Integral equation approach to atom-diatom exchange processes

INTEGRAL EQUATION APPROACH TO ATOM-DIATOM EXCHANGE PROCESSES

Michael BAER Department of Physics and Applied Mathematics, Soreq Nuclear Research Center, Yavne, 70600 Israel

I

NORTH-HOLLAND

-

AMSTERDAM

PHYSICS REPORTS (Review Section of Physics Letters) 178, No. 3 (1989) 99—143. North-Holland, Amsterdam

INTEGRAL EQUATION APPROACH TO ATOM-DIATOM EXCHANGE PROCESSES* Michael BAER Department of Physics and Applied Mathematics, Soreq Nuclear Research Center. Yavne, 70601.) Israel Received September 1988

Contents: 1. Introduction S 2. The Lippmann—Schwinger integral equation for exchange collisions (the BKLT equations) 3. The relation of the BKLT equations to other types of integral equations 3.1. The Faddeev—Lovelace integral equations 3.2. The Fock coupling scheme 4. Studies of special cases 4.1. The arrangement channel permuting array structure 4.2. The angle-dependent W matrix elements 4.3. The symmetric generalized potential 5. The coordinate representation of the BKLT equations

101 102 106 106 107 108 108 110 Ill 115

5.1. The application of the distorted wave Green functions 5.2. The collinear configuration 5.3. The three-dimensional configuration 6. The infinite-order sudden approximation (IOSA) 6.1. Introduction 6.2. The derivation of the IOSA—BKLT equations 6.3. The IOSA and the angle-dependent W matrix 6.4. Derivation of the IOSA T matrix elements ~ Numerical results 8. Concluding remarks References

115 118 123 130 130 131 135 136 136 140 141

Abstract: This review presents a discussion of integral equation approaches to exchange collisions between an atom and a diatomic molecule which result from a straightforward extension of the Lippmann—Schwinger equation originally devised for inelastic collisions. The extension to N arrangement channels is done via a N x N matrix Wwhich is responsible for the explicit couplings between the various arrangement channels. Different choices of the W matrix elements lead to different sets of integral equations; several of these are well known, whereas others are new. In addition to the theoretical derivations and analysis we refer to their coordinate representation, discuss numerical aspects and also present accurate results of the three-dimensional three-channel system H + H 2. In addition, an approximate set of integral equations is derived within the framework of the infinite-order sudden approximation. These equations are numerically tractable for most systems of interest and are known to yield relevant results.

*

This work was supported by the United States—Israel Binational Science Foundation, Jerusalem. Israel, under grant 85-0000 2/1.

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M. Baer, integral equation approach to atom—diatom exchange processes

101

1. Introduction

Molecular dynamics is one of the fields in physics in which various theoretical approaches are used extensively to study collisional processes among three or more particles (atoms). In general, one can distinguish between two “traditions”: one based on the differential equation approach, according to which one considers the Schrödinger equation and solves it either by propagation or by finite-difference (element) techniques; and the other based on the integral equation approach and solving the Lippmann—Schwinger equation. The main approach employed in the study of atom—diatom exchange processes was based on solving the Schrödinger equation [1—8].The theoretical background for this approach is relatively simple and straightforward and therefore has led to much activity. Progress was rapid mainly from the mid-sixties to the mid-seventies, during which period many numerical techniques were developed and a wealth of interesting results (which were then followed by beautiful models and approximations) were produced. The emphasis during this period was on collinear systems where the three particles were restricted to moving along a straight line throughout the reaction process (see fig. 1). By definition, such a system could not have more than two arrangement channels. The transition to two and three dimensions, usually accompanied by a transition from a two-arrangement-channel to a three-arrangement-channel system, encountered difficulties so severe [9—12]that theoreticians had to try other approaches, including approaches based on the operator formalism, which led to the integral equation methods. Many integral equation approaches to treat exchange processes are generally known, but only a few of them were tried for molecular dynamics. Among these we distinguish the Faddeev equations [13—15] and the more recent Fock scheme [16,17], which is closely related to a similar integro—differential scheme [18—21]. In the present review, we shall be interested mainly in a third approach, namely, the integral equation version of the Arrangement Channel Coupling Array, better known as the Baer— Kouri—Levin—Tobocman (BKLT) coupled T equations [22—91].According to this approach, the ordinary Lippmann—Schwinger (LS) equation derived for one arrangement channel is extended to a multi-channel system by forming a set of coupled integral equations one for each arrangement channel. The coupling is done via a general matrix W of dimension N x N, where N is the number of arrangement channels. The W matrix elements are to a large extent unrestricted and therefore different choices lead to different coupled systems of integral equations. In this composition we shall deal with several choices of these matrix elements and with the features of the resulting equations. —

R~

B(2)

C(3)

A(1)

Rv

B(2)

C(3)

rv

Fig. 1. The collinear system. Note the definitions of R~and R~.

A(1)

102

M. Barr, integral equation approach to atom—diatom exchange processes

At this stage a comment must be made. There is one major difference regarding exchange processes taking place between nuclei and those taking place between molecules; whereas for nuclei the existence of three-body continuum states is expected to have a significant effect on the exchange process, for most molecular systems of interest these continuum states are too far away to have any effect on the final outcome. This fact not only helps us to sort out the more appropriate methods, but also justifies the use of certain integral equations, without their potentials (and therefore also their kernels) necessarily being connected (a discussion related to this subject is given at the beginning of section 4.1).

2. The Lippmann—Schwinger integral equation for exchange collisions (the BKLT equations) A concept that will be mentioned frequently in this review is the concept of arrangement channel. In the case of three particles, we distinguish among three arrangements, designated as X, v and k (as mentioned before, the continuum states will be ignored). From fig. 2 it is noticed that the X arrangement channel describes the two clusters (A, CB) [or (1,32)1, the v arrangement describes the clusters (B, AC) [or (2, 13)], and the k arrangement describes the clusters (C, BA) [or (3, 21)]. Unless otherwise specified, the X arrangement is the initial arrangement. One of the more fundamental operators considered in scattering theory is the transition operator Ta~ between an initial state in the X arrangement and a final state in arrangement a(=X, v, k). In general, one distinguishes between two forms of the transition operator: the post form, designated as ~ and the prior form, designated as Ta~•If x1~.)and ~ are the X-unperturbed solution and the X-total solution, respectively, of the Schrodinger equation [as for the definitions of (2.4)], then ~ is defined as [92]

xx)

and

~i)

T~jx~~

see eqs.

(2.1)

=

where V~is the a-perturbation potential. In the same way, if

(xJ and

~IJ~Jare the a-unperturbed

A(1)

r~,

~/

HV(\

K

R

K

RV

B(2)

C(3)

Fig. 2. The three-dimensional system. Note the three orientation angles y~,y~and

Yk

as well as the angle .i,~.

M. Baer, integral equation approach to atom—diatom exchange processes

103

solution and the a -total solution, respectively, of the Schrädinger equation, then Ta~is defined as

[921

(2.1’)

(XaV~ak=(i!~lVX,

where V~is the perturbation potential in the X-arrangement channel. If H is the complete Hamiltonian and Ha is the a-arrangement unperturbed Hamiltonian, then

(2.2)

H=Ha+Va~

where Va fulfills the requirement (2.3)

lifflVa(Ra,.•.)=0.

The two functions Xa) and HaIXa) = EIXa),

i!;)

are solutions of the equations

H1I1J) = El~I).

We now consider the LS equation for

Ii!

i!’) [931,

lXa)+G 1’~*lXa),

and the LS equation for

(t/I

(2.4a,b)

(2.5a)

I,

(kaH(XaH(XalI”aGl,

(2.5b)

where G ± are the Green functions of the full Hamiltonian, G~=(E—H±ie)’.

(2.6)

Multiplying both sides of eq. (2.5a) from the left by Va and employing the definition of Tax~we get the integral representation of Ta~~ ~

(2.7)

TaV+VaG~V~~

(2.8)

or

In the same way we find for T,~the equation ~

The main difference between Ta~and Ta~is, as can be seen, in the inhomogeneity term. Next we consider eq. (2.8) and do the following:

(2.9)

104

M. Baer, integral equation approach to atom—diatom exchange processes

(a) Apply for any a’the identity

E—H+it/-=E—H+ir—V.,

(2.10)

which leads to an integral equation for G

G~=

G:. +

~,

GtV.G~.

(2.11)

(b) Introduce a matrix W with the elements W,ja [25] fulfilling for each c~the condition ~ W~ = 1

for any

(2.12)

~.

(c) Multiply eq. (2.11) by ~ [25]

and sum over a’ to obtain a more global equation for G~,namely

G~=>. W-G~. +~ W~aG~tVaG~•

(2.13)

(d) Substitute eq. (2.13) in eq. (2.8) to obtain Ta~=

+

= Va +

Va(~w~a.G:.+ ~ ~ ~

= Va + Va ~

W-.Gt(V~+ VaG~V~) w- G~(V~ V~)+ V ~ w-~Gt(V -

+ VaG~V~),

or ~a~a(1+~

(2.14)

W~aGat(V~_Va))+~VaW&aGatTa.~.

To continue, we consider the first term on the rhs of eq. (2.14), 1

+

~

=~

w,~a.G:~(VxV,,~)= 1 —

+

~

w,~a~G:.(G:)~’ = W~ +

w,~a,G:’[(G~)~’(G:.y’] —

~

W,jaG(G)~

The last term on the rhs becomes zero when it operates on an initial non-break-up state Consequently the final form of the equation for Ta~is [29] Ta~= V~W~ + ~ VW-,G~TX.

lxx)

[13,94].

(2.15)

In a similar, but more straightforward way, one may show, employing eqs. (2.9) and (2.13) that [22—25]

M. Baer, integral equation approach to atom—diatom exchange processes

=

V~+ ~ ~

105

(2.16)

The two sets of equations are expected to yield similar results; however, it is known that eq. (2.15) is more convenient for numerical applications [29]. It is noted that the index a has not yet been assigned a value, and different choices may lead to different equations. In all our applications a was assumed to be equal to a. However, in order to obtain the Faddeev—Lovelace equations [13, 95], for instance a, as will be shown later, is chosen differently. The main advantage of this choice is that it enables us to define a generalized potential matrix V with elements (2.17)

~Taa1/’a~~aa

Consequently, eq. (2.15) becomes Ta~= Va~+ ~ VaaGTaX~

(2.18)

which can also be written as a matrix equation,

T=V+VG~T,

(2.19)

where (2.20) In a similar way eq. (2.16) will be written as

T=UV+VGT,

(2.21)

where U is the matrix /1

.

1\ (2.22)

\i~~•iJ This compact way of writing the BKLT equations is reminiscent of the single-arrangement LS equation. In what follows we will consider only eq. (2.19) [oreq. (2.15)] and consequently the sign over T will be omitted. An important feature of the BKLT equations as given in eq. (2.19) is the fact that the global reactance operator K, defined as [29]

K=V+VG~K,

(2.23)

where G~is the principal value of G, satisfies the Heitler damping equation. The proof will not be given here because it is very similar to the one given for the elastic case. Thus

106

M. Baer, integral equation approach to atom—diatom exchange processes

T=K—iirK~5(E!—H0)T,

(2.24)

8(E!

(2.25)

where —

HO)IAA.

=

~AA.~(E — HA).

Like the global transition matrix T we may now introduce the global scattering matrix S, which is defined as

S=I+2iri6(E!—H0)T.

(2.26)

Consequently the relation between S and K becomes S=[I—iirô(EI— H~)K][I+iir6(EI— H~)K]’

(2.27)

.

In all numerical applications the global K matrix and not the global T matrix elements are calculated, because it avoids the need to deal with complex numbers. Comment. In what follows, all Green functions G will be assumed to be G + unless otherwise specified. 3. The relation of the BKLT equations to other types of integral equations In this section we briefly discuss the relation between the BKLT equations and other equations, known in the literature as the Faddeev—Lovelace equations [13,95] and the Fock Coupling Scheme [18]. 3.1. The Faddeev—Lovelace integral equations In order to derive the Faddeev—Lovelace equation, we must consider in more detail the perturbation potential Va written in the form (3.1)

where V~is the two-body potential defined for arrangement channel urn V= V~

a.

Thus (3.2)

.

Substituting eq. (3.1) in eq. (2.8) leads to T~= Va

+

~

(1

—

6-)V~GV~.

