DYNAMICS OF ATOM-DIATOM4 REACTIONS

AT LOW ENERGY

BY

JOHN ALBERT OLSON

A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF

THE UNIVERSITY OF FLORIDA

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1982

Copyright 1982

by

JOHN ALBERT OLSON

ACKNOWLEDGEMENTS

I would like to convey my sincere appreciation to my advisor, Professor David A. Micha. He played an essential role not only in selecting this problem but also in developing the formalism used to solve it. His patience, encouragement and support during this period of research is also gratefully acknowledged.

I would like to thank the other faculty members and

graduate students in the Quantum Theory Project. The many seminars and discussions have been of great educational value. In particular I would like to thank Professor Yngve Ohrn. His courses on Quantum mechanics greatly stimulated my interest in this field. I also appreciate the help of Dr. Eduardo Vilallonga in some of the numerical aspects of this work.

I would like to express my sincere gratitude to

Professor Per-Olov L6wdin for providing me the opportunity to attend the summer school in Sweden and Norway. His yearly organization of the Sanibel Symposium has also been of great educational value to me. His kind interest in me while at the Quantum Theory Project is warmly appreciated,

iii

I would also like to thank my family in Jacksonville, Florida, for providing support and for furnishing a place to go for an occasional rest.

Finally, I would like to thank the secretaries and

staff of the Quantum Theory Project for both their typing skills and their organization of numerous social events.

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS ......................................iii

ABSTRACT .............................................. vii

CHAPTER

INTRODUCTION ......................................... 1

1-1 General Problem .................................. 1

1-2 Electronic Motions ............................... 3

1-3 Nuclear Motions .................................. 6

2 ELECTRONIC REPRESENTATIONS .......................... 12

2-1 Introduction .................................... 12

2-2 Electronic State Representations ................ 16

2-2a The Adiabatic Representation ............... 19

2-2b The Strictly Diabatic Representation ....... 20 2-2c The Nearly Adiabatic Representation ........ 23

2-3 The Minimization Procedure ...................... 25

2-4 The Two Electronic State Problem in One

Dimension ....................................... 30

2-5 A Model Calculation ............................. 34

2-6 Discussion ...................................... 47

3 GENERAL FORMALISM OF THE DYNAMICS .................... 49

3-1 The One Electronic State Problem ................ 49

3-2 General Time Independent Formalism .............. 54

3-3 The Eikonal Approximation ....................... 58

3-4 The Short Wavelength Approximation .............. 62

3-5 Time Dependent Equations ........................ 66

3-6 Solutions of the Equations ...................... 72

3-7 Expressions for Observables ..................... 81

4 THE ELECTRONIC PROBLEM FOR H3+ IN THE

ADIABATIC REPRESENTATION ............................ 84

4-1 General Considerations .......................... 84

4-2 Method of Diatomics in Molecules ................ 87

4-3 The Eigenvalues ................................. 95

4-4 The Eigenfunctions ............................. 104

4-5 Non-Adiabatic Couplings ........................ 107

5 THE NUCLEAR PROBLEM FOR COLLINEAR H3 ............... 112

5-1 Hyperspherical Coordinates ..................... 112

5-2 The Hamiltonian in the Almost

Adiabatic Representation ....................... 119

5-2a The Electronic Transformation ............. 121

5-2b The Equations of Motion ................... 124

5-3 The Hamiltonian in the Diabatic

Representation ................................. 128

5-3a The Electronic Transformation ............. 128

5-3b The Equations of Motion ................... 130

6 CALCULATIONS FOR H3 ................................ 134

6-1 Electronic Results ............................. 134

6-la The Diatomic Potentials ................... 134

6-lb Adiabatic Potential Energy Surfaces ....... 138 6-1c Non-adiabatic Couplings ................... 143

6-ld Model of Non-adiabatic Couplings .......... 152

6-2 Trajectory Calculations ........................ 154

6-2a Test Cases ................................ 161

6-2b H3+ Results ............................... 187

6-3 Experimental Studies ........................... 234

6-4 Theoretical Studies ............................ 240

6-5 Comparisons ..................................... 243

6-6 Conclusions ..................................... 254

APPENDICES

1 HYPERSPHERICAL COORDINATES ......................... 259

2 COMPUTER PROGRAM .................................... 265

3 TOTAL ELECTRON TRANSFER PROBABILITY ................ 270

REFERENCES .............................................. 276

BIOGRAPHICAL SKETCH ...................................... 280

Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the

Requirements for the Degree of Doctor of Philosophy

DYNAMICS OF ATOM-DIATOM REACTIONS AT LOW ENERGY

By

John Albert Olson

August 1982

Chairman: David A. Micha Major Department: Chemistry

This work focuses on two fundamental problems in scattering theory. The first is centered around the electronic basis used to expand the solution of the time independent Schr6dinger Equation. The other consists of how to handle the problem for the nuclear degrees of freedom when more than one electronic basis function is included in the expansion.

In chapter two it is shown that, in the adiabatic electronic representation, the Hamiltonian is not hermitean if more than one electronic basis function is included in the expansion. It is pointed out that in such

vii

cases alternative electronic representations may be more suitable. The adiabatic and diabatic representations are reviewed and the "nearly adiabatic" representation and a representation based on a minimization procedure are introduced. Calculations are done to compare the results from the diabatic representation and. the representation arising from the minimization procedure.

Chapter three considers the problem for the nuclear degrees of freedom in multi-surface systems. A "common" eikonal is used for the nuclear wave function satisfying the time independent Schr6dinger equation. This "common" eikonal is obtained from a modified Hamilton-Jacobi equation that involves an average potential. Implementing the short wavelength approximation and transforming to a time dependent picture leads to a set of first order differential equations in time that determine the expansion coefficients. It is shown that these equations along with the differential equations obtained from the Hamilton-Jacobi like equation for the nuclear positions and momenta form a coupled set of first order differential equations in time that are formally equivalent to Hamilton's equations of motion.

This procedure is applied to the collinear H3+ system. Hyperspherical coordinates are used for the nuclear degrees of freedom and the adiabatic potential energy surfaces and electronic coupling terms are obtained from the method of Diatomics in Molecules. A transformation to the diabatic representation is made and the trajectories are calculated

viii

in this representation. Typical trajectories are presented in chapter six and a comparison of the total electron transfer probability with those from a quantum mechanical study is made. The results are encouraging.

CHAPTER 1

INTRODUCTION

1-1 General Problem

This study primarily addresses the problem of molecular reactions where more than one electronic state is energetically accessible. These processes are usually referred to as non-adiabatic collisions. The collisions are assumed to take place between combinations of atomic or molecular fragments that can be either charged or neutral. Since the origins of the formalism to be presented are based in quantum mechanics, a few comments on the general quantum mechanical treatment of the problem seem appropriate.

The general solution to the Schr6dinger equation

involving both nuclear and electronic degrees of freedom is a function of the nuclear and electronic coordinates. Until programs are available to numerically solve these multidimensional equations, some form of approximation to the solution must be made. Generally, the solution is expanded in an electronic basis. This in itself does not simplify the problem but with the approximation to be discussed next, the electronic problem can be solved independently of the dynamics of the nuclei.

The approximation referred to above is the well-known

Born-Oppenheimer approximation (Born and Oppenheimer, 1927).

This approximation is physically based on the fact that the electron's mass is orders of magnitude smaller than the mass of a nucleus and hence that the electronic motions are much faster than nuclear motions. Then one would expect that the dependence of the wave function describing electronic motion on nuclear variables could be neglected. In other words, this approximation assumes that the nuclear kinetic energy operator can treat the electronic wave functions as constants and leads to solutions of the electronic problem that depend only on the positions of the nuclei.

With this approximation, the electronic problem is

solved with the positions of the nuclei held fixed. Since the electronic problem must be solved for each internuclear configuration, the electronic functions, energies, etc. are said to depend parametrically on the nuclear coordinates. Since the dependence of the nuclear wave function on the electronic coordinates has been ignored, the solution to the problem is in the form of a linear combination of products of electronic functions that depend parametrically on the nuclear coordinates and nuclear functions that depend only on the nuclear variables.

This approximation is an extreme simplification to the most general solution in that the electronic problem is solved without a full dynamical knowledge of the nuclei. In other words, this approximation assumes that the electronic problem can be solved independent of the nuclear velocities. This approximation may not be valid when the nuclear speeds

are high (comparable to those of the electrons). But in this study, the nuclear speeds will be slow compared to electronic speeds so that this approximation will be assumed to hold to a high degree of accuracy.

From the preceding discussion, it is apparent that there are two essential steps to solving the problem. The first is to choose an electronic basis that is convenient for the calculation and to solve the electronic problem within the Born-Oppenheimer approximation. This will be the subject matter of the next section and chapter two. The second step is to expand the solution in this basis and solve for the nuclear expansion coefficients. This will be the topic of the last section of this chapter and chapter three. 1-2 Electronic Motions

As was discussed in the previous section, the solution

of the full Schr-dinger equation is expanded in an electronic basis and the electronic problem is solved with the nuclei held fixed. In general, the expansion in a set of n electronic states will result in a square nxn matrix represestation of the electronic Hamiltonian, i.e. the full Hamiltonian without the nuclear kinetic energy operator. Couplings between the nuclear expansion coefficients will occur through the off-diagonal matrix elements of the electronic Hamiltonian and through the electronic matrix representation of the nuclear kinetic energy operator. Different electronic representations give rise to some or all of these couplings and a brief discussion of some of

the more familiar representations and their properties follows.

One of the most widely used electronic representations is the adiabatic representation (Born and Oppenheimer, 1927; Born and Huang, 1954). This representation has both been extensively used in bound state calculations, i.e. in calculations where the dynamics of the nuclei can be ignored, and in scattering calculations. It is essentially characterized by giving a diagonal matrix representation of the electronic Hamiltonian.

The (diagonal) matrix elements, often referred to as eigenvalues, of this representation give rise to potential energy hypersurfaces. For adiabatic reactions, the reactants and products are both on the same potential energy hypersurface while for non-adiabatic reactions the surface for the reactants is different from that of the products. A consequence of this representation is that surfaces of the same electronic symmetry do not cross (Moiseiwitsch, 1961), i.e. couplings, in most cases, do not occur through the electronic Hamiltonian.

These surfaces are essential in many formulations of

the problem. As mentioned in the previous section, the eigenvalues of this representation depend parametrically on the nuclear coordinates so that the electronic problem must be solved for each nuclear configuration. Since scattering calculations normally require many nuclear configurations, an ab initio treatment of the electronic problem would almost surely be prohibitive. This offers motivation and support for using

more approximate procedures such as Diatomics in Molecules (Ellison, 1963) or Polyatomics in Molecules (Tully, 1977) which make use of either experimental or theoretical knowledge of the diatomics or polyatomics to construct the surfaces.

