Modern methods for multidimensional dynamics computations in chemistry 9789812812162, 9812812164

Table of contents :
Computational methods for polyatomic bimolecular reactions / George C. Schatz, Marc Ter Horst, and Toshiyuki Takayanagi --
Nonadiabatic dynamics / John C. Tally --
Methods for gas-surface scattering / Bret Jackson --
Molecular dynamics methods for studying liquid interfacial phenomena / I. Benjamin --
Direct dynamics simulations of reactive systems / Kim Bolton, William L. Hose, and Gilles H. Peslherbe --
Mapping multidimensional intramolecular dynamics using frequency analysis / Jan von Milczewski and T. Uzer --
Quantum generalized Langevin equation approach to multidimensional dynamics / H. Keith McDowell --
Quantum molecular dynamics simulations of processes in large clusters: methods and applications / R. B. Gerber, P. Jungwirth, E. Fredj, and A. Y. Rom --
Theoretical investigations of chemical and physical processes under matrix isolation conditions / Lionel M. Raff --
Macromolecular dynamics / R. V. Stanton, J. L. Miller, and P. A. Kollman --
Molecular dynamics simulations of carbohydrate solvation / J. W. Brady --
Computational simulation and modeling of molecular-based materials / Bobby G. Sumpter, Robert E. Tuzun, and Donald W. Noid --
Molecular simulation of detonation / Betsy M. Rice --
Monte Carlo methods in chemistry: a tutorial / J. D. Doll and David L. Freeman --
Monte Carlo methods for rate processes / Alison J. Marks --
Testing the accuracy of practical semiclassical methods: variational transition state theory with optimized multidimensional tunneling / Thomas C. Allison and Donald G. Truhlar --
A multidimensional semiclassical approach for treating tunneling within classical trajectory simulations / Yin Guo and Donald L. Thompson.

Citation preview

Modern Methods for Multidimensional Dynamics Computations in Chemistry

Editor

Donald L. Thompson

World Scientific

Modern Methods for Multidimensional Dynamics Computations in Chemistry

This page is intentionally left blank

Modern Methods for Multidimensional Dynamics Computations in Chemistry

Editor

Donald L. Thompson Oklahoma State University

TM$ World Scientific Singapore 'New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

MODERN METHODS FOR MULTIDIMENSIONAL DYNAMICS COMPUTATIONS IN CHEMISTRY Copyright © 1998 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof may not be reproduced in anyform or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-3342-6

Printed in Singapore.

CONTENTS vii

Preface Computational Methods for Polyatomic Bimolecular Reactions George C. Schatz, Marc TerHorst, and Toshiyuki Takayanagi

1

Nonadiabatic Dynamics John C. Tally

34

Methods for Gas-Surface Scattering Bret Jackson

73

Molecular Dynamics Methods for Studying Liquid Interfacial Phenomena /. Benjamin

101

Direct Dynamics Simulations of Reactive Systems Kim Bolton, William L. Hose, and Gilles H. Peslherbe

143

Mapping Multidimensional Intramolecular Dynamics Using Frequency Analysis Jan von Milczewski and T. Uzer

190

Quantum Generalized Langevin Equation Approach to Multidimensional Dynamics H. Keith McDowell

201

Quantum Molecular Dynamics Simulations of Processes in Large Clusters: Methods and Applications R. B. Gerber, P. Jungwirth, E. Fred], and A. Y. Rom

238

Theoretical Investigations of Chemical and Physical Processes Under Matrix Isolation Conditions Lionel M. Raff

266

Macromolecular Dynamics R. V. Stanton, J. L. Miller, and P. A. Kollman v

355

VI

Molecular Dynamics Simulations of Carbohydrate Solvation J. W. Brady

384

Computational Simulation and Modeling of Molecular-Based Materials Bobby G. Sumpter, Robert E. Tuzun, and Donald W. Noid

401

Molecular Simulation of Detonation Betsy M. Rice

472

Monte Carlo Methods in Chemistry: A Tutorial J. D. Doll and David L. Freeman

529

Monte Carlo Methods for Rate Processes Alison J. Marks

580

Testing the Accuracy of Practical Semiclassical Methods: Variational Transition State Theory with Optimized Multidimensional Tunneling Thomas C. Allison and Donald G. Truhlar

618

A Multidimensional Semiclassical Approach for Treating Tunneling Within Classical Trajectory Simulations Yin Guo and Donald L. Thompson