(3.3)

Recalling the global representation of G given in eq. (2.13) and identifying the undetermined index ~ with the summation index of eq. (3.3) leads to the following expression [see derivation of eq. (2.15)] [77]:

M. Baer, integral equation approach to atom—diatom exchange processes

Ta~= ~

Fixing the ~ (1

(1

—

W~a in —

~

+

~

(~ (1

—

~a~)~’W*ia)GaTax.~

107

(3.4)

such a way that

~ail”14’äa=

(1

~aa’)~’°~’

(3.5)

[this choice is compatible with eq. (2.12)] yields finally the expression Ta~=~(1~ ~aX)~” +~

(1— ~aa,)~~TaGa,Ta,~,

(3.6)

which is one form of writing the Faddeev—Lovelace equation [77] and is also close to an extended off-shell equation known as the Alt—Glassberger—Sandhas equation [96]. 3.2. The Fock coupling scheme In contrast to the Faddeev—Lovelace equations the Fock coupling scheme is much less known and therefore deserves short introductory remarks. This system of integral equations was recently introduced by Zhang et al. [17] to calculate atom—diatom exchange S matrix elements. The Fock coupling

scheme was shown to be equivalent to that applied in the multi-configuration Hartree—Fock method, which is used for electron scattering [19] and for atom—molecule reactions [20, 21]. We will not derive this integral equation following the original paper [17] but present a derivation which follows directly from eq. (2.15) [86]. Summing both sides of eq. (2.15) with respect to index a yields ~ T~= ~ VaW~~ + ~

where

a may be equal to a or

(~

VaW~a)Ga~TaX~

(3.7)

to some function of a. If now, again, W&a are chosen in such a way that (3.8)

we get ~

(3.9)

Equation (3.9) can also be written as ~

Ta~•

(3.10)

Equation (3.10) is the basic equation employed in the Fock coupling scheme [17]. To obtain from eq. (3.10) three independent equations, each of the amplitude density functions

I~~) TaXI4X) =

is expanded in an L2 basis set within its own arrangement channel.

(3.11)

108

M. Baer, integral equation approach to atom—diatom exchange processes

4. Studies of special cases In this chapter various systems of integral equations are introduced which follow from different choices of W matrix elements and some of their features will be investigated. 4.1. The arrangement channel permuting array structure The simplest and most straightforward choice is to assume the Wmatrix elements to be constant (i.e. pure numbers). This choice was made by Baer and Kouri [22—25]when they first presented their coupled T equations, which later became known as the BKLT equations. Moreover, they found that for a two-arrangement-channel system the best choice for W would be

w= (~~).

(4.1)

This structure was later generalized by Tobocman [30], as will be shown. One of the main questions one encounters in the theory of integral equations is whether the kernel is connected or not. When the LS equation is considered, the question is asked differently because the kernel consists of a product of Green functions which is assumed to be not necessarily connected and a potential function for which connectivity is required. Connectivity implies that either the potential or one of its powers becomes zero when any of the interatomic distances goes to infinity. The proof of the connectivity for the present case will follow the one given by Tobocman [30]. For this purpose we consider eq. (2.18) (with a = a) and recall that the kernel contains not just the perturbation potential but also a W matrix element. The proof is as follows. We consider the three-particle potential V(r,1, r13, r23), which will be written in the form (r~1is the distance between particles i and V=

va

+

j)

Vx~k,

(4.2)

where the VXVk potential is a three-body potential which becomes zero whenever one of the interatomic distances becomes large and the V” potential is defined as [seealso eq. (3.1)] VaVjJ+V,.k,

a(i,jk).

(4.3)

Here the two two-body potentials Vq and V~kare assumed to be dependent on the interatomic distances r11 and rlk, respectively. The perturbation V~is defined as V~=V—V73,

(4.4)

which becomes, following eq. (4.2), V~= VA

+

Vx~k.

(4.4’)

Since V~’is a function of two distances only, i.e. r15 and r13, it will not become zero when r23 becomes infinity and so the kernel of the corresponding integral in eq. (2.18) seems to be not connected.

M. Baer, Integral equation approach to atom—diatom exchange processes

109

However, whether the kernel is connected or not depends on whether following a finite number of iterations the resulting kernel does or does not vanish when any of the three interatomic distances becomes infinite. In order to carry out the iterative procedure we consider eq. (2.19); it can be seen that after n iterations eq. (2.19) becomes ~ (c~G)’~+ (~G)”T.

(4.5)

Thus, the kernel is now (VG)”, and to check connectivity we examine V”, i.e.,

V

=

(VW)’1.

(4.6)

For the three-particle system, it is enough to consider the case n = 3 (in the more general case, but still with one reaction coordinate per channel, n is equal to the number of possible clusters). Thus (“3’ Jaa’ I~V ~ aa” a’a” Va”a’ In order to evaluate this expression we considered the case a = a’ and a ~ a’ separately. aa’: + a

3)aa=V~a+ ~ ~“aa~1 a~~/’a~a+21”aa ~ a’a aa”

(V

~ VaV~aVa~~ aa”#a’~a

a’:

(V3)~~. = (V~a+

+ Vaa(

(4.8a)

‘~aYl’aa + VaaVaa~~aa+ (~aa+ ‘aa)~a,V~,a~ + V~,aVa~aV.*,~a~

VaaVaa~+ VaaVa~a+ Vaa~Va~a),

(4.8b)

where a ~ a’ ~ a”. In eq. (4.8a) the first three terms contain diagonal elements, whereas the other kinds are included in the fourth term. In eq. (4.8b), the first four terms contain diagonal elements and the fifth contains products of symmetric terms. It can be shown that except for the last term in eq. (4.8a) all the others lead to disconnected kernels. Thus, in order for the whole system of integral equations to be connected, the first three terms in eq. (4.8a) and all the terms in eq. (4.8b) have to be made identically zero and this can be achieved by assuming the W matrix to have a permuting array structure. Thus

fi,

j=i+1,i=1,...,N—1,

W.=~1, j=1,i=N, I~0, otherwise.

(4.9)

For the three-channel case W becomes /0 1

0\

/0 0 1\ W=(o 0 iJ or w=(i 0 0). \i 0 0! \0 1 0!

(4.10)

S

110

M. Baer, Integral equation approach to atom—diatom exchange processes

The permuting array structure followed from the assumption that the Wmatrix elements are constants. In numerical applications it was verified that this choice led to correct results for any two-channel system it was applied to but then failed once it was tested for a three-arrangement-channel case [82]. All numerical studies were carried out without including three-body continuum (the energies are usually low enough that these continuum states are not expected to affect the final results) and therefore the failure of the BKLT equations to produce the results for a three-channel case is attributed to the fact that the coupling scheme proposed by the channel permuting array structure is too weak and three-body continuum states have to be included in the calculations in order to obtain the correct results. This weakness seems to be due to the high asymmetry built into the W matrix so that each channel is explicitly coupled only to one other channel. 4.2. The angle-dependent W matrix elements [83, 84] From the discussion in the previous section, it seems that the failure of the BKLT equations to produce relevant results for a three-channel system is related to the assumption that the W matrix elements are pure numbers. Recently, Baer and Shima [83] considered W matrix elements which are functions of the internal orientation angles Ya of the three-body system defined as (see fig. 2) a=X,v,k,

~

(4.11)

where Ra and I~aare the a-arrangement translational and vibrational unit vectors, respectively. From eq. (2.15) it can be seen that W~ais a matrix element which couples arrangement a’ to arrangement Thus, for c~= X, the three matrix elements which couple the arrangement channel X to itself and to the other two are ~ W~,, and WAk, respectively. Since the channel is. stands for the arrangement (A, BC), the coordinate according to which it will be decided whether the channel is coupled to any of the three channels is the angle ~ (see fig. 2). Thus, when ‘~is small, the channel is expected to be coupled more intensely to the v (=B, CA) channel, and when ~ is close to ir, it is expected to be coupled more intensely to the k (=C, AB) channel. For the Y~interval where the is. channel is only weakly coupled to the other channels (usually for y~, i~/2),it will be assumed to be coupled to itself. One could think of many functional forms for the W matrix elements that would comply with the above description [and satisfy eq. (2.12)]. However, if again one requires the kernel to be connected, then these functions have to be chosen with some care. The simplest choice are step functions, ~.

-~

11, 10,

W~~(y~) =

°
WAk(YA)

ill, ~0, —

Y
_J1, [~,

—

~ otherwise. (4.112)

The proof for the connectivity was given by Neuhauser and Baer [87]and is as follows. We consider the three times iterated kernel, as given in eq. (4.8). First, we prove that the diagonal element VAX becomes zero if any of the three interatomic distances becomes infinite. V~is presented in eq. (4.4) and it is seen that it becomes zero when r12 (=rAB) and r13 (‘rAC) become infinity, but it may stay finite when r23 (rBC) becomes infinite. Let us assume that particle 2 is now moved to infinity such that the distance between particles 1 and 3 remains finite (if not, then V~will become zero). However, while we do that,

M. Baer, Integral equation approach to atom—diatom exchange processes

111

the angle ~ becomes close to zero (see fig. 3a), namely urn

r.,

(4.13)

YK°~

5—’~,r13<~

Recalling eq. (4.12), it is noticed that, as a result of that process (and assuming y ~ ~ 0), ~ becomes zero, thus making zero the corresponding diagonal element of the generalized potential matrix V. Next we must show that any product of two symmetrical elements of the generalized potential matrix, i.e. VXVVVA, also becomes zero when any of the three distances becomes infinite. The product V~V~ becomes zero when either particle 1 or particle 2 is moved to infinity but differs from zero when particle 3 is moved to infinity (assuming r12 is finite). However, moving particle 3 to infinity causes y~to become ir (see fig. 3b) and consequently W~,,,which is only nonzero for angles much smaller than IT, will become zero, thus making the product ~ zero. So far, we have proved that three out of the four terms in eq. (4.8a) and all terms in eq. (4.8b) will become zero when any of the three particles is moved to infinity. The fourth term in eq. (4.8a) will go to zero because at least one of the three ~a ‘s will be zero when any of the three distances is infinity, like in the channel permuting array case. Thus, the generalized potential with the new choice of W is connected and consequently the whole kernel is connected. As will be shown later, this choice of W matrix elements produces a coupled system of integral equations which was found to yield the correct transition probabilities for a three-dimensional three-channel system [84,88]. 4.3. The symmetric generalized potential The fact that the previous choice of W is able to yield correct results for a three-channel system whereas the channel permuting array structure failed to do so was attributed to the fact that in the former case the resulting generalized potential V is much more symmetric. This brought us to the idea

2~

~ Fig. 3. The three-particle system in extreme situations: (a) The situation when particle 2 is moved to infinity such that r,3 stays finite. The angle y~ becomes zero. (b) The situation when particle 3 is moved to infinity such that the distance r,2 stays finite. The angle y~becomes sr.

S

112

M. Baer, Integral equation approach to atom—diatom exchange processes

of imposing full symmetry on V[87]. In other words, we require that the W matrix elements fulfill the conditions -~

(4.14)

VW~=Va~W~a~

Equation (4.14) together with eq. (2.12) are not enough to determine uniquely all the elements of W Thus, in the case of two channels the number of unknown W matrix elements is four with only three equations to determine them, and in case of three channels, the number is nine with only six equations to determine them. It turns out that the two-arrangement-channel case is more complicated than the three-arrangementchannel case, and therefore we start the derivations of the generalized potentials for the threearrangement-channel case. However, before doing so, one comment has to be made. The symmetric choice will never yield a generalized potential which is entirely connected and therefore a system where the three-body continuum plays an important role cannot be treated with these equations. However, we will show that for all cases where the~three-body continuum can be ignored (like in most cases of interest in molecular dynamics) the kernel is connected, the missing connectivity being introduced by the Green functions themselves. The three-arrangement-channel case In this case we have, as mentioned earlier, nine unknowns and six equations. Consequently, we shall express the six off-diagonal elements in terms of the three diagonal ones. Thus, if 4.3.1.

Waa=1~F,~~ aX,v,k,

(4.15)

it can be shown that

V.

=

~(FV

+

TV.

—

F~V.),

(4.16)

where a a’ ~ a”~a and the ~a may or may not be constants, but they are assumed to be compact. Next, we prove that I~= f~.= I~,..,a ~ a’ ~ a”. To do that we consider the kernels of the equation [see eq. (2.15) with a = a], ~

=

(4.17)

+ VXXGX TAX + ~XVG~TVX + VXkGkTkX,

and analyse their behavior in the three asymptotic regions, i.e., Ra a = it, v, k. Before continuing, we shall introduce the assumption that none of the three unperturbed Green functions Ga~a = A, v, k, contains a contribution of the continuum other than the one along its own translational coordinate Ra• Thus, ~

fora~a’.

(4.18)

In other words, contributions from the bound two-particle continuum are ignored. In this context, it should be reiterated that our aim is to find W matrix elements (or generalized potential matrix elements) such that the neglect of these portions of the Green functions will not affect any of the results of the computations. This assumption is known to be relevant for molecular systems where for low

M. Baer, Integral equation approach to atom—diatom exchange processes

113

energy (chemical energies) many of the closed two-particle discrete states, not to mention the corresponding continuum states, can be ignored. Let us now consider each kernel separately. (i) The kernel ~ This kernel decays to zero along all three asymptotes, either by definition [see eq. (2.3)], 1irnV~=0,

(4.19)

or by eq. (4.18), 1irnG~(e~0)=0fora=v,k.

(4.20)

(ii) The kernel ~ This kernel decays to zero when R~ or Rk cc because G~(e~ 0) does so according to eq. (4.18). As for R~_~ao, we have to consider V~,,given in the form —*

=

~(1~V~ + F,,V~ 1V~)

—~

(4.21)

—

in greater detail. Recalling the definitions of Va~a = A, v, k, and considering the finite v case, we get (see fig. 3a) urn V—+ V13,

(4.22)

lirnV12=lirnV23=0.

(4.23)

It can be shown that ii

V~= ~V13([~ .1).

