielectric theory of Resat et al. [49], qA ρA (k) + qB ρB (k),

(5.151)

in terms of phase since zA and zB are opposite in sign and so are qA and qB (only the relative sign has importance here). That the descriptions of the optical mode in terms of Eqs. (5.150) and (5.151) are nearly the same can also be understood by noting that the rotational motion gives rise to the local charge-density fluctuations whilst the center-of-mass motion does not due to the charge-neutrality of the molecule, and it is the former (the rotational motion) that is responsible for the optical mode.

314


In this connection, it is also instructive to note that the following linear combination, which is found to describe the acoustic mode in the small-k region, (5.152) ρA (k) + ρB (k), is again ‘nearly’ the same combination as that of the local mass-density fluctuations of the system mA ρA (k) + mB ρB (k),

(5.153)

in terms of phase since masses are always positive. In fact, both of the above combinations can give essentially the same description as far as the dynamics of the acoustic mode in the small-k region is concerned [9]. So far we have employed the exponential model for the memory kernel appearing in the GLE for the density correlation functions. In [60] the model based on the mode-coupling theory described in Sec. 5.2.4 is applied to the calculation of the longitudinal current spectra of the same diatomic liquid as discussed here. It is found that the essential features of the results remained the same as far as the collective dynamics is concerned. It is also demonstrated that the results are in fair agreement with those determined from the molecular dynamics simulation.

4.

Ion dynamics in diatomic liquids

An ion in polar liquids is under continuous Brownian motion, and the frictional force exerted on the Brownian particle is proportional to its velocity. The proportional constant, or the friction coefficient, has been a focus of intensive research for almost 100 years both in experimental and theoretical studies [61-65]. According to the simple Stokes’ law which is based on the hydrodynamic theory, the friction should increase proportionally to ionic radii. However, the experimental observations for small ions such as alkali-halide ions in water show the ion-size dependence which is just opposite to the Stokes law [61-64]. Concerning the origin of the peculiar behavior, two models have been proposed and coexisted for long time, which attribute quite different physics to solvent response to a solute displacement. The first model, often referred to as the solventberg model, maintains the classical view of Stokes’ law but with an ‘effective’ ionic radius, or the Stokes radius, which takes into account the effect of solvation: solvent molecules are regarded firmly bound to the ion, and the radius of the solvated ion plays a role of the Stokes radius. The Stokes radius in this model decreases with increasing ionic radius since the ion-solvent interaction is weakened due to the increased ion-solvent distance. The solventberg model has


315

been successful in explaining many physicochemical processes in solutions [62, 63]. The other model for the ionic friction concerns the dielectric response of solvent to the solute perturbation. When an ion is fixed in polar solvent, the solvent is polarized according to the electrostatic field from the ion. If the ion is displaced, the solvent polarization is not in equilibrium with a new position of the ion, and the relaxation of the polarization should take place in the solvent. The energy dissipation associated with this relaxation process may be identified as an extra friction. The extra friction, called the dielectric friction, decreases with increasing ionic radius, thereby, with decreasing electrostatic field from the ion. The dielectric friction model developed by Born [66], Fuoss [67], Boyd [68] and Zwanzig [69, 70] has taken a complete theoretical form due to the work by Hubbard and Onsager [71, 72] who proposed a set of continuum electrohydrodynamic equations in which the electrostatic as well as hydrodynamic strains are incorporated. As far as qualitative aspects of the ion-size dependence of the friction coefficients are concerned, which has a minimum with increasing ionic radius, both models (the solventberg and dielectric friction models) explain the experimental observations to a certain extent. Then one might ask “which model of the two is more faithful to the real physics of the ionic friction?” and/or “how do the two effects interplay if both are coexisting?” For instance, use of the same parameter for the effective ionic radius will not be justified since increase in the effective radius should give rise to increase in the Stokes friction but decrease in the dielectric friction. The question should be answered by microscopic theory, not by treatment based on the continuum model. The first attempt toward the microscopic theory of the ionic friction in polar solvents has been put forward by Wolynes with his insightful paper in which ion-solvent interactions are decomposed into a shortrange repulsive and a long-range attractive parts [73]. The ionic friction coefficient ζ, related to the fluctuations of the random forces exerted on an ion, then splits into components arising from the correlations of the hard repulsive (H) and soft attractive (S) parts of the random force: ζ = ζHH + 2 ζHS + ζSS .

(5.154)

In Wolynes’ treatment ζHH is identified as the Stokes friction and is assumed to be given by the Stokes law with the hydrodynamic radius equal to the crystallographic radius. A second approximation is to ignore the cross term ζHS . Wolynes’ theory then focuses on the time dependence

316


of the fluctuations of the soft force, leading to ζ = ζ0 +

1 F 2 τF , 3kB T S

(5.155)

where ζ0 = ζHH is given by Stokes’ law, FS2 is the static mean-square fluctuation in the soft force, and τF is its characteristic decay time. An important feature of Wolynes’ theory is that his theory reduces to the solventberg and dielectric friction pictures in certain limiting situations: in the case of a strong, short-ranged, attractive interaction the drag on an ion reduces to that on a solvated solventberg ion, whereas in the limit of a weak, long-ranged, attractive ion-solvent interaction case, the dielectric friction picture recovers. It should be noted that in Wolynes’ theory both of these pictures manifest themselves through the second term in Eq. (5.155), and it is not clear how the two different mechanisms interplay when both are co-existing. Nevertheless, Wolynes’ theory has attracted considerable attention, motivated a number of further studies, and still constitutes a basis for theories of the ionic friction. Colonomos and Wolynes [74] tried to implement this molecular theory, but their attempt was severely limited by the lack of available theories for structure of polar liquids at that time. Realizing such a situation, Chen and Adelman [75] went back to the continuum model, but with proper account of the solvation in their generalized Hubbard-Onsager theory. Their theory properly treats the interplay between the Stokes and dielectric frictions within the continuum model, and leads to those frictions in the limit of small and large ionic radii, respectively. However, the theory could not answer, in its nature, any question concerning the microscopic solvent relaxation caused by the solute perturbation. Bagchi and co-workers took the microscopic solvent relaxation into consideration based on the mode-coupling theory and claimed good agreement with experiments [76]. However, their theory does not answer the question of how the Stokes and dielectric frictions interplay, because it disregards the microscopic nature of the Stokes friction. Here we address the problem from a different point of view, namely, in terms of a response of collective excitations in solvent to the ionic field. In Sec. 5.3 we have succeeded in abstracting the collective excitations in a model diatomic liquid which can be identified as acoustic and optical modes. The two modes arise essentially from the translational and rotational motions of solvent molecules. Since the Stokes and dielectric frictions originate basically from the energy dissipation due to the translational and rotational motions of solvent molecules, respectively, it is reasonable to ask how the ionic field couples with the collective excitations and/or how the two drag forces are related to the two col-

317


lective modes. Since the translational and rotational motions of solvent molecules are inherently coupled with each other in our atom-based description of solvent dynamics, the theory is free from artifacts associated with the decoupling of those motions, which has been inevitable in the earlier theories based on the explicit orientational coordinates.

4.1

Theoretical formulation

4.1.1 Mode-coupling expression for the friction coefficient. We begin with the definition of the velocity autocorrelation function of a monatomic solute Z(t) = vu,z vu,z (t),

(5.156)

where vu,z (t) denotes the z component of the velocity of the solute (specified with the subscript ‘u’) at time t. The memory function equation for Z(t) reads (cf. Sec. 1.1)

t

˙ Z(t) =−

dτ K(t − τ ) Z(τ ),

(5.157)

0

where K(t) denotes the memory function, which is given by the random force autocorrelation function K(t) = βm Rz (0)∗ Rz (t).

(5.158)

Here m denotes the mass of the solute, Rz represents the z component of the random force acting on the solute, and ∗ means complex conjugate. To evaluate K(t), we invoke the mode-coupling approximation. A slight generalization the formulation in Sec. 2.4 yields the following expression for K(t): ρkB T ∞ dk k4 cuλ (k)cuµ (k)Fu (k, t)Fλµ (k, t), (5.159) K(t) = 2 6π m 0 λ,µ

in which cuλ (k) denotes the site-site direct correlation function between the solute and the solvent site λ, Fu (k, t) the solute intermediate scattering function, and Fλµ (k, t) the solvent site-site intermediate scattering function defined in Eq. (5.85). To relate Z(t) with the friction coefficient we use the Green-Kubo formula and the Einstein relation [18, 19, 20]: ∞ dt Z(t), ζ = kB T /D, (5.160) D= 0

318


where D and ζ denote the diffusion and friction coefficients, respectively. Using Eqs. (5.157), (5.159) and (5.160), the expression for the friction coefficient in the mode-coupling approximation reads ∞ ρkB T ∞ dt dk k4 cuλ (k)cuµ (k)Fu (k, t)Fλµ (k, t). (5.161) ζ= 6π 2 0 0 λ,µ

This equation expresses the friction coefficient in terms of the solutesolvent equilibrium correlation function (cuλ (k)) and the nature of motions of solute (Fu (k, t)) as well as solvent molecules (Fλµ (k, t)). The site-site direct correlation functions cuλ (k) can be obtained from the extended version of the RISM integral-equation theory [58, 59, 77], and the site-site intermediate scattering function Fλµ (k, t) can be calculated as we have done in Sec. 5.3. For Fu (k, t) the diffusion approximation is invoked since here we are interested in the diffusive motion of the solute: Fu (k, t) = exp(−Dk2 t) with

D = kB T /ζ.

(5.162)

Thus Eq. (5.161) is a nonlinear expression that involves ζ on both sides. By solving Eqs. (5.161) and (5.162) self-consistently one obtains ζ. Equation (5.161) is very similar to that used by Bagchi et al. in their recent study [76]. However, since their expression is written in terms of the longitudinal ion-dipole direct correlation function and the orientational intermediate scattering function of the solvent in place of cuλ (k) and Fλµ (k, t) in our formula, its application is limited to the calculation of the dielectric friction. As we have clarified in Sec. 5.3, both the translational and rotational motions of solvent molecules manifest themselves in Fλµ (k, t), and Eq. (5.161) can be applied to the calculation of the friction coefficient which comprises the hydrodynamic as well as dielectric contributions. Thus Eq. (5.161) can be regarded as a more general microscopic expression for the friction coefficient.

