PH YSICAL RE VIE% A VOLUME 17, NUMBER 3 MARCH 1978

Semiclassical perturbation theory of electron-molecule collisions

W. H. MillerDepartment of Chemistry, University of California, Berkeley, California 94720

F. T. SmithSRI International, Menlo Park, California 94025

(Received 25 July 1977)

The theory of the classical or semiclassical S matrix is combined with the use of perturbation dynamics toderive an approximately unitary expression for the scattering matrix for a very general class of potentialinteractions. The S matrix takes the form of a sum over products of Bessel functions whose orders arerelated to the changes in quantum numbers occurring in the transition, and whose arguments depend on thedynamical variables of the problem, including the unperturbed quantum numbers. In the general case, thesearguments can be expressed as simple integrals over the unperturbed trajectory, and for well-behaved

potentials they can be explicitly evaluated in terms of the modified Bessel functions Ko and K&. Theconnection between the semiclassical perturbation scattering theory and other approximations, such as theBorn and eikonal approximations, is demonstrated. The general theory is illustrated by applications toelectron- (or ion-) polar-molecule scattering, including quadrupole as well as dipole interactions andincluding coupling to vibrations in both harmonic and anharmonic approximations, The more complicatedinteractions involve lengthier products of Bessel functions-in the sum-and-product representation, but theseare easily and systematically evaluated, and they. reduce smoothly to the appropriate simpler expressionswhen the coupling coefficients of the higher-order terms become small. More complicated potentials,including interactions between polyatomic molecules, can be handled by a simple systematic extension. of thesame principles. For electron-molecule scattering, these expressions can be used in their present form sincethe sums are dominated by' Bessel functions of comparatively low order which can be evaluated directly;extensions to molecule-molecule scattering and ion-molecule scattering are equally valid formally, but theirpractical application will often require the use of asymptotic approximations to the Bessel functions.

I. INTRODUCTION

One of the most important aspects of the class-ical S-matrix theory introduced in 1970 is that itshowed how dynamically exact classical mech-anics can be combined with quantum mechanics ina consistent way for describing complex (i.e. ,

inelastic and reactive) collision processes. Exactclassical mechanics for an inelastic collision, how-ever, means numerically computed classical tra-jectories, and although these are readily obtain-able nowadays, ' simpler' models for the dynamicsare often of sufficient accuracy and provide moreinsight into the mechanism of the process. Theyare also easier to apply and may even lead tosimple analytic expressions for the cross sectionsof interest.

This paper explores the form taken by class'icalS-matrix theory when the collision dynamics istreated by perturbation theory. In this limit thetheory becomes essentially equivalent to the per-turbative models from other approaches by Cross, 'Beigman, Vainshtein, and Sobel'man, Levine andJohnson, ' and Percival and Richards. ' A recentreview by Clark, Dickinson, and Richards' givesan extensive summary of the approach of Percivaland Richards and applications that have been madeby them and their co-workers. An important dif-

ference between our work and that of Percival andRichards for the case of rotational excitation isthat we have taken correct account of total angularmomentum conservation by applying the theory ina total angular momentum representation. Thus,our. 8-matrix elements are labeled by the (con-served) total angular momentum quantum numberand the full complement of other quantum numbersand energy variables, while they have worked with-in an impact parameter representation that m~esno reference to total angular momentum. Thismakes it possible, for example, for us to calculatedifferential scattering cross sections, momentumtransfer cross sections, etc. , simply by sub-stituting our S-matrix elements into the appropri-ate quantum- mechanical expressions.

Section II shows how semiclassica1. perturbation.scattering theory is obtained in a very simple wayfrom the more general classical S-matrix theory,and its relation to some other theoretical modelsthat have a very similar structure is discussed inSec. III. In particular, we demonstrate the sim-ilarity between the semiclassical expressions andvarious exponentiated forms, including-the dis-torted-. wave Born approximation; since .the gen-eral semiclassical theory leads naturally to anexponential form, it provides some justification forsome of these ad hoc exponential models.

939

94(} W. H. MILI. KR AWD F. T. SMITH

The initial applications we have in mind are torotational and vibrational excitation of moleculesby electrons, and Sec. IV gives some specifics ofhow the theory proceeds for these applications.It is of considerable practical importance that oneobtains closed-form expressions for the S-matrixelements for these inelastic processes. In gen-eral, these expressions take the form of sums overmultiple products of Bessel functions. They aresuited to the rapid evaluation of cross sections,and they have convenient and transparent con-vergence properties, that make it possible easilyto assess quantitatively the effects of additional in-teraction terms that are usually assumed smaQand dismissed with a quali. tative argument. %eshow, as a specific example, how vibrational androtational interactions can be incorporated to-gether, and how the expressions converge smoothlyto the pure rotational form at low energies. It iseasy to treat the consequences of other interactionssuch as higher multipole terms and anharmonicvibrational effects by the same procedure.

Our S-matrix expressions show that the per-turbation results are justified in two limits thatare in principle independent, though the domainsof convergence they define may sometimes over-lap. There is a domain of low-energy convergence,related to the fact that inelastic transitions becomeimpossible when the available energy is less thanthe next excitation threshold. There is also ahigh-energy convergence, reflecting the need forsome matching between the collision time and thecharacteristic response time of the internal modewhose excitation is being examined. The dimen-siOnless parameters characterizing these con-vergence properties" depend strongly on the re-

duced mass, among other factors, with the resultthat the details of the application of our formulaswill be quite different for electron-molecule. col-lisions and for heavy-particle collisions. In parti-cular, a very important domain of electron-molecule collisions, at energies up to 10 or 20 eV,can be examined with the use of Bessel functionsof comparatively low order, but heavy-particlecollisions require very high order and necessitatethe use of asymptotic approximations. %e focusour attention here mainly on the application toelectron collisions, and the following papersdemonstrate the detailed application of the methodto a study of total excitation and momentum trans-fer cross sections in electron-polar-molecularcollisions.