(4.24)

—

Thus, to guarantee the connectivity of the kernel VXVGV (employing only the discrete portions of the Green functions), one must choose 1~and 1 such that f~= i. In the same way, we can prove that I~and therefore also F~= 1. Thus, we have (4.25)

Consequently, the generalized potentials have the form VaaT(V~/~Va~),

aa’~a”;

Vaa(11’)Va,

(4.26)

where F is a constant not yet determined. 4.3.2. The two-arrangement-channel case In this context, we present two versions. (i) One version is identical to the three-arrangement-channel case where the third term in the first two equations and the third equation of eqs. (2.19) are simply ignored. Thus, we have

114

M. Baer, Integral equation approach to atom—diatom exchange processes

Vaa=(1~E)Va~

aX,v;

~F(VX+VPVk).

(4.27)

The neglect of these terms is justified by the fact that the third channel either is energetically closed (as is frequently encountered in molecular dynamics) and therefore Ga (r $ 0) is zero in all asymptotes, or it cannot be reached as, for instance, in the collinear case. (ii) The representation of the off-diagonal term, as given in eq. (4.27), is somewhat odd, because of the appearance of Vk, a perturbation related to an ignored channel. If can be shown that any representation of this kind, ignoring Vk and assuming F to be a constant, will lead to a non-connected kernel. We shall therefore write W

=

1— I~,

(4.15’)

where F is assumed to be a function of the coordinates. Recalling eq. (2.12) and eq. (4.14), we get Vaa=(1~F~)Va;

V~=V~a=FaVa~

a~X,v;

~

(4.28)

We analyse the term V~~G,(E$O). 3 From previous discussions we know that both Vap (=I~,Va)and G~(e ~ 0) decay to zero when Ra cc but they will not become zero when R,.~ cc unless F is defined in such a way that —*

—~

lim .f, = 0.

(4.29)

R~—‘

similar representation of eq. (4.28) for the generalized potential elements can be obtained by taking W~= 1 and assuming 1 to be dependent on the coordinates. However, in order to ensure connectivity we need to assume that A

—

lim F =0.

(4.29’)

R~=oc

In order for the two choices to lead to the same set of integral equations, we must assume

FpVpTV.

(4.30)

The requirements (4.29) and (4.29’) can now be fulfilled if

F,~=FV~,a~f3,

(4.31)

where F may or may not be a constant. Combining eqs. (4.28), (4.30) and (4.31), we get 1”XX =

V~ FV~V~, —

V~=

FV~V~, v~=

—

rV~v~,

where, like in the three-channel case F is yet undetermined.

(4.32)

M. Baer, Integral equation approach to atom—diatom exchange processes

115

5. The coordinate representation of the BKLT equations In this chapter the coordinate representation of BKLT equations for the atom—diatom system will be treated. Thus, we consider the process

I AB + C A+BC~~*lAC+B where A, B and C are atoms, and distinguish between two cases. (a) The collinear system, where the three particles are restricted to move along a straight line. Due

to this constraint, the number of arrangement channels is limited to two. (b) The three-dimensional three-arrangement-channel case. Before discussing these two cases still one general topic has to be addressed which is related to the coordinate representation of the Green functions. The Green functions can be presented in terms of wave functions which are solutions either of a free Hamiltonian or of a perturbed one. We tried both and found out that although the free wave Green functions are much simpler and easily handled the resulting coupled LS equations are numerically unstable [79]. Thus, one has to add to the free Hamiltonian a distortion potential such that regular solutions of the extended Hamiltonians are forced to go to zero when penetrating the classical forbidden region in the vicinity of R = 0 [58, 62]. The employment of distorted wave Green functions does not affect the final solution of the coupled LS equation. However, more care has to be taken regarding the calculation of the transition matrix elements. In the next section we will derive the relation between free wave and a “distorted” wave T-matrix elements for the inelastic and the rearranged collisions. 5.1. The application of the distorted wave Green functions The Hamiltonian H can be written in the form H=H

arA,v,

0a+Wa~

(5.1)

where H0~= ~

a = A, v,

(5.2)

is the ath kinetic operator and = Va + ua

,

a

= K,

W~is

v.

the potential, which will be presented as (5.3)

Here Ua is the distortion potential and v~is the perturbation. In what follows Ua is assumed to be a potential which does not permit rearrangement. Consequently, the inelastic (unperturbed) Hamiltonian is introduced, Hua=Hoa+Ua~

aX,v.

Next we define three wave functions:

(5.4)

116

M. Baer, Integral equation approach to atom—diatom exchange processes

(a) the free wave function, which describes either an initial state or a final state 4(a, q~)),where q~,, a = A, v stands for a set of quantum numbers needed to describe the specific state; (b) the distorted (non-reactive) wave function I X(a, q,,)); (c) the full solution of the Schrödinger equation for the initial state q~0,i.e. tfr(K, q~o)). Given I j.(a, q~))the function I ~(a, q,~,)) can be presented as an integral of the form ~

aX,v,

(5.5)

where Gua is the distorted wave Green function, 1, aA,v.

(5.6)

Gua=(E+jE~Hua)~

The full solution of the Schrödinger equation can be written, via the LS equation, in various forms. (a) The most straightforward representation is in terms of the free state employing A-arrangement potentials and Green functions, i/i(X, q~ 0))= ~(X,

q~0))+ G0~w~j~I(A, q~0))

(5.7a)

,

where GOX is the free Green function, 1 GOX=(E+ie—HoX) (b) The solution can also be represented in terms of an initial distorted wave function, .

i!i(x,

(5.6’)

q~ 0)) =

x(A, q~0))+ GUXvXIIfr(A,

(5.7b)

q>~0)) .

(c) A less common representation is the one employing the rearranged potentials and Green functions. Thus it can be shown that k(A, can also be written as ~XO))

iIi(A, q~0))= ~(A, q~0))+ Go~(wX w~)I~(A, q~0))+ G0~w~IiIi(A, q~0)) —

,

(5.8a)

or in terms of the distorted wave function,

li!(A, q~0))= IX(X, q~O))+ G~~(v~v~)IX(A,q~0))+

-

(5.8b)

q~0) ~4(a, q~)IT~~j~(A, ~Xo))= (4(a, q~)~w~~tfr(A, q~0))

(5.9)

—

G~~v~ItIi(A, q~~))

Next are introduced three kinds of T-matrix elements. (a) The ordinary (free wave) T-matrix element r(a, q,, A, r(a, ~

defined as .

(b) The distorted T-matrix element T~(a,

~XO)

Td~(a,~

~Xo)

defined as

q~fX,q~0)=(~(a,q~)~T~~x(A, q~0)= (X(a, ~

q~0))

(5.10)

(c) In addition to these two we also introduce the (inelastic) unperturbed T-matrix element q~IA,q~0)defined as

TIe(A

M. Baer, Integral equation approach to atom—diatom exchange processes

117

q~IX,q~0)= (4(X, q~)IT~I~(X, q~0))= (~(K, q~)~u~~x(X, q~0))

Tie(X

(5.11)

.

To continue, we distinguish between the non-reactive channel and the reactive one. The non-reactive channel We start with eq. (5.10) and replace (~(X,~~)Iby the integral given in eq. (5.5),

5.1.1.

rdis(K q~IX,q~0)=

(~(x,q3Iv~~çfr(K, q~0))+ (~(K,~

qXo))

.

(5.12)

Next, we replace G~~v~Iip(K, qAO)) appearing in the second term of eq. (5.12), by eq. (5.7b). Thus T~h~(K, q~IX,q~0)=

(4(x, q~)lv~l~i(X, q~0))+ (çb(X, q~)~u~l~i(X, q~o)) —

(~(K,q~)Iu~I~(X, q~0))

(5.13)

.

Recalling eqs. (5.9) and (5.11) it can be seen that eq. (5.13) leads to the relation r(X, q~IK,q~0)= r~(X ~

q~o)+

Tie(X,

q~1K,q~0)

(5.14)

.

This expression is the generalization for the case of (elastic) scattering by two potentials. The rearranged channel

5.1.2.

lx(a, qa))

Again we start with eq. (5.10) and again replace T~~is(v, ~

q~0)= (t~(v, q~)~u~~i/s(X, q~0))+

by the integral given in eq. (5.5),

(~(v,~

qxo))

.

(5.15)

Again, as before, G~~v~Ii!;(K, q~o))is replaced, but this time by the expression given in eq. (5.8b). Consequently eq. (5.15) becomes rd~(v ~

q~0)= (çb(v, q~)~v~~çfr(X, q~0))+ (~(v, q~)~u~~i/i(X, q~0)) +

(5.16)

(~(v,q~)Iu~Z~Ix(X, q~0))

,

where Z,, is the operator (5.17)

~

Here I is the unit operator. Recalling the definition of Ga,, [see eq. (5.6)] as well as eqs. (5.1), (5.3) and

(5.4), Z,, can be evaluated as follows: Z~,= GUV(E + ie H~ + —

—

GUV(E

+ ie

—

v~)= GUV(E

+~

and consequently

Z~Ix(K,q~0))=

I

—

H~~)Ix(K, q~0)) .

However, X(X, q~0))is the solution of the equation

—

H~),

118

M. Baer, Integral equation approach to atom—diatom exchange processes

(E—H~~)~x(A,q~0))=0,

(5.18)

so that Zjx(A, q~0))= G~irjx(A, q~0)),

(5.19)

which becomes zero when E goes to zero (the Lippmann identity) [94]. Thus the contribution of the third term in eq. (5.16) can be ignored and consequently r~(v,~

~Xo) =

~~(v, q~w~I~(X, qXo)),

or, recalling eq. (5.9), r(v, q~A, q~0)= r dis (v, q,, A, qXOY

(5.20)

It is noted that, in contrast to the non-reactive T-matrix elements, the reactive ones do not have to be modified when presented in terms of distorted solutions of the Schrödinger equations.

5.2. The collinear configuration 5.2.1. The Schrödinger equation The description of the collinear configuration is given in fig. 1. The Hamiltonian which describes the motion of these particles may be written as 2I2/2a)t3I2/ôr~ ~ + V(ra, Ra), 13, a = A, v, (5.21) H= _(h where Ra and ra are the translational and vibrational coordinates, respectively, V(ra, Ra) is the potential which governs the motion of the three particles and .ç and p~are reduced masses given by —

~Xm+m~ mBmc

~X~m+m+m~ mA(mB +

m~)

mA+mC mAmc

‘

~vX=m+m+m~ mB(mA +

m~) (5.22)

Here m~,mB and m~are the masses of the three interacting particles. Scaling the coordinates, r~=aara

,

R~=aa’Ra

(5.23)

,

where 1/4

(5.24)

aa = (/~a’/~aj3)

the Hamiltonian in eq. (5.21) becomes H~—(h2/2~)(~2I~R~ + 92Ic3r~)+ V(Ra, ra) where we deleted the prime sign from Ra and

ra

,

and j.~,given by

(5.25)

M. Baer, Integral equation approach to atom—diatom exchange processes =

[mAmBmCI(mA + m~+ m~)]112,

119

(5.26)

is defined as the reduced mass of the system. 5.2.2. The coordinate representation

The integral equations that will be treated in this section are those given in eq. (2.15), ~ = VXGVTVX, ~ = V~+ VvGaTaa ,

(5.27)

where W was assumed to be of the permuting array structure given in eq. (4.1). Introducing now the amplitude density I ~ 7nXO\T

~ aX

n~ 0/

where I ~

is an elastic wave function belonging to a vibrational state n~in the K-arrangement, and

substituting eq. (5.28) in (5.27) yields the system of integral equations for the amplitude density functions,

~~°)

= VXGVI~~O),

VjO~0)+ VVGXI~~0)

(5.29)

.

Next we introduce the coordinate representation of eq. (5.29). Now since (Ra, r~I~~0) = ~aX(’~a, (Ra,

r~In~0) ,

(Rh, r~I~) = O~(R~, rKInXO)

raIVaIR,~,r,~)= V~(R~, r~)8(R~ R,~)6(r~ r~) —

—

(Re, raIGIR~,r~)= G(R~<,TaIR, ri),

(5.30)

,

I=JdRa dTa

IRS, ra)(Ra, r~I,

and interpreting the scalar product (R~,r~ IRA, r~)as an extended type of Dirac s-function, (Re, r~IR~, r~)= i5[(R~, r~) (Ra, re)] —

(5.31)

,

which implies R~= Rp(Ra, r~) ,

rp = r~(R~, ra)

,

a, /3

= K, v,

a

/3

(5.31’)

,

we get for eq. (5.29) ~ ~VX(RV,

r~In~0) = VX(RX, rX)

JJ

dR~dr~G~(R~, r~IR~, ~

r~InAO), (5.32).

rVInXO) = V1.,(RV, r~)O(R~, r~In~o) + VV(RV,

JJ

r~)

dR~ dr~GX(R~,r~IR~, ~

Here R and R~stand for the lesser and the greater of Ra and R~.

r~In~0).

120

M. Baer, Integral equation approach to atom—diatom exchange processes

5.2.3. Derivation of the Green functions To calculate the Green functions G(Ra<~ralRa>, r~)as well as Hamiltonian Ha introduced in eq. (2.2),

O(R~,rXInXO)

we consider the

92It9r~,)+ VOa(Ra, ra).