4.1.2 Relaxation functions. Now we would like to understand or interpret the expression for ζ, Eq. (5.161), in terms of the solvent response to a solute displacement. To this end we define the microscopic solvent site number density of type λ at position r relative to the monatomic solute as ρu;λ (r) ≡

N

δ(r − ru,iλ ),

(5.163)

i=1

in which ru,iλ ≡ riλ − ru with riλ and ru denoting the position of the atom λ in the ith molecule and that of the solute, respectively. The

319


total Hamiltonian of the system (here we are considering infinitely dilute solution systems) is (5.164) H = Hv + Hu + Euv , where Hv represents the Hamiltonian of the pure solvent, Hu that of the solute, and Euv is the solute-solvent interaction part. In terms of the site-site solute-solvent interaction potential φuλ (r), Euv takes the form Euv = dr ρu;λ (r)φuλ (r) =

1 (2π)3

λ dk

ρu;λ (k)φuλ (−k),

(5.165)

λ

where we have introduced the spatial Fourier transform of ρu;λ (r), −ik·ru

ρu;λ (k) = e

N

eik·riλ = e−ik·ru ρλ (k),

(5.166)

i=1

and that of φuλ (r). We next introduce a perturbation which displaces the solute position. This perturbation disturbs the spatial correlation between the solute and the solvent, and can be formulated in terms of the time-dependent external field Φext (r, t) that couples to ρu;λ (r). Thus the Hamiltonian which represents the perturbation takes the form ρu;λ (r)Φext (r, t) H (t) = − dr 1 = − (2π)3

λ dk

ρu;λ (k)Φext (−k, t).

(5.167)

λ

Let us consider the following relaxational process. At times t < 0 the solvent is in equilibrium with the solute: the average solvent structure around the solute is characterized by a set of solute-solvent site-site pair correlation functions (5.168) ρu;λ (r) = ρ huλ (r), with an equivalent relation in k space. At t = 0 the external field Φext is applied and disturbs the solute-solvent spatial correlation, and at t > 0 the system relaxes toward the new equilibrium state. During the relaxation process the solute may also move. The relaxation continues until the solvent is in equilibrium with the solute after the disturbance.

320


We define the following relaxation functions of the solute-solvent structure and energy: 8 ∆ρu;λ (k, t) ∆ρu;λ (k), (5.169) Rstr (k, t) = λ

8

E

R (t) = ∆Euv (t)

λ

∆Euv (0),

(5.170)

where ∆A(t) ≡ A(t) − A and the bar denotes the nonequilibrium ensemble average [78]. Assuming that the response to the perturbation is linear one has according to the linear response theory [78] Rstr (k, t) = C str (k, t) / C str (k, 0), RE (t) = C E (t) / C E (0),

(5.171) (5.172)

with C str (k, t) =

1 δρu;λ (−k)δρu;µ (k, t), V

(5.173)

λ,µ

C E (t) = δEuv (0)δEuv (t),

(5.174)

where δA(t) ≡ A(t)−A, and the ensemble average is taken with respect to the unperturbed Hamiltonian. In evaluating Eq. (5.173) we notice that C str (k, t) is essentially a fourbody correlation function, and an approximation must inevitably be invoked. It follows after the factorization approximation (see Eq. (5.135)) Fu (k, t)Fλµ (k, t). (5.175) C str (k, t) = ρ λ,µ

In order to evaluate Eq. (5.174) it is useful to rewrite the expression as 1 E dk δρu;λ (−k)δρu;µ (k, t)φuλ (k)φuµ (−k), C (t) = (2π)3 V λ,µ

(5.176) where we have used Eq. (5.165). The time-dependent part in the expression can be evaluated as in Eq. (5.175). To incorporate the solute-solvent correlation into Eq. (5.176) we resort to the surrogate Hamiltonian description [16, 17], which amounts to replacing every φuλ (k) by the renormalized potential −kB T cuλ (k), the final expression being ρ(kB T )2 E cuλ (k)cuµ (k)Fu (k, t)Fλµ (k, t). (5.177) dk C (t) = (2π)3 λ,µ

321


The relaxation functions can be obtained by substituting Eqs. (5.175) and (5.177) into Eqs. (5.171) and (5.172), respectively: str

R (k, t) =

λ,µ Fu (k, t)Fλµ (k, t)

E

R (t) =

λ,µ

λ,µ χλµ (k)

,

(5.178)

dk cuλ (k)cuµ (k)Fu (k, t)Fλµ (k, t) , (5.179) λ,µ dk cuλ (k)cuµ (k)χλµ (k)

where we have used Eq. (5.90) and Fu (k, 0) = 1. As a result of the factorization approximation the time dependence of the relaxation functions in Eqs. (5.178) and (5.179) is all in the time-correlation functions of the solute and solvent dynamics. Thus, within the linear response regime and the factorization approximation, the solvent response part due to the solute perturbation is determined by Fλµ (k, t). As demonstrated in Sec. 5.3, Fλµ (k, t) in the small-k regime comprises contributions from collective excitations of acoustic and optical modes of solvent, and these modes are responsible for the density and dielectric (or polarization) relaxations, respectively. We note the integrand in the energy relaxation function, Eq. (5.179), is essentially the same as that in Eq. (5.161). This feature can be understood by noting (1) RE (t) describes how the energy of the system relaxes after the solute perturbation and (2) the energy dissipation associated with the relaxation process may be identified as the origin of the friction. Thus Eq. (5.161) essentially expresses the friction as an amount of the dissipated energy during the relaxation process.

4.2

System description

We consider the following infinitely dilute solution systems. For the solvent, we consider the same diatomic liquid as in Sec. 5.3 The number density is 0.012 molecules ˚ A−3 and the temperature is 250 K unless specified otherwise. To investigate the solute-charge effect on the friction coefficient, we consider a cationic solute (abbreviated as CU, which comes from ‘Cationic solUte’) and a neutral solute (NU). We also deal with solutes of various sizes to see the solute-size dependence of the friction coefficients. These solutes are denoted as CUi (and NUi) where i A, varies from 1 to 10. σ in LJ parameters for CUi (and NUi) is σi /2 = i ˚ while is kept fixed to = 0.44 kcal/mol for all the solutes. Note that, since here we are concerned only with the diffusive motion of a solute, it is not required to specify its mass.

322

4.3


Solute viewpoint

In this subsection we investigate the effect of the ionic charge on the friction coefficient. This will be done by calculating ζ for ionic and neutral solutes and by comparing them. Thus we manipulate the charge on the solute to obtain some insights into the nature of ionic friction coefficients, and this will be referred to as ‘the solute viewpoint’. We solve Eqs. (5.161) and (5.162) twice to obtain the friction coefficient for an ion, ζ, and the one for a neutral solute, ζneut . An extra drag due to the charge on a solute, denoted as ζcharge , is defined as the difference between ζ and ζneut : ζ = ζneut + ζcharge .

(5.180)

It should be noted that by our definition the cross term as in Eq. (5.154) does not appear in this treatment. Figure 5.7 summarizes the results of the decomposition based on Eq. (5.180) for solutes CU1 to CU10. The solid line in Fig. 5.7 gives the solute-size dependence of the ionic friction coefficients ζ in Eq. (5.180). (Here we take the radius of a solute to be given by the LJ radius, σ/2.) The dashed and dash-dotted lines present the solute-size dependence of ζneut and ζcharge , respectively. It is seen that ζneut increases with increasing the solute size. The solute-size dependence is almost linear, as predicted by Stokes’ law, even for a solute which is smaller in size than the solvent molecule. ζcharge , on the other hand, shows the opposite trend, and it decreases rapidly as the solute size increases. The resulting solute-size dependence of ζ shows a minimum at the intermediate solute size due to the two opposite trends in ζneut and ζcharge . Concerning the solute-size dependence of ζneut and ζcharge , the results shown in Fig. 5.7 are quite reasonable and qualitatively in accord with the prediction of existing theories. However, at this level of our treatment, it is difficult to establish a picture for large ζcharge of small ions: e.g., it is not clear which of the two classical pictures for ionic friction coefficients, the solventberg and dielectric friction pictures, is responsible for the large ζcharge of small ions. In the next subsection we present another decomposition of ζ that overcomes this limitation and gives us deeper insight into the physics of the ionic friction coefficient.

4.4

Solvent viewpoint

In this subsection another decomposition of the ionic friction coefficient is investigated, which turns out to be quite similar in spirit to Eq. (5.154). To this end we manipulate the solvent site-site intermediate scattering functions in Eq. (5.161). This standpoint will be referred

323

Dynamical processes in solution -21

x 10 [g/ps] 14 12 10

ζ

8 6 4 2 0 1

2

3

4

5

6

7

8

9

10

σ / 2 [Å]

Figure 5.7. Decomposition of ζ based on Eq. (5.180) for solutes CU1 to CU10. Solid line, ζ; dashed line, ζneut ; dash-dotted line, ζcharge .

to as ‘the solvent viewpoint’ in which it turns out that the friction on an ion is described in terms of collective excitations of the solvent.

4.4.1 Formulation based on the solvent viewpoint. We turn the description of the solvent dynamics from that in terms of each solvent density, ρA (k, t) and ρB (k, t), to that in terms of their linear combinations: ρN (k, t) ≡ ρA (k, t) + ρB (k, t), ρZ (k, t) ≡ ρA (k, t) − ρB (k, t).

(5.181) (5.182)

ρN (k, t) is the total number density of the solvent whilst ρZ (k, t) is essentially the (polarization) charge density due to the charge neutrality of the solvent molecule, qB = −qA . In terms of these new quantities, timecorrelation functions of number-number (NN), number-charge (NZ), and charge-charge (ZZ) densities can be defined: FXY (k, t) =

1 ρX (k, 0)∗ ρY (k, t), N

(5.183)

where X, Y = N or Z. The site-site intermediate scattering functions Fλµ (k, t) in Eq. (5.161) can be expressed in terms of these time-correlation functions. Accord-

324


ingly, ζ of Eq. (5.161) is decomposed into ζ = ζNN + 2 ζNZ + ζZZ , where each term on the right-hand side is given by ∞ ρkB T ∞ dt dk k4 cX (k)cY (k)Fu (k, t)FXY (k, t). ζXY = 24π 2 0 0

(5.184)

(5.185)

Here cN (k) and cZ (k) are the following combinations of the solute-solvent direct correlation functions, which come from our adoption of the new basis set in describing the solvent dynamics: cN (k) = cuA (k) + cuB (k), cZ (k) = cuA (k) − cuB (k).