713

PREFACE An interesting and challenging aspect of chemical processes is that they usually involve the collective behavior of a large number of degrees of free­ dom. The task for the theoretician is to describe these processes, in all their dynamical complexity, in molecules, liquids, and solids. One of the denning characteristics of theoretical chemistry over the past few decades has been the rapid progress in the development of methods for treating the dynamics of complex processes in multidimensional systems. This volume is a collection of some of the key theoretical techniques that have resulted. The selection of topics reflects the fact that currently all manner of multidi­ mensional dynamics are being actively explored, that practical computational methods are now available for almost all areas of chemistry, and that innova­ tive approaches are being explored to further extend their applicability. There is such a large number of methods and applications that fall within the range of the title of this volume that it is not possible to thoroughly cover the en­ tire spectrum of topics in a single volume. However, I hope that my choice of topics, though arbitrary, is sufficiently wide to provide the reader with a good overview of the theoretical methods available for treating chemistry problems which involve many degrees of freedom. The choices were made with beginning students, mature researchers in other fields, and experts in chemical dynamics all in mind; the chapters cover both basic theory and practical methods, estab­ lished and evolving methods, and esoteric and general subjects. The methods described in this volume are based on classical, semiclassical, and quantum me­ chanics for applications that span the range from reactions of isolated, small molecules to complex processes in the condensed phases. The goal was to produce a volume that would enjoy more than the brief, transitory "life" that can be the fate of such collections of chapters and that would, at the same time, contribute to the further evolution of the field since there is still much to be done.

Donald L. Thompson

vn

COMPUTATIONAL METHODS FOR POLYATOMIC BIMOLECULAR REACTIONS

GEORGE C. SCHATZ, MARC TER HORST Department of Chemistry, Northwestern

University, Evanston IL 60208-3113

USA

TOSHIYUKITAKAYANAGI Advanced Science Research Center, Japan Atomic Energy Research Institute,

Tokai-mura,

Naka-gun, Ibaraki 319-11, Japan

This paper describes computational methods that are commonly used to describe bimolecular reaction dynamics at the state-to-state level for reactions involving four or more atoms. Only purely quantum and quasiclassical methods are considered, with the bulk of the discussion pertaining to classical methods for defining state-resolved .processes in polyatomic molecules and to reduced dimensionality quantum coupled-channel methods based on hyperspherical coordinates. For each of these methods, we give a detailed description of the theory and involved, and numerical procedures that are used in making applications. The methods are illustrated through an application to the reaction CN + H2 - HCN + H, including comparisons of classical and quantum thermal rate constants, product state distributions, and average product quantum numbers.

1 Introduction This chapter is concerned with theoretical methods that have been developed to describe gas phase bimolecular chemical reactions in which at least four atoms are involved. A commonly studied example is the reaction between two diatomics to give an atom plus a triatomic, i.e., AB + C D - A + BCD

(1)

and we shall emphasize this class of problems in much of our development. After significant early interest in reactions involving three atoms,' the field of reactive scattering has moved in recent years to consider more complicated reactions, and the purpose of this article is to discuss the methods that are commonly used for this purpose. Our development will be restricted to two kinds of methods, the quasiclassical trajectory (QCT) method, and time independent quantum scattering (TIQS) methods.

1

2

Other methods, such as semiclassical methods and time dependent quantum scattering methods are described in other chapters in this volume. We also confine our discussion to reactions that proceed on a single Bom-Oppenheimer potential surface. Most of our development will consider methods that are capable of providing a full state-to-state description of the reaction dynamics, so we do not consider methods that directly determine only averaged information like the thermal rate coefficient or cumulative reaction probability. At the end of this article we illustrate the methods we discuss by presenting results for the reaction CN + H2 - HCN + H. This one of the most thoroughly studied reactions at the state-to-state level,2"" and it is possible to present detailed comparisons between QCT and many different kinds of TIQS methods for this reaction, all for the same potential surface. There are, however, several other reactions that have been treated with a comparable level of theory, as recently reviewed by Bowman and Schatz,12 and by Schatz.13 Most of these are diatom + diatom - atom + triatom reactions like OH + H2, OH + CO, NH + NO, 0 2 + 0 2 , Hj+ + H2 and OH + HC1, but there have also been a few comparable studies of other types of reactions such as C2H + H2 and CH3 + H2. One of the challenges associated with studying the bimolecular reactions just mentioned is that there is significant variation in the types of dynamical processes which can occur during reaction. OH + Hj, for example, is a direct exothermic reaction involving a simple barrier, while OH + CO involves the formation of a long lived complex, and there are barriers on both sides of the complex. NH + NO also involves a complex, and this can decay to two different kinds of products, one of which involves breaking both of the reagent diatomic bonds. Although most reactions have been considered at energies where breakup to produce three or morefragmentscannot occur, the reaction H2+ + H2 has been the subject of very extensive studies of breakup dynamics,14 and this occurs in parallel with atom transfer, proton transfer and 4-center exchange. The only dynamics method of the ones that we consider that can routinely describe all possible bimolecular rearrangements is the QCT method, so in many respects this is the "work horse" of the reaction dynamics field. TTQS methods, by contrast, are more limited due to computational complexity. As a result, the TIQS applications have always been implemented with one or more dynamical approximations, aimed at reducing the dimension of the active degrees of freedom to be described in the reaction dynamics. Approximations to the inclusion of angular momentum coupling in the TIQS calculation are also included to reduce computational effort, but as a result of these approximations it is often not clear how accurate are the TIQS results. One way to find out involves comparisons with QCT results, although these comparisons are subject to uncertainty due to the unknown influence of quantum effects on the QCT results. Comparisons with experiment can also be used, but these comparisons are subject to uncertainty due to