2I2j~)(c2I~R~ +

Ha = —(h

(5.33)

We choose T’~~ (Ra, ra) to be of the form (5.34) where va(ra) is the asymptotic two-body potential of the a-diatomic, and ua(Ra) is the distortion potential, which monotonically increases when Ra 0 and satisfies the condition —~

urn uX(RX) = 0,

urn u~(R~) = d.

(5.35)

Here d is the exothermicity (endothermicity) of the reaction. A relevant choice for U~(Ra) is Ua(Ra) =

V(Ra, ra

=

rao)

(5.36)

,

where rao is the equilibrium distance of the a-diatomic. Consequently the perturbation potential is given in the form Va(Ra, ra) = V(Ra, ra)

—

u~(R~) Ua(ra) —

(5.37)

.

The solution of eq. (5.33) is Oa(Ra, ralna) = ~/i~(RaIna)4~a(raIna)

where

4~(ra I n~)is the

eigensolution of the equation

+

and

t//(Ra Ira)

(5.38)

,

ua(ra)

—

en]4a(rajna) = 0,

(5.39)

is the regular solution of the equation + Va(Ra)

—

k2~]~/sre(R In

) =0.

(5.40)

Here k~ is the a wave number, k~= h1~2 1.i(E

—

(5.41)

er).

Equation (5.40) is solved subject to the following asymptotic conditions: 2 sin(knRa + ô~) for open states urn ~re(RIn) = k” ,

urn

~ (Ra I ~a)

= ~ K~’2

e K,.Ra —

,

for closed states,

(5.42)

M. Baer, Integral equation approach to atom—diatom exchange processes

121

where &,, is the phase shift caused by the elastic scattering process and open states are defined as those (vibratjoi~tal)states for which E> e,, Closed states are defined as those for which E < s,~ and consequently .

E), for s~>E. K~=l11\/2,u(e~— The Green function Ga (R

(5.41’)

, r,, I R , r~)is given in the form (5.43)

Ga(Ra<,raIRa>,rr)=~~~~

where g~(R~In~) is defined as (5.44)

g~(R~In~)= ~,re(RIn)

The irregular solution ~Ii’,~(RaIna) is obtained from solving eq. (5.40) subject to asymptotic conditions which produce [together with l/l~(Ra I na)] a Wronskian equal to unity. Many functions comply with this requirement. However, the simplest choice is tim

~/I(RaIfla)

=

k~2Cos(k~Ra + ~),

for open states, (5.45)

urn cl?~(RaIna)= ~K~112 e’~”, for closed states.