(5.186) (5.187)

Before embarking on the analysis of the ionic friction coefficient based on Eq. (5.184), we explore main features of the direct correlation functions and of the solvent time-correlation functions which appear in Eq. (5.185) and clarify the physical meaning of the decomposition.

4.4.2 Direct correlation functions. In integral-equation theories for charged systems it is conventional to separate direct correlation functions into short-range and Coulombic parts [19]. In r space the site-site direct correlation function assumes the form [58, 59, 77] cuA (r) = cshort uA (r) − β

qu qA , r

(5.188)

where the first term denotes the modified direct correlation function which is short-ranged, and the second term the Coulombic part. Similar equation also holds for cuB (r). Due to the charge neutrality of the solvent molecule (qB = −qA ), it follows from Eqs. (5.186) through (5.188) that short cN (r) = cshort uA (r) + cuB (r),

qu qA short . cZ (r) = cshort uA (r) − cuB (r) − 2β r

(5.189) (5.190)

Thus cN (r) comprises only short-range contributions whereas cZ (r) contains the long-range Coulombic term. It should be noted at this point that, although the bare Coulombic part is absent in cshort uλ (r), its renormalized effect is contained in this function, and cshort uλ (r) should not be taken as a quantity that is determined solely from the short-range (such as LJ) interactions. We define uN (r) ≡ −kB T cN (r) and uZ (r) ≡ −kB T cZ (r), which can be regarded as effective (or renormalized) interaction potentials. uN (r)

325

Dynamical processes in solution [kcal/mol]

(a)

[kcal/mol]

10.0

(b)

100.0 CU4

CU4

0.0

0.0 100.0

10.0 CU3

CU3 0.0 U Z (r)

U N (r)

0.0 10.0 CU2 0.0

100.0 CU2 0.0

10.0

100.0 CU1

CU1

0.0

0.0

- 10.0

- 100.0 2.0

6.0 r [Å]

10.0

2.0

6.0

10.0

r [Å]

Figure 5.8. The effective potentials as defined in the text for solutes CU1 to CU4 calculated from the extended version of RISM. (a) Solid lines, uN (r) ≡ −kB T cN (r); short dashed lines, contribution from cshort uA (r) term; dotted lines, contribution from cuB (r) term. (b) Solid lines, uZ (r) ≡ −kB T cZ (r); dashed lines, contribution from the Coulombic term.

and uZ (r) for CU1 through CU4 are presented in Figs. 5.8(a) and 5.8(b), respectively. (The results for CU5 to CU10 are omitted for brevity since they are very similar to those for CU4 if we appropriately rescale the abscissa of the plots according to the solute size.) From these figures it is seen that uN (r) and uZ (r) are really short and long ranged, respectively, and that in the large-r region uZ (r) is essentially given by the bare Coulombic contribution. From Fig. 5.8(a) it is seen that uN (r) for CUi with i ≥ 3 are essentially repulsive with the shallow energy depth while for CU1 and CU2 the depth of the energy well is much larger than the thermal energy kB T . A consequence of the deep attractive part of uN (r) for CU1 and CU2 will be discussed below. In view of these results it is understood that the decomposition of Eq. (5.184) is very similar in spirit to that of Eq. (5.154) in Wolynes’ theory where the decomposition is made according to the separation of the random force on an ion into short-range repulsive and long-range iondipole interaction parts. More importantly, all the terms in Eq. (5.184)

326


have their own microscopic expressions, Eq. (5.185), and no such simplifying assumptions as made in Wolynes’ original theory are required.

4.4.3 Solvent time-correlation functions. We next consider the solvent time-correlation functions appearing in Eq. (5.185). As we have clarified in Sec. 5.4.1, these time-correlation functions describe, within the linear response regime and the factorization approximation, the solvent response to the solute perturbation. We summarize here main features of FNN (k, t) and FZZ (k, t) in the small-k region where collective dynamics of the solvent molecules are probed. In Fig. 5.9 we report FNN (k, t) and FZZ (k, t) for selected wavevectors, which are normalized according to their initial values. Corresponding longitudinal current spectra, ω2 CL,XX (k, ω) = 2 k

∞ −∞

dt eiωt FXX (k, t),

(5.191)

for X = N and Z are plotted in Fig. 5.10. It is seen from Fig. 5.9 that in the small-k regime the decay of FNN (k, t) is slow and monotonic whereas FZZ (k, t) decays very rapidly and shows an oscillatory feature. These behaviors can be understood, using the results in Fig. 5.10, in the following way. As we have clarified in Sec. 5.3, there are two kinds of excitation modes (which are collective in the smallk region) in our solvent model: the acoustic and optical modes, which arise essentially from the translational and rotational (librational) motions of solvent molecules, respectively. Their characteristic frequencies within the non-damping approximation in the k → 0 limit are given by Eqs. (5.145) and (5.148), respectively. It is important to notice that ωopti (k) does not vanish in the k → 0 limit. Thus in the small-k region the characteristic frequency of the acoustic mode is lower than that of the optical mode: the lower-frequency peak of the longitudinal current spectra presented in Fig. 5.10 corresponds to that of the acoustic mode, whilst the higher-frequency peak is that of the optical mode. From Fig. 5.10 it is seen that both the acoustic and optical modes are involved in CL,NN (k, ω), and these contribute to the time evolution in FNN (k, t). It is the low-frequency nature of the acoustic mode in the small-k region that is responsible for the slow decay of FNN (k, t). On the other hand, it is seen from Fig. 5.10 that only the higherfrequency optical mode is involved in CL,ZZ (k, ω). This is because only the rotational motions give rise to the local charge-density fluctuations. (The translational motions do not due to the charge-neutrality of the solvent molecule.) Thus it is the higher-frequency nature of the optical

327


1.0

1.0

k = 0.7

0.5

k = 0.7

0.5

0 1.0

1.0

0.5

0.5

0

C L (k,ω)

F (k, t)

k = 0.5

k = 0.5

1.0 k = 0.3

0.5

1.0

k = 0.3

0.5

0 1.0

1.0

k = 0.1

k = 0.1

0.5

0.5

0 0

0

1.0

2.0

t [ps]

Figure 5.9. Normalized FNN (k, t) (solid lines) and FZZ (k, t) (dashed lines) at the indicated k (in ˚ A−1 ) values.

0

25

50 -1

ω [ps ]

Figure 5.10. Longitudinal current spectra, CL,NN (k, ω) (solid lines) and CL,ZZ (k, ω) (dashed lines) at the indicated k (in ˚ A−1 ) values, in arbitrary units.

mode that is responsible for the rapid decay of FZZ (k, t) in Fig. 5.9, and its oscillatory feature reflects the librational nature of the mode involved.

4.4.4 Decomposition of friction coefficients based on the solvent viewpoint. Now we present the decomposition of ζ based on Eq. (5.184). As discussed above, this decomposition is similar in spirit to that of Wolynes, and ζNN , ζNZ and ζZZ in Eq. (5.184) correspond respectively to ζHH , ζHS and ζSS in Eq. (5.154). Combining the theoretical consideration in Sec. 5.4.1 and the results discussed above, it is understood that FNN (k, t) and FZZ (k, t) respectively describe, within the linear response regime and the factorization approximation, the collective density- and dielectric-response of solvent to a solute displacement, and they are responsible for the density and polarization relaxations after the solute perturbation. Correspondingly, it seems reasonable to regard

328

Molecular theory of solvation -21

x 10 [g/ps] 14 12 10 ζ

8 6 4 2 0 1

2

3

4

5

6

7

8

9

10

σ / 2 [Å]

Figure 5.11. Decomposition of ζ based on Eq. (5.184) for solutes CU1 to CU10. Solid line, ζ; dashed line, ζNN ; dotted line, ζNZ ; dash-dotted line, ζZZ .

ζNN and ζZZ as the hydrodynamic (or Stokes) and dielectric frictions, respectively, and ζNZ as their cross term. The solid line in Fig. 5.11 gives the solute size-dependence of the total ζ, and the dashed, dotted, and dash-dotted lines denote that of ζNN , ζNZ , and ζZZ , respectively. Several observations can be made concerning the theoretical results presented in Fig. 5.11. First, the cross term ζNZ (the dotted line), which was ignored in Wolynes’ theory, is rather large for small ions. As the solute size increases, however, ζNZ decreases, and the neglect of this term may be a good approximation in dealing with larger ions. Another situation in which ζNZ can be neglected in our treatment is when the solvent consists of homonuclear diatomic molecules: FNZ (k, t) in Eq. (5.185) becomes identically zero in this case. However, this case should be regarded as an exceptional one since most of existing solvents do not possess such symmetry. Hence a proper theory for the friction coefficient should incorporate the effect from the cross term if one wishes to treat small ions as well as large ones. The solute-size dependence of ζZZ (the dash-dotted line in Fig. 5.11) is in accord with that of the dielectric friction picture, which also justifies our convention to call ζZZ the dielectric friction. On the other hand, the solute-size dependence of ζNN (the dashed line in Fig. 5.11) shows a peculiar behavior. One might expect, if ζNN were indeed the hydrodynamic friction, that ζNN would obey a linear solute-size dependence


329

(Stokes’ law) to a certain degree. The dashed line in Fig. 5.11, however, reveals that the expectation does not apply to small ions: ζNN are larger than would be expected on the basis of the Stokes law. It should be remembered that, as we have seen in Fig. 5.8, uN (r) ≡ −kB T cN (r) for small ions (CU1 and CU2) has a strong short-ranged attractive part, and consequently the solvation of these solutes will be much more significant than that of larger ions. The picture for the large ζNN of small ions which emerges from the strong solvation is that of a solvated solventberg ion. To confirm our conjecture we further investigate the peculiar solute-size dependence of ζNN in the next subsection by analyzing its temperature dependence: if the solventberg is actually formed, the temperature dependence of the frictions for solvated ions should be qualitatively different from that for unsolvated ions.