H(P, R, n, q)=P'/2m+ e(n)+ V(R, n, q); (2 1)

R ~nd P are the coordinate and momentum for rela-tive translation of the two collision partners, nand q are the action-angle vari. ables for the inter-nal degree of freedom, arid &(n) is the WEB eigen-value function for the internal degree of freedom.The collinear collision of an atom with a dia-tomic molecule is an example of this type of sys-tem. The reader should see Ref. 8 for a moredetailed discussion of this model and appli. cation,of the general classical S-matrix approach to it.

The initial-value representation for the classicalS matrix is given by'

II. GENERALTHEORY: PERTURBATIVE LIMIT

OF THE CLASSICAL S MATRIX

It is simplest first to consider a system with onlyone internal degree of freedom, for which theclassical Hamiltonian is

$ = (2v) dq,' . exp(i{4'(q(, n, )+qt(qI, nq)[ny(qq, n()-ny]f) (2.2)

(units being used so that 5= 1), where n&(q„n&) and

q&(q&, n& ) are the final values of the action andangle variables, respectively, as determined byclassical trajectory with initial values n, and q,for these variables (q, and qz are, more precisely,the differences of the angle variables from theirunpertuibed values; see Ref. 8 for more details);n& and n, are integers specifying the initial andfinal quantum states of the internal degree of free-dom. The action integral 4(q„n, ) is defined inRef. 8; for present purposes it is only necessaryto note that its derivative with respect to q, isgiven by

V(R, q, n) = Vo(R) + &V(R, q, n), (2.4)

and the zeroth-order trajectory is determinedusing the interaction Vo(R). With this Eeroth-order Hamiltonian,

H, (P, R, n, q) =P'/2m+ V, (R)+ a(n), (2 5)

the translational and internal motions are un-coupled, and it is easy to see that the zeroth-order approximation to the trajectory is

To introduce the perturbation approximation, theinteraction term in the Hamiltonian, V, is dividedinto two parts,

ec (q„n,), , en&(q„n, )qfkq, , n, j

oq ~ Qq . (2.3)(2.6a)

(2.6b)

SEMIC LASS ICAL PERTURBATION THEOR Y OF. . . 941

q (t) = q, + ]dt,

where

(2.6c) 5= —+ lim -kR+4 s dR']2m[E, —V (8')] ]'~'),

m= &'(n, ), (2.6d)

ng (gg, n)): ng +

and where R,(t) is the translational trajectorydetermined by Vo(R). n&(q„n,.) is then given tofirst order in the perturbation by

(2.14)

E, =5'k'/2m being the translational energy .As itstands, the S matrix in Eq. (2.12) is not sym-metric (as the exact S matrix is), and it can besymmetrized in one of several ways. The usualprocedure is to use for the unperturbed parametersthe average of initial arid final values, making

or

8

eq,.

&V,(RO(t), q, + &dt, n, }eq

dt V, (R,(t), q, +&dt, n,},

25 6)+5~,n, -r'(n, +nq)-=n, E, = ,'(E, , +-E, t), (2.15)

where f6,), i = 1, 2 is given by Eq. (2.14) withE« E —z(n-—, ). This symmetrized version of theperturbation approximation for the classical 8 ma-trix is thus given by

e& (6&+6&) (2 Pny, ng

BA(q(, nq )n~(q, , n,. )=n, —

Bq]

where

(2.7)

with

d—e-f & n~ m] )q] e-f A (q], n) (2.16a)

A(q„n, }= dt V(( Ro(t), q(+(ot, nq} . (2.8) A(q, n) = dt &V(R,(t), q, +art, n} . (2.16b)

s4(q„n,. ) AB(q, , n,). (2.10a)

or

&'A(q, , n, )4(qg, n)) = 4o+ c@ q(

Bq](2.10b)

It is necessary to determine qz(q„n, ) only tozeroth order:

qt(q( ~ng) =qg ~

Using Eq. (2.3) with Eqs. (2.7) and (2.9) gives

This perturbative result has been obtained by anumber of other workers using a variety of formal-isms, most notably by Cross, 2 by Beigman, Vain-shtein, and Sobel'man, by Levine and Johnson, 'and by Percival and Richards.

In this section of the paper, and in Sec. III, weuse a detailed notation suitable for deriving thegeneral formulas' for the semiclassical perturba-tion S matrix, Eqs. (2.16a) and (2.16b), and otherapproximations. In Sec. IV we adopt a simplifiednotation, based on Eq. (2. 15) and the followingfurther substitutions:

where 40 is the phase if ~V—= 0. Integrating Eq,.(2.10) by parts gives the action function as p rif rif ~ri

y (2. 17)

BA(q, , n, )4 (q „n,) = 4, —A (q, , n, ) + q, 8'q ~

(2.11)

ed%&(2 )| 2ff

g" ))q]e 4A&q], n])

r

Substituting the perturbation approximationsfor q&(q„n, ),n&(q, , n, ), and 4(q„n, )—Eqs. (2.9),(2.7), and (2.11)—into the initial value representa-tion, Eq. (2.2), gives

Sn n- =sat ~f i (2.18)

(See the Appendix for

further�'discussion.