Substituting now eqs. (5.43) and (5.44) in eq. (5.32) yields the following integral equations:

~~~(r~In~)J

~Xx(RK, rXInXO) = VV(RX, r~)

dR~dr~~

r~In~

0), (5.46)

~VX(RV, rjn~0)= VX(RV,

+

r~)O(R~, r~In~0)

~

V~(R~, r~)

J

t~~(r~In~) dR~dr~~

r~InXO).

o

In the next section we show how these equations are solved. 5.2.4. Derivation of the T-matrix elements Considering eq. (5.28) a matrix element Ta

is obtained by multiplying it with the ket (O~I.Thus

(t1aITa~k~0) = (2~//12)(O~I~~0).

(5.28’)

Defining now the coordinate representation of the amplitude density function, ~a~U~a’r~jn~0) = (Ra,

weget

r~I~~) ‘

(5.47)

M. Baer, Integral equation

122

(~~aITaXkXO)=

~ ffdR~ dra

approach to atom—diatom

exchange processes

Oa(Ra~ ralna) ~a~(Ra,raInXo).

(5.48)

The main advantage in treating amplitude density functions over treating scattering wave functions is the fact that by definition [see eq. (2.1)] ~aX(Ra~raInXo) becomes equal to zero whenever either l/It~(RX,rXInXO) or Va(Ra~ra) does so. Thus !1m~aX(Ra,raInXO)—0.

(5.49)

As for the dependence on ra, it is seen from eq. (5.46) that the Green functions project out values of the amplitude density functions which are located along a finite interval of ra. Consequently, only this interval of r,, is relevant for the numerical treatment. It is of importance to mention that this argument holds as long as no three-body continuum states have to be included in the calculation. The nice outcome of the above analysis is that for those cases where three-body continuum states can be ignored the amplitude density functions can be expressed in terms of integrable L2 functions. Whereas the basis set along the translational coordinate Ra can be chosen in various ways it seems that the simplest choice for the basis set along the vibrational coordinate ra is the one employed for expanding the unperturbed wave function 0,, (R,~ra I na). Thus ~aX(Ra~

ralnao)

=

~

aaX(n,,ItaInXo)Xa(RaIta)~a(raIn,,).

(5.50)

Here aa A (‘ia I ta I flXo) are constants to be calculated and X,, (R I ta) is a translational basis function. Substituting eq. (5.50) in eq. (5.48) yields the final expression for the T-matrix elements, (naIT,,XjnXo)

= /12

~ a,,X(n,,It,,InXO) w,,(n,,,

(5.51)

ta),

where wa(na,ta)=fdRa

e(RaIna)Xa(Ralta).

(5.52)

To calculate aax(naltjnxo) eq. (5.50) is substituted in eq. (5.46). Multiplying the first equation by XX(RXItX)~X(rXInX)and integrating over RX and r~and the second equation by ~ and integrating over R,, and r~yields a coupled system of algebraic equations for the two sets of unknowns, i.e. aaX(naItaInAo), a=A, v. Thus aXX(nXItXInXO) = ~ C~(n~, tXInV, t~)a~~(njtjn~ 0) (5.53) aVX(nVItVInXO) =

bVX(nVItVInXO) + ~ C~(n~, tVInX, tX)aXX(nXItAInXO) ~

,tk

where b~X(nrItvInXo)

ff

dr~dR~ ~

rV)~A(rXInXO)~X(RXInXO),

(5.54)

S

M. Baer, Integral equation approach to atom—diatom exchange processes

Cap(na,

talnp, tp)=Jjdra

dR,,

123

~

~

to). (5.55)

Here F~ (R~I n~,t~)is defined as Fp(RpInp,tp)JdR,~gp(R~IRp>lnp)xp(R,’~Itp). It is well noticed that none of the Cap (n,,,

I

t~ n~,t~)depend

(5.56)

on n,,0 and so the coefficient matrix C will

be the same for any initial state n~0.Moreover C will apply also for any initial state in the v-arrangement, namely n~0.Thus, we define the following matrices: ~—~°

~

b—~°

~

0 )‘

~CVX

b~~\

(a~~ a~~’\

0 )‘

~a

557

(

a~~)’

.

)

so that

a=b+Ca.

(5.58)

It should be noted that none of the partial matrices CVX or Cx,, has to be a square matrix. Thus a different number of basis states can be applied in the two arrangements. Once C is calculated the solution is substituted in (5.51) and the various T-matrix elements are calculated. 5.2.5. Conclusions In the previous sections we have described in detail how eq. (2.15), which is an equation for operators, has been converted into a set of algebraic equations to calculate the T-matrix elements for the collinear two-arrangement-channel case. The derivation was carried out for the channel permuting array structure of the equations, i.e. when a = a and for W given in eq. (4.1). A similar derivation can be carried out for T,,~given in eq. (2.16). 5.3. The three-dimensional configuration 5.3.1. The Schrödinger equation The description of the system and the definitions of the relevant coordinates are given in fig. 2. The

Hamiltonian of the three-particle system will be written in terms of the a -arrangement coordinates, 2/2~s)(V~ + V~)+ V(R,,,r,,), a = K, v, k, (5.59) 11 —(h where R,, and r,, are the mass-scaled [see eqs. (5.23) and (5.24)] translational and vibrational coordinates, ~ is the reduced mass of the system given in eq. (5.26) and V~and V~are Laplacians defined as (here and in what follows the index a will be omitted) 2

2_i

tI

RR

t9R

2

2

1 /12R2’

2 ~ rr~’~

“

.2

J

.

124

M. Baer, Integral equation approach to atom—diatom exchange processes

Here I and j are the orbital and internal angular momentum operators, 2

27

1

o

~

-

~2

~

.i

.~

1 =—/1~—~_-_0~slnO~+.2—lj,

i

~

~

~2

~

(5.61) The angles 0 and p are the spherical angles associated with R and Or and ~,those associated with r. Next we introduce J, the total angular momentum defined as J =j + I, and M, its z-component in a given space-fixed system of coordinates, J. = M. If m1 and m are the z components of 1 and j, respectively, then M = m1 + rn. Given the Hamiltonian in eq. (5.59) the solution of the corresponding Schrödinger equation for a given J and M can be written in the form ~P(R,rIJ, M) = ~ i/i(R, rh, j, J, M)F(& i~Il,j, J, M)

(5.62)

,

1.1

where F(A, i~Il,j, J, M) is the angular part of the wave function [hereR = (0, q’) and i~= (Or,

~‘~)], which

may be represented in terms of the space-fixed coordinates in the following form: F(E, ~Il,j, J, M)

(5.63)

1’imj(Or~q’r)~

C(l, m1, j, m.IJ, M)Yim(O~‘P)

~

m

1+m1=M

are the spherical harmonics and C(l, rn1, j, mIJ, M) are the Clebsch— Gordan coefficients [97].In what follows the Rose phase convention [97] will be employed. The function F(1~,111, j, J, M) can also be described in terms of the body-fixed angles (0, ~,y, where 0 and ~ are as before and y and ~ are the two angles which replace Or and ‘Pr when the system of coordinates is rotated in such a way that the z-axis points in the direction of R. Thus it can be shown that Here

Y~m(O, p)

and

1’imj(0r, ‘Pr)

cosy=cos9cosOr+sinOsinOrcos(’P—’Pr),

cot ~ = sin 0 cot O~cosec(’P

—

‘Pr)

—

cos 0 cot(’P

(5.64) —

(5.65)

‘Pr)~

The angle y is recognized as the polar angle between R and r, y = cos ‘(k ~),and the angle ~ together with 0 and ‘P form the three Euler angles. Consequently F(1, Ph, j, J, M) can also be written in the

form F(k, P~l,j, J, M) =

~/2~1

~ C(l, 0, j, (1IJ, 11)1’~(y,0)D~(’P,0,

~),

(5.66)

where the summation is over those .11 that fulfill lilt
S

M. Baer, Integral equation approach to atom—diatom exchange processes

[—~ (~ ~

+

R+! ~

r)+~-~_ 1(1+1)

125

J(j+l) —E]i/i(R,rll,j,J,M)

~ V(R, rll, j, 1’, j’, J)~/i(R,rh’, j’, J, M) = 0 ,

(5.67)

I,,’,

where V(R, rh, j, I’, j’, J) is defined as V(R,

rh, j, 1’, j’, J) = (F(k, Phi, j, J, M)IV(R,

r, 7)IF(ft,

Pci’, j’, J, M)) ,

(5.68)

and can be shown to be equal to V(R, rh, J ~ f J)

= \/(21

x

where (j,

+1X21 +1)

~

C(l, (1,

j, —12~J,0)C(i’, (2, j’, —I2IJ, 0)

(j, [1~V(R, r)hj’, 12) ,

(5.69)

~lV(R, r)Ii’~[2) is defined as

J

(I, ilIV(R, r)lj’, (2) =2~ ~(y,

0)V(R, r, y)~,~(y, 0) dcos y.

(5.70)

This will complete the treatment of the angular part of the Schrödinger equation. 5.3.2. The coordinate representation of the BKLT equations in three dimensions The equations to be treated in this section are those given in eq. (2.18). (In what follows, the wiggle sign above T will be deleted.) Like in the collinear case we introduce the amplitude density functions =

Ta~hO~) ,

(5.71)

where 0~, is a non-reactive wave function belonging to an initial state characterized set where of quantum 0~astands for 0a (n,,, by la~aIa)’ ~‘~a iS o’~~ in the K-arrangement Thus introduced in general before. The treatment here and in all that anumbers vibrational quantum number andchannel. (1,,, Ia) were follows is assumed to be carried out for a given J and M and therefore they may be occasionally

omitted. Like in the collinear case, the coordinate representation of the various functions and operators is introduced [77], (Ra,raIc:~°)=~ax(Ra,raIux 0), ~ (Ra,ra,i/aIVaIR,~,rc~,y,~)=V(Ra,ra,ya)3(Ra_R,’r)ô(ra ~T~)6(CO5ya

(Ra,raIGIR~,r~)= G(r,,<, TaIR,,>,T~)~

3r,, IRa, Ta)(Ra,

—cosy~), Tat,

I=Jd~Rad

and we interpret the scalar product (Re, r~ IRa, Ta) as an extended type of Dirac 5-function,

(5.72)

126

M. Baer, Integral equation approach to atom—diatom exchange processes

(5.73)

(Rp,rpIRa,ra)=3JJ~6MM~3[(R$,rp,yp)~(Ra,ra,ya)],

which means: RpRp(Ra,ra,ya),

rp—rp(R,,,ra,ya),

o~)and using eqs. (5.72) and (5.73) we get

Multiplying eq. (2.18) from the right by

~

r,,Io~0)= T7,,~(Ra,r,,,

(5.74)

y~=y~(R,,,r,,,y,,).

rAt cIA

0(’~~’

0)

Ya)

(5.75)

~

To continue, we will need the explicit forms of the various functions and this will be done in the next section. 5.3.3. Derivation of the Green functions To calculate the Green functions G,, (R, ra R~,r’) as well as the initial channel wave function 0~(R~, r~Io~0) we consider the Hamiltonian H introduced in eq. (2.2), 2/2~)(V~ +

V~)+

V

Ha = —(h

0a(Ra,

The unperturbed potential V0a(Ra,

r,,, Ya)

=

V0a (R,,, ra, Ya)

u,,(R,,,

Ya) + V,,(ra)

ra, yj.

(5.76)

is written in the form

,

(5.77)

where Va (r,,) is the asymptotic two-body potential of the a -diatomic and u,, (R,,, ~y~)is the distortion potential which monotonically increases when Ra 0 and satisfies the asymptotic conditions —~

hrnu~(R~,y~)O, )irn~u,,(R~,~,,)=d,,,a~X.

(5.78)

Here d,, is the exothermicity (endothermicity) of the reaction in the channel a. A choice usually made in the calculation is u,,(R,,, y,,) = V(Ra, ra

=

r,,0, Ya) ,

(5.79)

where r,,0 is the equilibrium distance of the a-diatomic. Once

V~a(Ra~ ra,

y,,) is determined the

perturbation potential is defined as VJRa,

r,,, ~y~)= V(R,,,, r,,, y,,) u,,(R,,, ‘ye) —

—

v,,(r,,)

(5.80)

.

The solution of eq. (5.76) will be written as ~

O(Ra~r,,IcI,,0) = ara

~e(RaIUa,

cI,,o)~a(r,,In,,,j,,)F(y,,, A,,IJ,,,

la),

(5.81)

M. Baer, Integral equation approach to atom—diatom exchange processes

where

127

stands for a set of quantum numbers which describe the initial state in the a -arrangement channel, the function (R,, I ETa’ cIa0) is the regular translational part of the total wave function [in £TaO

i~i~

what follows we will also refer to ~/i’~(RaI 0a~ £Tao), which is the irregular translational part of the wave function], F(y~,~‘ta I I a’ la) stands for the angular part of the wave function and takes the form given in eq. (5.66), Aa is a symbol which describes the three Euler angles, Aa (‘Pa, 0,,,, na)’ and 4a(Ta na, Ia) are the vibrational wave functions of the a-diatomic,

(—~~

The translational functions cli”(Ra I o-~,,,cIa0), q

7

/12

~2

~

/12

la(la +

1)

R~

~

+

(5.82)

+va(ra)_enj)cba(ralna,ja)=0.

~

=

re, ir, are solutions of the equation

2

—k~1)~(RaI(Ta,cIa0)

~ ~

q=re,ir,

(5.83)

‘~a= ‘~a0 and V0a(RaIla, Ia’ l~,j~) is given in eq. (5.69) and k~1is defined in a similar way as in the collinear case [see eq. (5.41)]. In case the distortion potential u(Ra, )‘a) does not depend on Va eq. (5.83) becomes

where it is assumed that

(_~-

la~~~) +Voa(Ra)_k2nj)~/J~(RaIcra,cIa0)=0,

~

q=re,ir.

(5.84)

Equations (5.83) [or (5.84)] are solved subject to the following asymptotic conditions:

tim iIi~(RaIUa,o~ao)= 5a.a. J~(k,,,1R,,)

+

Ni(knjRa)(O~aIK~hUa0),

(5.85)

where (cIa I K°aI cIa0) is an inelastic (elastic) reactance matrix element. J~(k~1Ra) and N1(k~1Ra) are defined as follows: 1~ (k~

Ra)

=

k~ Ralt (k~ Ra),

open states, (5 .86a)

Jta(knaj,. Ra) =

I k~