4.4.5 Temperature dependence of friction coefficients. In this subsection we analyze the temperature dependence of the friction coefficients: friction coefficients at 300, 350, 400, 450, and 500 K are calculated in addition to those at 250 K. However, since densities for a given pressure at higher temperatures are not available for our solvent model, the density is fixed to the value at 250 K. (This also forces the solvent to remain in the liquid state even at 500 K.) Thus, strictly speaking, our results at higher temperatures presented below are those for systems at elevated pressures. The temperature dependence of the total ζ is plotted in Fig. 5.12(a). It is seen that the temperature dependence of the total ζ is somewhat complicated: for the solutes CU3 to CU10 the temperature dependence is subtle, whereas for small cations, CU1 and CU2, it is significant. To understand features in Fig. 5.12(a), especially for small ions, we plot the temperature dependence of ζNN , ζNZ and ζZZ in Figs. 5.12(b)– 5.12(d), respectively. Concerning ζNZ and ζZZ , the temperature dependence is subtle for CU3 to CU10 whereas it is considerable for CU1 and CU2. However, the solute-size dependence is preserved in ζNZ and ζZZ : the larger the solute size, the smaller ζNZ and ζZZ at all temperatures, i.e., there is no crossing among the curves for ζNZ and ζZZ . The temperature dependence of ζNN shows a peculiar feature, and the solute-size dependence is not preserved in this case: for example, ζNN for CU2 is larger than that for CU3 at 250 K, but the opposite is true at higher temperatures, and the resultant solute-size dependence at higher temperatures is in accord with the prediction based on the Stokes law. ζNN for CU1 shows more dramatic decrease as the temperature is increased. As we have seen in the previous subsection, the solventberg picture seems to be adequate in interpreting ζNN for CU1 and CU2.

330


(a)

-21

x 10 [g/ps]

(c)

-21

x 10 [g/ps]

15 2 10

CU10

CU7

NZ

CU8

5

ζ

ζ

CU9

1

CU6 CU5 CU4 CU3

CU3

I 0

CU10

0 250

300

350

400

450

250

500

300

(b)

-21

x 10 [g/ps]

350

400

450

500

TEMPERATURE [K]

TEMPERATURE [K]

(d)

-21

x 10 [g/ps] 6

10 CU10

5

8

CU9

4

CU6 CU5

2

CU4 CU3

ζ

NN

ζ

CU7

4

ZZ

CU8

6

3 2 CU3

1

I CU10

0

0 250

300

350

400

450

TEMPERATURE [K]

500

250

300

350

400

450

500

TEMPERATURE [K]

Figure 5.12. The temperature dependence of the friction coefficients for solutes CU1 to CU10. In all the figures solid lines denote results for CU3 through CU10 as indicated in the figure, dashed lines the results for CU1, and dash-dotted lines those for CU2. (a) The temperature dependence of the total ζ. (b) The temperature dependence of ζNN . (c) The temperature dependence of ζNZ . (d) The temperature dependence of ζZZ .

In view of this we attribute the origin of the dramatic decrease in ζNN for CU1 and CU2, which is absent in data for CU3 through CU10, to the ‘melting’ of the solventberg. In the high temperature limit where the solventberg is completely melted away, ζNN will obey, to a certain degree, the solute-size dependence which can be predicted on the basis of Stokes’ law. Summarizing the results in this section, we have found that both the solventberg and dielectric friction mechanisms are responsible for the large ζ of small ions, and they are reflected in ζNN (the Stokes or hy-


331

drodynamic friction) and ζZZ (the dielectric friction), respectively. The coexistence of the both mechanisms does not lead to any contradiction since it is impossible to realize the perfect rigid solvation of an ion. In the high temperature limit where the solventberg is completely melted away, ζNN will obey, to a certain degree, the solute-size dependence as predicted on the basis of the Stoke law. The collective response of the solvent to the solute perturbation plays a major role in determining the ionic friction, and it is demonstrated that the acoustic and optical modes of solvent are responsible for the Stokes (ζNN ) and dielectric (ζZZ ) frictions, respectively. Another important feature achieved from the present analysis is that one can not in general neglect the cross term (ζNZ ) in dealing with small ions, which has been disregarded in most of the theoretical considerations in the past.

5.

Collective excitations and dynamics of ions in water

By extending the theory employed in Secs. 5.3 and 5.4 for the analysis of the collective excitations and ion dynamics in a model diatomic liquid, we consider here those dynamical processes in liquid water.

5.1

System description

In this section the following aqueous solutions at infinite dilution are considered. For the solvent we employ a three-site water model whose molecular structure and parameters defining the intermolecular interaction were taken from the SPC/E model [79]. The model comprises three sites, oxygen (O) and two hydrogens (H1 and H2 , or simply H) as depicted in Fig. 5.13. The site-site interaction potential consists of the Lennard-Jones (LJ) and the Coulomb interactions, with parameters A, O = 0.156 kcal/mol and qO = −0.8476 e for the oxyσO = 3.16 ˚ A, H = 0.046 kcal/mol and qH = 0.4238 e gen atom, and σH = 0.4 ˚ for the hydrogen atom. The number density of the solvent is 0.03334 molecules/˚ A3 and the temperature is 298.15 K. The solutes we consider are the alkali metal ions (Li+ , Na+ , K+ , Rb+ and Cs+ ) and the halides (F− , Cl− , Br− and I− ). The site-site potential between the solute and the solvent also consists of LJ and the Coulomb interactions. The parameters for ions are chosen to be the same as those used by Rasaiah et al. [80] in their molecular dynamics (MD) simulation study in order to facilitate the comparison with their simulation results. To extend the size of ions, we have also considered artificial ions, I+ , CU3, CU4 and CU5. The ion I+ has the same parameters as I− except that it has the positive charge. The latter three are artificial monovalent

332


ions (whose abbreviation comes from ‘Cationic solUte’), and σ in LJ parameters is σ/2 = i ˚ A for CUi whilst its and mass are kept fixed − to those of I . The corresponding artificial anions (denoted as AUis) are also considered which have the same parameters as CUis except for the charge. In this way, we extend the size of the ions beyond the usual range to study the behavior of large ions. In the following subsections we first analyze the collective excitations in water, and the ion dynamics in water is discussed next in terms of these excitations.

5.2

Collective excitations in water

5.2.1 Eigenmodes of the system. As discussed in Sec. 5.3.2, the eigenmodes of the collective density fluctuations can be obtained by diagonalizing the frequency moment matrix ω 2k , whose explicit expression for a three-site water model in Fig. 5.13 can be found in [53]. The eigenfrequencies as obtained by diagonalizing ω 2k for water are plotted against the wavenumber k in Fig. 5.14(a). Since ω 2k is a three-bythree matrix, there are three eigenfrequencies. One of them vanishes in the k → 0 limit, and can be identified as that for an acoustic mode. The asymptotic form of the eigenfrequency of the mode becomes in the

(0, - yH, zH)

Z

(0, yH, zH) H1

H2

θ Y O

(0, 0, zO)

Figure 5.13. Schematic representation of the three-site SPC/E water model. Indicated are the principal axes of the body-fixed reference frame (the x axis being normal to the molecular plane) and the coordinates of constituent atoms. The values of the A, yH = 0.8165 ˚ A, zH = 0.5127 ˚ A, θ = 54.74 ◦ . parameters are: zO = −0.0646 ˚

333


(b)

1.0

XO

0.0 X H1 , X H2 -1.0

(c)

1.0

X H1 , X H2 0.0

(a)

XO

-1.0

120

-1

ω [ps ]

160

(d)

1.0 80

X H1 0.0

XO

40

X H2

-1.0

0 0.0

1.0

2.0

3.0

4.0

5.0

0.0

1.0

2.0

3.0

4.0

5.0

k [Å-1]

k [Å-1]

Figure 5.14. (a) Eigenfrequencies as evaluated by diagonalizing ω 2k . Solid, dashed and dash-dotted lines give the eigenfrequencies of the acoustic mode, OM-I and OMII, respectively. (b) xO (solid line), xH1 (dashed line) and xH2 (dash-dotted line) defined in the text corresponding to the acoustic mode. Notice that xH1 = xH2 holds in the whole k-region. (c) xO (solid line), xH1 (dashed line) and xH2 (dash-dotted line) corresponding to OM-I. Notice that xH1 = xH2 holds in the whole k-region. (d) xO (solid line), xH1 (dashed line) and xH2 (dash-dotted line) corresponding to OM-II. Notice that xH1 = −xH2 holds in the whole k-region. From (b) to (d), xO , xH1 and xH2 are normalized such that x2O + x2H1 + x2H2 = 1.

non-damping case [53] 2 (k → 0) = ωacou

kB T k2 , M χ(k = 0)

(5.192)

where M is the total mass of a molecule, and we have noticed that all the site-site static structure factors coincide in the k → 0 limit, denoted as χ(k = 0). Like for a model diatomic liquid as considered in Sec. 5.3.2, the acoustic mode stems from the translational motion of molecules. The other two eigenfrequencies, on the other hand, do not vanish in the k → 0 limit (optical modes). These modes are concerned with the relative motions of atoms, whose characteristic frequencies do not vanish even in the k → 0 limit. The limiting behaviors of the two optical modes, to be called OM-I (the optical mode I) and OM-II (the optical mode II),

334


are respectively given by [53] 2 (k → 0) = ωOM−I

2kB T 3χ (k =

(zH − zO )2 (1/Ix + 1/Iy ),(5.193) 0) 2 (k → 0) = kB T (1/Ix + 1/Iz ), (5.194) ωOM−II where Ix , Iy and Iz are the moments of inertia around the principal x, y and z axes, and zO and zH are the z coordinates of O and H atoms of a molecule in the body-fixed coordinate (see Fig. 5.13). The optical modes are related to the rotational motion of molecules as can be inferred from the appearance of the moments of inertia in the above expressions. OM-I involves the second moment of the density correlation function,

1 χ (0) + χH1 H2 (0) , (5.195) χ (0) ≡ χOO (0) − 2χOH1 (0) + 2 H1 H1 where χαβ (0) = limk→0 d2 χαβ (k)/dk 2 , and is closely related to the dielectric constant of liquid [81, 82]. It also indicates that OM-I has a collective character. On the other hand, OM-II is essentially a single-molecule mode since the expression is not associated with any collective density correlation function. The contributions from each atom to the mode can be extracted in the same way as we have done in Sec. 5.3.2 in terms of the linear combination of the density fluctuations: xO (k) ρO (k) + xH1 (k) ρH1 (k) + xH2 (k) ρH2 (k),