) Inthis notation, Eqs. (2.16a) and (2. 16b) become

and on' an alter native notation for the S-matr ixelements that segregates unperturbed (average)values of quantum numbers from the changes inthem,

(2.12)Sn e2$6(2&)-1

2ge-fg4n e-fA (q, n) (2.16a')

The unperturbed phase 40 is twice the phase shiftfor the potential V, (R),

(2.16b ')

40=26,

where

(2.13)This notation is easily generalized to apply tosystems with multiple quantum numbers; in Sec.IV we shall generally label these with the con-

942 W. H. MI.L LEE AND F. T. SMITH

I

ventional symbols (J,j, / for angular momenta, v for vibration), thereby liberating the proteanletter "n" for other uses.

III. COMPARISON WITH OTHER PERTURBATIVE MODELS

It is illuminating to compare the perturbative limit of classical () -matrix theory, Eq. (2.16),with some. other similar models.

Consider first the semiclassical version of the distorted wa-ve Born aPProximation (DWBA) for theS matrix':

. 2mn n e ~ f ~n n 2

0 0f nf i t n. (3.1)

where the semiclassical internal wave functions

(t)„(q) aresin —+

RdR'k (R') sin —+ dR'kr (R'))

f

(t)„(q) = (2)[) 'i 'e'"', (3.2)

B= —cos dR [k, (R ) k, (R )], (3.4)

and the translational wave functions f(R) are givenby their WKB approximation,

kq(R) —k, (R) =— [5(nq) —5(n, )])&t' k(R)

rf, (R) = sin —+RdR ' k, (R ')) [k; (R) )

' 'where

m(np n, )(—u

@2k(R)(3 6)

with

k, (R) =(2ii[E —&(n, ) —V,(R)]/k']]'~'. (3.3)

k(R) = [ki (R)ky(R)]

and with

(3.6)

With the following rather standard set of approxi-mations,

dR=dt,k k(R

it is not hard to show that Eq. (3.1) becomes

Z23'

2 " e'"I'r' 2 —=— dt(rrr) ' d ''"r "r'" "k(r(R (t) q ng).n pn& nf, n& qe 0f' a t)O 0(3.8)

Changing the integration variables q to q„q=q, +~t,

then gives (setting tl= 1)

(3 9)

SBWBA i (62+ 5i)(2 )-2nf, n

2fi'

x dq, e (("f "!"([1-iA(q,:,n)] .0

(3.12)

Since

ei (5(+ 6y) 6 i(2w)-2nf, n&

x dq, e''"r "rr'rr((qr, n)). .0

(3.10)

2g

g &)w ei (5( +55& 6 5(2w) 2 f d -i (n -n )qnf5 n~ nfp n~

0

x, dt ~V R0 t, q& +(dt, n

1 —iA(q„n)- e '""i"', (3.13)

one sees that' the perturbative classical S-matrixresult, Eq. (2.16), is obtained. The procedure ofthe DWBA, joined with a particular type of expo-nentiation, thus'leads back to the perturbationlimit of classical S-matrix theory.

Th(2 exponentiation indicated in Eq. (3.13), which

gives the perturbative classical S-matrix ex-,pression, is different from the usual ad koc ex-Ponential aPproximation. " The usual approachis

If one now exponentiates the integrand in Eq. (3.12),

2g

6„„=(2w)'

dq,. e ' '"p "i"i,0

the result may be written as

(3.11) gExp. Approx. —ei (6 (+6y) (e-iA)nf, n&

where A is the matrix

(3.14)

SEMICLASSICAL PERTURBATION THEORY OF. . . 943

A„„=(2)T) 'dq,.e ' "t "» )6)A(q„n); (3.15)

i.e. , the usual approach exponentiates the matrixof A, while Eq. (3.13) is the exponentiation of the

integrand of the matrix element.Another kind of exponential approximation can

be obtained as follows. Returning to the DWBAexpression in Eq. (3.8), one sees that it can alsobe written as

$ Dw»)A e» (5»e 5y ) 1 (2»»)-1nI, n&

2g OQ

dqe ''"t "t ' dte' t"'' 'ttq(tt (t), q n))m Oo

e »(6»+'6y ) (2 )1 dqe ' "~ "»" 1- — dt e''"t "t'"'»( V(R, (t), q, n)

1(3.16)

If one here exponentiates in the following indicated way,OO OD'

1 — dt exp]t(nt —n, )tet] tip(R (.t), q, n), -nxp —— dt exp[i(nt —nt)tet ] ttp( R, (t), q, n)), (3.17)

then one obtains a result which may be thought of asan extension of the eikonal approximation" to in-elastic scatter ing:

$„'"'„' = exp [i (6» + 6~)] (2»»)'

2'x dq exp[-t(n~- n, )q]

0

~ ~ (e

x exp —— dt exp i n& —n, ut-AJ „

x ttp( tt, (t},q, n)) . (3.18)

Although similar to the perturbative classical S-matrix expression, Eq. (2.16), Eq. (3.18) is clearlydifferent.

Finally, the sudden aPProximation is obtainedfrom Eq. (2.16), or Eq. (3.18) by setting (d = 0,i.e. , the eikonal. approximation and the perturba-tion limit of classical S-matrix theory becomeidentical in the sudden limit.

The perturbative limit of classical S-matrixtheory, the exponentiated distorted-wave Born.approximation, and the semiclassical eikonalapproximation all have a family relationship aris-ing from the use of some quantity related to aclassical action integral. The semiclassical per-turbative S matrix, being obtained from the gen-eral classical S-matrix formalism, rather thanby an ad hoc exponentiation, is in general con-ceptually preferable. The semiclassical expres-sions are known generally to represent an asym-ptotic approximation to the true solution of aquantum-mechanical problem. As such, 'they mayoscillate about the true solution, as various param-eters are changed. A systematic quantitative studyof their accuracy in comparison with numericalquantal calculations and with other analyticalmodels wiJ, 1 be illuminating; in companion papers

we report such a comparison for the problem ofelectron collisions with polar diatomic molecules.