1I_h/2 exp( I k~1IRa) ,

N1 (k~ Ra) = k~’~ Raft

(k~ Ra)

,

closed states

open states, (5.86b)

nt(knjRa) = ~ ~

_1/2

exp(— hknj

h’~a),

closed states .

Here It,,,V’n,,,j!~a)and ft(knjRa) are the spherical Bessel and Neumann functions. As for the irregular translational wave function i/s (Ra I 0~a,~Ta~ we require its asymptotic form to be

lirn

(RahcIa,

crao)

= 5,,.jNt(knjRa).

(5.87)

128

M. Baer, Integral equation approach to atom—diatom exchange processes

It can be shown that this choice of ~(RaIoa, tJaø) guarantees that the Wronskian is unity in both the elastic and the inelastic case. The corresponding Green function is

G(R:,raIR:,r~)=-~ /1

1 ara

a’~a ~

x ~a(rahna, ja)~a(r~In~, J~)F(ya,AaI Ia’ la)F(y~~ A~Ij~, li),

(5.88)

where like in the collinear case R and R are the lesser and the greater of Ra and R~and the function g(R~R~Iu~, o~’,)is defined as

g(R~Iu~, cIi) =

cI~11’~(RhcIa,o~,).

~frre(R
(5.89)

In what follows, we will be interested only in case when the distortion potential ua(Ra, ra) is not dependent on Ya and consequently O(Ra~r,, I cIa0) as well as Ga (R, Ta 1R, r~)become much simpler. Thus O(Ra,

t~(Ratuao)øa(raInao,Ja~)F(ya,Aat Ja~~ lan)

Tab cIa0) =

1

G(R~,TaIR:,r~)=—~ /1

(5.81’)

ara

ara

if,,

(5.88’)

~

This completes the derivation of the Green functions.

5.3.4. Derivation of the T-matrix elements The derivations to be carried out for calculating the T-matrix elements are very similar to the ones done for the collinear case. Consequently only a few of them will be presented. Following eq. (5.71) an ordinary T-matrix element is obtained by multiplying the equation from the left by a final a-state (O~j,

(uIT~Io-~) = (2,~Ih2)(oa.I~°)

(5.90)

.

From the definitions of the coordinate representation of (0,,,I and

I,~)[see eq.

(5.72)] we get (5.91)

3Ta Oa(Ra, TaIcIa)~~(Ra, rahuxo).

(UaITa~IcI~0) = ~

ffd~Rad

Again, like in the collinear case, we expand ~a~(Ra,rat cI~o)in terms of the a-channel basis states, ~a~(Ra~ TaIcI~o)=

~ aax(cIaItaIcIxo)Xa(RaIta)~a(rajna,Ia)F( Ya~AaIla, Ia)’

a’~a a.,,.l,,,

where Xa (Ra I ta) is a translational basis function. Recalling the explicit form of

eq. (5.81’) we get the final expression for the T-matrix element,

°a (Rae

(5.92)

ra I cIa) given in

M. Baer, Integral equation approach to atom—diatom exchange processes

(O•aITa~It7~

129

aax(cr,,It,,Icrxo)wa(cra, ta) ,

(5.93)

TC(RaIcIa)Xa(RaIta).

(5.94)

0) = /12 ~

where Wa(Ua,

ta)

=

JdRa ~b

Our next task is to construct the set of algebraic equations to determine a,,, ~(cIa I tat cI~o).This is done by substituting eqs. (5.81’), (5.88’) and (5.92) in eq. (5.75), multiplying from the left by (Rara)~1Xa(RaIta)4.~a(raIna, I,,)F(ya, AaIla, Ia) and integrating over all variables. Thus aaX(craItaIoAo)=baX(uaItaIuXo)+~~ Cap(ua,taIcIp,tp)apx(o~pItpIO~xo).

(5.95)

p

Here the inhomogeneity

ba X(0a

I I~

111 ~

ba~(tTaItaIcI~O) =

is

ta

dRa

dra dcos Ya ~a(’~aIta)~a(raka,

x (Ia’ lalV~P(Ra,ra, ya)IIxo’

and the C-matrix element

~

Ia) JXO)cb

(Rjo~XO),

(5.96)

Cap(cIa, taIO~p,t~)is

Cap(cIa,taIcIp,tp)=JJJ~~dRadradcosYaXa(Rahta)4~a(raIna,Ia)

(5.97)

X(Ia,laIV~p(Ra,ra,ya)IIp,lp)4:;p(rpIfp,jp)Fs(RpIcIp,tp),

where F~(R~Iu~, t~)=

—

~

J

dR~

~

(Ia,laIp(Ra,ra,ya)IIp,lp)2~

(5.98) \/(2i +1)(21 +1) ~C(la,[la,Ia,~[2aIJ,0)

x C(l~,f2~,J~—12~hJ,0)Y

1~

0)~ap(Ra~ ra, 7a)1~np(7p~ 0)d~i,,np(L1ap).

(5.99)

In eq. (5.99) we encounter the function d~n~ (nap), which is defined as ~

= D~~(0,

L1ap~0),

and the angle zl,,~,which can be shown to be (see also fig. 2) ~ap = 0,~— 0,,,. When a

(5.100)

/3 the function

d~2(0)becomes 5ç~.

d~mnn(L1ap= 0) =

(5.101)

130

M. Baer, integral equation approach to atom—diatom exchange processes

Equation (5.95) can be written as a matrix equation. To do that the following matrices are introduced: (1) The square homogeneous matrix C, ~

c=

Cx,,

CXk

C,,~ C,,,

CVk Ckk

CkX Ck,,

,

(5.102)

where the elements of each submatrix are designated by the row indices (cIa, t~) and the column indices (a-, ta)~ (2) The inhomogeneous matrix b,

~ bXV bxk b= bVX b,,,, b,,k bkX bk,, bkk

(5.103)

,

where the elements of each submatrix are designated by the row indices (op,

t~) and

the column index

U.

(3) The solution matrix a, which has an identical structure to b. In both matrices b and a, each column stands for an initial state in a given arrangement and therefore the columns are designated by one index only (which in fact stands for the three quantum numbers (n,,, Ia’ la). Having introduced C, b and a, eq. (5.95) can be written as

a=b+Ca. One way of solving it is by inverting the matrix

(5.104) (I

— C).

Thus (5.105)

This completes the derivation of the algebraic equations. 6. The infinite-order sudden approximation 6.1. Introduction The infinite-order sudden approximation (IOSA) can be described classically as an approximation in which the treatment of a collisional non-reactive atom—diatom system is carried out by “freezing” the angle y defined as y=cos’(1~.P),

(6.1)

where R and P are the translational and vibrational unit vectors, respectively. Once a calculation for a given value of y is completed it can be repeated for a series of y values thus yielding y dependent magnitudes. To obtain one single (relevant) value, integration over y is carried out; thus if P(y) is such a quantity then the relevant “averaged” quantity is

~i~h~

~or))

1t6M’eih~inke

a

f dcosyP(y). 2~

P=-1

~

~

S

5(j)

cit ~nt

~

~

Jc

~

.

f3J~

~SS~e~c

S

-‘,“:~~)i~i p~

.~•.‘

~•

-

(6.2)

S

t~n flio ha~lirth~I1Q$ ec~m~e~at ~ PQ~I~MI ;q~i~ly pflOflti~l~ ~ jt~~Rtt

‘T((huLtnstã~4,dep~fld qlt,i) This approximation was suggested by Takayanagi [98] ~ai~cL~b~ ~ ab4~~ern~ini{99J1l extended’) h~i Curtiss [100],studied by Pack and coworkers [101,102] and presented in a beautiful form by Secrest [1fi~..A few years later the IOSA was e~end,e~.t t&r.e~ct~l~re sys~h~3 [1Q44 The lextension..is~neby assigning to each arrangement channel an angle; thus the angle Y~is attached to the K-arrangement channel and the angle y,, to the v-arrangement channel. In this w~onefor~q~uan~ie~ ~(y~,Q

i~4.

1which

depend on two angles. To obtain one single valuéP’~f~é ~ll~’1dg”thrhiul~ is’emf~lb’y’ed? +1

~=~J 1

/

I

+1

dcosy~f dcos~VQ(~X,y,,)P(y~,yV), ~

~.)

-

(6.3) \\

SS~

~

~

where ~ y,,) is a weighting function still to be determined. In numerical~ studies [111;]it was. found that, the, best choicp fdr ~ ry,~)is the Dirac 3-function, which implies ~ th.te~&~oné ~re~a’tion ~b~t~eên ‘ahd y,,~As~ires~t[t, on’e ii~t~egrationban.~è done analytically [106] and consequently P becomes ~

+1

P=~

~

‘1l~~,’~’:..

f dcosyXP[yX,yV(yX)],

‘S.

(6.3’)

—1

S~

SS

~,:.

‘V

which is very similar to the non-reactive expression. In the next sectic~it is shdwn~ how’ b~J~intioduci1gone s~J~gle assumption i~gad1n~ the) ‘Green function the BKLT equation becomes an equation where the three angles ~, -v~and Yk are not variables but parameters, th~~ fprmii~g~e/ framework fqr)the~reac~iyeIO~A~ 6.2. The derivation of the IOSA—BKLT equations

-

-

.

S

.-H’~, , Recalling eq. (5.88’) the Green function for an elastic perturbation potential, G,, (R~, be written as ‘.ft~

.:

~‘~/~tt,,’

‘)~‘.1lt

.,t

I

.‘

~..tt.-

.‘.~I~

‘/,j~

~JI~R~ ,‘~j4A~i’fr

S

~

~

,,~f

,~-

G(Ra,TaIRa,Ta)=~~~ /1 RaraRara ,,

L.V~) ‘I’

~

.1

“.f;~

g(Ra,r~IR,,,rjl,,,j~)

b~’~i.t .ix1t.~t~1~j,/ F(ya, i’tahja, j,,)F(y,’,, n~fia,Ia)’

St

~

where I

I

g(R,,<, raIR,,>, 1~t~ II

il:t(t~)’q’i

r~Il,,,Ia) = ~ j’~()~

~re(R
,‘?~-~)S’

Here, like before, u~stands for

~

‘A,

,

‘‘~~‘

ç,j~,

J~).

t~tl

~

S

S

~

it~i

~

f

Ia)(Ia-a)~fr~kia,

~)r~~ (ii

t~

.‘~

-/

~(~t

1~).

.~)i ,;‘~ c~t [i

~‘

(6.5)

t~i ~ ‘~ t~

~

t/~) .~.) ~iL~

132

M. Baer, Integral equation approach to atom—diatom exchange processes

We now introduce the IOSA by making the assumption that (6.6)

~ where l~and Ia

are parameters. Any quantity that will be calculated employing this theory will depend on Ia and Ja~Consequently, to obtain the corresponding physical quantity additional assumptions have to be made. If now Z( Va’ ‘4a I y,~,, A~)is defined as Z(ya,AaIY~,A~)= ~

(6.7)

F(Ya,AaIla,Ia)F(Y~,A~hla,Ia),

ta Ia

then the IOSA

GI(R,

Ta

IR~r,~) will be written as

GI(Ra<,TahR:,T~)=—~RR1,rri~

(6.8)

Recalling eq. (5.66), the function Z(y,,, AaIy~,A~)can be expanded in the form Z(Ya,

21~+1

(~

(~

Ia’ ~

jt,,Iy~,A~)=

C(1,,,

C(1~,0,

0, Ia’

,,I~,

0))

~

0)),

fl~)D~~(A~) ~

(6.9)

which can be rewritten as Z(Ya,AaIY~,A~)=1 ~

D~(Aa)Yfl(Ya,0)D’Mfl(Aa)Yjfl(Y~,0)

Ia~t~a~fla

x~

C(l,,,

0,

a

Ia’

a

a

a

a

a

u2,,I.1, u1a)C(~~a, 0, Ia’ i2~,I~ [2~)(2l,,+ 1)

However, the last summation yields the Dirac 3-function multiplied by (2J

+

.

(6.10)

1) and consequently the Z

function becomes Z(Aa, y,,IA~,~

2J+ 1 ~

± D~ (A,,)D~ (A~) ~ 0a5,

~

~ (y,,,0)~~ (y~,0).

(6.11)

J,,>It1,,I

Since the Y~ey,0) for each (1 form a complete set, the second summation (over Ia) in eq. (6.11) yields

~

~afla(Ya’0

ia>I~aI

J7(y~’0)=

3(cosy,, —cosy~).

(6.12)

Substituting eq. (6.12) in eq. (6.11) and eq. (6.11) in eq. (6.8) yields the final IOSA representation for the Green function,

M. Baer, Integral equation approach to

Gi(Ra<,TaIRa>,Ta)=~~4 R

atom—diatom exchange processes

133

R’ 2J+1 g(R,,<,r,,~R,,>,r~~J,,,i,,) aa a,, 8ir

x 5(cos 7,,

—

cos y~,)~ D(A,,)D~,(A~).

(6.13)

Next we consider the amplitude density function C,r~(Ra~ T,,IcrAO) which was introduced in section and presented explicitly in eq. (5.92). It will be written now in a slightly different way, 5.3.2 [see eq.

(5.71)]

CaA(Ra~TaI0~XO)R

~ ~ ,,r,, Ia’’,,

(6.14)

or, introducing the function e,,x(Ra, ra, y,,IfI,,, oAo) and recalling eq. (5.66), ~

T,,,~/aI(Ia~a-~0)

~

~

=

r,,Ij,,, l,,Iu~0)

‘a Ia>10a1 X

we get for C,,~(R,,,Tat ~

C(l,,~0~ j,,, (1JJ, (1a)Y~(y,,,O),

(6.15)

the expression

±

~,,~(R,,,T,,IU~0)=_L aTa

(6.16)

D’~(Aa)~ax(Ra,ra,7.,,Iila,cIxo).

iJa=_J

The final function to be considered is the unperturbed elastic wave function 0,,, (Ra,

Ta

I tr,,,), which is

presented (in three dimensions) in eq. (5.81), O(Ra,raIo•a)=~-~——ik(Ra,raIo~a)F(ya,Aalja,1a).

Introducing now the function ~I’a (R,,, r,,,

I

(6.17)

2a~oS,,),

Va “

*,,(R,,,r,,,y,,Ia-,,,il,,)=~J2~ ~a(R,,,raIa-a)Yjn(ya,0)C(l,,,0,ja,ImaIf,I1a),

(6.18)

it can be shown that 0,,, (Rap r~a-,,) takes the form

±

O(Ra,TaIO~a)=~~~

aTa D,,J

a(u1a)~I1a(1~a,Ta,yaI0a,h2a).

(6.19)

a

The three functions, namely, the unperturbed elastic wave function 0(R,,, T,, a-a) given in (6.19), the amplitude density function ~ r,,Io~o) given in eq. (6.16), and the Green function G

1 (R,,<, T,, IR~ r,,) given in eq. (6.13) are in the appropriate form for the application of the BKLT equations. Thus employing eq. (5.75),

134

I~S~&~

~

Q,,~J

v~~

L

=

pC1,I~Xg~l~q~gç

j~JS

~

V,,~(R,,,ra, VaY

~

~:

.L

,,~,

~

~

S

~it

;~.Jtji~1~(a’’

‘1 ~~)l~)j

~

~i)iL~y~

~

~

~

a5~

x g(R~,r~IR,~

(cos

x (~D~n

~s

~: 1_,~- ,j~’

j~ijJr5J t~

)fr)

~

~

.H.’’.~

~

~~9~(A~))

~

~

S

a-~)

X~X ~

p~eSse,~

,.

5 5

-‘

~

‘S.

(6.20)

.~ S

where the summation over /3 runs over all three arrangement channels~j3 = K, v, k. The integration over cos and can be carried ø~t~~a1ytiba4iy,~ th~~ sinij~lifyi~g e~(6~20) At this stage, two comments will be made. ~ (at)~The connection between Aa~ntnd..4,, (f~c~r ç~., a i~)is-, 5

S

+

= ~

where ~a

‘a

(0, ~

‘a’

0), 5

,~‘

,-

S

is defined as

1(~a.~1~a),

5

~aac05

~i~’ñ~~

‘~k,

‘a

~

(6.22)

5

S

=

~

D~

,

(Aa)~t,,sui,,(~aa)~

where d~Qfl(~iaa)is defined as [2 .V (~a’a?

=,D~5Q(0,

0).

5.

gi~,e1~ihLthet f’orin

iS’

(6.23) ,

S

~a’a~

5.

S

~S

(b) The relation between the two matrix elements D’M~Aa) är~dFi~~

-

.~ ,

‘.

5 ~‘5S

:

..

.

S

5(6.24)

5

.

S

-

Multiplying eq. (6.20) by D~~~,(Aa) and integrating over Aa while recalling eq. (6.23) yields ~ ara