(5.196)

where xO (k), xH1 (k) and xH2 (k) are the components of the eigenvector corresponding to the mode. Thus by analyzing the sign and magnitude of xα (k)s, it is possible to obtain the information on how each atom contributes to the mode. Plotted in Fig. 5.14(b) to 5.14(d) are the contributions of each atom to the acoustic and two optical modes, respectively. It is seen from Fig. 5.14(b) for the acoustic mode that xO (k) ≈ xH1 (k) = xH2 (k) holds well in the small-k region, which is consistent with the sound mode steming from the center-of-mass translational motion of the molecules, i.e., each atom in the molecule equally contributes to this mode. On the other hand, as can be seen from Fig. 5.14(c), OM-I is governed by the lighter hydrogen atoms over the entire wavelength range, because the rotational motion of a molecule is dominated by the motion of the hydrogens which are located further from the center of mass. Figure 5.14(d) shows that OM-II is just related to the rotational motion in which two hydrogen atoms move in the out-of-phase fashion with oxygen atom fixed. More detailed analysis of the eigenvectors revealed that OM-I and OM-II for

335


(a)

Z

H1 (H2)

X O

(b)

Z

H1

H2

Y O

Figure 5.15. Schematic representation of the molecular motions associated with (a) OM-I and (b) OM-II. These motions are respectively the pitch and roll librational motions using the terminology of [83].

small wavenumbers are respectively associated with the pitch and roll librational motions as depicted in Fig. 5.15. Since OM-II is strictly the single-molecule reorientation and does not contribute to the ion dynamics to be described in the next subsection, it will not be considered any more in the present article.

5.2.2 Longitudinal current spectra. Here we also employ a simple exponential model for the memory kernel described in Sec. 5.2.3 to account for the damping effect on the dynamics. Figures 5.16 and 5.17 exhibit the results based on this theory along with the MD simulation data performed on the same system. The results are reported in the

336


1.0

n=2

n=4

n=1

n=3

C L (k,ω)

0.5 1.0 0.5 0.0 0

100

0 ω [ps -1 ]

100

200

Figure 5.16. The theoretical (solid lines) and MD simulation (circles) results of the longitudinal current spectra for the acoustic dynamics, CL,MM (k, ω). The results are A−1 and n = 1 ∼ 4, plotted as a function of ω for k = n kmin with kmin = 0.3185 ˚ in arbitrary units. kmin refers to the minimum accessible wavenumber from MD simulation.

form of the linear combination of the longitudinal-current spectra cα cβ CL,αβ (k, ω), X = M or Z, (5.197) CL,XX (k, ω) = α,β

since one can separately discuss the acoustic and optical (OM-I) modes based on these combinations [84]. Here CL,MM (k, ω) and CL,ZZ (k, ω) denote the longitudinal-current spectra of the mass and charge, respectively, which can be obtained by setting cα = mα (mass of atom α) or cα = qα (partial charge of atom α) in Eq. (5.197). It is seen from Fig. 5.16 that the theoretical results for the acousticmode spectra are in fair agreement with MD simulation results; especially, the peak positions of CL,MM (k, ω) are well reproduced. (The discernible lower frequency peak in MD simulation result for n = 4 in Fig. 5.16 can be ascribed to a single-molecule excitation, and not to the collective acoustic excitation, as exhibited by Miura [9].) Compared in Fig. 5.17 are the theoretical and MD simulation results for the longitudinal current spectra of the optical dynamics, CL,ZZ (k, ω). Although the overall shape of the spectra are well reproduced by our theory, the theoretical results for the peak frequencies are considerably lower than those of the MD simulation. Thus, although some quantitative disagreements can be seen between the theoretical and MD simulation results,

337


1.0

n=2

n=4

n=1

n=3

C L (k,ω)

0.5 1.0 0.5 0.0 0

100

0 ω [ps -1 ]

100

200

Figure 5.17. The theoretical (solid lines) and MD simulation (circles) results of the longitudinal current spectra for the optical dynamics, CL,ZZ (k, ω). The other notation is the same as in Fig. 5.16.

the qualitative nature of the collective density excitations in liquid water is well captured by the present theory. Some other features of collective excitations in water, such as the high-frequency sound velocity and the dispersion relations, have also been discussed in [53] and [84], showing that all the essential features, reported previously by neutronscattering experiments [3], MD simulations [4, 5, 7, 8, 9] and dielectric theories [49, 50] are well reproduced.

5.3

Dynamics of solvated ions in water

Additional questions are raised when the moving ion is immersed in water instead of a simple dipolar liquid discussed in Sec. 5.4. The hydration phenomena observed by many different experiments show a remarkable variety in its dependence on size and sign of ions, which may not be characterized by a simple chemical model such as ion-water complex formation, or a solventberg model. Such a model seems to apply to very small monovalent ions like Li+ and F− and multi-valent ions which make a stable hydration shell with substantial lifetime. However, water molecules in the first hydration shell around ions with greater size are more mobile and disordered than those in bulk water. The behavior which has been referred to as ‘structure breaking’ by Frank and Wen [85] and as ‘negative hydration’ by Samoilov [86] is due to the competition between two forces acting on water molecules in the shell, the

338


isotropic electrostatic field from the ion and the hydrogen-bonding with other water molecules. Such characteristics of ion hydration is expected to produce non-linearity in the response of the electrostatic potential fluctuations by solvent to the ion field, which has strong dependence on size of ions [87, 88]. Recently, such hydration phenomena have been characterized theoretically based on the RISM theory in terms of the ion perturbation to the pair-correlation functions [89]. The following questions may be naturally raised in conjunction with the ion dynamics in water. Does the dynamics of ions in water reflect such characteristics of the equilibrium structure of hydration? If it does, how and to what extent? Is there any qualitative difference in dynamics of ions in water compared to that in simple polar solvents, and what is the difference if any? In this subsection we try to answer some of these questions. Here we discuss the ion dynamics in water in terms of the diffusion or friction coefficients, which can be described as the response of the solvent collective excitations to the ionic field, as we have discussed in Sec. 5.4.4 (called the solvent viewpoint). Another direction to which we extend our theory is to further refine the memory function for the velocity autocorrelation function of ions: our previous expression for the memory function, Eq. (5.159), is based solely on the mode-coupling theory, and consequently its short-time behavior is not adequately taken into account. If one is only interested in the long-time or the low-frequency phenomena like diffusion processes as in Sec. 5.4, it is not essential to take seriously into account the short-time behavior of the memory kernel. However, if one is interested in the velocity autocorrelation function itself rather than its time integral, it is important to adequately incorporate such contributions. In this subsection this will be done by exploiting a simple Gaussian ansatz for the short-time variation of the memory function [30, 90]. The two approximations for the memory function, the Gaussian ansatz and the mode-coupling approximation, will be connected according to the prescription described in Sec. 5.2.4.

5.3.1 Memory-function formulation of ion dynamics. Following the procedure described in Sec. 5.2.4 we write the memory function for the velocity autocorrelation function (cf., Eq. (5.157)) as K(t) = Kfast (t) + Kslow (t),

(5.198)

where Kfast (t) denotes the rapidly decaying portion of the memory function due to the binary collisions, whilst Kslow (t) represents the slow portion arising from correlated collisions. It shall be assumed that the fast portion can be well represented by the Gaussian ansatz, Kfast (t) ≈ K(0) exp(−t2 /τ 2 ),

(5.199)

339


with the decay constant defined by 1/τ 2 = −

¨ 1 K(0) . 2 K(0)

(5.200)

In this expression K(0) is the so-called ‘Einstein frequency’, which can be calculated exactly from the information of the site-site intermolecular ¨ potential as well as of the density pair-correlation functions. K(0) is also an equilibrium quantity but with the three-particle correlation functions, whose approximate expression has been derived in [91]. For the slow part of the memory function, we follow the prescription described in Sec. 5.2.4, which leads to [91] ρkB T ∞ dk cuλ (k)cuµ (k) Kslow (t) = 2 6π m 0 λ,µ

× (1 − fu (k, t))Fu (k, t)Fλµ (k, t). (5.201) This expression can be regarded as the generalization of Eq. (5.159) so as to exhibit the proper short-time dynamics. For an adequate description of the short-time dynamics of the solute, one also has to generalize Fu (k, t) given in Eq. (5.162) which is valid only in the long-time regime. For this purpose, we adopt the Gaussian approximation for Fu (k, t),

t 2 dτ (t − τ ) Z(τ ) , (5.202) Fu (k, t) = exp −k 0

which is exact in the short- and long-time regime [18, 19, 20]. In Eq. (5.201) fu (k, t) is an auxiliary function defined by Fuid (k, t)/Fu (k, t), where Fuid (k, t) = exp[−(kB T /m)k2 t2 ] denotes the intermediate scattering function of an ideal gas. The velocity autocorrelation function Z(t) and its memory function K(t) can then be obtained as a self-consistent solution of the above equations. The diffusion and friction coefficients can be obtained as their time integrals (cf., Sec. 5.4.1) ∞ ∞ dt Z(t), ζ = m dt K(t). (5.203) D= 0

0

5.3.2 Velocity autocorrelation functions. The velocity autocorrelation functions (VACF) of alkali and halide ions in water obtained from the theory just described are plotted in Fig. 5.18. The general behavior of VACF is highlighted by the pronounced oscillation seen in the small ions, which disappears with increasing ion size. The

340


1.0 Cs +

0.5

Normalized Z (t)

+ K + Rb

0.0 Na+ Li +

1.0

0.5

IBr -

0.0

Cl F-

-0.5 0.0

Figure 5.18. in water.