Since the electronic Hamiltonian is diagonal in this

representation, the couplings between the nuclear expansion coefficients arise from the matrix representation of the nuclear kinetic energy operator and gradient in this basis. The inclusion of these terms in a scattering calculation leads to two main difficulties. First, efficient computer programs are not available for solving sets of equations that contain these terms and second, as has been noted (Smith, 1969), if all coupling terms are included, the Hamiltonian in this electronic basis is not hermitean. Although the first difficulty is a computational one that could be conceivably overcome, the second difficulty introduces theoretical problems such as nonconservation of energy and flux or the need to introduce a second set of nuclear coefficients that satisfy the adjoint operator of the original problem.

As can be surmised,this representation is probably not the most useful for processes that exhibit non-adiabatic affects. This representation is however still essential in that the solution of the electronic problem in this basis is well understood and theoretically sound. Other representations don't have this property so that they must be related to the adiabatic basis by way of unitary

transformations in order to form a matrix representation of the electronic Hamiltonian. That is, the eigenvalues of the adiabatic representation are convenient to form a matrix representation of the electronic Hamiltonian in any electronic representation.

There are a variety of representations other than the

adiabatic one that can be used in a scattering calculation. They are based on either completely or partially eliminating the couplings due to the nuclear kinetic energy operator or gradient or on minimizing the couplings in a way to be discussed later. These schemes have one thing in common in that coupling terms present in the adiabatic representation that are eliminated or minimized are replaced by electrostatic terms that give rise to a non-diagonal matrix representation of the electronic Hamiltonian.

The electronic problem will be the subject matter of chapter two. Four representations including the adiabatic one are discussed and some of their advantages and disadvantages are pointed out. It will be shown that care must be used in selecting the representation that would be most suitable for a scattering calculation. 1-3 Nuclear Motions

Once the electronic representation has been chosen

there remains the problem of solving for the nuclear expansion coefficients. Generally they are solved for quantum mechanically, semi-classically or classically. Since quantum mechanical and semi-classical treatments of this

problem are not topics of this study,only a brief discussion of these methods follows. Comments on these subjects will be restricted to be general and provide motivation for using the method to be developed.

Since the electronic problem was treated quantum

mechanically, it would seem natural to treat the nuclear problem quantum mechanically too. Indeed, this does give dynamically consistent formalisms but all quantum mechanical approaches to this problem have a serious disadvantage. Since the nuclei have vibrational, rotational and translational degrees of freedom, a quantum mechanical approach even for inelastic collisions would require expanding the nuclear expansion coefficients in at least internal states corresponding to the vibrational and rotational degrees of freedom. Since normally many internal states are energetically accessible, this expansion leads to a large number of coupled differential equations to be solved simultaneously. This is actually one of the simpler cases in that if a partial wave expansion is made a normally much larger set of equations is obtained.

The problem becomes even more difficult for reactive

collisions. In this case there is more than one asymptotic Hamiltonian to be considered. Expansions in internal states for each asymptotic Hamiltonian must be made. Not only does the number of coupled equations increase but basis elements corresponding to different asymptotic Hamiltonians are not

necessarily orthogonal and can lead to problems such as overcompleteness.

It would seem then that even though a quantum

mechanical approach is theoretically appealing, it leads to some very fundamental difficulties. Even if approximations were introduced to reduce the number of coupled equations, it would be helpful to introduce a formalism that would avoid these difficulties as much as possible. Before pursuing this however a few comments on the semiclassical approach to non-adiabatic collisions without nuclear rearrangements will be made.

Semi-classical approaches to non-adiabatic processes

without nuclear rearrangement have been used extensively and many good reviews are available on this subject (McDowell and Coleman, 1970). These approaches normally start with the time dependent Schr6dinger equation for the electronic Hamiltonian and assume that the trajectories of the nuclei are known. One then needs to solve a set of first order differential equations in time for the expansion coefficients of the electronic wave function. Simultaneous integration of these equations gives these coefficients as a function of time and their values at the final time are related to the probability of non-adiabatic transitions for the collision.

As mentioned in the preceding paragraph, the nuclear trajectories in these approaches are assumed known. These lead to different approximations such as the impact parameter

method (high velocity approximation) or the perturbed stationary state method (low velocity limit). Besides the limitations caused by not determining the trajectories from ab initio considerations the extension of these approaches to include nuclear rearrangement is not trivial.

Since the masses of the nuclei are so much heavier than those of the electrons, it would seem possible that some or all of the nuclear degrees of freedom could be treated classically. This approach has been successful for the case of one potential energy surface, i.e. in the absence of couplings between nuclear and electronic degrees of freedom. For such systems, the nuclei evolve according to classical equations of motion on a quantum mechanical potential energy surface. The initial conditions of the trajectories are obtained by taking a Monte Carlo sampling of the possible initial states of the system. This approach is however not as straightforward when more than one surface is included.

A primary concern in this study is to develop a

formalism that extends the classical treatment of nuclear degrees of freedom to systems that are characterized by more than one electronic state. This avoids the difficulty of expansions in internal states required by quantum mechanical treatments. Also assumptions about the nuclear trajectories are not needed since they are determined from classical equations of motion. This also has the advantage that individual trajectories give a clear conceptual picture of the collision event.

Briefly, the approach to be used here starts with the

time independent Schr6dinger equation. The wave function is expanded in a set of electronic states. The nuclear expansion coefficients are written as products of an amplitude and a common phase which is proportional to the eikonal. The gradient of the eikonal is required to satisfy the Hamilton-Jacobi equation whose potential is determined from the quantum mechanical equations. Upon implementing the short wavelength approximation and making a transformation into time, it is found that the nuclear expansion coefficients satisfy first order differential equations in time and that the gradients of the eikonal become the classical momenta of the nuclei. If a convenient form for the nuclear expansion coefficients is chosen, it is found that not only the nuclear positions and momenta but also the expansion coefficients satisfy Hamiltons equations of motion.

Although other approaches (Meyer and Miller, 1979) have obtained similar results, the author considers this treatment to be on a more sound theoretical foundation. The development of this formalism will be the subject matter of chapter three.

The formalism of chapters two and three is applied to the collinear H3+ system. This system was chosen partly because of the presence of large non-adiabatic coupling terms. Also, due to the relatively light masses of the nuclei, this system should provide a good test of the theory. It is also interesting from the viewpoint that not

only do elastic and inelastic processes occur but it also exhibits reactions and or rearrangements. Finally H3+ is adequately described with two electronic states which is a natural starting point for an application of the theory.

The electronic and nuclear parts of the problem for H3+ will be developed in chapters four and five respectively. The electronic problem will be solved with the method of Diatomics in Molecule and the nuclear problem will be solved in hyperspherical coordinates. This convenient choice of coordinate system will be discussed at length in Appendix one.

This study will conclude with a presentation of the

calculations in chapter six. A brief background on previous experimental and theoretical studies for the H3 system will also be given. A comparison with a quantum mechanical calculation (Top and Baer, 1977) for collinear H3 + will be made and this work will close with some comments and conclusions.

CHAPTER 2

ELECTRONIC REPRESENTATIONS

2-1 Introduction

The adiabatic approximation has played a central role in the study of molecular processes since its introduction by Born and Oppenheimer (Born and Oppenheimer, 1927; Born and Huang, 1954). The approximation introduces a basis of electronic states that provide an adiabatic representation for electronic operators. Non-adiabatic collisions, originally studied by Landau, Zener and Stueckelberq, (Landau, 1932; Zener, 1932;Stueckelberg, 1932; Nikitin, 1970) require information on the momentum couplings of the adiabatic representation. Many models have been developed to incorporate the couplings (Child, 1979; Tully, 1976; Garrett and Truhlar, 1980; Delos, 1981).

In cases where the adiabatic couplings can not be ignored, a different electronic basis and corresponding representation may prove to be useful. One such alternative is a diabatic representation (Smith, 1969) defined so that momentum couplings are exactly eliminated and transitions occur only through Coulomb interactions of electrons and nuclei. This is done by introducing the eigenstates of the momentum operator. In its original version this representation was criticized because the electronic states

of the new representation could not change with intermolecular distance, except for phase factors (Gabriel and Taulbjerg, 1974). Working however with finite bases one can define a diabatic representation by requiring that the matrix of the momentum operator is zero. This introduces a matrix unitary transformation from the adiabatic to the diabatic set of states which does change with intermolecular distance. Given this matrix, one can transform the matrix of Coulomb interactions to the new basis.

Although this procedure is mathematically rigorous, it may lead to complications in the physical description of collisional processes. As we shall see, depending on the magnitude of the momentum couplings in the original adiabatic representation, the diagonal elements of the Coulomb interaction matrix in the new diabatic representation may be far from physically meaningful. For large momentum couplings these diagonal elements may repeatedly cross; while for small momentum couplings they may be far from adiabatic potentials in regions where these are physically meaningful. The latter problem can be particularly significant in studies of reactive atom-diatom collisions because the new representation may give the correct description of the reactant potentials but a completely unphysical one for the product potentials.

These difficulties result from using finite bases and from the differential equation satisfied by the unitary transformation, which is of first order in the intermolecular

position variables. It follows that given the known boundary conditions for large distances the transformation and diabatic potentials are mathematically determined for all shorter distances, which does not leave any room for physical considerations.

The aims of the present chapter are to introduce a new diabatic representation which leads to physically well behaved potentials (in the sense to be described), and to show how it is constructed around pseudocrossings. It starts with the adiabatic potentials and couplings which are obtained in electronic structure calculations. The procedure is based on the minimization of coupling terms, and provides a criterion to determine the range of kinetic energies over which it is justifiable to neglect couplings altogether.

Other diabatic representations are possible and have been introduced by means of physical arguments (O'Malley, 1967). For energetic atom-atom collisions, several of the representations have been extended to incorporate electron translation factors in order to satisfy asymptotic conditions (Delos and Thorson, 1979; Delos, 1981). These extensions shall not be considered because the immediate aims refer to thermal and hyperthermal collisions. Adiabatic and diabatic representations have also been introduced for atom-diatom collisions (Baer, 1975; Top and Baer, 1977a.

Numerical studies of electronic states in various representations include calculations of potentials and

their couplings for atom-atom (Redmon and Micha, 1974; Nimrich and Truhlar, 1975; Evans, Cohen and Lane, 1971) and atom-diatom systems (Rebentrost and Lester, 1977; Tully, 1980). A great deal of related work has also been done on diabatic molecular orbitals and their energies (Lichten, 1963; Briggs, 1976; McCarroll, 1976), to which the developments in this chapter could also be applied.