-

-

~

~..1I~i.k~IJ)I1tt

II

)flfflX

ra, yaIa” a-AO) = ~ xrx ~ ~ap(Ra, ra, ya)JJ

(t~)’I!2t

~

~,,

~5~u ~

iJ>iH -:~rfi ~‘t /1,~t;.-,~h(~ ‘~

~~(Ra,ç,

,.

~

~

,.~

~

11

~

rx,yjQx;~o)

~ .

dr~g(,r~R~,~

q~~j) ‘I~

~

(IA

c~’;~c;! ;

~

Ya)~

~)

I.

-S

.

S ~

5

where the functions q,~(R,,r,, y,I~,,,cr~

5 ~

0),t = a, a’,

r’,.,

/3,

-‘

-~~.i;.

S

SS

~

are defii~d;~as

5:,,

5_5

-

~Z~)

S‘ S

5

S

,,i~

,

SS~-.~

5~5

‘

‘/ftá~

ton

~

~

r,, ~(1,,, ~

~ww)desI1s

YaIula’ cIxc~’)~~

~ ~

~

~i1t~~)i

1~ ~

~3~i~çompletesthe treatment of ~

~b

6.3. The IOSA and the angle-dependent W matrix hni~(~)1~

i~r’tt~J~du~i),) S~L

~kI\~ ~c~tri~ü ~ ~ih niiJcfo 01

Ta•~

4i~Jt

.

t5,it~.~

S

5

hu~(O I ~) p~iii n.wc~~ (~is

5

~

~

The angle-depend~t~W~ &~W~ntk ~,lit~f ‘~j~(~~l!1~JS~ applied to continue the incorporation of the IOSA within the arrangement channel approach. The idea ~

behind the y-angle de~pepdentW-matrix elements is tl~a~, fixing ~ in~tw~ce v~’deteripines the arrang~m~’nt:~ñné1 w~i~h~4 ~os 1~l(e to o~n~e1h~~ i~~~hO~1iih ~uch a M~’~1~at the arrangement channel v is the one most likely to couple to~then the equation for a = K

takes the form

~ ~11(jIbflIJ.I ~ict?b!3W (V~~) .?.p~)to

1~

Jud

A

i—

tioi~uJo~ ~uiT 3bfiTi .id

(Ii

~I1I~fl~)fflifl0~) hnil ,u()

q~(R~, rx, YAI~P’q~ 0)=

— -~-

x

i’k ‘~)

W~,,(y)~f~(~~gt~

__i~~_ I IdI R R J J

~r1

Vg

1~(~)

~T

~

p~ft ~II.ti1u1~ibIi~ir{1nioi~

~

v ‘ v v’ Jvi’lvA\ v’ v’ Iv v’ cTxo ~fl 10 n ~.FPIn~b ~)th 1~iqIrto- ~r1T~ .p~i~o ii~qqi;luili [IOiJ~1iLt1 ~di ~ A~O1~v~i~ii~i It should be mentioned that in obtaining this equation we also assumed that a’ v. Since on the r.h.s. of eq. (6.27a) the function ~ r~,yJ12,,, a-~0)is encountered for a given y~,it is natural to choose y,, in such a way that the v-arrangement will couple to the ~ second equation is v

vi hi~~t~ui ~d.Iiii V ‘

(CII .~) ~ ~ ~1 n.h;up~t~l3) flJJ0u1--~Id~cIi~0 10 J~’iii’il) )th ~J~fl1~ S~1C~JI~ ~! J~u1 ~sII1~IIIiIJU ~)13fIN~ .~~ 8.A ~— .)U ~ ~iI ~10 ~ l~c1ak-~n~th~nfmE-owI lcA’fljo b~mu~ Hi; ~“ ~4~i,f~H 7~ ~ ~(II1 ~ mcfl± lo .ili~m ~ i ~tiih~ i ~urh ~ .~4imlnj~d til ourwr~o~ ?‘ ‘i ~j,crso,i,,ib owJ )cf I ot nornrno rr,bmW)W~R~rs, ~) o~4~iU~~~< 1ti;J~il~c~ ~ Jt1~1~ 1u9l{~E!u3 Vf1 ~f)fl0i;0~)Ilufi01d1~!flib-’2..uirlit)[11 101 ~f10i ~ ~ iIwI1~itibbnE b~t(qqn0?.li~ 31011 ?f;t;-~fl~~JçY;S4~10(11 ici J 011i0111 i~1ItU~d 120 i1;~d0o~oitithi~,ti~w di;no~i;oi6~iI~ *w~*~ ~2~)rOrei suni1~rtttootM~ç ~ l~(5.32x~eçi4lfothtka ~pp~an~e, ~fttW~.j(~j~at~d~ oi4n~p1i~tedrfi!1h4fi1dgencityi&iInIo o~oilidciw iIoa’~.~ th~stag~ ~ddi~oiMotl 1ptiinrniconi~rnngithatiela*in~ bei~qth~h~)iaird 10

S~S~f

b~h~J1 ~

~fldj(i,~-ij~

.Sf

~

~orft~iand j~ h~ll~a~iag&ih

dd~rib~di;in ~

~

[7itpthuy

~xoha o~1~imis1&nd arU IhrW l~*i~Ont~ theisdo~,f;~r~eg~hp

all [thèrbfote~~lli~ ~th~th rW4th~it~onotl~ ~aklii~i~iflia~c1Ee1~vdnt1~e~thts ?an~b~ ~s~ikmià~j~on~4tcr4tmebrelation~be~eeñu~1 aii4l ,~u)/J~ i~4(~i] ~

established in a symme~oc&~çtw~in~ ~ ~q$Ii4612~3in4l~s1thkthW~(c~i ~mibW~J~3)ai~âbloqua}~o ~n~II~MI~fthhtkmii(~qs. ~ ~ifio~’~cIañi~t/ e4 fiJ~il~g~ of ~ ~ oni~od 01 (~,,~k)~ born ir~’i;ovi ~oihu1~~rn otinil iuo io J~omni .[~

,~

136

M. Baer, Integral equation approach to atom—diatom exchange processes

6.4. Derivation of the IOSA T-matrix elements

In order to obtain the IOSA T-matrix elements we employ eq. (5.91), (a-VITVXIa-XO)/1

(6.28)

2JJdRVdTV0v(RV~TvIcIV)~VX(RV~TvIa-XO)~

rVI a-~0)is given in eq. (6.16) and 0~(R~,T~Io~~) in eq. (6.19). Substituting eqs. (6.16) and (6.19) in eq. (6.28) and performing the integration over A,,, yields the expression

where

~

,‘

2

(I (a-VITVXIo~XO) = ~ 8ir ~ .~ -c-’i U 5 I

hJ

cc ii

dcos YvJ

T

V

j

dR~ dr~i~ç(R~, r,,,

*

yj(2,,, a-~)~VK(RV,

~

YVI’~V~ a-~0).

(6.29)

One final comment has to be made. The solution of eqs. (6.27) yields the functions q>, and ~ but from their definition in eq. (6.26) it can be seen that ~

r,,, y~If2~, a-~ 0)= ~

r,,,

~

a-~0)

(6.30)

which is the function that appears in the integral of eq. (6.29). This completes the derivation of the reactive IOSA. 7. Numerical resuhts During the last 18 years, since the derivation of the Baer—Kouri (BK) equations [seeeqs. (2.16) or (2.21)], the BKLT equations were exposed to several numerical treatments. The first applications were all carried out for two-arrangement-channel systems of the kind A + BC—* AB + C, where B, the atom

common to the two diatomics, was assumed to be infinitely heavy (thus yielding a skewing angle of irI2). Baer and Kouri calculated transition probabilities for the cohlinear wave guide model [23]and integral and differential cross sections for the three-dimensional version of it [24].The collinear results fitted reasonably well with those obtained by other methods [112].The BK equations were also applied to the low-energy e—H scattering process [25]and the calculated phase shifts were in good agreement with those of Schwartz [113]. The Kouri—Levin version [see eqs. (2.15) and (2.19)], which, as mentioned previously, is more convenient for numerical application, has been applied since then. Their first applications were again carried out for the above mentioned colhinear wave guide models [35]and the e—H scattering process [27,28, 54]. More recently, the BKLT equations were applied to a number of collinear atom—diatom systems with smooth potentials and all three atoms having finite masses. The following systems were studied: the H + FH system [58,62], the H + H2 system [63,64, 79] the D + H2 system [72]and the F + H2 system [72].In all cases, the results were compared with those obtained by more established methods and were found to be in good agreement with them. Before continuing with this subject we would like to refer to one ignored detail of the calculation, 2 translational basis set namely, the functions Xa(Ralta), introduced in eq. (5.50), which form the L [58,62]. In most of our finite mass studies we assumed Xa (Ratta) to be sine functions of the kind

M. Baer,

2

Xa(1~aIta)

p

.

/

approach

to atom—diatom exchange processes

137

Ra~Ria\

SiflktaIr

— p

“Ia

Integral equation

~

—

2a

“Ia

~

“Ia

),

(7.1) S

where [R2,,, R1,,] is the R,, interval along which the amplitude density functions ~ r,,, In~0)are assumed to be different from zero. All the above mentioned systems, whether collinear or three dimensional, where two-channel systems and the corresponding 2 X 2 W matrix given in eq. (4.1) was found to yield the relevant coupling between the two arrangements. Difficulties were encountered once the BKLT equations were applied to three-dimensional threechannel systems. In the first application the channel permuting array structure of W, given in eq. (4.10), was employed for the study of the exchange process H+H2(nI,IIIJ=0)—*H2(nf,jfIJ=0)+H,

(7.2)

and no relevant results could be obtained. More recently, the present author [84, 89] applied the orientation dependent W matrix elements given in eq. (4.12); the results of this numerical study, which was carried out applying the Porter—Karplus potential (designated as PK2 [114])are presented in tables 1 to 5. The results in tables 1 to 4 are given mainly to show their dependence on y ~. The interval [0,y ~]is

the range of the orientation angle ‘Y~for which the strongest coupling between the initial A-arrangement and the final v-arrangement is expected (see fig. 2). In the same way the interval [yr, IT] is the range of 7~for which the strongest coupling between the K-arrangement and the final k-arrangement is expected. Due to the high symmetry of this reaction not only Y~= IT — y~,but also all three ~ a = A, v, k, are equal. Thus the final results will depend on one single undetermined parameter Yo~ In tables 1 and 2 are shown transition probabilities for reaction (7.2) as well as for the reaction H+ H

2(n, = 0, j1IJ=0)—*H2(n~,E~IJ=0)+ H,

(7.3)

as calculated for the total energy E~0~ = 0.6 eV. The calculations in table 1 were done for the vib—rotational basis set (12, 12, 6) and in table 2 for (12, 10, 8, 5). The set of numbers (12, 12, 6) means that in the calculations w~employed three vibrational states with 12 rotational states in the first and second vibrations and six in the third. It is important to note that for E~0~ = 0.6 eV one encounters only seven open states, all of which belong to the ground vibrational states. Thus, in the case of table 1, 30 Table 1

Three-dimensional probabilities for H + H2(n1 = 0, j~ = 0~J = 0)—~ H2(n, = 0, 1,11 = 0) + H for E,,, =2 0.6 eV. The calculations are carried out with (12, states 12,6) vib—rotational states and 8 translational L cosy, 0.156 0.309 0.454 0.588 0.891 ref. [115]

0—+0

0—’!

0—.2

Total

i—*0

i—al

i—~2

Total

0.068

0.029 0.037

0.048 0.054

0.033

0.125

0.045

0.063

0.049

0.071

0.049

0.078

0.029 0.028 0.030 0.041

0.133 0.145 0.155 0.173

0.044

0.077

0.040

0.168

0.015

0.028

0.021

0.037

0.026 0.027

0.044 0.049

0.016 0.017 0.016 0.017

0.025

0.047

0.021

0.094 0.097 0.095

0.025

0.044

0.023

0.097

0.083

M. Barr, in~ç,~ral equq(ion q~oproachto qtqin-~diat~m. exchqr~gepro cesse,s

l3S

Table ~ Same as table 1 (for E,,, = 0.6 eV) but with

S

(12,states 10, 8~5) ~ib—rotationaIstat~and -

COSYQ

, 0.309

0—~1

r

1

0.454 S

~ ‘

S

5

SSS

5’

.0.707, 0.809

. ‘

.0~588 --.

1.

‘~

0.022 0.027

0.035 0.041 9.048.~

0.0l~ 0.018 v.0209.02~ 0.021

9t~~~S~S ~49. 0.027 0.047

.

i-’’

Total

1—+0

1—+.1

0.~5.

0,

~‘‘‘.‘-

0.086 0.093

1—~2

0.057. 0.063 o~Io~5,55 0~~~0,’1Q3.‘ ~0~1 “~‘ 0.085 0.098 0.048 Q0~4

0213.5 ‘

0.037 0.042

(197

S

S

I

‘~

~03U.

-~

0—~2

2

-

10 trpnslation~lL

‘

,

,

Total

0.0~.

.

0.139.

,,

.

O.o~1 ‘ 0.145 ~ ‘~ 0~66’

~44,0~$)77.~

S0~(34(i,

-55

5

s’

5

;5

S

-

5

S,0.~S5

S

S

555

9.l~3~ ‘1J;lj7’7~ Q~0~7,

5~~5555S5

S -

S

5’

~ 5

55

vib—rotational states are employed although only seven ~ere’~pe’i~.In~th~ same way, the ~esults~iver~in t~I~1~ 2 were calculated with 35 vib—rotational state~In adqition we 2 1 ~ha4these to ~mpl9y, ~i1S9 ~the L traii~lationalbasis set. The results in table 1 were obtained u~lhgei~fitof fuh~t1onsfor each vib—rot~tiona1~t~te in 1each ~rrang~m.e~t.Ij’3~i~,, altog~t~ei ~ 3Q,~ 9,~l~c~ic e~t~p.ps.,~aç1,to ~sdrv~ order tb ~lI~ i4t~es~inc9~e ~ i~ ~ 2 ç~e~vçd ~ st~ t~,çresqtt~,gi ~i t~i~ t~h~ç were Obtained lollowlflg t e so ution of luDO a~gebraicequations. . ~e~ults in t~bles1 (and 2) are ~omnared with those of Schwenk~et al.. {J ~51,, )yhiçh - were or ~ 13~i ‘‘ud- -3 0 ~ ‘lH~ 1 . oltàined’ by t~ç ‘All ~ 13 ~1ti, ~ ~pt~4,th~ ~ S~Sa~9~.e~ ~US ~ia~Ia~ik~2, 1,16, ~ a~every~sin~iIar t~each otJ~er.(øi1~rdnce~ ~r.e5 rnarnly,in ~ ~eco~d l-~ I 3 3 ‘ ~ 1 0 — ~igni can ig t e main reaso~is or the s ight diffrrenc~scou d be inaccuraç~es rel~t~dto

~fp

-

,‘.

‘

-‘‘‘~: 5~5

‘

!?-~S

--~~

3,SS~

S••~

-,

.

‘

7

Ii

‘

‘1

I

I

cacula ions o eige~va1uç~, ,e1ge~1i~inctions,trans ationa,! waye,~q~ctipns,.~tc., a~ .as ~i.g t~rrpr~ In the a~pp~ication of physical cons4nts From the ~ ~~fó~i~s ~oti~: 1th~~ ~t~ie ~ i~esults i.e., the clo~rthey come to r1~i ~hiaffer ‘ ~i~ ~‘ (~hêiar~~’ ~ ‘~y,’ ~hQseobtained by other methods. . ~b) For practical purposes, most of the results 1for j~= 0 thta~edfor ‘cos -y 0 in the are~ rãii~e0891> 45f are ,IIev~r,~ s~~g~pcqmes smaller wh~nthe. /~, = 1~ ~ereç1 ct~ y9~’ O . ihtab~et —‘‘lU * — ‘~‘ :i~3 ..~. -~o .— I~. — ~i ~enetai, me iesuttsm t,a 1The ‘~o~h~o~ t~i~shldy i~that Yo ~no~ fit c~n o~i~fofc’ed ~e 1bi~~ -~eemt9 better~ro~r1yt, the correqt, resulis thapi&e’~e ~ in ~,~eof ~, ~ ~iii~~er ~o’ac~ii&v~d èr~è’ltlc~ T hu~f& . mnstaii~d’ ~ rçsul~s~or &~ CU® - “as 3 .1 1’ , 1 sho’~,’nin ta Ic 1, are not the fina ones. n or er to obtain for this value of cos y 0 the correct results, 2 translational states or both until convergence is attained. oneResults has to for addthevib—rotational same system states but fororE,L 0~= 0.7 eY~represented in table 3. This time, however, the -