3

errors in the potential surfaces that are always present. A third approach is to examine convergence of the reduced dimensionality results as the dimension is gradually increased. We will consider all of these approaches in the results presented in Section 4. Here is a summary of the rest of this paper. Section 2 considers the application of classical trajectory methods to the description of state-resolved bimolecular reactions involving 4 or more atoms. The corresponding TIQS treatment is given in Section 3, with emphasis on reduced dimensionality methods with 3 or 4 degrees of freedom. Our applications of both QCT and TIQS methods to reaction (1) are presented in Section 4. 2 Classical Trajectory Methods 2.1 General Description of the Quasiclassical Trajectory Method Classical trajectory methods and particularly the quasiclassical trajectory method have played a dominant role in the theoretical description of bimolecular reactions from the beginning of the field. This method has been well described for atom-diatom reactions,15 but its more general implementation is still incomplete. The well known code VENUS'6 provides a fairly general implementation that allows one to describe bimolecular reactions between molecules of arbitrary complexity, but its treatment of the initial vibrational states of the reagent molecules is limited to harmonic normal modes for molecules larger than diatomics, and there is no analysis of product states. Schatz17 has previously reviewed trajectory methods for describing state-to-state dynamics in polyatomic molecules, but the methods for determining vibrational actions that are described in this review have now been superseded. The present discussion is in many respects an update to this earlier article. A related discussion, but from a different perspective, has previously been given by Schatz.18 The basic idea underlying a QCT calculation is relatively simple. One wants to integrate the classical equations of motion simulating bimolecular collisions, with the reagent molecules initially taken to mimic whatever quantum states are of interest, and with appropriate averaging over impact parameters, orientations, and vibrational phases. Also, if product state-resolved information is of interest, then the vibration/rotation action variables associated with each product molecule need to be calculated. These action variables are the classical analogs of quantum numbers, and they can be used to define quantum states by a variety of algorithms, of which the simplest involves rounding the action off to the nearest integer (in units of h). This will be discussed further below. The classical equations of motion for the collision are most conveniently integrated using space-fixed Cartesian coordinates. Thus if rio denotes the ath component

4

(a = x,y,z) of a vector r, which locates the ith atom (i = 1 N) relative to a space fixed coordinate system, then Hamilton's equations of motion are:

dt dpig dt

mi =

_5V " dria

i = 1,2,...,N, a = x,y,z (2) i = 1,2,...,N,

a = x,y,z

Here p„ is the momentum conjugate to r^, rr^ is the mass of atom i, and V is the complete potential surface describing all the atoms in the reacting system. Eq. 2 defines a system of 6N coupled equations that can be solved by many different methods as has been discussed elsewhere.19 Motion of the center of mass, or of overall rotation, are normally not removed from the system as the reduced computational effort (comparing 3N-6 coordinates to 3N coordinates) is not worth the added notational complexity for N greater than about 4. To solve the equations of motion, it is necessary to specify initial conditions (initial values of rio and pia), the procedure for which is described below. These initial conditions usually refer to separated reagents that are moving towards one another with a specified velocity and impact parameter, and with the reagent molecules in specified quantum states. The integration is carried out until product molecules have separated sufficiently that they no longer interact. In many cases it is necessary to do a careful testing of several internuclear distances to determine which products have formed. It is quite easy to define the product relative translational energy using the Cartesian momenta, but other aspects of the product distributions require the determination of the vibrational and rotational action variables for each product. This is the subject of the section 2.4. 2.2 Defining molecular initial and final conditions: Normal coordinates In this section we consider the relationship between the Cartesian coordinates that are used in integrating the classical equations of motion (Eq. (2)) and the action variables that are used to define the initial and/or final states of the molecules that are undergoing bimolecular reaction. We will base this development on normal coordinates, which is the conventional way to describe small amplitude vibrational motions in polyatomic molecules. Other coordinate systems are possible, and are in some cases are essential, but normal coordinates are adequate to describe a large fraction of the chemical reactions of interest to the gas phase kinetics community. In this discussion we assume that we have a molecule with M atoms, and with vectors r, which locate each atom (i = 1,...,M), i.e., the instantaneous configuration. In