To illustrate some of the numerical aspects, some

+

results are briefly mentioned for H3 and FH2 in the collinear conformations. For a basis of two electronic states (the two lowest states of 1Z symmetry for H3 +; the lowest 1E and 3Z states for FH2), the transformation from the adiabatic to a diabatic basis depends on the integral S dXTa x)

SD (X

where Ta is the momentum coupling in the nuclear variables X, and D is their domain. As shall be seen in Section 2-4, the standard diabatic representation (Smith, 1969) works well when the integral equals w/2 but not when it differs appreciably from 7/2. For atom-diatom mass-weighted Jacobi variables (Z,z) for the intermolecular and internal coordinates, a transformation to polar coordinates r = (Z2 + zz) and 0 = tan-](z/Z) leads to values of 0<

which for H3+ go from 0.3 to 1.6 as r varies from 4 a.u. to 12 a.u. (Tully, 1976), while for FH2 they stay around 0.7

as r varies from 8 a.u. to 12 a.u. (Tully, 1980). Other numerical examples can be found in the recent literature on adiabatic-diabatic transformations for slow nuclear degrees of freedom, where the integrals are instead larger than 7/2 and multiple crossings occur (Baer, Drolshagen and Toennies, 1980).

Given the wide variety of problems where pseudocrossings may occur, this chapter shall not concentrate on a given physical system but shall instead construct a model of potentials and couplings with parameters that will be varied around physical values. The shape of the potentials and couplings are similar to those calculated (Tully, 1976)

+

for H3 and the physical parameters relate also to this system.

The adiabatic, diabatic and nearly adiabatic representations will be briefly developed in Section 2-2. This will be followed in Section 2-3 with a general development of the minimization procedure. Section 2-4 will give a detailed treatment of the two electronic state problem in one dimension. A comparison of the results for two of the diabatic procedures will be given in Section 2-5 and the chapter will close with a discussion. 2-2 Electronic State Representations

We consider to begin with a molecular system with n nuclei, in a center-of-mass reference frame. Introducing cartesian coordinates and the nuclear position vectors

17

{yi' i=l to n}, the nuclear kinetic energy operator is expressed as

n

T = nu i=l

(2mi)-1 a2/y?

where m. is the mass of the ith nucleus and we have used

1

atomic units (l=1). Introducing a change of variables,

xi = (mi/M) Yi (2-2)

where M is the total mass, gives

^1 n T = - (2M)i=l

D2a2Ix?

(2-3)

Defining next a nuclear momentum operator in vector form with n orthogonal components and written as

P

n - i a x , 1 , n

-nu x1 x2 ~n -x

(2-4)

where X = (x , . . x ), the kinetic energy operator becomes

~ ~ ~n

Tnu = (2M)- Pnu * P = -(2M)-I2/SX2

. (2-5)

For the special case of systems with two nuclei, the momentum operator in the center-of-mass coordinate system would simply be

(2-1)

P =-i 3/aR , (2-6) ~nfu ~

where R is the relative position vector between the nuclei. For the three-nuclei system, mass weighted Jacobi coordinates also satisfy the above conventions.

Solutions of the time independent Schr6dinger equation satisfy

(H-E)jIT(X)> = 0 , (2-7)

where

H = T nu(a/X) + H e(X) , (2-8)

AA

Tnu and the nuclear positions X were defined above and Hey, is the electronic Hamiltonian including nuclear-nuclear repulsion terms. The bracket notation refers to the electronic coordinates and involves an integration only over electronic coordinates. Invoking a separation of electronic and nuclear variables leads to the solution having the general form

IT(X)> = EI'i(x)> iM (2-9) where one requires that the electronic functions, Oi. form a complete, orthonormal set at each X.

Substituting Eq. (2-9) into Eq. (2-7) and multiplying from the left by <(D (X)I leads to

.[

i(X) + i(X)] = EPW(X)

(2-10)

A

where Tnu operates on all factors to the right. The development thus far is completely general but not of much use. More useful representations may be obtained by requiring the electronic functions to obey additional properties besides those of completeness and orthonormality and these will be briefly discussed below. 2-2a The Adiabatic Representation

This highly useful and widely used representation

(Born and Oppenheimer, 1927; Born and Huang, 1954) is based on the requirement that the electronic functions satisfy

al l = va (x) (2-11)

where the superscript "a" denotes the adiabatic representation. Using this basis in Eq. (2-10)leads to

a, A a a a (X) a a

Ei(X) + Vjj=Ejm(X) (2-12) i nu i1- J

A

where Tnu operates on all factors to the right. As is well known, adiabatic potential surfaces V a of the same symmetry JJ

do not cross. They can however exhibit "avoided crossings" which can lead to a breakdown of descriptions of nuclear motions.

2-2b The Strictly Diabatic Representation

The diabatic representation (Smith, 1969) is based on the requirement that the electronic basis elements satisfy

= 0 for all i,j (2-13) i1 nu

where the superscript "d" indibates diabatic quantities. This representation will be referred to as "strictly diabatic" when compared with our representation, or as diabatic when the meaning is obvious. A general form for the basis which would satisfy the above requirement around a chosen point Xo would be {I(Xo)>I. This basis can be related to the adiabatic basis by a unitary transformation. In matrix notation one would have

q()o)> (a(X) A(X, Xo) (2-14)

where I'k> =(I4k>...I&k>) k=d,a and A is a nxn matrix. From Eq. (2-13) one finds that the transformation must satisfy

P A(X,Xo) + <= PI'a>A(X,X 0 (2-15)

nuw - -0udauoZ Z - ,0 z

with the boundary condition A(X ,X ) =A . Expanding the Z -0 0 :ZO

solution of Eq. (2-10)in this basis and performing a little algebra (see Baer for details) (Baer, 1975; Top and Baer, 1977a)leads to a final result, in matrix notation, of the form

d d d d

T id(X) + vd (X) = E (x) (2-16)

nuT ~ - ~

where

V d(X) = A (XX )V (X)A(X,X (2-17)

d d0

id (X) = [d(X)] , a column matrix, (2-18)

and the elements of the diagonal matrix Va are defined by Eq. (2-11).

As was mentioned previously, this procedure rigorously eliminates the momentum coupling terms between electronic states. It does not however guarantee physically well behaved potentials corresponding to those electronic states. The following example should help clarify what is meant by physically well behaved potentials.

Consider the hypothetical case depicted in Fig. (1) of a one dimensional two electronic state system. In regions

(a) and (c) the momentum coupling between electronic states should be small since the potentials are relatively far apart. One would therefore expect that the adiabatic

Vd

11

b

Vd Vi2

R< Rx R> R

Fig. () Schematic representation of a pseudocrossing. Solid curves correspond

to adiabatic potentials, dashed lines intersecting at Rx correspond to

diabatic potentials. The superscripts "a" and "d" signify adiabatic and

diabatic, respectively.

approximation would be valid in these regions. In region

(b) the potentials do approach each other relatively closely so the adiabatic approximation would no longer be valid and one should either include the momentum coupling terms in the adiabatic basis or use a diabatic representation. Then physically well behaved diabatic potentials should fulfill the following conditions. In region (c), as shown in Fig.

(1), the diagonal elements of the diabatic potential matrix

d d

V1i and V22 should coincide with the adiabatic potentials a ada d d V i and V22, respectively. In region (a) V11 and V22 should coincide with Va2 and Va1, respectively. In region (b) the diagonal elements of the diabatic potential matrix should vary smoothly and exhibit a single crossing. Finally, the

d d

off diagonal matrix elements V12 and V2, should vanish in regions (a) and (c) far from the crossing.

As is seen in Eq. (2-15), the transformation matrix

A(X,X ) satisfies a first order differential equation and one therefore has only one boundary condition at one's disposal. This will ensure proper behavior in one of the regions in Fig. (1), usually chosen to be region (c), but the behavior in the other regions will depend on the coupling terms so that this procedure does not in general guarantee diabatic states that satisfy the above conditions.

2-2c The Nearly Adiabatic Representation

As was pointed out in Section 1-2, one of the

difficulties that arise when using an arbitrary electronic

representation is that Eq. (2-10) is not necessarily Hermitian. The non-Hermitian components arise from the first term on the left hand side of Eq. (2-10). It can be shown (Smith, 1969) that for a complete electronic basis that is real and orthonormal

= (2M)-1(P <(IP IP > + <(DIP 4>.<4IP nu>)

nu nu z nu z nu z nu

(2-19)

AA

Since P is imaginary and <(IP I > is an imaginary

nu nu a

antisymmetric matrix, the first term on the right hand side of Eq. (2-19)is real and antisymmetric while the second is real and symmetric. Thus the first term is non-Hermitian and would cause difficulties if it were not neglectable.

An obvious way to restore Hermiticity would be to choose an electronic basis that satisfies

P n<( nLP nu4,n> = O (2-20) nu nu

where the superscript "n" stands for nearly adiabatic. Relating this to the adiabatic representation via a unitary transformation, i.e.

Sn a aC (2-21)

leads to, with some simplification,

C P2C + (P C )-P C + C (P n)c = 0. (2-22)

nu, nu, nu, nu flu

With the form of Pnu given by Eq. (2-4) one sees that the transformation satisfies a second order differential equation. Since there are two boundary conditions, use of this representation could ensure proper behavior of the electronic potentials on both sides of the pseudocrossing-.

It should be noted that since second order partial

differential equations are usually difficult to solve, the usefulness of this representation may be somewhat limited. It should however provide an alternative in cases where the non-Hermitian term in the adiabatic representation can not be ignored and the strictly diabatic representation gives unphysical results.

2-3 The Minimization Procedure

Physically ill behaved diabatic potentials can be a serious drawback in studies of scattering processes. The procedure to be developed here starts from a different point of view. Instead of completely neglecting the momentum coupling terms, our procedure requires well behaved diabatic potentials. It introduces a unitary transformation with parameters to be variationally chosen, and then minimizes the momentum coupling in the pseudocrossing region.

Since the electronic Hamiltonian is usually only known in the adiabatic representation, a unitary transformation between the representations is necessary. In this procedure, a transformation B willbe chosen to guarantee physically

well behaved potentials and will in general depend on a set of parameters {ai i = 1 to k} to be determined. The two bases can be related through

Pm (x;c... ) = (a (X)B(X;az... a (2-23)

where the superscript "m" denotes the electronic representation obtained through a minimization procedure. Using this basis in Eq. (2-10)and suppressing the arguments lead to

numm I + < e >m =(2-24)

where the unitarity of B has been used and T operates on Z nu all factors to its right. From the discussion in the preceding section one has T = p 2 /(2M) (2-25) nu nu

so that by using Eq. (2-23)in Eq. (2-24)one has

-1 in -l m m m

(P 2/2M) + M P mP + K mp + m= Epm (2-26)

nu nuT Z

where

p B< aIPnu a>B (2-27)

A

Km = (2M) -B B (2-28) zZ Z nu '

Vm= B VaB (2-29)

and Pnu operates on all factors to its right.