IV. UNIFORM CLASSICAL S-MATRIX ELEMENTS

FOR SPECIFIC SYSTEMS

A. Ion-dipole collisions

We have considered the case of rotational ex-citation of a dipole by a charged particle pre-viously. " The perturbation potential is

V„= (et»:/R') cosy, (4.1)

where y is the instantaneous angle between theorientation of the dipole and the line from it tothe other particle,

cosy= —cosqy cosq, +N sinqy sinq, ,

and tt= (j +f' —d')/2jt =cos)( is the cosine of theangles between the planes of the rotationaltranslational motions. Cross sections for variousprocesses are given in terms of the S-matrix

(4.2)

One extremely convenient feature of using theinitial-value representation for the classical Smatrix is that it effectively produces the uniformclassical S-matrix elements directly, particularlyso for the perturbative situation con.sidered here.

For inelastic but nonreactive processes, it isoften legitimate to treat the dynamics with a per-turbation approximation. If we make maximal useof rotational coordinates, sinusoidal functions,and Fourier expansions in the description of theunperturbed motion and the perturbation, the Smatrix turns out to be expressible, in general as asum over products of Bessel functions, with veryuseful convergence properties. We shall illu-strate the procedure first with some simple ex-amples, and then discuss the general form of theexpressions.

W. H. MILLER AND F. T. SMITH

elements Si r i r (d), where J is the total angularmomentum of the system, and j and 1 are the

ff' i

rotational angular momentum of the dipole and theorbital momentum of relative translation, re-spectively. Equation (2. 16a) of Sec. II must thusbe generalized to the case of two internal degreesof freedom, and for this case the unperturbed po-tential Vo(R) is zero.

Equation (2.16a') then gives

Sayrrr =Sr&i&,j,ri(d. )

2t ', ~f'

=(2&r) 'dq dq e ' "re ' i're '""r'i'

f0 0

(4.3)

where

For the ion-dipole interaction we have shownearlier" that

A(q„q ) =-(2e&«m~/l)([Kr(&u) NKo(u&)] cosq, casque

+ [Ko(«&) NKr(«&)] sinq, sinqi], (4.4)

where e =Blj /E„K, and K, are the usual modifiedBessel functions, l =-,'(I, +lI), j = z(jr+ jz), lr, is thedipole moment, m is the reduced mass for transla-tion, and e is the electronic charge of the ion.Since

function, so that

Si,rr„r——g„(Z,)g„(Z,) (4.11)

with &rr and Z, , i = 1,2, given. by Eqs. (4.10) and(4.7), and

(4.12)

Equation (4.11) is our result obtained earlier, "modified by inclusion of the phase factor i"~'"2 toinsure symmetry and approximate unitarity of theS matrix (unitarity is approached rapidly as j andl i.ncrease). It is also obtained if the "primitive"expression for the classical S matrix is used,rather than the initial-value representation, withsubsequent "uniformization" via the Bessel func-tion scheme. " Use of the initial-value representa-tion thus provides a simple, direct route to thedesired uniform semiclassical expression.

The quantities j„l„jf,lf,j, and l as we have in-troduced them are classical angular momenta, andwe have assumed units such that A=1. For com-parison with full quantal calculations (or with ex-periment), we use the usual equivalence

j„-(j,„+—,')lr, l„-(l,„+—,')5,which is strictly valid in the asymptotic limit oflarge quantum numbers and approximately trueeven for small ones. We now recapitulate thesalient results using the dimensionless parameters

cosqr cosqy =. z cos(q, + qi ) + —,' cos(q, —qr ),

sinq, sinq = ——,' cos(qr+qi)+-,' cos(q, —q ),

Eq. (4.3) takes the form

&t -'I

S ' =(2&r)' dq dq e ' "re ' &'ikgb, l

0 0

{4.6)ps = ml&, e/I', cr = 8'/2IE& = B/E, , (4.13)

and assuming hereafter that j's and l's are quantalvariables in the semiclassical limi. t. Then Eq.(4.10) is unchanged and we have

(4.14)

where

iZ ccs & sr+oi& iZz ccs & a&-cr& (4 6)(4.16)

Zr = (e lcm/I ) (1+N)«& [K&(«&) —Ko(«&)], (4.7)Z, = (eiim/1 )(1—N)«&[K&(«&)+Ko(e)] .i

Changing integration variables from q, and q& toQ, and@, ,

Q, =q, +q&, Q, =q, —q&,

gives

(4.8)

yq -jK~ Qg -fx2Q22

&& eiZr cosqrei ZcosQS (4 9)

where

&&i= z (&j +&l), &&, =s (&l —&j ); (4.10)

the. integrals in Eq. (4.9) factor and are recognizedas integral representations of the regular Bessel

Z, = s (1 N)&us[Kr(~s)+K, (&u~)] .l+&

(4.16)

~~ can be recognized as the ratio of the angularvelocity of the colliding pair at closest approachto that of the rotor, and the functions of „appear-ing in Z„Z, are of the nature of resonance inte-gl als.

The semiclassical S matrix for ion-dipole scat-tering is completely defined by Eqs. (4.10) to(4.16). The interaction strength parameter Ps canbe used as the basis for a power-series expan-sion of the S or T matrices and the cross sec-tions. From the expansion formulas for the Besselfunctions it is seen thatthe lowestpower of P~ in

S~z» is P", where

SEMICLASSICAL PERTUQ, BATION THEORY OF. . .