5

order to define normal coordinates, it is necessary to define an equilibrium configuration of the atoms; let us denote this using the vectors a,. It is also necessary to locate this equilibrium configuration relative to the instantaneous configuration. This is customarily done using the Eckart condition:20 ^m^xr,) = 0

(3)

The three Cartesian components of this equation can be used to define three Euler angles20 which rotate between the space and body-fixed coordinate systems. The latter coordinate system is assumed to be attached to the equilibrium configuration of the atoms. If the molecule has three atoms, then two of the three Euler angles can be defined by the constraint that the plane associated with the instantaneous configuration of the atoms be the same as that associated with the equilibrium configuration. Even with the Eckart constraint, there is still ambiguity in the coordinate definition, but usually this can be resolved by requiring that the Cartesian components of the r, match those of a, in the limit that the molecule is at equilibrium. Once the Euler angles are defined, the transformation of space-fixed components of the vectors rs to body-fixed components can be accomplished by simple matrix multiplication as described in Ref. 20. Once of the body-fixed frame has been defined then it is possible to define normal coordinates by diagonalizing the 3M x 3M matrix:20

where v is the intramolecular potential, and the matrix of second derivatives is evaluated at the equilibrium configuration. The 3M-6 nonzero eigenvalues associated with M (for a nonlinear molecule) are just the squares of the normal mode frequencies a)k (k = l,...,3M-6). The remaining eigenvalues are zero, corresponding to overall translation and rotation of the molecule. If we label the normal coordinates by Xk and the corresponding eigenvectors by 1^, then the Cartesian to normal coordinate transformation is accomplished using

X ^ E m ^ i 2f c A r ,

(5)

6

where Ar^ is the displacement of r,0 from its equilibrium value a^. Since the matrix M is symmetric, the eigenvectors are orthogonal, so Eq. (5) is readily inverted. This inverse formula is useful in defining trajectory initial conditions once the normal coordinates are determined as further described below. 2.3 Normal Momenta Eq. (5) provides a transformation back and forth between Cartesian and normal coordinates, but to transform the corresponding momenta requires several additional steps that we now describe. Starting from space fixed Cartesian coordinates, one first transforms the time derivatives dr/dt to the body-fixed frame using the Euler angle transformation defined above. Next we need to separate this velocity into one component that arises from overall rotation of the molecule, and another that describes motion internal to the rotating molecule. This decomposition is accomplished using the equation dr.

=

dr,

IT IT

+ uxr

(6)


is determined, Eq. (6) can be evaluated to determine the internal motion velocities, and then the time derivative of Eq. (2), i.e., I

^=£m^% at

i

(10)

at

determines the normal coordinate time derivatives dXk/dt. These time derivatives are not the same as the normal momenta Pk, but rather20

Pk = - ^

+

(20)

where the vibrational angular momentum is defined by Jvib

2 ^ H'k^k°k I*'

(21)

13 In these formulas, the instantaneous moment of inertia is (Dap = £ m i ( r i \ p - VjP>

(22)

i

and the vibrational moment of inertia is

(Wop

=

Y, CkCk»kXk.Xk„ •

(23)

kk'k"

Note that Ck\ is the ath component of the vector Ckk defined by Eq. (12). The least detailed approach to the specification of j , such as would be used for simulating a canonical or microcanonical ensemble of rotational states, is simply to choose the magnitude of j from the desired distribution (say, a Boltzmann distribution) and to choose the direction of j randomly in space. This procedure will work for most molecules, but there are limitations with linear molecules (where j must be perpendicular to the molecular axis). Even for nonlinear molecules, this procedure may run into trouble if the total energy of the molecule is also constrained, as some values and directions of j may lead to a rotational energy that is larger than the desired total energy. The most detail specification of j involves defining the good action associated with motion about the body-fixed z axis, in addition to the magnitude of j . To do this we need to calculate the semiclassical eigenvalue, following one of the procedures outlined in Section 2.4. The root trajectory that defines the appropriate good actions is then sampled to determine coordinates and momenta that have the desired rotational angular momentum. The direction of j will usually be fixed in this trajectory, so to insure random sampling of the space fixed direction in a QCT calculation, it is necessary to rotate all molecular coordinates and momenta using randomly chosen Euler angles prior to simulating the collision. For linear molecules and symmetric tops, there are some simplifications to the sampling of the rotational angular momentum that can be used. The sampling of both vibrational and rotational coordinates for diatomic molecules has been described in some detail (at least for Morse potentials) by Porter, Raff and Miller32, and a related discussion is given by Karplus, Porter and Sharma'5. This treatment can also be used for linear polyatomic molecules, except that one needs to realize that the rotational angular momentum in this treatment does not include the vibrational angular momentum of the