Since the momentum coupling terms in this representation are not identically zero, the usefulness of the representation will depend on whether they can be neglected. To make them as small as possible our procedure determines the coefficients, ai, so as to minimize the positive expression tr(P mP m), where tr indicates the trace operation, in a domain D. Explicitly for each a one requires that

f dX tr (pmt.pm) = 0 i=l,2 ...,k (2-30)

aa i D

where the domain of integration, D, is the region where the adiabatic momentum coupling is significant. From Eq. (2-27) and remembering the assumed form of P discussed in nu

Section 2-2, one has

m pa

BPB + BP B (2-31) ~ ~ ~ Z nuwhere

= (2-32) Z Z nu

and the orthonormality of the adiabatic basis set was used. The adjoint matrix is given by

mt t at A

P = Btp B + (P B) B (2-33)

;Zf :L - nu,

so that Eq. (2-26)becomes

Da. dXtr{[BtpaB + (P nB) B](BtPaB + B nu B)} = 0

D (2-34)

where P nu operates only on the first factor to its right.

It should be emphasized that this procedure only

minimizes the momentum coupling terms. Whether or not they can be neglected will depend on the particular system being investigated and the relevant collision energies. In general, these couplings can be neglected if the terms in the Schr6dinger equation involving couplings between nuclear

AM

functions by Pnu' i.e. lp . Pn ' are negligible compared to the couplings of the nuclear functions by the potential m -M

Vm. An estimate of P nm is given by /2mEr (X where

Znuz- rel -

E (X) is the local relative kinetic energy. Since P is rel Z a real antisymmetric matrix (provided the electronic functions 1(m> are assumed to be real),one has

S-ipf m/2SMErell<< ij IVij1(2-35)

as a condition for neglecting momentum couplings. This

expression should be helpful in determining the range of the relative kinetic energy where the momentum coupling can be neglected.

Neglecting the Km coupling terms should also be justified. Defining

Ka = (2M) -< alp 2 a (2-36) ~ z nu

and assuming the adiabatic basis is complete, Km can be related to Ka by Eq. (2-28) with Ka obtained by letting Pnu operate on to give

Ka = (2M) -l( PPa + paa (2-37) nu,

where the Pa . pa term comes from the completeness of ,a. Since the literature usually only reports couplings between the lowest electronic states, this expression may not be useful. If a valence bond, Diatomics in Molecules or other method,was used to obtain the X-dependent expansion coefficients for the electronic states, the K a terms could be obtained by numerical methods. Once K a has been calculated, it is straightforward via Eq. (2-28)to determine Km A justification for neglecting it would depend on whether [Km 1

for each i,j- where E is now the total energy. In the

examples given later in this paper, only a functional form for Pa will be obtained and the terms Km will not be considered. Normally, though, if the coupling from Pnu is small, one would expect that the coupling from p nu2 would be at least as small.

2-4 The Two Electronic State Problem in One Dimension

In the previous section a general development of the minimization procedure was presented. Since many systems that exhibit non-adiabatic effects can be treated as processes occurring on two electronic potential curves, a more detailed treatment of the two-state case will now be given. For simplicity it will be restricted to one dimension, indicated by the radial variable R.

A general form for the real unitary transformation matrix for this case is given by

Cos Ym(R;a) sin Y m(R; )

B(R;) = (2-39) Z i-sin Ym (R;ct) cos Ym(R;c)

with Ym (R> ;) = 0, Ym(R ;a) = and where the parameter a is to be determined. In one dimension, the momentum operator is given by

Pnu =-i d/dR (2-40)

and the antisymmetric matrix of the adiabatic momentum coupling terms has the form

p a a

z i Ta (R)

-iTa (R)\ 0/

a a (,a T (R) = <$illd 2/dR>

Using Eqs. (2-39) through (2-42) in Eq. (2-34) leads to

f dRTa (R) + Ym(R;a)']2 = 0

where the "prime" denotes differentiation with respect to R. The domain of integration in this case will be the interval [R< ,R>] (see Fig. (1)) where T a(R) is significant. In analogy to Eq. (2-41) one has

0

pm=

Z ~i -rm (R ; a)

-iTm(R; )

0

(2-44)

where, from Eq. (2-31),

Tm (R;) = Ta(R) + .m(R;a)'

(2-45)

Assuming the integrand in Eq. (2-43)to be continuous gives

R>

f dR[,a(R) +Y M(R a ]3 (R;c)/Da = 0 .(2-46)

where

(2-41)

(2-42)

(2-43)

A functional form of ym that varies smoothly from zero for large R to R/2 for small R (see the discussion in Section 2-1) was introduced by choosing

y m(R;a) = (7r/4) [l-tanha(R-R x)] (2-47)

where Rx is the point where the diabatic surfaces cross. Then

y m(R;)' = -(7/4) a Sech2a(R-R ) (2-48)

and

3y m(R;a)'/3a = (7/2)sech2c(R-R x)-[(R-Rx)tanha(R-Rx)-l/21 (2-49)

Using Eqs. (2-48) and (2-49) in Eq. (2-46) leads to

R<

f Rsech 2a(R-R X)I [a (R) -(r/4)sech 2a (R-Rx)]

(2-50)

x [a(R-Rx)tanho (R-R) - 1/2] = 0

Equation (2-50) does not give an analytical solution for the parameter a so that a numerical procedure must be implemented to determine the value of a such that the integral is less than a small positive number.

In the next section, the results of this procedure will be compared to those obtained from the strictly diabatic procedure presented in Section 2-2. For the two electronic state problem in one dimension, the transformation matrix for the strictly diabatic procedure can also be expressed as

A cos yd (R,RO) sin y d (R,R0)

A (R',Ro) = -i d(RR) d (2-5 1) 0-sin y d(R,R) Cos y d(R,Ro)

However, the angle yd is given by (Baer, 1975; Top and Baer, 1977)

R

7d(R,R) = f dRa(R)(2-52)

R

0

where y d(R,R) = 0. Equation(2-52)was obtained by multiplying Eq. (2-15)from the left by A (R,R ) which eliminates

z 0

the sine and cosine factors, and by solving the resulting differential equation for yd

One can easily determine from the form of the

transformation matrix that Vu is given by the elements

U va u + va u V11(R) = V1(R)cos'yu(R,c) + 22(R)sin2y (R,c) (2-53)

u = va 27U a u V2 2(R) 1 V1 (R) siny (R,c) + V22(R)cos2y (R,c) (-4

(2-54)

u =U Va _ a u u V12(R) = V21(R) = (V11-V22)sinyu(R,c).cosy (R,c) (2-55)

where (u,c) equals (m,a) or (d,R ), respectively.

Equation(2-52) emphasizes the point made in the last part of Section 2-2b that the behavior of the diabatic potentials Vd depend on the momentum coupling T-aR). Letting R equal R> in Fig. (1), one sees that unless the integral in Eq. (2-52) from R< to R> is equal to 7/2, the diabatic potentials will not coincide with adiabatic potentials in region (a). One also has that the diabatic V..d will cross whenever

ii

y d(R,Ro) = ï¿½(2n+1)7/4 , n = 0,1,2,... (2-56)

and that depending on TAr), they can cross more than once. These undesirable features do not occur in this procedure since it started with a proper form of ym. 2-5 A Model Calculation

In the following calculations T a(R) was given the form of a Gaussian and written explicitly as

= -b(R-R x 2 (2-57)

where Rx is in the region of the pseudocrossing. Using Eq. (2-57)in Eq. (2-52) leads to the result

d

y (R,R 0 T xYr/ (4b) {erf [ Y'E(R-R )]-erf[/ rb-(R-5R H (2-58)

where erf(x) (Abramowitz and Stegun, 1972) is the error function. To obtain the parameter in Ym, Eq. (2-50) was integrated numerically using Simpson's rule(Abramowitz and Stegun, 1972) and variable step sizes.

Morse potentials (Morse, 1929), defined by

Va (R) = Di{exp[-2a (R-R?) ]-2 exp [-a (R-R)]

i=1,2, were used for the adiabatic potential curves V1 and V2 . Vl awas chosen to roughly correspond to H giving a

2 2 * 2 well depth, D1, of .176 a.u. with a minimum located at

= 1.4 a.u. and a value of a, of .801 a.u. V2 was

rather arbitrarily chosen to give a well defined avoided crossing at a distance of 2.2 a.u. This is somewhat like the pseudocrossing between H+-H2 and H-H2 (Tully and Preston, 1971). The values used were .139 a.u. for D2,

0

2.1 a.u. for R2 and 1.8 a.u. for a2.

The value of b used in Eq. (2-57) for T a(R) was 50 a.u. and Rx was set equal to 2.2 a.u. With this choice of b, the Gaussian in Eq. (2-57) has a half width of approximately

0.2 a.u. This choice of b reflects the rather sharp avoided crossing of the adiabatic potentials. Three different values of Tx were used in this study all with b = 50 a.u. The value Of Tx = 10 a.u. roughly corresponds to the results shown in Fig. (2) in the paper by Tully and Preston (Tully and Preston, 1971). The values of Tx equal to 7.5 a.u. and 5 a.u. were included for comparisons.

For Tx equal to 10 a.u. the value of the parameter a satisfying Eq. (2-50) was found to be 10.89 a.u. The results of Eqs. (2-53) through (2-55) for u=m and d are shown

d

in Fig. (2). One notices that the diagonal curves, V i and

d

V22 cross twice and they don't coincide with the adiabatic curves for small values of R. Also as seen in Fig. (3) the off diagonal matrix elements V12 diverge. This procedure clearly avoids these disparities.

The momentum couplings Tm given by Eq. (2-45) and Ta from Eq. (2-57) are compared in Fig. (4). One sees that this procedure reduces the couplings by about a factor of three. Also shown in the figure is the absolute value of the derivative with respect to R of ym, which would be equal to Ta in the strictly diabatic case. FromEq. (2-35), the range of energy where these couplings can be neglected is given by

EreIï¿½(M/2) lV/m2/ . (2-60) m ad M

From Fig. (3) and (4), estimates of V12 and P12 are given by .002 a.u. and 3 a.u. Using that the reduced mass M of the H +-H2 system is 1836 a.u., and substituting the values in Eq. (2-60) justify neglecting the momentum coupling if Erel is much less than about .0004 a.u. (or .25kcal/mole). Then for very low collision energies our procedure would be useful whereas for higher energies an adiabatic representation may be more convenient. This requirement of very low energies is due partially to the small reduced mass for H+-H2, and should not be as severe for other systems with larger reduced masses.

-0.11

-0.12

-0.13

d

> -0.14

-o , - . . _ - . . - ... .

-0.15

-0.16- "

1.8 2.0 2.2 2.4

R (a. u.)

Fig. (2) The pseudocrossing region comparing the results for the two formulations

of diabatic states for Tx = 10 a.u. The solid lines correspond to the

adiabatic potentials of Eq. (59). The dash-dot lines correspond to the diabatic potentials of Eqs. (53) and (54) with u = m. The dashed lines correspond to the diabatic potentials of Eqs. (53) and (54) with u = d.