V= Kl + K2 (4.17)

Terms with v= 1 correspond to the first Born ap-proximation. The v= 1 and v=0 terms are confinedto the following combinations of ~l, ~j:

5 'A(q„q&) =Z, + gZ, cosQ, .f=1

(4.23)

The S-matrix elements of Eqs. (2.16) and (4.3)are then

f 2f

&j +1 +1 —1 —1 0

~l +1 . -1.+1 -1 0

S~~", ~', = (2n)'

dQ, dQ,

x exp [.-i(-,'- A Q, + —,'

&jQ,)]Because pure dipole scattering does not producea phase shift. for &=0., elastic scattering does notenter until the &=2 terms; correspondingly, itis absent in the first Born approximation. Terms.with &=2 (&j =0, +2), v=3 (&j=+1,+3), and higherare easily evaluated.

B. Ion-quadrupole collisions

x.exp(-iA) .

Intr'oducing the identities21/

dQ3 ~(Q, —Qg —Q2),0

6(Q. —Q, —Q.)=-2K

Q ~on3

(4.24)

The interaction potential for this case is

V, (R, cosy) = (eQ/2R') ~ (3 cos'y —1)

= (eQ/2R')-,' (3 cos'y+ 1), (4. 18)

together with similar identities involving6(Q~+ Q, —Q,) and k„and interchanging the orderof summation and integration, we can integrateEq. (4.24) as a sum of Bessel function products:

where Q is the quadrupole moment and y is againgiven by Eq. (4.2). The function A(q„q&) can beexpressed thorough the use of the auxil. iary vari-abl|.s

Q, =2q„Q.=2q, , Q. =2(q, +q, )=Q, +Q. ,

where

, ~$ +k, +k, ,

(4.26)

Q4=2(q, —q()=-Qi+Q2 (4.19) v, =-2 ~j k, +k4)

Because of the symmetry of the potential, thecharacteristic frequency for the quadrupole inter-action is twice that for the dipole case,

3--k3&4= -k4. (4.2V)

co =2~ ) (4.20)

and the interaction strength is reflected in thedimensionless parameter

The symmetry of the potential admits only transi-tions with even values of &j and &E, and theindices v,. are a.ll integers. The index

em'Q 2E(4.21)

(4.28)

These parameters appear in some resonance in-tegrals:

P 3N 1O (f+ &)2

identifies the leading power of P, in a powerseries expansion. For v = 0 and v = 1, the indices&j,hl, k„k4 appear in the following combinations:

. P, 1 —N(l+-')' 8

0

~l 0

2 0. -2 o -2 2 2 -2

0 2 0 -2 -2 2 -2 2

Z P, . 3(1-N}(f —'}2 8»» x(~ }» (4.22)

The function A(q„q~) is then found to be

[,(1 —2,)&,(,)+2 ', IC,(,)),P, (1+N)'+2

Z, =(

', , [ar,(l+ 2(u, )K,((u,)+ 2(u', K,((o,)].p (1 —N)+2

k, 0 0 0 0 0 0 0 1 1

k4 0 0 0 0 0 1 -1 0 0

The nonvanishing Zo phase-shift term in Eq. (4.26)produces elastic scattering in the quadrupole caseunlike the dipole; expanding the exponential. l onegets the first-Born elastic contribution. The v= 1terms give the inelastic selection rules for smallP, : &j =+2, -2(&l =0, +2).

946- H. MILLER AND F. T. SMITH

C. Vibrational inelasticity in ion-dipole collisions manner. The interaction potential is

V(R, r, cosy) = ep(r) cosy/R', (4.28a)

We treat vibrational inelasticity in ion-dipolecollisions as an example of the general case.Vibrationally inelastic transitions may be includedin the charge-dipole model in a straightforward

where g(r) is the dipole moment as a function ofthe vibrational coordinate r. The perturbation limitof the classical S matrix is given by the obviousextension of Eq. (4.3):

S = (27)) ' dq dq dq e '~"~e '~J' e '"" e)'""~" ")'Agan/b, v V g 1 ~ s

0 0 0(4.29)

where v, q, are connected with the action-anglevariables for the vibrational degree of freedom+v = vy —V. and

d(q„, qt, q)=lt ' f dty(tt(t), r(t), ocsy(t)).a CQ

a power series in cos(q„+ 27) v„t).Another simplifying approximation, which is

often adequate but which can also be easily ex-tended if necessary, is to expand the dipole mo-ment about the equilibrium value of the vibrationalcoordinate,

(4.30)u(r) = ~(r.,)+ t '(r.,)(r- r.,)+ (4.32)

R(t) and cosy(t) are the same functions of t asfor the rigid-rotor case. r(t), the unperturbedmotion of the vibration coordinate, is given inthe simplest approximation —a harmonic oscilla-tor —by

and retain only the linear term. The expressionfor the action of Eq. (4.30) is then

"dt ep(r„) cosy(t)qqtt q f t q j R (t)2

r(t) = r„+ cos(q„+ 2)) v„ t ),h(2v+ 1)2'1TPl „&

(4.31)+ dt ept (r.,), [r(t) —r„]cosy(t)

where r, is the equilibrium vibrational coordinate,2~v„ is the vibrational frequency-, mv is the re-duced mass of the oscillator, and v is taken to bethe average of the initial and final vibration quan-tum numbers, v=~ (v, +v,). For highly excitedvibrational states this harmonic approximationmay not be adequate, but it can. be readily ex-tended: the right-hand side of Eq. (4.31) becomes

(4.33)

A„(q, , q&)+A. (q-., q„q,); (4.34)