14 diatomic (the angular momentum associated with rotation about the molecular axis). This extra angular momentum is usually small in magnitude compared to the rotational angular momentum, but the energy associated with each unit of angular momentum is large (equal to the bending quantum) so the sampling of vibrational angular momentum (say, in a stateresolved calculation) needs to be done with great care. This issue has been discussed by Schatz.13 The sampling of rotational angular momentum for symmetric top molecules has been discussed by Augustin and Miller.31 3 Time Independent Quantum Scattering Calculations for Polyatomic Reactions 3.1 Introduction In this section we discuss several methods for performing time independent quantum scattering calculations for reactive bimolecular collisions. Most of our discussion will refer to diatom-diatom collisions which result in atom transfer to produce atom-triatom products, but the concepts can easily be generalized to many other kinds of atom transfer reactions. Our discussion will be specifically restricted to reduced dimensionality quantum scattering methods, as these have received the most extensive application to reactions with four or more atoms, but it should be noted that in the last year there have been several reports33,34 of full dimensional (or nearly full dimensional) quantum applications for diatom-diatom reactions There are three important components to each of the methods we consider: 1) Choice of coordinates to be treated actively in achieving a reduced dimensionality calculation 2) Treatment of the inactive degrees of freedom 3) Method for describing reactive collisions with the actively treated coordinates In the following sections we discuss each of these points for several of the commonly used methods. 3.2 Choice of Coordinates and Treatment of Inactive Degrees of Freedom for Reduced Dimensionality Calculations Figure 1 shows the Jacobi coordinates that are commonly used to describe the collision of two diatomics, and which serve as a reasonable reference point for the introduction of reduced dimensionality approximations into reactive collisions. There are six internal coordinates associated with a four atom system, of which one describes out-ofplane motions (the torsion angle T, which is not pictured), and the others describe bends (Y and 5) and stretches (R', r' and z'). The specific reaction we consider is given in Eq.

15

Figure 1.

Jacobi coordinates associated with diatom-diatom collisions.

(1), and involves transfer of the atom B to produce a product triatomic BCD, so this means that the CD bond z' is not broken during reaction. We will denote this as the "spectator" bond, although it should be pointed out that such bonds can and do play an active role in the reaction dynamics in some cases. Let us now use the coordinates in Fig. 1 to introduce four different reduced dimensionality approximations. 3.2.1 RD-AB [Reduced dimensionality with adiabatic bends]. Active variables: R', r' and z' This three degree-of-freedom approximation was originally introduced by Bowman and coworkers35, and involves the active treatment of the stretch coordinates: R , r' and z', and the adiabatic treatment of the bending angle coordinates y, 8 and T. In the adiabatic treatment, the vibrational wavefunctions which describe the inactive degrees of freedom are assumed to evolve adiabatically (i.e., with fixed quantum numbers) while the reaction occurs. This means that the effective potential governing the active variables is the average of the full dimensional potential surface over the quantum state chosen for the

16

inactive variables. The final scattering results are thus labeled by this inactive quantum state, and to generate physically meaningful information it is necessary to average over the inactive quantum numbers, weighted by the distribution of quantum numbers appropriate to the reagents. The RD-AB approximation works most straightforwardly for reactions where the three atoms A,B and C are nearly collinear while reaction occurs, as then R', and r" can be used to describing the bond breaking and formation steps, and the transformation to active product coordinates is decoupled from the inactive coordinates. If the spectator bond z' is only weakly coupled to motion along the reaction path, then the active treatment of this coordinate may not be necessary, in which case a two degree of freedom model is obtained. 2) 3.2.2 RBA [Rotating bond approximation]. Active variables R', r' and 6. This three degree-of-freedom approximation, which has been extensively used by Clary and coworkers36, uses the spectator angle 6, instead of the stretch coordinate z', as an active variable along with R'and r'. The inactive degrees of freedom, y, z' and T, generally are treated in the sudden approximation in applications with this method. This means that they have fixed values in each calculation, and then the final results are averaged over these variables, weighted by a probability distribution that is determined by the asymptotic wavefunction for each degree of freedom. 3.2.3 RD3D [Reduced dimensionality in three dimensions]. Active variables: R', r' and y. This three degree-of-freedom approximation, which comes from Takayanagi et a/8, includes the reactive bond angle y rather than the spectator angle 5 in the active group, along with R' and r'. The remaining degrees of freedom have been treated in both the sudden and adiabatic approximations in applications done to date. RD3D is the ideal approximation for a spectator bond that can be completely ignored in the reaction, for in this method the four atom-dynamics is reduced to an equivalent atom-diatom reaction. The only effect of the spectator bond is in the potential energy surface, where a different surface is obtained for each value of the inactive coordinates in the sudden treatment and for each spectator internal state in the adiabatic treatment. 3.2.4 RBA-4D [Rotating bond approximation in four degrees of freedom]. Active variables: R\ r', z'and 8. This four degree-of-freedom method, which has been used by both Clary" and Takayanagi and Schatz910, adds the spectator bond z'to the list of active coordinates in