0.005- -10.0

Ta

0.004- 8.0

0.003- 6.0

0.002- 4.0

0 . .o o \ 2 .0 P'

>

0.0 -0.0

d d

I 12

\I

-0.002 \I

-0.003- /

1.6 2'.0 2 4 28

R (a.u.)

Fig. (3) A comparison of the off diagonal potential matrix

elements for the two formulations of diabatic

states for TX = 10 a.u. The dash-dot and dashed

curves correspond to Vm and vd in Eq. (55) 12 12inE.(5

respectively. Ta given by Eq. (57) corresponds

to the solid line and is included for comparison.

10.0"

8.0

I I

I I

I I

6.0-/

-I I

d 4.0.-2 0

0.

i \. / ' \ /1 \%.

0.0 o

1.6 2.0 2.4 2.8 R (a. u.)

-2.0

Fig. (4) A comparison of the momentum couplings in the adiabatic and diabatic

representations for TX = 10 a.u. The solid curve corresponds to Ta

in Eq. (57). The dash-dot line corresponds to the absolute value of

Ym(R;a) ' in Eq. (48).

For T equal to 7.5 a.u. the value of the parameter a

x

was found to be 9.172 a.u. The diagonal elements of V and V are compared in the crossing region in Fig. (5). Although the strictly diabatic curves v. do not have multiple crossings in this case, they still don't give the correct behavior for small R. As shown in Fig. (6), the off diagonal elements of Vd again diverge.

The momentum couplings Ta and T are compared in

Fig (7). One notices that the momentum coupling is reduced by a factor of about seven. With a value of .002 a.u. for V12 and 1 a.u. for P12, the energy range where the momentum coupling can be neglected is given now by Erel <<.007 a.u. (or 4.4 kcal/mole). Thus for this case the energy range has considerably increased.

Figures )through (10) show the results for Tx = 5 a.u. In this case a is 6.840 a.u. Fig. (8) again demonstrates the

d

improper behavior for small R of the Vil's. One also notices that as Tx is decreased the crossing from this procedure occurs over a larger region. This coincides with the larger spread in v12 shown in Fig. (9). One also

d

notices the divergence of V12 in this figure. As noted previously, this will always occur if the integral of T a(R) doesn't equal 7/2.

The coupling terms are compared in Fig. (10) where one sees that this procedure reduces couplings by a factor of

M m

around ten. With V1 and P12 equal to .002 a.u. and .5 a.u., respectively, the energy must be much less than .03 a.u.

-0.11

-0.12-0.13

-0.14

-0.15- - . -'

-0.16

.1.8 2.0 2.2 2.4 R (a.u.)

Fig. (5) The pseudocrossing region comparing the results for the two formulations

of diabatic states for Tx = 7.5 a.u. The solid lines correspond to the adiabatic potentials of Eq. (59). The dash-dot lines correspond to the diabatic potentials of Eqs. (53) and (54) with u = m. The dashed lines correspond to the diabatic potentials of Eqs. (53) and (54) with u = d.

-.I \\-t

"I I

I

0.0

0.0

-0.01 .,- .

" V2 /"

\ t /

-0.002 ' / \l /

i

I.vi

-0.002 '\

I.

I \ *

I

-0.0031.6 2.o 2.4 2.8 R (a. u.)

Fig. (6) A comparison of the off diagonal potential matrix

elements for the two formulations of diabatic

states for Tx = 7.5 a.u. The dash-dot and dashed

curves correspond to VTm and vd in Eq. (55) 12 12inE.(5 respectively. Ta given by Eq. (57) corresponds

to the solid line and is included for comparison.

6 4.0,

Fig. (7) A comparison of the momentum couplings in the adiabatic and diabatic

representations for Tx = 7.5 a.u. The solid curve corresponds to Ta in Eq. (57). The dash-dot line corresponds to the absolute value of

ym(R;a)' in Eq. (48).

-0.11

1.8 2.0 2.2 2.4

R (a.u.)

Fig. (8) The pseudocrossing region comparing the results for the two formulations

of diabatic states for TX = 5 a.u. The solid lines correspond to the

adiabatic potentials of Eq. (59). The dash-dot lines correspond to the diabatic potentials of Eqs. (53) and (54) with u = m. The dashed lines correspond to the diabatic potentials of Eqs. (53) and (54) with u = d.

0.001 a I.0 .

0.0 -GO\ /

-0.001 \ /

-0.002\. 12

-0.003

1.4 1.8 2.2 2.6 R (a.u.)

A comparison of the off diagonal potential matrix elements for the two formulations of diabatic states for TX = 4 a.u. The dash-dot and dashed curves correspond to VT2 and vd in Eq. (55) respectively. Ta given by Eq. (57) corresponds to the solid line and is included for comparison.

C

Fig. (9)

Fig. (10) A comparison of the momentum couplings in the adiabatic and diabatic

representations for Tx = 5 a.u. The solid curve corresponds to Ta

in Eq. (57). The dash-dot line corresponds to the absolute value of

ym(R;a) ' in Eq. (48).

(or 19 kcal/mole). Thus again the range in energy where the momentum couplings can be neglected has considerably increased.

2-6 Discussion

This chapter has proposed an alternative procedure to construct electronic states. This procedure is based on choosing a transformation from the adiabatic representation into another that will ensure the proper behavior of the potentials away from the crossing region. The new transformation depends on a set of parameters which are obtained through Eq. (2-30). It was shown in Section 2-4 that this procedure led to a rather simple treatment of the two state problem in one dimension while the results demonstrated that it gives well behaved potentials and reduced momentum couplings.

The transformation only depends on the momentum

couplings through the set of parameters {ai}. The problem is thus broken up into two parts: (1) the determination of the parameters (which gives also the momentum couplings), and (2) the use of the transformation matrix to obtain the diabatic representation. In contrast, in the strictly diabatic transformation a numerical procedure must be implemented to determine the transformation at each point X. The transformations in this treatment are always analytical so that numerical procedures need only be used once to determine the parameters.

As was emphasized earlier, whether the momentum

couplings can be neglected or not depends on the system being investigated and the collision energy. An appealing aspect of the present procedure is that it allows us to estimate the range of energy where it can be used. If the collision energy is not in this range, one could use either the adiabatic representation with its diagonal electronic Hamiltonian matrix or the diabatic representation, Eq. (2-15), with the possibility of unrealistic potentials.

Several related problems can be studied along the present lines. In particular, a numerical treatment of the similar problem for three electronic states in one dimension would be useful. Extensions of the formalism would also be helpful in describing reactive scattering.

CHAPTER 3

GENERAL FORMALISM OF THE DYNAMICS 3-1 The One Electronic State Problem

As an introduction to the more general formalism to be presented in the following sections of this chapter, a brief development of the simpler one electronic state problem is presented using this formalism. It is hoped that this will provide a background that will aid in understanding the more general cases to be considered later. This case should also provide a more transparent connection between the formalism and classical mechanics since the results lead to classical trajectories for the nuclear particles on an electronic potential energy surface, which is a well known method for handling such systems.

This procedure, for obvious reasons, originates from a time independent treatment of the problem. Explicitly, one seeks the solutions of the time independent Schr6dinger equation,

HIT(R)> = EIT(R)> (3-1) where

H=T (R,-) + H (R)(

nu aR el (3-2)

and the notation was previously defined in Section 2-2. Since there is only one electronic state, i.e. no couplings, the adiabatic representation will be used. Using the Born-Oppenheimer approximation, the solutions to Eq. (3-1) are written as

I = Y > (R)>P(R). (3-3)

Replacing this in Eq. (3-1), multiplying from the left with

T nu (R) + V(R)4(R) = Ei(R) (3-4)

where

V(R) = < (R)IHelID(R)>, (3-5)

i.e. the adiabatic potential energy surface. It has been assumed that the electronic functions are real and normalized. Equation (3-4) is the usual time independent Schr6dinger equation for one electronic state systems which is the starting point for many treatments of this problem.

Using the nuclear coordinate system described in Section 2-2, Eq. (3-4) becomes

- -V 2 4(R) + V(R) p(R) = E4(R). (3-6)

A form for the solution of Eq. (3-6) is apparent if one considers the case of V(R) equal zero. In this case the solutions are just traveling waves. A form similar to this will be chosen for the case of a nonzero potential. This is referred to as the Eikonal Approximation and the solution is written as

S (R)

f(R) = X(R)e f . (3-7)

where S(R) is the eikonal, assumed along with X(R) to be real. Using this in Eq. (3-6) leads to, after some simplification,

2 V2 + (VS) 2 X it (VS) _ i i(V2S) X +

2mX-- (V)(VX) 2mS +(-E)X = 0 mV 2m '~m 2m -(V (3-8)

where the R dependence has been suppressed. The essential need at this point is to find an auxillary equation involving the eikonal that will lead, on a transformation into time, to a straightforward classical interpretation of it. An expression that will be seen to fulfill this is obtained by requiring that the gradient of the eikonal satisfy a Hamilton-Jacobi like equation, i.e. define

2

S+ W = E (3-9) 2m

where W is to be determined. Using this in Eq. (3-8) gives

S2 V2X -if, (VS)_(VX) -- (V 2S)X + (V- W)X = 0 .(3-10) 2m m 2

An expression for W is obtained from the real part of Eq. (3-10) and one finds that

_ f 2 7 2

W =V I 2-- X (3-11) 2m x

The imaginary part of Eq. (3-10) would lead to an expression for the flux but since it is not necessary for this discussion, the treatment of it will be deferred to a later section.

The next step involves implementing the short wavelength approximation. This approximation essentially assumes that the nuclear wave functions, X, vary slowly so that terms involving V2X can be neglected. It can be shown (this will be treated in detail in Section 3-5) that if this approximation is used in Eq. (3-10) and a transformation into time is made, X will satisfy a first order differential equation in time. Even though the case being considered here is trivial in that no electronic transitions take place, similar results, i.e. first order differential equations, will be obtained in the general case. Further, as will be shown in Section 3-6, for X expressed in several convenient forms, these

differential equations will result in X being determined by Hamilton's equations of motion.

Using the short wavelength approximation in Eq. (3-11) and replacing the result in Eq. (3-9) leads to

(V)2

2+ V = E (3-12) 2m

On transforming into time, the gradients of the eikonal become the nuclear momenta, P(t), and the nuclear positions become functions of time, R(t), so that Eq. (3-12) becomes

+ V(R(t)) = E . (3-13) 2m ~

Defining the Hamiltonian,

H(R(t), P(t)) P(t)2 V(R(t)), (3-14) Pt)) 2 +

choosing the initial positions R(t.n) and momenta P(tin)

in in such that

H(R(tin), P(tin)) = E (3-15)

and requiring that

dR H (3-16) and

d DH R(3-17) dt 3R

i.e. that the nuclear positions and momenta satisfying Hamilton's equations of motion. result in conservation of total energy (this can be easily seen by taking the total time derivative of the Hamiltonian and using Eqs. (3-16) and (3-17)).