A.„ is precisely the function obtained for pure rota-tional excitation of a rigid rotor, Eq. (4.4), andA„ is the addition to the action due to vibration.Using Eqs. (4.30) and (4.31), A„ is

e tt'(r. ,) 'It(qos () )'t', qy qy tq&tV V

Evaluating this integral gives

cos(q„+ 2)) v„t)(b'+ 2m-'E, t ')"'

x {[ bcosq,-+ (2E, /m, )'t 't sinq, ] cos[q& + 2B(j + 2)S 't]+N[b sinq, + (2E,/m, )'t't cosq, ] sin[q, .+ 2B(j + ~)I 't]) . (4.35)

A„(q„,q&, q, ) = ' O1+N)~, [K,(&u, ) —K,(~, )] cos(q„+ q~ + q, )m, e y.t(r„) h (2v+ 1)2h2 (l + —,

') 27) m„v„

+ (N-1)m, [ K,(a, )+K,(w, )] cos(q„+qz —q, )

+(N 1)&u [K,((e ) K, (~ )] cos(q„—q&+q, )

+(N+1)&o [ K, (m ) K„(&u )] cos(—q„+q&+q,)j, (4.36)

where K, and E, are, as before, modified Besselfunctions, and

&o„= (l + —,')hv„/2E, (4.38)

CO = CO„+ CO„, (4.3V) and +„as in Eq. (4.15).The function A„(and A„) can be simplified by de-

fining

, SEMICLASSICAL PERTURBATION THEOR Y OF. . .

mre lr, '(r„) k25 gm~ V~

(4.39)6

(4.46)

I „(x)=x[Kr(x) +K,(x)],and taking

(4.40) of an expansion in P~ and P to examine the small-Pselection rules for this interaction. From (4.45)we find

and

Z, = 4 (1+N)I, (&o„),l+2

Z2=

Z4—

Z—

(1 N)I —(&u4),l+2

1 l/2(1+N)I (~.),

l+2P„(rr+ k)" '

l+21 l/2

(1 N)I, (~ ),l+2

1 l'/ 2

(1+N)I (~ ),l+2

(1 —N)I (rd, ),

Q, =q, +q, )

Q =qr —qr,Q, =q„+q& +q, ,

Q4= q, +qr —qr = —Qr+ Qr+ Q3,I

Q5= q. —qr + qr = -Qr —Q2+ Qs,

(4.41)

(4.42)

6V = V3+ V4+ V5 —V6 ~

6j= Vl+ V2+ V3+ V4 —V5+ V6,

Vl V2+ V3 V4+ V5+ V6 '

(4.47)

v 0

vl 0 +1 0 0 0 0 0

v, 0 0 gi 0 0 Q 0

0 0. 0 a1 0 Q 0

v4 0 0 0 0 +1 0 0

v5 0

V6 0

0 0 0 0 +1 0

0 0 0 0 0 +1

~v 0

bj QI

~l 0

0 0 +1 +1 +1 +1

+1 +1 +1 +1 +1 +1

~1 +1 ~1 +1 +1 +i

We can tabulate the available combinations forv=Q, 1:

Qs=-q +qr +qr =2Qr —Qs

Then we have6

~ --Q Zr cosQr

A4 =Q Zr cosQr (4.43)

6

A=A4+A„=gzr cosQ, .

S ~y'~'r', "r. = ~ D&. (~r»a4, a, , a6=-

where

v, =~(nj+ nl) —&v k, k, +2k, ,

v, = —,' (&j —&l )+ k4- k, ,

v, = &v+k4+k, —k6,

V4——-k4, V5 = -k5, V6= -k6.

Again. , we can use the index

(4.44)

(4.45)

Now the S-matrix elements can again be ex-pressed in the integral form of Eqs. (2. 16) and(4.3). Introducing identities like Eq. (4.25), wecan perform the integrations. The result is

To conserve parity, changes of one unit in j areaccompanied by changes in l. More significant isthe selection rule (for v=1: first order in P) re-quiring a change in j (and l ) to accompany any

change in v. This is, of course, a specific con-sequence of the potential we have assumed, which

couples the vibrational motion with rotation throughthe ion-dipole term with a varying dipole moment.For this potential, we expect the same result toappear in the first Born approximation.

At first sight, the multiple-sum-and-productform of the S matrix, as in Eq. (4.44), may seemintractably complex. Fortunately, it has verysatisfactory convergence properties, and it iseasy to show that (4.44) collapses smoothly tothe simple form of (4.11) both at low energies Erand when P„ is much smaller than P4. To demon-strate the behavior in these limits it is helpful toinspect the form of the functions I,(x), Eq. (4.40),that appear in all the arguments Z, of the Besselfunctions, Eq. (4.41). The functions I, are every-where positive, never exceed an upper bound

'

slightly greater than unity, and decline exponen-tially toward zero at large arguments. When thetranslational energy E, falls significgntlybelowthe vibrational spacing, co„and consequently ~,become large, the I, functions in Z, to Z, be-come small exponentially, and so do Z, to Z, them-

W. H. MII I, ER AIWD F. T. SMITH

selves. On the other hand, if p„«1 (and «p~),and the vibrational quantum number v is not ex-cessively large, Z, to Z, wil. l be small at allenergies (the factor 1+% always falls between 0and 2). Whenever the arguments Z, are verysmall, the functions $„(z,) approach zero exceptfor J,(Z, ), which approaches unity, i.e. , g„(Z, )—6(v, ) (i =3, . . . , 6), and consequently Eq. (4.44)-Eq. (4.11). Furthermore, when some of the Z&'sare small but not negligible, the corrections to(4.11) can be evaluated by a power series expan-sion of the appropriate ~'s combined with a directBessel function evaluation of the others.