17

what is otherwise an RBA calculation. The angles y and T are treated with the sudden approximation. This method has the advantage of letting the spectator diatomic be completely active (for motions within the plane of the four atoms), so any influence of this diatomic on the reaction will be properly described. Another feature is that all three vibrational modes of the product triatomic are treated actively, so product vibrational states are realistic. One can also think of this method as being derived from RD-AB with the addition of the angle 6 to the active variables, and using the sudden approximation rather than adiabatic. From a technological viewpoint, the RD-AB, RBA and RBA-4D methods in the above list are similar in that in all these methods, the non-spectator part of the active variables is just the two degrees of freedom R'and r'. These non-spectator coordinates are distinguished from spectator coordinates by the fact that completely different nonspectator coordinates are needed to describe the separated reagents and products. The RD3D method is qualitatively different in that there are three non-spectator coordinates: R', r' and Y- Adding Y to the non-spectator coordinates leads to a more complex description of the quantum scattering which makes RD3D more difficult to apply than the other methods that have three degrees of freedom. This issue is important in developing other kinds of reduced dimensionality approximations. One can imagine other types of reduced dimensionality approximations that are built out of RD3D in which one or more spectator coordinates are combined with R , r' and Y in the list of active variables. However the added complexity of RD3D without including any spectator coordinates makes the implementation of these higher dimensionality RD3D methods less practical, and so far there have not been reported applications. This point is also significant to the development of reduced dimensionality quantum scattering methods for reactions with more than four atoms. Such treatments will be straightforward provided that there are only two nonspectator coordinates, but if three or more coordinates are needed then the complexity is substantially higher. 3.3 Hamiltonian in Hyperspherical Coordinates In this section we write down the Hamiltonian associated with each of the reduced dimensionality quantum scattering methods described in the previous section. In all cases we begin by mass-scaling the coordinates, thereby converting R', r' and z'into R, r and z according to the general prescription: q=gq'

(24)

18

where q stands for R, r or z, and g is a mass scaling factor that has the general form

g = (-)1

(25)

with m being the reduced mass that is appropriate to the motion of the unsealed coordinate being considered (the AB reduced mass for the coordinate r', the CD reduced mass for z' and the AB-CD reduced mass for R'), and u an arbitrarily chosen mass that becomes the only remaining mass after scaling. Although the value of u is arbitrary, it is useful to give it a value similar to the other masses in the system so that the scaled distances will have familiar values. It is also convenient to choose u so that it is invariant to any rearrangement of the atoms. For a four atom system, an example of this is

(

m A n W

"

D

)* .

(26)

After mass scaling, we use the coordinates R and r to define a hyperspherical radius p using l

p = (R 2 + r 2 ) 2 .

(27>

If we also define a hyperangle 8 using the expressions R = p cos 5 and r = p sin 8, the kinetic energy operator corresponding to the two non-spectator coordinates R and r becomes

™> = lH? * 7sr>'

This is only part of the kinetic energy operator for each of the reduced dimensionality models, but is common to all of these models, and it dictates how the Schrodinger equation is solved. Note that in writing Eq. (28) we have assumed that the scattering

19

wavefunction has been multiplied by a prefactor such as rRp"2 or p3'2 (depending on how centrifugal sudden approximations are introduced) so that the three dimensional radial kinetic energy operators associated with r, R and/or p motion are simplified to their one dimensional counterparts. This also contributes an additional centrifugal potential (in some applications) that we combine with other centrifugal potentials in expressing the final Hamiltonian. Now let us give the rest of each of the Hamiltonians that define the models introduced in the previous section. 3.3.1 RD-AB: The Hamiltonian for this model adds the kinetic energy associated with spectator stretch motions to Eq. (28). In addition, the centrifugal energy of the four atom system is included in the centrifugal sudden approximation. This depends on the total angular momentum quantum number J. The result is: H = T(p,5) - £-JL ♦ - ^ - J ( J + 1 ) ♦ Veff(p,6,z) 2M dz2 2up 2