The preceding discussion has demonstrated that, for the one-surface case, this formalism leads to the well known classical trajectory method. That is, the nuclei are treated as classical particles that evolve on a quantum mechanical potential energy surface. This should aid in seeing through some of the complications that arise in the more general cases to be considered next. 3-2 General Time Independent Formalism

In the preceding section, development of the onesurface case was presented and it was shown that the formalism led to the method of classical trajectories. The rest of this chapter will develop the formalism for a system of n electron states. The formalism will be developed without reference to a particular coordinate system in order to emphasize the generality of the procedure and avoid cumbersome notation.

In order to avoid assumptions about the time dependence of the nuclear positions and momenta, this procedure originates in a time independent formalism. Specificially, one seeks the solutions of the time independent Schr6dinger equation which satisfy

A

(H-E) IT> = 0 (3-18) where I > refers to electronic coordinates, < I > indicates integration over electronic coordinates,

A T2+

H 7M - 2 + He (3-19)

and the terms in Eq. (3-19) were defined in Section 2-2. For brevity, the dependence on nuclear coordinates has been suppressed. As was discussed in chapters one and two, one expands the solution of Eq. (3*-18) in an electronic basis. In order to further clarify the problem of non-Hermiticity mentioned in Sections 1-2 and 2-2c, a specific electronic representation will not be used at this point. The basis will, however, be assumed to be real, complete and orthonormal. Expanding in this arbitrary electronic basis leads to

IT> = (D> i(3-20)

where Ij> is a lxn row matrix and i is a nxl column matrix. Using Eq. (3-20) in Eq. (3-18), multiplying from the left by < j and integrating over electronic coordinates gives

K p + v. -V2--- V 1Iv- E =0 (3-21)

en -V V. :e SE =

where

2m

- 1 (3-22)

v = <4 --- VJ > = v+ , (3-23)

A V

V = <=IH e,> V (3-24) e z

and one must remember that, due to the presence of the gradient, the matrix v must be treated as a vector. Obviously Eqs. (3-22) through (3-24) define nxn square matrices.

As was mentioned previously, Eq. (3-21) is not in

general Hermitian. The terms that give rise to this come from Eq. (3-22). Since

V2<) , (3-25)

z z

which gives

V. + = 0 (3-26)

it is not difficult to show, by making use of the completeness of the electronic basis, that

K= .t v + mvv) .(3-27)

- 21 z z z

As was discussed in Section 2-2c, this is not Hermitian because the first term on the right hand side of Eq. (3-27) is real and antisymmetric. If this term were present in the Hamiltonian, serious difficulties would arise since energy and flux would not be conserved.

In order to avoid the difficulties mentioned above, the electronic basis will be assumed to satisfy

Vv = 0 . (3-28)

This is the same electronic basis that was discussed in Section 2-2c. It was shown there that the unitary transformation relating it to the adiabatic basis satisfies Eq. (2-22), which in this notation (the superscript n used in Section 2-2c has been suppressed)

C V2C + VCi.VC + Ct(V.da)C = 0 (3-29)

where

da = <,aVDa>

Then in this electronic representation the nuclear wave functions satisfy

h z v.V + -h+ V - E

- + + + -. (3-30) Z 2 Z Z

This equation is Hermitian and provides the starting point for the developments to be presented in the next sections. 3-3 The Eikonal Approximation

As in the one surface case, the next step in this

treatment involves using the Eikonal Approximation. The form of the solution is chosen to be the product of an amplitude written as a nxl column matrix of complex elements and a common phase that is proportional to the eikonal. A common phase is used because it will lead to common momenta and positions for all electronic channels. Explicitly, the form of the nuclear wave function is chosen to be

iS(R)

p(R) = X(R)e f (3-31)

where S(R) is the eikonal. Using Eq. (3-31) in Eq. (3-30) leads to

_ 2 iti i_____ }

2m - - -(VS)-(VX) - --(V2S)X + (VS2 X

(3-32)

+ (VS) vX +h v'VX + m v'vX + VX - EX = 0

As in Section 3-1, the gradient of the eikonal is required to satisfy a Hamilton-Jacobi like equation which gives

(VS)2

2m + W = E (3-33)

where W is to be determined. Using Eq. (3-33) in Eq. (3-32) results in

f- V2 - -(VS) . (VX) - (VZS) X + (VS)_ VX

2m m. Zn- 2m ZZ

(3-34)

h

+ + v-vx + vx- Wx = 0.

An expression for W can be obtained by multiplying Eq. (3-34) from the left by X multiplying the adjoint of Eq. (3-34) from the right by X and adding the results. Doing this one finds that

- l(XtX)-I(2XtVX + it VS. ((Vxt)X - xtvx)

2 = = m 2: - m

'h t 2 x(

- XvX + W + -(x v - (Vx) "vx) t t=

+ mx v'vx + 2VS'X vx) (3-35)

This expression for W may seem to be a bit complicated and perhaps a few comments on its general properties may be helpful.

Considering an initial channel where one of the

amplitudes equals one and all others are zero, then if an initial time is chosen so that the interactions between the fragments and the couplings (v) are zero and the amplitudes are constant, the expression for W reduces to a single potential energy surface. This is certainly what

one would expect. As time increases and the fragments enter the interaction zone, the amplitudes are free to change. In this case there are three contributions to W. The first contribution comes from the first term on the right hand side of Eq. (3-35). This can be viewed as an average potential with the weights determined from the amplitudes. The second contribution comes from the next two terms in Eq. (3-35) and arises from the variations of X. If the elements of X are slowly varying, this second contribution may be "small" enough to neglect. The final contribution comes from the last three terms which depend on the electronic couplings. At a final time when the amplitudes are again constant and the electronic couplings and interactions between the fragments are zero, W is again given by the first term on the right hand side of Eq. (3-35). However q for the final state is now an average potential with the weights determined by the final values of the amplitudes. The implications of this will be discussed in a later section after a transformation to time has been made.

An expression for the flux is obtained by multiplying Eq. (3-34) from the left by t , multiplying the adjoint of Eq. (3-34) from the right by X and subtracting the results. Carrying out the algebra leads to

ifl[(v2s)t + (VS)x t.VX + (VS).(VX )X] + yj[(V2Xt)x

- V X1 + -( - )

Xt2X + ( + (Vxt "vx) = 0 . (3-36)

Defining

(VS) XtX (3-37)

ï¿½ m

3i m =

X t , (3-38) = xvx

and

_ih [x VX - (VX) tX] (3-39)

33 2m

and recalling that the electronic basis satisfies Eq.

(3-28) one obtains

V-j - Vj = 0 (3-40)

where

: + j .(3-41)

From Eq. (3-40) one sees that there are three terms in the expression for flux. The first term j defined by Eq.

-1

(3-37) is the most transparent. At the initial time when v is the zero matrix and X is constant, Eq. (3-40) reduces to the gradient of j equal to zero. j is just the product of a velocity term given by (VS)/m times the number density of particles given by X X. At later times when v is nonzero and X is changing, the flux has additional terms

involving electronic couplings and variations of . If x is slowly varying, it may be possible to neglect the last term in Eq. (3-40) so that the flux will be given by Eq. (3-41).

As has been suggested throughout this section, if use is made of the slowly varying nature of X, the expressions for W and j are considerably simplified. This will be the topic of the next section.

3-4 The Short Wavelength Approximation

In the previous section, the Eikonal approximation

was used and it was found that the nuclear wave function, X, satisfied Eq. (3-34). This section will make use of the slowly varying nature of X in order to introduce a physically motivated approximation to Eq. (3-34). This will in turn simplify the expressions for iW and the flux.

The approximation referred to above is the well known

short wavelength approximation (Newton, 1966). This approximation is based on the classical behavior of the nuclear particles. From Eq. (3-33) one has

IVSI = V2m(E-W) (3-42)

so that a characteristic wavelength can be defined as

_ -h (3-43) c t e o 2m(E-W)

In a full classical treatment of the trajectories, 'f would

be zero so that the wavelength would also be zero as it should. The short wavelength approximation assumes that

<< 1

(3-44)

which is normally valid except around turning points where E = W. Obviously, from Eq. (3-43), as the total energy is increased this approximation becomes better so that it is in actuality a high energy approximation.

In a similar fashion one can define "characteristic wavelengths" for the amplitudes as

j,i

IV2x.J -l

v2X.I = Xj,

for an arbitrary electronic channel j. If the amplitudes are assumed to be slowly varying functions, their "wavelengths" should be large compared to K so that with Eq. (3-44) one would have that

-i1

X,1 <<

(3-46)

and

(3-45,a)

(3-45,b)

and

64

-li -i

X,2 << "(3-47)

Using these conditions in Eq. (3-34) leads to

i_ -m2Sv+'vx.+ 1T+ V _W- =0

- -(VS) .(VX) (V2S)X + (S).vX +X X

m 2m 2' z Z

(3-48)

Carrying through the same procedure that was used to obtain Eq. (3-35) results in

1 (X tX) ~ (2xt VX + - (VS)ï¿½((Vxt)x - xtVX)

+ 2(VS).X tvx + mx v-vx) (3-49)

Assuming that the second term on the right hand side of Eq. (3-49) is negligible gives

- W t =- t (V + VS-v + r vv) X . (3-50)

The general properties of W discussed in the previous section still hold here. Suffice it to say that the approximations used to obtain Eq. (3-50) have considerably simplified the determination of W in that only a knowledge of the amplitudes and not their variations is needed.

The flux is determined by using the same procedure that led to Eq. (3-36). Carrying through the algebra

results in

(V2s) t (Vs).t (Vs) vt

m X X + X VX + (Vx)x = 0 (3-51)

m ~ m z

or

V.j = 0 (3-52)

-1

where j is given by Eq. (3-37). Equation (3-52) is an expres-1I

sion for the conservation of current. In the case of one electronic surface, Eq. (3-37) reduces to

2 VS (3-53) ~, Xm

which is a well known result (Messiah, 1966).

Equations (3-48), (3-50) and (3-52) are the main results of this section. One notices that Eq. (3-48) does not contain terms involving V2X. It will be shown in the next section that, because of this, the amplitudes will satisfy first order differential equations in time. This is a pleasing aspect since Hamilton's equations of motion are also first order differential equations in time.

Another point worth mentioning has to do with the

form of q. At a later stage in this development, W will be used to construct a "classical" Hamiltonian and one sees from Eq. (3-50) that 7 depends on the amplitudes.