D. General formula

&n,.q,. —A, (4.48)

where

with

&Vdt =Z, + g Z&(E, n, ) cosQ&, (m~n)g-1

(4.49)

Q, =q, (i&n) (4.50)

(but sometimes it is useful to make a linear trans-formation on the q, 's first), and

It is now sufficiently obvious that we are dealingwith a general procedure that can be appl. ied in agreat variety of atomic, molecular, ionic, andelectronic collisions. The principal limitationsare (a) that the transition is produced by a pertur-.bation, which should not be too large, and (b)that the transition. not involve a chemical reactionor the transfer of an electron, ion, atom, orother particle. With these conditions, when theunperturbed collision problem and the perturba-tion. potential are described by the use of angular(periodic) functions (including multipole expan-sions or expansions of anharmonic terms in Four-ier series), except for one radial degree of free-dom, the S matrix can be expressed as a sum ofproducts of Bessel functions whose orders v,.depend linearly on the changes in the quantumnumbers W, and whose arguments Z, are in.-tegrals over the unperturbed trajectory express-ible in terms of the unperturbed quantum. numbers8 f

Suppose there are n periodic degrees of freedomwith unperturbed (average) quantum numbers n,and changes ~n, , each associated with a coordinatex& =x& (t)+q& and a perturbation &V(E,n&, x&).The perturbation S matrix can be expressed withthe help of the phase function.

n

Q)——QX~q Qq (n& j ~m) . (4.51)

If we again use the convention

Sn &n /2y n+ b, n /2 (4.52)

and similarly for multiple indices, the S matrixcan be expressed as

S~„~„=(2n) " 2g 2g~ dq dq e1

0

e itS0

where

(4.53)

(4.55)

where 8„'(Z) = g.*„(z).By successive applications of this identity, it

can be shown that our approximate S matrix is ap-proximately un. itary —it would be strictly unitaryif all the Z~'s were constant (as functions of theaverage quantum numbers n&) and if all the n, 's(or their differences W, ) were allowed to runfrom -~ to ~. The derivation from these two con-ditions are only severe when one or more of thequantum numbers is very small, and the semi-classical S matrix approaches unitarity rapidlywhen the quantum numbers are large.

It is easy to apply these general formulas to ex-tend the results we have given earlier. For in-stance, rotational excitation with a combined po-tential having dipole and quadrupole terms re-quires six Q's, combining Eqs. (4.8) and (4.19),and the S matrix is a fourfold sum over a six-fold product of 4's. Similarly, an anharmonicvibrational term can be added to the rotating oscil-lator. If the classical motion of the anharmonicoscillator is described by a Fourier expansion,additional Q's and Z's can be defined extendingEqs. (4.41) and (4.42), and additional 8 's willappear in the product S matrix, When the effect

v, =&n, + Q kqXq, (i ~n),g=n+1

(4.54)v, = k~ (n& j ~m).

The proof can be carried out by changing variablesfromq&'s to Q&'s and inserting identities like Eq.(4.25) in the multiple integral.

The general formula has several useful proper-ties. The S matrix it describes is symmetric be-cause all the indices v& are integers, and it followsfrom the properties of the Bessel function that8, (Z) =g (Z). In the Appendix we show that

SEMICLASSICAL PERTURBATION THEOR Y OF. . .

of the new terms added to %he potential is small,,this will appear through the behavior of the newZ&'s that are arguments of the added 8„ func-tions: when we have Z, «1, it follows that J„(Z,)-6(v,). As we have seen in the vibrational-ro-tation ease above, the S matrix then collapsessmoothly to a simpler form with fewer terms underthese conditions, just as we should expect in-tuitively.

In general, the behavior of the arguments Z,of the Bessel. functions controls the rapidity ofconvergence both of the infinite sums in the8 matrix of Eq. (4.53) and of the sums over S thatdefine differential and total cross sections.Especially important are the conditions underwhich Z& 0.

In. the cases we have seen, and many others, theZ's ean be factored into several components:

Z, =[0,/(~+3))f, I, (~, ), (4.56)

where P,. is a dimensionless parameter measuringthe interaction strangth, f, is a function of someof the unperturbed quantum numbers such. asv, j,l, J [ typical factors in f, are 1 + N, (v+ 3)'~ 3],and I, 's are collision integrals involving the mod-ified Bessel functions Ko and E, with an argument+, that often represents a ratio of angular velocit-ies; .in particular, the I, 's-0 exponentially as (d&

If the unperturbed collisional motion is in asimple straight line, (d,. can be factored in turn:

00, =g, (f + 3)/E, = (2g, /h)(b/V, ), (4.57)

where E, is the translational energy, v, the trans-lational velocity, b the impact parameter, and g,depends on internal parameters and quantum num-bers of the colliding partners. Thus, ~, in-creases with f and with 1/E, . As a result, thefactor I&(sr&) in Z& approaches zero as E, de-ereasesandas l increases. This decrease in Z,at large l is reinforced by the direct factor (l+ —,') ''in Z] .

P, itself contains the reduced mass m, as a di-rect factor. For electron collisions, P, and theother terms in Z, seldom much exceed unity,and therefore the Bessel functions are to beeve. luated only in regions where their argumentsare small or moderate in size. If follows also thatwe need to consider only small to moderate valuesof the indices &, , because the Bessel functions de-cline exponentially toward zero for v, » Z, .