0.1

0.2 0.3 t [ps]

0.4

0.5

The normalized velocity autocorrelation functions of cations and anions

oscillation is apparently due to the vibrational motion of the ions. Similar pictures have been obtained by Rasaiah and co-workers based on the MD simulation [80]. This suggests the existence of a solvent cage around the ions, whose lifetime is long enough to support the vibration. Remember, the ions which have pronounced oscillation in VACF are those classified as of ‘positive’ hydration by Samoilov [86]. Therefore the dynamic picture is well in harmony with the static one in this respect. The oscillation disappears due to two physical causes; the size and mass. As the ion size increase, the ion-solvent interaction, whose essential nature is electrostatic, decreases, which loosens the solvent cage. Increasing mass makes the ion motion more ‘inertial’ with longer characteristic time. As a whole the behavior of VACF obtained from the theory is consistent with our intuitive picture for ion dynamics in water.

341


x 10 -5 [cm 2 /s] 3.0 Cations

2.0

1.0

D

Rb Li Na K Cs

I CU3

CU4

CU5

Br Cl I AU3

AU4

AU5

4.0

5.0

0.0 Anions

2.0

1.0 F

0.0 0.0

1.0

2.0

3.0

σ i / 2 [Å] Figure 5.19. The diffusion coefficients of cations and anions in water as a function of the solute size taken as σi /2. CUi and AUi denote artificial cations and anions, A. The circles represent the theoretical respectively, whose sizes are given by σi /2 = i ˚ results of the present work, whilst the asterisks (∗) denote the MD simulation results after Rasaiah et al. [80].

5.3.3 Diffusion constants. The diffusion constants calculated from Eq. (5.203) using VACF described above are depicted in Fig. 5.19 against the ion radius, which is taken as half of the Lennard-Jones σ parameter. The behavior is striking in a sense that it entirely breaks the Einstein-Stokes law, which predicts monotonic decrease of the diffusion constant as the ion size increases. (Though we have not shown it here, our theory, in fact, predicts the monotonic decrease, when the electrostatic interaction between the ion and solvent molecules is ‘turned off’. See [91].) Instead, our results exhibit just the opposite behavior to the Einstein-Stokes law when the ion size is small. We will discuss the phys-

342


ical origin of the anomalous behavior seen in the diffusion constant in detail later, because the diffusion constant contains essentially the same information as the friction coefficient. Here we would like to make just two points. By comparing the plots for the cations (upper panel) and for the anions (lower panel) one can readily find that those curves do not fall on top of each other. The asymmetry with respect to the sign of charges comes from the asymmetry of charge distribution in a water molecule. As has been addressed in many ways, the structure of water is characterized by the well-developed hydrogen-bond network, which has its origin in the unique molecular structure typically represented by the simple point charge model; a large negative (partial) charge located at nearly center of the spherical core representing the oxygen atom, and two positive charges on hydrogen atoms which are embedded near the surface of the core, asymmetrically in a sense that they do not have inversion symmetry. Equilibrium properties of ion hydration reflect such properties of molecular and liquid structures of water in one way or the other. The hydration free energy and its entropy and enthalpy components of alkali and halide ions exhibits marked asymmetry in terms of dependence on the sign of ion charges; when it is plotted against the ion size, the thermodynamic quantities for the cations and anions fall on different curves [92]. This behavior, which can never be explained by a continuum dielectric model such as the Born model, is attributed to the asymmetrical charge distribution in water molecule just described [93]. The other point to be made is the agreement of the theoretical results with those from the MD simulation, which is not very bad considering that no empirical parameters are involved in the present approach.

5.3.4 Friction coefficients. In [91], the following decomposition of the friction coefficient is proposed in terms of the response of collective excitations in water to a solute displacement: + 2ζNZ + ζZZ . ζ = ζNN

(5.204)

This decomposition is essentially the same as the one given in Eq. (5.184) which is discussed in Sec. 5.4.4 based on the solvent viewpoint:the only difference comes from the fact that here the short-time behavior of the and ζZZ denote the memory kernel also affects the friction. Thus ζNN terms which are essentially due to the couplings of the ion dynamics to the solvent acoustic and optical (OM-I) modes, respectively, and ζNZ represents their cross term. (Explicit expressions for these terms can be and ζZZ as the found in [91].) Therefore we shall also regard here ζNN ‘Stokes’ and dielectric parts of the friction, and ζNZ as their cross term as in Sec. 5.4.4.

343


x 10 -21 [g/ps] 6.0 Cations

4.0 2.0 0.0 -2.0

ζ

-4.0 -6.0 Anions

8.0 4.0 0.0 -4.0 -8.0 -12.0 0.0

1.0

2.0

3.0

4.0

5.0

σ i / 2 [Å] Figure 5.20. Decomposition of ζ based on Eq. (5.204) for cations and anions in ; squares, the water. Circles, the total friction ζ; triangles, the ‘Stokes’ part ζNN dielectric friction part ζZZ ; diamonds, the cross term ζNZ multiplied by a factor of 2, see Eq. (5.204). The solid and dashed lines are to guide the eyes.

The friction coefficients ζ are plotted against the ion size in Fig. 5.20 for the cations (upper panel) and for the anions (lower panel). Also , ζ depicted are the decompositions of the coefficient, ζNN ZZ and 2ζNZ , based on Eq. (5.204). The overall ζ first decreases with increasing ionic radius and then begins to increase after passing through a minimum. The behavior for the small ionic radii is just opposite to that predicted from the Stokes law, and is in accord with either of the physical models, the solventberg or the dielectric friction models. Incidentally, the size of alkali and halide ions fall approximately in the range where the friction decreases with increasing radii in harmony with the experimental observation for those small ions.

344


Amongst the decompositions, ζZZ shows relatively simple behavior; it decreases monotonically with increasing ion-radii. This term is related to the dielectric response of solvent to the ion displacement, and the monotonic behavior is ascribed merely to the reduced Coulomb interaction between ions and solvent with increasing ion size. The contribution, , exhibits striking complexity. The behavior for larger ionic radii is ζNN just what is expected from Stokes’ law; it increases as the ion size gets larger. On the other hand, the behavior for smaller ions does not seem to follow such simple physics; the friction decreases with increasing ionic radii for the smaller ions except for Li+ . In Sec. 5.4.4 we have attributed such a behavior to the formation of a ‘solventberg’ or a ion-water ‘cluster’. Although the bare Coulomb interaction has been removed (see [91]), the short-range renormalized infrom the expression for ζNN teraction between ion and water still includes attractive interactions of electrostatic character, which is strong enough to form such a cluster for the small ions [91]. The size of solventberg gets larger as the ion size decreases, since the ion-water interaction increases. The case of Li+ can not be an artifact, but is an exception to the general trend, since the results from the MD simulation exhibits qualitatively similar behavior. Our tentative explanation for the puzzling behavior of Li+ is due to its ion size which is too small to make a large cluster with solvent molecules. In fact, the experimental estimates reported for the hydration number of Li+ range from 4 to 6. Another remarkable observation in Fig. 5.20 is the non-negligible contribution from the cross term ζNZ , which has been neglected in most of the earlier studies. However, our results unambiguously demonstrate that it is not only non-negligible, but as large as the other contributions. One interesting point is that for small ions ζNZ behaves as if it cancels the Stokes part of the friction, and consequently the total friction for small ions in water seems to be to a large extent determined by its dielectric component, namely, ζ ≈ ζZZ .

6.

Concluding remarks

In this chapter we have described a theory for dynamics of polyatomic fluids based on the memory-function formalism and on the interactionsite representation of molecular liquids. Approximation schemes for memory functions appearing in the generalized Langevin equation have been developed by assuming an exponential form for memory functions and by employing the mode-coupling approach. Numerical results were presented for longitudinal current spectra of a model diatomic liquid and water, and it has been discussed how the results can be interpreted in

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S.-H. Chong and F. Hirata, J. Chem. Phys. 111, 3095 (1999). H. S. Frank and W.-Y. Wen, Discuss. Faraday Soc. 44, 133 (1957). O. Y. Samoilov, Discuss. Faraday Soc. 44, 141 (1957). S.-H. Chong and F. Hirata, Chem. Phys. Lett. 293, 119 (1998). R. Akiyama, M. Kinoshita, and F. Hirata, Chem. Phys. Lett. 305, 251 (1999). S.-H. Chong and F. Hirata, J. Phys. Chem. B 101, 3209 (1997). L. Sj¨ ogren, J. Phys. C 13, 705 (1980). S.-H. Chong and F. Hirata, J. Chem. Phys. 111, 3654 (1999). W. M. Latimer, K. S. Pitzer, and C. M. Slansky, J. Chem. Phys. 7, 108 (1939). F. Hirata, P. Redfern, and R. M. Levy, Int. J. Quantum Chem. 15, 179 (1988).

INDEX

anion 29-31, 40, 78, 135-137, 140-142, 145, 155, 156, 161, 181, 192, 193, 195, 322, 340-344 associating fluids 46, 54 atomic fluid 3, 47 auto-ionization of water 78

α-helix 120, 122, 123, 129, 133, 134 β-lactoglobulin 122 β-sheet 122, 134 3D-RISM integral equation 174, 175, 183, 189, 202, 203, 206, 230 ab initio molecular orbital (MO) 6163, 65, 70, 72, 73, 84, 91, 92, 94, 170, 174, 251, 252, 261-263 ab initio MO/3D-RISM-KH theory -- carbon monoxide in aqueous solution 256-258 -- effective potentials 254 -- Fock operator of a solvated molecule 253 -- transformation from the Gaussian basis to plane waves 255 acceleration of convergence -- DR method 266-267 -- initial vectors 269 -- Kohn-Sham residuals 269 -- MDIIS method 264-266 -- RISM residuals 267-268 acetylglycine ethyl ester (AGE) 136 acid strength 77 acid-base equilibria 62, 75, 81 acoustic mode 277, 295, 308, 311, 314, 326, 332-334 activation free energy 32, 86, 88 activity coefficient 15, 136 alcohol 102, 122-124, 126-129, 131135, 157, 171, 174, 175, 179 all-atom model 102, 187 amino acid 147, 149-151, 153, 154, 171 amino-acid sequence 101 amyloid 122

backbone 112, 117, 126, 130, 131, 188 basicity 75, 76, 80, 81 basis function 92, 159, 261, 262 bent hydrogen-bond model 23 biomolecule 101-106, 135, 162 Boltzmann constant 6, 104, 157 Boltzmann distribution 4 Boltzmann equation 285-287 Boltzmann factor 106, 107, 116, 182, 184, 221 bond length 19, 181, 295, 306, 345 Boyle temperature 47 bridge function 7, 8, 50, 120, 184, 203, 205 Carnahan-Starling approximation 47 cation 40, 136, 140-142, 155, 161, 192-194 central limiting theorem 31, 87 chain sum 11-13, 17, 18, 20 chain-rule theorem 7 charge distribution 2, 31, 36, 67, 69, 70, 87, 208, 243, 342 charge transfer 87, 251, 277 chemical bonds 2, 84, 251 chemical potential 6, 21, 22, 26, 27, 34, 36, 46, 50, 51, 89, 94, 180, 181, 183-189, 202, 203, 206-209, 222, 230, 231, 233, 239, 240, 243, 252, 257-260