(29)

where Vcff here stands for an effective potential that is obtained by summing the full dimensional potential evaluated at its minimum with respect to the bending coordinates, and the harmonic bending vibrational energy. The centrifugal potential given here is the simplest of several possible expressions which may be developed. This one is obtained from the rotating line approximation,38 and it has the feature that the centrifugal energy is independent of the hyperangle 6. This simplifies the asymptotic (p-x) analysis. 3.3.2 RBA: In the RBA, one adds spectator rotation to Eq. (28) using the rotational angular momentum operator j . The four-atom centrifugal energy, which is developed from the rotating line approximation,38 contains a term that depends on j and so does the 3-atom centrifugal energy involving the motion of atom B (using the notation of Eq. (1)) relative to the center of mass of CD. If we also include the centrifugal energy associated with the body-fixed z-axis projection of the total angular momentum Q (also the same projection associated with rotation of the product triatomic), we get36 2

H = T(p,5) ♦ - ^V- j 2Mz2

2

♦

2 2p.BCDRBCD(p,6,6)

-(j 2 -Q 2 )

2

+ _— i _[[Jj ( Jj ++il))--fQ i 22++4i] ♦ v eff (p,5,e). 2 2HP 4

(30)

20

Here pBCD is the B,CD reduced mass, and RBCD the distance from B to the CD center of mass. The effective potential Veff in this case is the full potential evaluated for fixed values of the inactive variables. Note that if the product molecule is linear, Q is the vibrational angular momentum. 3.3.3 RD3D: In this Hamiltonian, one adds the non-spectator rotational energy operator to Eq. (28). This depends on the AB diatomic angular momentum operator j ^ . The centrifugal energy associated with orbital motion of AB about CD has not been defined in past RD3D calculations, but it is not hard to derive an expression. If we exploit the similarity of the RD3D hamiltonian to that for atom-diatom reactions,39 we obtain:

H = T(p,5) ♦

*—j£, 2 M r(p,5) r "

+

>,2

^ ( J ( J 2MR(p,5)2

+

l ) + j ^ - 2fi2)

Veff(p,6,Y)

8HP 2

where the choice of Veff depends on whether the sudden or adiabatic approximations has been used, but otherwise follows the prescriptions given earlier. 3.3.4 RBA-4D: In this Hamiltonian, we start with Eq. (30) and add the kinetic energy associated with spectator vibration. In addition, the potential now depends explicitly on four active variables, yielding37

H = T(p,8) - i l l J l ♦ ^ - j 2ndz

2

2uz

2

*

2

1

3

2up

tf

2MBCDRBCD(p,6,z,0)2

_ Q2) (32)

[J(J + 1)-Q 2 + ^] + Veff(p,6,z,e) . 2 4

3.4 Hyperspherical Coordinate Coupled Channel Calculations To complete this section we discuss the solution to the Schrbdinger equation using the Hamiltonians given above. Since the complete details of each method could occupy several papers by themselves, our intention here is to make the development illustrative rather than exhaustive. The basic idea involved in all the coupled channel (CC) calculations is relatively

21

straightforward. One wants to take advantage of the fact that motion in all the variables except the hyperradius p is bound motion, so the complete wavefunction can be expanded in terms of fixed p eigenfunctions, with the expansion coefficients taken to be functions of p. If we use the symbol , (q) to denoted these eigenfunctions with t being a collective label for all the quantum numbers associated with thefixedp eigenfunctions, q a collective label for all the active coordinates except p, and C,(p) being the expansion coefficients, then the expansion has the form:

*(p,q) =5> t (q)C,(p).

(33)

If this expansion is substituted into the Schrodinger equation, this equation is converted into a set of coupled ordinary differential equations for the expansion coefficients, the solution of which can be done using a variety of well known methods. These CC equations have the general form:

- f = EU,Ct, dpz

(34)

t

where U„. represents matrix elements of all the terms in the Hamiltonian except the first one in the right hand side of Eq. (28), i.e.,

U„, = (%i

4 1 5p

t

> ■

(35)

Note that the basis functions $ t depend on the value of p chosen to define them. To generate a compact representation of the wavefunction, it is important to periodically change basis functions as the propagation of the CC equations proceeds from small to large p. Such a change in basis functions leads to a transformation in the coefficient matrices of the form: Ct - 2J s t t c i '

(36)

22

where the overbar labels the transformed coefficient matrix, and the overlap matrix has the form S„. = , refers to the eigenfunctions that replace 4>, is Eq. 33 after the transformation. The boundary conditions on the CC equations are that the coefficient C, should vanish in the limit of small p, corresponding to repulsive regions of the potential where all of the atoms are close together, and it should reduce to the following combination of incoming and outgoing waves in the limit of large p: 2 C n ,(p-») - ( ^ ) 2 ( e ^ \ , - e * ' \ , ) .