S

5

5

_s~_

‘S

——

‘

-

..

‘SSS

-

-

~-S5S

S

S

-S -

‘

5

-

‘s

.~

TabJe3 S

ss S

5

Same. as table 1 but for E,0 =0.7eV and with (12, 12,9) vib— 2 states , S rotational ~tatés and 9 translatioiial L COS)(’ ‘-0~0 t~—~’1 0~2’ 0_413 ~S 0—*4 TotaL 0 454 0.5~8 0,891

re,~.[116]

q (~46 QQ54 .Q.1)53 0.1)48

0 074 0.990

0 0~5 0 929 0 00~ 1)209 0.055, 0.021. 0.006, 0.225 0.Q~5 0.01~. 0.001 0.218

Q,09~

0.0~9

—

—

0.21

5

5

-

S

S

,_~

,

.

frJ.$~er~ I~gr~I-,eq44~t9c~t qppfpqçl

11.

3313 1; it -

3

HL0- . 8,

~ ,1333;

i3~s(10s’

—or;:03fJ

-;

1

5~5J

‘WJd 12,10)

~‘

‘~

~ 0,;-

‘~

1 (1~Sl.~~)41.)~L;~

~

1 0.037

.

.:i3li33fsI0~4~.l ~

t’ 0072

,

~ ~tIffl)i =

I

-

~

0” ~

~‘A0

L,3IS1UI

Oi;.’/

1T11,l0t’~3 )j~

1

~0’’,o~~’l’i

~ ot)4r3;3,’ o~
.jjs,J~~

,jS

33~0S I3r

‘‘5’S~

~ ~

.~

0t~jj~ji

.

.,.o

j,3,~

~

~

~

~

3’fli) o. 5-

‘11

31 1’i’li;o~.-’-

A1,,3,;~,) -s~

I

‘5

031; 0)

3~3)353, 3;;35q ‘4r’!’) ,~,fl1;I 0.

~u~-:.o~0i130ST3ffl’,’~ ~

V~S&~

40~~ ~

i~

-M~ 0,13

.

It

-

-;IJ

~~gth~Pe ~ g~1iu1~ ~ ii~abI~it Of )1I0 .~l’(i3;3.f33)) jl3,5”j0~,1i 1i,)1:Il SAOOI 1’Af,)I,°I k. i l~)%~0E~)huui ~‘V, ~dk~1a~6à .~6’~y ~ ~

-

-1 3

~ 3

LO~S..lf

“I’!

~

10r1(~,-/

.3 1.13333330

~ 0~~lj.I~

.

~5sA~’~

(~)Sj9~

~

S

~O

~‘~~‘~13

~f(iI0;J;.l’i

p~i~

S

11~ ,J’,3~1l3

1Ss~.’l~A4$3ii.~1

~

c

-

Al)

S

i~r 0.041n~’..’o~ ,Q.02O.~~ Q.046 0 110 0.20 W ~1il0o~3 ~lM?6O_SI 3 ~

0d~i1~

ti’

0) + H for E,0, =

S

~4

~

4’,; ~12r~2~1t:~ ‘.-i0~ If5 i0~043 1tI0~9111I ~9~1 0~ ~0~8 ‘~f~0f~fl~~3 (12, 12,9, 11 0,051-, 0.098 0.061 0034 0.016 Q 27 ~(~Z ~10~ ~) ~~-)W:’ I(10~°S21~Y3°1° 0808 (12 12 10 7) Ill ~ ~Wr1~’ ~ 3

~ -;

~

;.~0,

l~39

diatct~ne*cch~.p~g~ pr.xess~s

Table 4 Three-dimensional probabilities for H + H 2(n = 0, jl.! = 0)—~H2(n, = 0, JfjJ 1j 0,1 11 ,i r3 uW t~i 0.9eV 0 31 1 il Vib-rot. -- Number of .,~ ~ 1 1 .o-j! oo~-j ieb~4r-Iio ~ dWo tIi0i0j1~3i~0P52t .

S

3 oil

st9a

lj

~

11j~

o~c&,~m4;~

11.

because for ~ = 0.9 ~he4th~ke ~ y is ~i~d sm~ll~t ~&d~d~y~ re~siiits~ Prdi?n3~e~lts 4.dj~o)f1èiei ~hd~ ~o~’~W ~L ~ ~ th~~ ~ ~S~ increasingly relevant. All our results are summarized in table 5 as calcu~te4!or,the eneij~ies.~tot i 30.6, 0.7 O.~,, td.0.9, 1.00 eV and compared with those obtained by other methok~s.Th~ä~F~eemen~ ~s r~~1il~ goo ~31I .~tofIi oill .r,o jr10 0 br -~-i10t3 Iolt,ijo)o~1)310 130 i~1,3f~1t ~L~001 Oil) 0313 3303; 0 -(1033; ‘~l!i ~L’;’.,r~tiA .dfI’)iliIL; .0311031 “N iiiOl)fi0r13l’ :313)00 0~il~1~0nr,Ill) OiO’-rOftlfflO Lon~81;)o ‘f,1i’ ‘‘

.

‘

,~

5~’

.‘‘,w’il:’/i

?3U,01’’’.,ifl ~ ~‘,i~l •idt .i J’J’~~.’5~uori~ ,,~j ) /—011331) 1031,-c 0f;(,OA’’t)l ,‘f,lO 1J 0_U 01.0) ~ .o4~,l;(1’44q L-’,lo-;3’1lo-3 .01) 003 “ ‘i if AtOll ;f~3 1 (33, ),Li~ 1 3 1 1 3 Al ‘(1 0 ,; E 0—-pO 0—al 0—az 0—*3 0-÷4 Total , ;~i0fI .t’ui!’1i0°’0I1~1” --i’Ou’73-: ~3f ofO 0 h;~lJ ‘;;il’ - 0 ~,..)fr!!’,0 :3)0013.0”.

‘/)10 : -

-

1

1 -

-1)1

,~.-no:;rl(; JO

330.3

“i)i’~0i

1~3~,1~

_0~21

-

5Q.t2~3,.3~1O ~ ob,.i’;00l’,~i Oil~30) ~~-;r1-~

,~

1~

I

~

0 ‘

It

_j(A~4~/01~92 ‘o0~60

0.8

-

0 3 .1

-

-.

30i11,

J’. -0’:-~it’0? 1)0ItJO.000):L~,i01I ,~3

i-ill~- ~t11

0.034 3; °“ff~ ~ 0.037 , ~‘‘ 0’’00’I ~

,C.’3;l1l3-P1 ~

3;, f3~ -oJOJI) ,l1lll ,,‘0 I 10’) III 1\’ 01(13 )‘ 91(11) S ‘ ~‘11 01 I(011110O~0‘jOl -

-)1r30

-c

0.077 0.063 35~~~’1 0.079 - ~ p.068

~i.

. , .1 ~

~—‘

,

0.039

—‘

11’

~21

.s;,(,(3

¶1

~

~.t103i’t

5O~.5S0~0~5 ~~ss 0.23

,Pft~t; 0~it0.27’;.~. 0-orP59,3’,5 ioP 1mio ~tli

S

‘30~YS

, 01 ~003’ 5

~~JS

~Oii31~25

‘l13

10 ,li

‘3,

1,

-‘.0~

1~/(0.~l,1/1

1’1I’.410 33,113 00

8’jt’0ii~6,

q.9~(1SO

~

~

113cr 1)?0-~ ,,i~l’ft”111033301113 br:, 001 tOcltufl JUo

abihties from. ref.-11161 f ~eif&ies E=A).6 0.7 0.8 1.0eV and IOi~~J~j~ ~ °-~

~

‘•:

lo 1~~(tC1091f3rJ /111;

~‘0lIi

UJ~10

~

mlrtl ~nc-( o 23 03~13:$LI0X1;~1’7}~i13 .0 03031.) 33131 ;r

1’~ioOto;09 i000Ifl’j iblw

S

~

~

b?~ap~~t

(33

$s~e—~ ~

‘{

‘3) (31 03 ~

?i;1)11031301fl3 on’) -IL-01i3OfT1 S ~ 110330! oil)

ml)j~OlitA’) 7331010,1 ‘1)13 U1!,,lA10 of, oil ~0 01)015 ;fiJ 1’ 5’~000311 t,i 1(110

34

140

M. Baer, Integral equation approach to atom—diatom exchange processes

8. Concluding remarks In this review we described a theoretical approach to studying atom—diatom exchange collisions which is based on a straightforward extension of the Lippmann—Schwinger integral equation. The extension is done in an ad hoc fashion via a N X N W matrix (N being the number of arrangement

channels), which is responsible for the coupling of the various arrangements. We have shown that this approach is quite general in the sense that one can obtain not only different kinds of known equations

(e.g. the Faddeev—Lovelace equations and the Fock coupling scheme) but also new sets of equations, which may be numerically tractable and /or theoretically attractive. In this review we discussed two such systems of integral equations. One of them follows from the assumption that the generalized potential matrix VW is symmetric. This assumption led to an interesting set of integral equations, which deserves further theoretical study. These newly found equations are not

only guaranteed to yield a unitary S matrix but their structure as a whole is interesting. In the case of three arrangement channels the W matrix contains nine unknown elements, which have to fulfill three linear algebraic equations (the “completeness” requirements). Supposing the generalized potential to be symmetric yields another three linear algebraic equations and the requirement concerning the

connectivity of the kernel yields another two, so that we are left with a system ofthree coupled integral (and for that matter also three coupled Schrödinger) equations which depend on a single parameter F, which may or may not be a pure number. The physical meaning of F is not entirely clear yet and its effect on numerical results is not always predictable. This system of equations was exposed to a numerical study and we found that only for certain F values correct results were obtained. Since this behavior was not consistent, no definite statements regarding F can be made.

In this review we presented three-dimensional transition probabilities obtained for the three-channel system H+ H2(n,, j1IJ=0)—>H2(n~,jfJJ=0)

+

H,

where n, and j, are the initial vibrational and rotational states and n~ and j~are the final ones. The results were obtained employing the orientation angle dependent W matrix elements. Although their intro-

duction may look somewhat clumsy (“too many words”), they have two important features: (i) they yield connected potentials which guarantee the connectivity of the kernels, and (ii) they are numerically not only very tractable but form, to date, one of the more efficient numerical methods for treating exchange collisions. It is true that in the final structure of the equations there appear a few undetermined parameters, namely, the angles 7:’ a = X, v, k, but correct results can be obtained for any combination of these, although the rate of convergence may depend on the choice made. However, since the various Va have a clear physical interpretation, they can be chosen a priori in such a way that a fast rate of convergence is guaranteed.

Moreover, the integral equations are usually solved by converting them into algebraic equations. To do that one integrates over all variables (coordinates) over their whole range of definition. Some of the integrations, like those with respect to the Euler angles, can be done analytically. The others have to be carried out numerically and most of the computer time is consumed by this process. In all available methods the integrations over the angles 7~,‘ç and Vk are carried out each time from zero to ir. In our case the integration over the angles ‘y is done by parts. Thus, for instance, for V~,the integration in the case of the Xv exchange term is done from zero to -y ~ in the case of the Xk exchange it is done from y to -ir; and in the case of the elastic term (non-exchange) it is done from y ~ to ‘y ~. This fact alone can

M. Baer, Integral equation approach to atom—diatom exchange

processes

141

reduce the consumption of computer time by one order of magnitude as compared with the other available methods. The coupled Lippman—Schwinger equations were introduced about two decades ago. However, the fact that the W matrix can be different from the channel permuting array structure and yield relevant results in a very efficient way, makes them more interesting than ever. Acknowledgement I would like to thank Dr. Daniel Neuhauser from Soreq Nuclear Research Center for the most successful collaboration that led to most of the new ideas presented here, to Professor D.J. Kouri from

the University of Houston for many fruitful discussions regarding the three-dimensional aspects of the coordinate representations, to Professor Y. Shima from Soreq Nuclear Research Center for his devoted collaboration in developing the numerical solutions and finally to Professor H. Nakamura from the Institute for Molecular Science at Okazaki, Japan, for his encouragement in the early stages of this review. References [1] M. Baer, in: The Theory of Chemicai Reaction Dynamics, Vol. 1, ed. M. Baer (CRC Press, Boca Raton, FL, 1985) p. 91. [2] M. Baer, in: Advances of Chemical Physics, Vol. 49, eds I. Prigogine and S.A. Rice (Wiley, New York, 1982) p. 191. [3]R.D. Levine, Quantum Mechanics of Molecular Rate Processes (Clarendon, Oxford, 1969). [4] R.E. Wyatt, in: Atom—Molecule Collision Theory, ed. RB. Bernstein (Plenum, New York, 1979) Ch. 17. [5] A. Kuppermann, in: Theoretical Chemistry: Advances and Perspectives, Vol. 6a, eds D. Henderson and H. Eyring (Academic Press, New York, 1981) p. 79. [6]M.S. Child, Molecular Coilision Dynamics (Academic Press, London, 1974). [7]R.D. Levine and R.B. Bernstein, Molecular Reaction Dynamics and Chemical Reactivity (Oxford Univ. Press, New York, 1987). [8]E.M. Mortensen and KS. Pitzer, Chem. Phys. Soc. (London), Spec. PubI. 16 (1962) 57; [9] A.B. Elkowitz and RE. Wyatt, J. Chem. Phys. 63(1975) 702. [10] A. Kuppermann, G.C. Schatz and M. Baer, J. Chem. Phys. 65 (1976) 4596. [11] R.B. Walker, J.C. Light and A. Aitenberger-Siczek, J. Chem. Phys. 64 (1976) 1166. [12] R.T. Pack and GA. Parker, J. Chem. Phys. 87 (1988) 3888; see also: (a) G. Hauke, J. Manz and J. Romelt, J. Chem. Phys. 73(1980)5040; (b) A. Kuppermanii, J.A. Kaye and H.P. Dwyer, Chem. Phys. Lett. 74 (1980) 257. [13] L.D. Faddeev, Soy. Phys. — JETP 12 (1961)1014. [14] K.M. Watson and J. Nuttal, Topics in Several-Particle Dynamics (Hoiden Day, San Francisco, 1967). [15] D.A. Micha, in: The Theory of Chemical Reaction Dynamics, Vol. II, ed. M. Baer (CRC Press, Boca Raton, FL, 1985). [16] K. Haug, D.W. Schwenke, Y. Shima, D.G. Truhlar, J.Z.H. Zhang and D.J. Kouri, J. Phys. Chem. 90 (1986) 6757. [17] J.Z.H. Zhang, D.J. Kouri, K. Haug, D.W. Schwenke, Y. Shima and DO. Truhlar, J. Chem. Phys. 88 (1988) 2492. [18] M.J. Seaton, Philos. Trans. A 245 (1953) 469. [19] P.G. Burke, H.M. Schey and K. Smith, Phys. Rev. 129 (1963)1258. [20] D.A. Micha, Ark. Fys. 30 (1965) 411. [21] W.H. Miller, J. Chem. Phys. 50 (1969) 407. [22] M. Baer and D.J. Kouri, Phys. Rev. A 4 (1971) 1924. [23] M. Baer and Di. Kouri, J. Chem. Phys. 56 (1972) 4840. [24] M. Baer and D.J. Kouri, J. Chem. Phys. 56 (1972) 1758. [25] M. Baer and D.J. Kouri, J. Math. Phys. 14 (1973)1637. [26] D.J. Kouri and F.S. Levin, Phys. Left. B 501 (1974) 421. [27] D.J. Kouri, M. Craigie and D. Secrest, J. Chem. Phys. 60 (1974)1851. [28] D.J. Kouri, F.S. Levin, M. Craigie and D. Secrest, J. Chem. Phys. 61(1974)17. [29] D.J. Kouri and F.S. Levin, Phys. Rev. A 10 (1974)1616.

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