351

352 chemical reactions 2, 28, 29, 61, 63, 72, 74, 79, 83, 85, 86, 92, 147, 345 chemical shift 2, 62, 88, 96 closure approximation -- 3D-BPGG 181 -- 3D-HNC 178, 180, 182, 190, 202, 205, 209, 211 -- 3D-KH 203, 205, 209, 211 -- 3D-HNC-RBC for hydrophobic hydration 183 -- site-site HNC-RBC 187 -- site-site KH 204 closures 7-9, 19, 50, 107, 119, 138, 178-182, 184, 191, 205, 207, 222, 226, 239 cluster expansion 12 coarse variable 158, 159, 162 coexistence curve 48, 51, 53-55, 179 combination rule 109, 110, 192 configurations 3, 4, 12, 22, 24, 26, 28, 31, 62, 73, 104, 108, 142, 197 conformation 80, 81, 101, 102, 104108, 110-126, 128-134, 136, 146, 147, 163, 164, 188-190, 193 conformational change 80, 81, 108, 111, 114, 124, 133-135, 149 conformational energy 104, 106, 107, 11, 124, 133-135, 145 conformational entropy 106, 107, 117 conformational equilibrium 81 conformational sampling 102, 111113, 163, 164 conformational stability 101, 102, 107, 109, 111, 113, 114, 120, 122, 123, 126, 129, 132, 135, 145, 188 constraint 19, 73, 92, 148, 149, 153, 155, 163, 181, 215, 220, 221, 226, 228, 263 continuum dielectrics 12

continuum model 28-30, 37, 62, 65-67, 69-72, 91, 107, 122, 123, 135, 174, 200, 212, 315, 316 convolution integral 11, 14, 17-20 coordination number 23, 24, 126, 128, 131, 132, 187, 238, 256 correction for the 3D-FFT supercell artifacts 191, 231 correlation function 1, 3, 5, 6, 8, 10-13, 15-17, 19-26, 28, 36, 45, 49, 50, 72-74, 77, 95, 110, 145, 147-149, 151, 152, 155-157, 160164, 169, 170 Coulomb potential 108, 157, 162, 164, 232, 252, 253, 266 Coulombic 8, 64, 73, 95, 114, 122, 123, 3224, 325 C-peptide 113-116, 120, 121, 123, 129-133 critical phenomena 8 Crout's method 162 C-terminus 162 de Broglie thermal wavelength 46 Debye screening constant 14, 15 Debye-Huckel limiting law 15 denaturated structure 145, 146 denature 1 density 1, 3-6, 10, 11, 14, 20, 21, 26, 28, 29, 31, 42-49, 52-55, 72, 80, 85, 86, 89, 92, 97, 103, 110, 114, 122, 123, 126-128, 131, 133, 172, 178-181, 183, 187, 207, 222, 226, 231, 242, 243, 248, 253, 258, 278, 282, 290, 292, 296, 297, 299, 304, 306, 307, 313, 310, 321, 323, 327, 329 -- field 3-5 -- fluctuation 3-6, 277, 284, 288, 311, 314, 332, 334 -- functional theory 6

353 density-density pair correlation function 3 dielectric constant 12, 29, 36, 66, 122, 123, 136, 149, 156, 191, 192, 199, 203-205, 232, 234, 334 dielectric continuum model 65, 66, 71, 91, 200 dielectric friction 2, 296, 315, 316, 318, 322, 328, 330, 331, 343, 345 dielectrically consistent RISM integral equation (DRISM) 203-205 Diels-Alder reaction 83, 85 diffusion constant 39, 341, 342 dimethyl sulfoxide 50, 82 Dirac delta-function 3, 4, 12, 153, 155 direct correlation function 6-8, 1013, 17, 19, 20, 49, 50, 73, 110, 147, 148, 155, 161, 173, 175, 177, 181, 182, 202, 206, 207, 226, 266, 286, 317, 318, 324 dissociation 77, 78, 151, 211, 216 dynamic Stokes shift 2 dynamic structure factor 281, 297, 306 electrolyte solution 2, 12, 15, 202 electronic reorganization 34, 79 electronic structure 2, 35, 61, 62, 67, 71, 72, 74, 83, 88, 91, 94, 97, 170, 714, 175, 192, 240, 249, 251-253, 262 electrostatic field 28, 29, 31, 33, 66, 212, 217, 235, 315, 338 electrostatic interaction 2, 45, 66, 71, 73, 110, 111, 114, 124, 134, 138, 139, 145, 151, 171, 191, 192, 199, 228, 341 electrostatics 2, 29, 175, 206 electrostriction 149, 151 energy gradient 67, 74, 96

energy surface 22, 36, 37, 74, 87, 116 ensemble average 4, 7, 103, 104, 106, 108, 279, 320 equation of state (EOS) 47, 48 -- by the RISM/KH theory 50-51 -- Carnahan-Starling-RedlichKwong 47 -- van der Waals 46 ethanol 123, 124, 126-129, 132134 excess free energy 27, 105, 106, 202 excluded volume 8, 46, 53, 116, 119, 153, 155, 182, 226, 229 extended RISM theory (XRISM) 2, 36, 48 extended simple point charge (SPC/E) model 109, 136, 149, 208, 253, 331 external field 6, 9, 319 fine variable 158, 159, 162 Fourier transform 14, 15, 17, 148, 155, 157, 159, 161, 170, 176, 177, 188, 229, 261, 265, 266, 281, 285, 297, 302, 319 Frank-Wen model 38 free energy functional 10, 96, 240 free energy in the 3D-RISM/KH approximation 257 frequency moment 282, 283, 291, 300, 303, 332 friction coefficient 296, 314, 315, 317, 318, 321, 322, 324, 327-330, 338, 339, 342-345 functional differentiations 6, 243, 252 functional Taylor series 10 generalized ensemble 103, 106, 120

354 generalized kinetic theory 284-287, 290, 292, 294 generalized Langevin equation 296, 344 geometry 2, 16, 19, 51, 74, 75, 82, 192, 222, 249, 295, 345 Gibbs free energy 26, 27 grand partition function 6 hard sphere interaction 12 hypernetted-chain (HNC) 8, 9, 11, 13, 19, 20, 24, 48-50, 52, 96, 157, 161, 173, 178-181, 184, 189, 193, 204, 205, 226, 230, 252, 260 -- closure 8, 19, 22, 36, 48-50, 94, 95, 107, 119, 138, 157, 170, 178, 179, 181, 182, 187, 190, 191, 203, 204, 228, 258, 265 Hofmeister series 135, 136, 138, 140, 145 hybrid algorithm 158, 160-164, 188 hybrid approach 62, 90, 102, 107109, 11, 114, 116, 117, 121 hydration of -- ion pairs 214-220 -- met-enkephalin 189 -- rare gases and alkanes 185 -- simple ions 238, 239 -- water molecule 212, 214, 217, 248 hydrocarbon group 122, 124, 127129, 131, 133, 134, 187 hydrodynamic friction 328 hydrogen bond 23-25, 45, 52, 53, 77, 84, 116, 118, 119, 123, 129, 132-135, 215, 220, 234, 235 hydrogen fluoride 50, 51, 53-55, 77, 179 hydrogen-bond network 23-25, 45, 53, 342 hydrophilic atom 108, 116, 121, 126 hydrophobic atom 108, 116, 121, 126, 128, 129, 131, 132

hydrophobic hydration 114, 138, 145, 182, 184, 187, 188, 226 hydrophobic solute 45, 119, 184, 235 hypervertices function 12, 13 ice-like tetrahedral coordination 23, 45 ideal fluctuation volume 149, 153 interaction site 2, 16, 51, 92, 93, 95, 102, 109, 110, 119, 149, 155, 161, 164, 169, 176, 193, 226, 230, 252, 255, 277, 294-297, 304, 345 interaction site model (ISM) 16, 51, 193, 294-296 interfacial liquids 2 intermediate scattering function 281, 282, 290, 291, 294, 297, 299, 302, 305, 317, 318, 322, 323, 339 intermolecular potential energy 6 interstitial model 23 intramolecular correlation 2, 19, 95, 97, 149, 155, 161, 163, 177 intramolecular correlation functions 19, 95, 163 ion hydration 28, 38-41, 145, 338, 342 ionization 111, 151 isothermal compressibility 28, 49, 52, 148 Jacobian matrix 113, 118, 158160, 162-164, 261, 263 KH 9, 49, 50, 52, 178-180, 191, 204, 228, 257, 258, 260 Kirkwood coupling parameter 21, 22 Kirkwood-Buff theory 147 Kohn-Sham DFT/3D-RISM-KH theory 240 -- procedure of convergence 244