(38)

Here we have added an additional index t' to the coefficient Ct to label the asymptotic initial state, vk and vk. are the asymptotic velocities in channels t and t', k, is the wavevector associated with channel t, 6 ir is a Kronecker delta, and S„. is the scattering matrix connecting channels t and t'. The scattering matrix elements may be used to define all the physically relevant information about the collision process, such as the reaction probabilities, cross sections and rate constants. Note that we have developed the asymptotic boundary conditions in terms of the hypersphericai coordinate p (i.e.,p - R) will

De

of order

\la and the correction term jh2 M'1 Z>«(R) will be of order h2/(2Maa2)

;

i.e., of the order of a rotational energy splitting. Thus the correction term is quite small and can usually be neglected. A possible exception is when the wave function, «&/°(r,R), varies very rapidly with small change of R, e.g., near an avoided crossing. In such cases it may not be valid to neglect DJJ(R), but in these cases it is likely that the adiabatic approximation itself is inadequate.

4.2

Mixed Quantum-Classical (Conventional Molecular Dynamics)

The physical picture underlying the adiabatic (Born-Oppenheimer) approximation, Eqs. (4.1) and (4.2), (neglecting the D'JJ term) is that the fast particles adjust instantaneously to the motion of the slow particles, so that the

45

latter are governed by the adiabatic potential energy surface £/°(R). In addition to the adiabatic approximation, the conventional molecular dynamics simulation approach assumes that the slow particles are governed by classical mechanical equations motion; i.e., rather than solving for the time-dependent slow-particle wave function Q(R,f), classical trajectories are integrated on the potential energy surface 5/°(R). Following Messiah27, the classical limit of Eq.(4.2) can be obtained by first expressing the slow-particle wave function as Q(R,f)

=

A(R,t) exp

jS(R.t)

(4.5)

where /4(R,f) and 5(R,f) are taken to be real-valued. Eq.(4.2) and separating real and imaginary terms gives dS dt

+

Substituting this into

ft2 fc2 ___ V2*« A = - I ^ - M a - ' D « ( R ) + Ia a 2 " »" ' „ 2 A '

„,.._,/ \2 _ _ l\M?(v +S/°(R) Ras) a2 " \ ** f '

(4.6)

^lA/

a

-'V,A.V^

+

l X A V ^

= 0.

(4.7)

Equations (4.6) and (4.7) are entirely equivalent to the Schrodinger equation, Eq.(4.2). Multiplying Eq. (4.7) from the left by 2A, we obtain27 —A2

+

M;lVRa.(A2VRas)

= 0.

(4.8)

This is the equation of continuity of flux. A2 is the probability density, as can be seen directly from Eq. (4.5), and M~x A1 V R S is the current density. Thus we identify the velocity as R«

=

M-'VRaS.

(4.9)

Planck's constant does not appear in Eqs. (4.7) or (4.8), and only quadratically in Eq. (4.6). The classical limit is obtained by taking the h —> 0 limit on the right hand side of Eq. (4.6), yielding the Hamilton-Jacobi Equation of classical mechanics28:

46

^■ + l\M;y(vRas)2+£f°(R)

= 0.

(4.10)

Equations (4.8) and (4.10) describe a fluid of non-interacting multidimensional classical particles, i.e., a swarm of independent trajectories moving according to the potential energy function g/°(R). 2 7 Note that, consistent with the h -» 0 limit, the correction terms %2 M"' £>? (R) disappear in the classical limit. It has not been established whether re-inserting these terms in classical mechanical simulations generally increases or decreases accuracy. Substituting Eq. (4.9) into (4.10), we obtain -^ + l{MaRa+gf°(R)

= 0.

(4.11)

Since the second and third terms on the left hand side of Eq. (4.11) are the classical mechanical kinetic plus potential energy, we have 5(0 = -\E(t')dt'. o

(4.12)

Taking the gradient of Eq. (4.11), we obtain Newton's equations of motion,27 pa

= MaRa

= -V^S/°(R),

(4.13)

where pa is the classical mechanical momentum of slow particle a. This establishes the equivalence of Eqs. (4.8) and (4.10) with classical mechanics. It is instructive for all of the mixed quantum-classical methods described here to examine conservation of energy. In a proper self-consistent theory, energy must flow between the fast and slow particles in such a way that the total energy of the system is conserved. This property is not satisfied by the Redfield and classical path methods, since they do not include feedback in both directions. While energy conservation is readily apparent for the adiabatic method, we demonstrate it here in anticipation of the more substantive cases to follow. For the classical adiabatic method, the total energy is given by E

= J X ' P "

+ j0f(r,RR(r,R)0?°(r,R)dr.

(4.14)

47

Conservation of energy requires that the derivative of E with respect to t be equal to zero. Carrying this out, using the chain rule and recalling that we are using Cartesian coordinates, we obtain ^

+

V^f