Then the "classical" Hamiltonian will not only depend on the traditional variables but also on the nuclear amplitudes as well. This suggests, already at this point, that all the nuclear dynamics could perhaps be determined through Hamilton's equations of motion. 3-5 Time Dependent Equations

The developments of the preceding sections of this

chapter were done with a time independent formalism. This section shall be concerned with transforming these previous results into a time dependent picture. The use of the word time is somewhat arbitrary but it will be seen later that the parameter, which the nuclear variables are a function of, plays the role of a "classical" time.

In making a transformation into time one has

R + R(t) , (3-54)

VS - VS(t) = P(t) (3-55)

and

X(R(t)) A(t) . (3-56)

Matters are somewhat simplified if a coordinate system is chosen such that one of the orthogonal unit vectors, say s, is in the direction of P (see Fig. 11). Then one has

Fig. (11) Schematic of coordinate system assumed in making a transformation

into time.

v2s = V.P = dP

~ ds

dt dP m dP ds dt P dt

P =lIE

d

V2S = m -(InP)

Furthermore,

ds d at ads=

dA

ds dta

dt ds dt

so that Eq. (3-48) becomes

dA

+l ( P-Wd

(V + P*v + m vev)A = 0

. (3--1)

By the now standard technique, one can show that

W=(A A)-I A(V + P.v + m v-v)A

(3-62)

A somewhat simpler differential equation that Eq. (3-61) results if one defines

A(t) = C(t)exp-{

t 'K d I/2}

dt'(W(t') lin P2)/

ti P)

, (3-63)

where

(3-57)

(3-58)

(3-59)

P

VX

dA dt

(3-60)

where ti is the initial time, so that

dCm

-dt + (V + P-v + r v.v)C = 0 (3-64) i dt 2 - z :

It is also relatively easy to show that

W = C)- C (V + P-v + M v.v)C (3-65)

By multiplying Eq. (3-64) from the left by C and the adjoint of Eq. (3-64) from the right by C and adding the results, one finds that

d(CtC) = 0 (3-66)

i.e. the probability is conserved. Equation (3-63) can also be written as

t

A(t) = (-1-)ePt

A ( C(t) (3-67)

which shows that the amplitudes have the correct asymptotic form. The amplitudes C are more convenient to work with because they avoid the singularities that occur in A at the turning points where P is zero.

The main result thus far is given by Eq. (3-64). If the trajectories and momenta are known functions of time,

the amplitudes are determined from a set of coupled first order differential equations in time. The remainder of this section will deal with how to determine the trajectories and momenta which will also lead to some general relations between the real and imaginary parts of the amplitudes.

As was mentioned in Section (3-3), requiring that the gradient of the eikonal satisfy Eq. (3-33) would be suggestive when making a transformation into time. Using Eq. (3-55) in Eq. (3-33) gives

2 + =E (3-68) 2m

Defining the "Hamiltonian" as

H(R,P,C) = p + W (3-69) 2m

and requiring that

dH

-= 0 (3-70) dt

would result in the conservation of energy. The dependence of the "Hamiltonian" on the amplitudes is a manifestation of its non-classical nature. Recalling that the amplitudes were assumed to be complex and writing the real and imaginary parts as CR and C. respectively which are J J

independent variables leads to dHf af dR + f dP i 7 ((3H_ )h + (3H_ )h dC3) (371 dt = (R + U)t+ K j 3CR dt + I dt (3-71)

3 )

where the partial derivatives are evaluated holding everything but the variable in question constant and H is assumed not to depend explicitly on t. If one requires that

dR

d- = 0, (3-72)

dP H

dt DR (3-73)

dCR

Sa (3-74) dt a I

and

h dCH

dt = CR(3-75)

then energy would be conserved. Equations (3-72) and (3-73) are in the form of Hamilton's classical equations of motion and provide a prescription for determining the trajectories and momenta. Equations (3-74) and (3-75) are interesting in that if the real and imaginary parts of the amplitude are

chosen to be conjugate variables, then with trajectories and momenta satisfying equations (3-72) and (3-73) respectively, energy is conserved.

It is not difficult to show that for the amplitudes expressed in terms of their real and imaginary parts Eqs. (3-74) and (3-75) are equivalent to Eq. (3-64). The advantage offered by Eq. (3-64) is that the form of the amplitudes is not specified. It will be shown later that the flexibility in choosing the form of the amplitudes will lead to other sets of canonical variables that have a straightforward physical interpretation.

The next section shall be concerned with finding

solutions of Eq. (3-64). Several different forms for the solution will be developed and it will be seen that they satisfy Eqs. (3-74) and (3-75). Having more than one form is certainly not surprising. It is in a way equivalent to the problem in classical mechanics of having more than one choice of coordinates some of which lead to a simpler treatment of the problem.

3-6 Solutions of the Equations

This section will develop solutions to Eq. (3-64) for two forms of the amplitudes. It will be shown how these forms can be used to construct canonical variables that satisfy Eqs. (3-74) and (3-75). It will also be emphasized that care must be used in selecting a set of canonical variables in that some of them can lead to inherent difficulties that arise from the form of the Hamiltonian.

The matrix C is in general a nxl column matrix of

complex numbers. A general way of writing such a matrix is to let

t

C(t) = ec(t) (3-76)

where y is a real square diagonal nxn matrix and c is a real nxl column matrix. Using Eq. (3-76) in Eq. (3-64) and rearranging terms give

i _i d i d i ei M-K

C = (V + Pv + v-v)e c (3-77)

It is not difficult to show that the jth component satisfies

i

i i n

= yjc - e (V + P-v

k=l k ï¿½ jk

i

+ m (v-v) jk)e-f k C (3-78)

An equation for c. can be found by multiplying Eq. (3-78)

3

from the left by cj, multiplying the complex conjugate of Eq. (3-78) from the left by cj and adding the results. Doing this results in

dc. = i cj-Yj n 2c j c.eh

jdt h 3 k=l

i

i e, Yj

- c]e

J ,

i

Ukj e -Fk ck n fi k= Ujke Ck

(3-80)

Uj = Vj + Pvk + m v '0 vk jk jk - k jk k

With the form of C given by Eq. (3-76), one has that

i i

= (c c) ceU e c

(3-81)

where the elements of U were defined in Eq. (3-80). It is straightforward to show that

DYj

i k

cj- iY cj e c h j 2. C2ï¿½

(3-82)

Comparing Eqs. (3-79) and (3-82) gives

where

(3-79)

2c. dc. Ec? dt jI

(3-83)

aY.

3J

or by defining

C2

p. =

j 3

(3-84)

and recalling from Eq. (3-66) that

d - c =0 at j j

(3-85)

one obtains

dP.

dt ay.

(3-86)

One would expect at this point that yj and P. are canonical variables and that they should satisfy equations similar to (3-74) and (3-75) respectively. The other relation can be obtained by sutracting the results that were used to obtain Eq. (3-79). This results in a differential equation for yj and one finds that

dy.i I i n dt" 2cj z U ke

J k=l J

jn -k ck)3

C k + e- j E U kj_ e ï¿½ (3-87)

k=l

Since, from Eq. (3-84),

n

cj = ( Ck)Pj k=l

W as a function of P. and y. becomes

3 ]

i i W = (e Pi..e

i

p n)

n

i

e-fin

n

Then

i i

w=p eij n -k aP 2 k=ie Ujk e Pk

i i

1 -y jn eF-ï¿½k +1-Pe Tz U e P 2 j k-i kj k

i i i i

_W 1 (ejYj n k- n eiYk

- -(e E Uje-'ik c + e-f-tj E U k j 2 k=l jk k k=lkj

ck) ï¿½

(3-90)

(3-88)

(3-89)

Comparing equations (3-87) and (3-90) gives the desired result in that one has

dyi

j _ H- (3-91) dt -0

I

Then as was expected, yj and P. form a set of canonical J J

variables. Expressing the total time derivative of the Hamiltonian in this set of variables and using Eqs. (3-86) and (3-91) lead to energy conservation.

An interesting aspect of these variables is seen by

considering Eq. (3-89). Since the terms on the right hand side contain the quantity Pj ,there is an essential singularity in this equation if P. is equal to zero. This can be very troublesome since normally one starts in a pure electronic state where all probabilities, Pj, are zero except one of them which is set equal to one. Trying to integrate this set of coupled equations that contain singularities at the boundary conditions is clearly not meaningful.

Even though this set of variables has. the difficulty that was mentioned above they are easily understood in a physical sense. Indeed, it is gratifying to see that the quantum mechanical probability, P., and phase, Yj, turn out to be canonical variables in a "classical" description. Although other sets of canonical variables which avoid these difficulties are used to do the calculations, it is

convenient to construct the probabilities and phases as a guide to a straightforward physical interpretation of the collision process.

Another form for the amplitudes is obtained by letting

C = aX + iSY . (3-92)

Using this form of the amplitude in Eq. (3-64) gives

dX dY

ah Su h

+ dt at- + aUX + iUY = 0 (3-93)

where the elements of U are defined by Eq. (3-80) and X and Y are real nxl column matrices. It is straightforward to show that the jth component satisfies

dx. dY.

h j + ha d + k IUjk k i UjkYk =0 (3-94) kdtk k

A differential equation for X. is obtained by subtracting

J

Eq. (3-94) from its complex conjugate which gives

dx.

-a (U.-U.)X +~ (U +U)y . (3-95)

dt 2ah kjk jkk k jk jkk

In a similar fashion, the differential equation for Y. is

3

found by adding Eq. (3-94) to its complex conjugate and one finds that

dY .1*

{a (Uj + U~kX + i8 )' (U.k - Ujk)Yk (3-96)

dt 2hn k jk jk)k k jk )kk

Equations (3-95) and (3-96) provide first order differential equations for determining the amplitudes but the task still remains to determine sets of canonical variables. In order to find sets of canonical variables, it is necessary to find an expression for W. To somewhat simplify matters, Eq. (3-66) will be used and it will be assumed that

C = 1 . (3-97)

The results of this analysis do not depend on this assumption in that if Eq. (3-97) was not satisfied, then new variables weighted by the inverse of the square root of Eq. (3-97) could be formed that would also satisfy Eqs. (3-95) and (3-96). Using Eqs. (3-97) and (3-92) in Eq. (3-65) gives

= (cxk - iSYk) J'Ukz(aXï¿½ + i8Yz) (3-98)

k k.

It is not difficult to show that

y -i8{c k (Ujk -Uk)Xk + i3 I (Ujk + Uj)y (3-99)

JY k k-Ukk k Ik j k

and

YW- = {c (UU k jk U )y (3-100)

j k k k i j

where use was made of the property that Ujk = Ukj . (3-101) Equations (3-99) and (3-100) are similar to Eqs. (3-95) and (3-96) respectively and for certain choices of a and B X. and Y. will play the role of canonical variables.

By comparing Eq. (3-95) to Eq. (3-99) and Eq. (3-96)

to Eq. (3-100) one finds that X. and Yj will form a canonical set of variables if

2 1 (3-102) Two obvious choices of a and B that satisfy Eq. (3-102) are given by letting

= B 1 (3-103) and

= 1 , (3-104) = 1 ï¿½ (3-105) These, of course, are not the only possibilities. The case

where a and 8 are given by Eq. (3-103) will be referred to as the symmetric form and this form was used in doing the calculations.