355 -- free energy functional 244, 247 -- Kohn-Sham equation 244, 247 -- effective potentials 242-245, 251 -- metal-water interface 246-253 Kronecker δ-function 5 K w 79 Lennard-Jones interactions 12, 109, 331 Lennard-Jones potential 47, 159 liquid dynamics 278, 280, 295, 296 liquid structure 4, 23, 72, 77, 172, 265, 342 liquid-gas phase transition 46 liquid-vapor coexistence curve of -- dimethylsulfoxide 50 -- hydrogen fluoride 50 -- methanol 50, 52 -- water 50, 52 local activity 6 longitudinal current correlation function 297 longitudinal current spectrum 306, 310-314, 326, 327, 335-337, 344 lower triangular and upper triangular (LU) decomposition 162, 164 Marcus theory 37 Maxwell construction 46, 50, 51 mean spherical approximation (MSA) 8, 49-51, 171, 172, 178-180 -- closure 9, 49, 50, 171, 180 mechanical instability 51 melittin 122 memory function 279, 280, 284, 287-290, 293-296, 300-302, 317, 338, 339, 345 -- formalism 278, 279, 283, 284, 299, 344 metastable states 51 Met-enkephalin 189

methanol 48, 50-55, 122-124, 126-134, 179 mixture model 23, 24 mode-coupling theory 290, 292, 302, 304, 314, 316, 338 molecular dynamics 3, 24, 61, 62, 71, 108, 171, 172, 186, 209, 244, 249, 262, 277, 284, 292, 295, 302, 314, 331, 335-337, 340-342, 344, 345 molecular liquids 1, 2, 20, 22, 24, 169, 171, 173, 174, 204, 226, 229, 277, 278, 283, 294-296, 300, 302304, 344 molecular orbital (MO) 61, 89, 249 molecular polarization of solvent 36 Monte-Carlo (MC) 24, 62, 71, 101-103, 106-108, 111, 112, 114, 116, 121, 163, 164, 172, 188 -- simulated annealing 102 MSOZ integral equation 230 multicanonical 102, 103, 116-118 native structure 122, 135, 145 negative hydration 39, 41, 45, 142, 145, 337 neutron diffraction 23 Newton Raphson method 110, 158, 164, 172, 261 NMR chemical shift 2, 62, 88, 90 non-equilibrium free energy 31, 32, 34, 36, 37 nonlinear simultaneous equations 107, 158 N-terminus 104, 118, 151, 152 number density 55, 110, 123, 126128, 131, 133, 136, 137, 149, 155, 156, 161, 175, 202, 204, 245, 280, 281, 307, 318, 323, 331

356 OPLS-UA force field 51 optical mode 277, 295, 296, 308313, 316, 321, 326, 331, 333, 334, 345 Ornstein-Zernike (OZ) equation 2, 6, 7, 119, 155, 230 P - V isotherm 46 pair correlation function 1, 3, 6, 16, 22-26, 28, 36, 72, 77, 95, 145, 147, 148, 151, 152, 319, 338 partial charge 72, 92, 109, 126, 131, 138, 141, 149, 151, 192, 230, 252, 307, 336, 342 partial molar volume 2, 119, 147151, 153-155 Pauling’s crystal radii 29 peptide 101, 102, 106-117, 119-121, 126-130, 132-136, 138-147, 163, 164, 190 Percus' trick 9 phase diagram 8, 9, 48, 52 Picard iteration 110, 157, 160, 163, 164, 261, 264, 340 pKw 79, 80 Poisson equation 244, 253 polarization 31-36, 62, 69, 71, 76, 80, 140, 170, 208, 243, 245, 247, 253, 315, 321, 323, 327 polyatomic molecules 5, 15, 50 positive hydration 41, 45, 142, 145 potential energy 6, 9, 81, 93, 104, 108, 109, 113, 116, 188, 189, 240, 252 potential of mean force 40, 81, 102, 190, 198, 200-202, 212, 213, 232 -- in solution -- -- between molecular ions in acetonitrile 198 -- -- between simple ions in water 209

-- -- correction for the 3D-FFT supercell artifacts 191, 209 -- -- orientationally dependent from 3D-RISM/HNC 192 pressure in the 3D-RISM/KH approximation 260 pressure isotherms of water 51 pressure-induced denaturation 147 primitive model 12, 65, 209 product 3, 14, 19, 20, 32, 35, 36, 61, 74, 78, 80, 84-87, 91, 263, 267, 279, 281, 291-293, 297, 304 projection operator 279, 289, 293, 299 protein 1, 16, 22, 29, 81, 88, 101, 102, 107, 108, 116, 120-122, 135, 136, 145-147, 149, 151, 154, 155, 164, 171, 188, 190 -- folding 2, 101, 102, 122 PY 8, 10, 11, 13, 19, 20, 50, 171, 179, 180 QM/MM method 62, 71 , 72 quantum tunneling 32 radial distribution function 8, 44, 47, 72, 77, 90, 94, 203, 212, 228, 238, 280 reactant 32, 35, 36, 74, 84-86, 91 reaction coordinate 22, 32, 34-36, 40, 74, 84, 86, 87 redox pair 31, 32, 34, 36, 87 renormalization 2, 11, 12, 157, 162, 265, 266 reorganizaton energy 32, 34, 35, 74, 79, 231 replica exchange 102, 103, 121 repulsive potential 109-111, 131, 183, 232 residence time 41

357 residue 109, 113, 121, 129, 131, 132, 149-151 RHNC 24, 53, 54, 172, 173 RISM theory 2, 15, 17-20, 22, 24, 27, 28, 30, 31, 36, 37, 44, 48-55, 62, 72, 87, 89, 91, 92, 94-96, 101, 102, 106-108, 110, 111, 113, 114, 116121, 123, 135, 136, 138, 146-149, 153-155, 158, 163, 164, 169, 170, 174-196, 198, 200, 202-213, 215, 216, 220-223, 226-240, 243-245, 249, 251-253, 255, 257-262, 266, 298, 318 RISM-SCF 35, 62, 72, 75, 79, 82-85, 88-94, 96, 174, 251, 252 RISM-SCF/MCSCF 62, 72, 75, 91, 92, 94 roof basis function 159 root-mean-square distance 112, 117 salt 109, 111, 121, 135, 138-141, 143-146, 155-158, 160-164, 186, 209, 221, 222 salting-out 136-139 secondary structure 122 self-consistent 3D-RISM integral equation -- corrections for 3D-FFT supercell artifacts 231 -- solute-solvent 239 -- solvent-solvent 239 side chain 112-115, 126, 130-132 simple liquids 1, 6, 22, 24, 26, 28, 47, 181, 277, 278, 282-284, 296, 302, 306 Singer-Chandler formula 36, 110 site charge 110, 111, 116, 124, 126128, 130-132, 134, 138-140, 149, 160, 163, 184, 191, 192, 205, 232, 233, 245, 252, 266, 267 site-site direct correlation functions 17, 110, 148, 181, 298, 317, 318, 324

site-site Ornstein-Zernike (SSOZ) equation 155, 157, 161 SN2 reaction 22, 83-85 solubility 86, 135, 137, 145, 146 solute 20-22, 31, 33, 34, 36, 37, 45, 62, 65-67, 69-77, 79, 82, 83, 87, 90-96, 119, 120, 122, 123, 135, 137, 138, 140, 147-149, 153155, 160-164, 169-171, 174-176, 181, 182, 184-193, 197, 199, 202, 203, 205-208, 212, 221, 226, 231, 239, 240, 251-255, 257, 266, 267, 277, 296, 314-322, 325-332, 339, 341, 342, 345 solute-solvent interaction 20, 61, 71, 74, 94, 95, 151, 180, 183 solvation 2, 20, 21, 32, 37, 50, 62, 63, 67, 69, 71, 74, 76, 78, 79, 8185, 90, 91, 122, 135, 136, 141, 145, 169, 171, 174, 175, 180, 181, 183, 184, 186, 192, 196, 202, 203, 206-209, 222, 225, 226, 230, 231, 233-235, 238, 239, 243, 252, 258260, 277, 314, 316, 329, 331 solvation chemical potential -- 3D-RISM/KH 180, 257 -- 3D-RISM/HNC-RBC-TPT 183 -- in the 3D-RISM/HNC approximation 180 -- of an ionic cluster in electrolyte solution 202 -- of simple ions 238, 239 -- SC-3D-RISM/HNC 180, 228, 229, 232, 234-239 -- SC-3D-RISM/KH 180 -- site-site RISM/HNC-RBC-TPT 188 -- thermodynamic decomposition 211, 216 solvation free energy 21, 29, 30, 34, 36, 66, 68, 70, 73, 74, 76, 78, 79, 87, 91, 105-107, 110, 113,

358 114, 119-122, 124, 129, 131-139, 145, 162, 202, 206, 260 solvent 20-22, 29-37, 39-41, 44, 45, 61, 62, 65-67, 71-78, 80, 82-83, 8588, 90, 91, 94-97, 102-111, 117, 119124, 126-128, 130-132, 134, 135, 140-142, 147-149, 151-154, 157, 163, 169-178, 180-184, 188, 190196, 198-200, 202, 203, 205-209, 211, 212, 217, 221, 222, 226-235, 239-245, 247, 248, 251-253, 257260, 263, 265-267, 277, 296, 314324, 326-331, 338, 340-342, 344, 345 solvent coordinate 34 solvent fluctuation 31, 34, 37, 87 solvent reorganization energy 32, 35, 231 solvent-modified potential 104, 106 solventberg model 314, 315, 337 solvophilic atom 110, 133, 134 solvophobic atom 110, 133, 134 SPC 31, 51, 52, 109, 234, 245 SPC/E 109, 110, 113, 123, 136, 149, 208, 239, 253, 331, 332 spherical invariant 15, 16 spinodal line 51 square-root law 15 steric hindrance 127, 131, 134 Stokes friction 315, 316 structure breaking 37, 39, 337 Tautomerization, 82 tertiary structure 101 thermal average 3-5 thermal fluctuation 4, 31 thermal motion 3, 34 thermodynamic perturbation theory 47, 120, 183 TIPS 31, 245 torsion energy 104, 110, 114

total correlation function 6, 73, 95, 110, 155, 161, 175, 184, 202, 205-207, 244, 252, 266 total energy 62, 63, 71, 74, 75, 106, 107, 111, 112, 115, 121, 132, 145, 188, 189, 242 transition state 32, 35, 41, 74, 85 translational invariance 5, 8, 293 transmission coefficient 32 transversal current correlation function 281 transversal current spectrum 282 trifluoroethanol (TFE) 123, 134 van der Waals (vdW) theory 46 van der Waals-Maxwell theory 47 van Hove correlation function 280, 281, 295 velocity autocorrelation function 317, 338-340 viscoelastic model 283, 284, 292 viscosity 39 water 1, 2, 23-26, 28, 31, 37, 3945, 48-55, 77, 78, 84-86, 90, 91, 103, 108-117, 119-124, 126-137, 139-146, 149, 151, 152, 155-158, 160-164, 170-175, 178-190, 192, 208, 209, 212-217, 220-222, 225, 226, 234-239, 243, 245, 247-251, 253-256, 295, 299, 302, 312, 314, 331, 332, 337-344 X-ray diffraction method 23 zwitterion 150

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