One notices that Eqs. (3-99) and (3-100) do not

contain singularities in the variables Xj and Y. so that the difficulties that occurred when using the probability and phase as canonical variables are avoided. As was mentioned earlier, the probability and phase do have the advantage of being easy to physically interpret and of being formed quite easily by using the relations

P. = c2Xj + 2Y (3-106)

and

=- tan-i( v--)

7j - ax (3-107)

The solution of the problem will then consist of integrating the coupled equations given by the expressions (3-72) and (3-73) and the corresponding ones for the amplitudes. The amplitudes will be chosen to have the form of Eq. (3-92). The probabilities and phases are found by using Eqs. (3-106) and (3-107).

3-7 Expressions for Observables

As was pointed out in Section 3-3, one of the

curious features of this formalism is that the trajectories end up on an average potential energy surface with the

weights determined by the probabilities P.. This may seem

J

to be a serious drawback because experimentally, the system begins and ends up in a definite electronic state. This feature is not unique to this formalism in that this problem also occurs in other semi-classical treatments of multi-surface systems.

Trajectories that do go from one surface to another can be found by considering the quantum mechanical expressions for the transition matrix or scattering matrix which can be expressed in terms of the initial and final states of the system. The problem then consists of finding trajectories that go smoothly from the initial to the final state. Several procedures have been developed for finding these trajectories but they are quite complicated and not easy to implement. A further discussion of this problem will be presented in chapter six.

The approach to be used in this work is based on

accepting average potentials for the trajectories and on using the P.'s to construct a total electronic transition

J

probability. The problem is that there are many "classical" trajectories that correspond to the same quantum mechanical initial state. One would then expect that some sort of an average of these "classical" probabilities, Pi, would correspond to the quantum mechanical probability.

This becomes more clear if one considers the case of acollinear collision between an atom and a diatomic molecule. The "classical" initial state would consist of

specifying a total energy, E, which is the sum of the vibrational energy, En, of the diatomic and the relative kinetic energy, the distance between the atom and the center of mass of the diatomic, the vibrational coordinate of the diatomic, the direction of the diatomic momentum and the initial amplitudes. The quantum mechanical initial state, however, would represent the diatomic by a wave function, n' corresponding to the nth vibrational state. The total electronic transition probability can be written as

Pj, j (EEn) = f dX Pj,.j (E,E nX) In (X)12 (3-108)

where X is the vibrational coordinate. Equation (3-108) provides a prescription for determining the total electronic transition probability. Final probabilities for different values of X are calculated and the total probability is obtained by averaging the final probabilities with the weights given by Eq. (3-108).

The procedure for determining total probabilities in the general case is essentially the same. For variables that are treated quantum mechanically by probability distributions, one determines the final probabilities for a number of values of the variable and averages them with the weights dependent on the probability distribution.

CHAPTER 4

THE ELECTRONIC PROBLEM FOR H3

IN THE ADIABATIC REPRESENTATION 4-1 General Considerations

As has by now become apparent, the potential energy surfaces play a crucial role in the formalism that has been developed. It was pointed out in chapter two that it is convenient to solve the electronic problem in the adiabatic representation because there exists a number of theoretically sound procedures for solving the problem in this representation. These procedures can in general be divided into three categories based on whether they are ab initio, semi-empirical or empirical in nature. The ab initio approach is of course a first principles procedure while the other two make use of experimental information.

Naturally, an ab initio approach would be theoretically more appealing. There are however two main limitations that must be considered. The first has to do with the number of electrons. Roughly the time for a calculation increases as the fourth power of the number of atomic basis functions. Thus the cost could become prohibitive for systems with many electrons. The second limitation comes from the number of nuclear configurations that are needed to obtain a reasonable potential energy surface. If ten configurations were necessary for the case of a diatomic

molecule, the surface for a triatomic system would require in the neighborhood of a thousand nuclear configurations. Even if the electronic calculation were to only take ten seconds per configuration, the surface would still require nearly three hours of calculation time. These limitations are inherent in the ab initio approach and from the second limitation alone it is unlikely that this approach would be feasible for obtaining the surfaces.

At the other extreme of the spectrum lie empirical

approaches to the problem. These approaches are based on using arbitrary functions to construct a potential energy surface. The function's parameters are adjusted until the surface passes through either experimentally known or theoretically calculated points. Even though these approaches bypass the need to do a quantum mechanical calculation, quite a bit of guesswork is involved. If these approaches were used on systems that didn't contain a wealth of experimental or theoretical information there would exist a significant risk of obtaining unreliable results.

The final class of approaches to be considered here are the semi-empirical ones. These approaches have the advantage of starting with the time independent Schradinger equation. This can not only lead to a better understanding of the approximations used but also offer some insight into its limitations. The number of semi-empirical approaches

is staggering and in a way reflects the great diversity of physical systems being studied.

As is well known, most of the effort in an electronic calculation is spent in evaluating the two center electron integrals. The common trait of semi-empirical approaches is that they either simplify or reduce the number of those integrals. A well known approach that simplifies the exchange integral is the Xa method (Slater, 1971). There are a number of other methods such as CNDO that simplify the problem by reducing the number of electron integrals.

One of the possible difficulties that arise in using most semi-empirical methods is that they usually employ only one electronic configuration, i.e. a single antisymmetrized product of molecular orbitals. This may give a good description of the surface for some regions of the internuclear coordinates but usually does not adequately describe the entire surface, which is necessary for the scattering calculation. Although there are semi-empirical methods that employ configuration interactions, the number of internuclear configurations needed to determine the surface would more than likely make using these procedures too costly.

The method that will be employed here is called Diatomics in Molecules. It was introduced by Ellison (Ellison, 1963) and later generalized to include directional bonding (Kuntz and Roach, 1972; Tully, 1973). It has fairly recently been used to construct a number of

potential surfaces for triatomic systems. Two of the many examples are the surfaces for LiH2 and FH2 (Tully, 1973a).

This method utilizes the fact that the electronic

Hamiltonian can be written in terms of atomic and diatomic Hamiltonians. This leads, as will be seen in the next section, to a solution which is expressed in terms of an overlap matrix and atomic and diatomic energies. The ground and excited energy levels of the diatomics are obtained through either experiment or theory. Whether or not the electronic problem can be solved with this approach depends on the availability of the diatomic energy levels.

Another advantage of this method comes from the form of the electronic basis functions. Since the basis functions are expressed as an antisymmetrized product of atomic orbitals, a little manipulation of the antisymmetrizer leads to a valence bond description of the electronic problem. Thus, this method has built into it the correct electronic description of the reactants and products.

In the following section the formalism for Diatomics in Molecules will be briefly developed. This formalism will be applied to the H3+ system and the eigenvalues and eigenfunctions will be the topic of Sections 4-3 and 4-4 respectively. Section 4-5 will focus on the non-adiabatic couplings which are a convenient by-product of this method. 4-2 Method of Diatomics in Molecules

The goal of Diatomics in Molecules (DIM) is to make use of the energies of the diatomics that comprise the

polyatomic to construct the polyatomic energies. If the diatomic energies are all known experimentally, this procedure eliminates the necessity of doing an electronic calculation except perhaps for overlaps. In most applications the internuclear dependence of the overlaps is neglected so that the electronic calculation is avoided altogether and the electronic problem is solved with only a knowledge of the diatomic energies.

Since there are already a number of detailed treatments of this subject available in the literature (Ellison, 1963; Tully, 1973b), only a brief outline of the method will be presented here. This will hopefully have the advantage of displaying the main conclusions without resorting to the rather cumbersome formalism that is inherent in more detailed treatments. The development presented here will somewhat follow the one given by Tully (Tully, 1980).

The basis functions in this procedure are chosen to be antisymmetrized products of atomic functions. For a system composed of n electons and N nuclei where at least three nuclei are assumed to be present, the basis functions can be written as

D (l,'-,n) = A nm (l,-,n) (4-1)

where An is the n electron antisymmetrizer and the m are defined as (Moffitt, 1951)

m(l,..., n) T= (A) (l'''''nA)n(B) (n + 1,000,n + n m m (, nA~f m AA B

0 ..1(N) (n N + l,.* ,n) (4-2)

mN

where the atomic functions n are assumed to be antisymmetric. The notation has been somewhat changed here in order to emphasize the importance of assigning electrons to particular nuclei. The bracket will still be used to indicate integration over the relevant electronic coordinates. Expanding the total wave function in terms of the basis of Eq. (4-1) gives

T (1,--.,n) = E (D (l,.--,n)r m (4-3)

k m

where rpm are the expansion coefficients. Using Eq. (4-3) in the time independent Schr6dinger equation leads to

Hr = SEE (4-4)

where the adiabatic energies are given by the diagonal matrix E,

Hmm, = (4-5)

and

Smm, = <(mI(mI>

(4-6)

Equation (4-5) was obtained by using the property that the antisymmetrizer commutes with the Hamiltonian.

The next step in this procedure is to partition the Hamiltonian into components that are atomic and diatomic in nature. This can be accomplished by writing (Ellison, 1963)

A N N A N

H(KL) - (N-2) I H(K) (4-7) K=l L>K K=l where H(KL) is the Hamiltonian for the isolated diatomic which is comprised of the nuclei K and L and the electrons assigned to those nuclei. A similar definition holds for H(K) In the electronic basis defined by Eq. (4-1), the operator H has the following matrix representation

A N N N

H = X H(KL) - (N-2) H(K) (4-8)

K=l L>K = K=l z where

HIM (4-9)

and

HmmL= <4m IAnH(KL) IDmv> (4-10)

Since the total antisymmetrizer does not commute with the fragment Hamiltonians, HL and ( Eq. (4-8) is not term by term Hermitian. The sum will however remain Hermitian if a complete set of electronic states are used.

Equation (4-8) is of utmost importance in this theory. One sees from Eqs. (4-9) and (4-10) that the total Hamiltonian is now expressed in terms of fragment Hamiltonians whose operators correspond to isolated atomic and diatomic systems. The solution to the problem will consist of making transformations of the electronic basis given by Eq. (4-1) to bases that diagonalize the fragment Hamiltonians, i.e. whose eigenvalues correspond to the experimental or theoretical energies.

In order to show how these transformations are done, a brief development for the operator H(K) is presented here. For clarity, the basis functions are written in the form

#(K) = 0(R)-(K) (4-11) m m m

where (K) is the product of all atomic functions not centered on atom K and n (K) is the atomic function centered

m

on atom K. Of course, the electrons are still assigned according to Eq. (4-1). Assuming that the electronic basis is complete, one has

H (K) (K) (K)(K) (4-12) mK P, P, m