In the case of heavy-particle collisions, m, isvery large (in atomic units), and correspondinglylarge values of P, and Z, are often attainable.Consequently, the Bessel functions need to beevaluated in a very different region of Z, , and alsoto v, , 'and different approximating schemes mustbe used. For this reason, we concentrate our at-

E. Anharmonic oscillator

Using the general expressions we have just de-rived, it is simple to extend the expressions wehave derived earlier for the -harmonic oscillator-rotator to include anharmon. ic terms, rotation. —

vibration interaction. , higher multipoles in theangular dependence, and other intermolecularforces. The case of the anharmonic oscillatorserves to illustrate the effect of the increasedcomplexity that such an additional potential termintroduces, and the way in which the more completeexpression converges to the simpler prototypewhen the magnitude of the new potential term be-comes negligibly small. . %e do not give detailsof the derivation, but we present the results as afurther example'of the use of the general formulas.

. The inclusion of a simple anharmonic componentin the motion requires inclusion of Z7 to Zyo,like Z, to Z, but with new frequencies,

(4.59)

and a new factor P, incorporating anharmonicparameters from the vibrational potential and thedipole moment function p(3'- r0). ,

Four new &&'s

are added, and the first three v's are replaced by

2ka 2kg +4kzo~

p2 —p2 +2k8 2kg ~

p3 —p3 +2k 7+2kB +2kg 2kyo p

V7, ...,.o--kv, ...,.o.

Four new Q's are also needed:

Qv= 2Q3 ~ Q3

= -2Qz +2Q3+2Q3,

Q3 = -2Q~ —2Q3+2Q3 & Q10 4Q1 2Q3.

(4.60)

(4.61)

tention initially on electron collisions.For heavy particles, it is useful to express some

of the criteria in terms of classical quantities suchas the impact parameter b arid the collision velocityn, rather than l . Thus we note that

p, ep,. m, mp,

tB) @($ + 3) Bl( AVt

Through this factor, Z, depends inversely on bv, .Through +, , on the other hand, Z, ultimately de-creases as b/v, increases. The p,.(l+3) term thuscauses Z, to decrease at high energy, and theI,(~, ) term causes Z, to decrease also in the op-posite limit of low energy. The low-energy be-havior reflects the fact that a threshold amountof energy must be available before a given trans-ition is feasible. The high-energy behavior isprobably connected with the mismatch between thecollision time and the response function of themode being excited or deexcited.

950 H. MILLER AND F. T. SMITH

These anharmonic terms appear as additionalBessel functions J~,(Z, }.~ ~ ~ 8u»(Z») and additionalsummations in the formula for the S matrix. Thereare no convergence difficulties, As long as the an-harmonic effects are small, Z7 ~ ~ ~ pZyp 0, theadditional g functions-g, (0) =5(v), and the wholeS matrix collapses to the pure harmonic oscillatorform. The corrections needed for high vibrationallevels can be evaluated directly from the fuller ex-pression.

From the relations

(4.66)

» r,-=A, +P [A„cos2))'nv,.„(t—to)n=&,

The relation between energy and vibrational quan-tum number is given by

I

(, +-', ))-= (dr(2m„(E —rr, (~)]P*,/

We can invert to get E(j,v), 7'(j, v), and r(j, v;t).Since» is periodic in t with the period 7 (j, v), itcan be expanded in a Fourier series:

+v —v3 +v4 —v5 —v6 +2v7 + v8 +2vg vip p

&j =vi+v2+v3+v4 v5+v6+2v7+2vs 2vg+2vio~

4l =v, —v2+v3 V4+v5+v6+2v7 2v8 2vg+2vjp

(4.62)

by tabulating all the possibilities for the indexv =P» ~ v, ~

= 1 we find the first-order perturbationselection rules for transitions given by the follow-ing table:

+g„sin2mnv, .„(t—t, )],mhere the coefficients are

1dt [~(t) -», ],

0

2A, „=— dt[r(t) -r, ] cos27»nvt,

0

2B„=— dt [»'(t) -j",] sin2»»nvt .0

(4.67)

. (4.68)

. gv 0 0 +1+1~1~1+2+2 +2 ~2

&j 1 1 1 1 -1 -1 2 2 -2 -2

gl 1 -1 1 -1 1 -1 2 -2 2 -2

As is to be expected, the anharmonic terms infirst order can introduce changes of ~v =+2, butthese are associated with angular momentumchanges of +2 in ~j and gl. This connection isagain a special consequence of the potential wehave assumed. Since the anharmoniciiy is usuallysmall, the first-order terms in P, will ordinarilycompete with second- and higher-order terms in

P~ and P„, so these first-order selection rules arenot likely to be observable in practice to any sig-nificant extent. However, the full Bessel functionexpressions mill be very useful in assessing therelative magnitude of the anharmonic effects in thecross sections for various inelastic transitions.We therefore examine the relevant equations lead-ing to the coefficients needed in the explicit evalu-ations. ~4

Consider the general potential V(r) and the effec-tive potential

Comparing with E»l. (4.31), it can be shown that theangle variable conjugate to v is

q„=-2r v, „to. (4.69))

When the action integral'„analogous to Eq.(4.36) is constructed, its Fourier expansion willcontain all harmonics of q„ leading to the intro-duction of additional Q, 's such as nq„+q,.+q, . Fur-thermore, this Fourier expansion will contain, ingeneral, sine as well as cosine terms just as in Eq.(4.67). We need examine only one such pair. Con-sider .

A,. =Z„cosQ, +Z~ sinQ,. =Z,. cos(Q, -c(»),0

Z, =(Z„'+Ze)'~', tann, =Z~/Z„.

In constructing the S matrix this mill lead to theintegral

1e e-i Zi cos(Qi-ni) -i&igi d~

It 'p

-»a» -iz»~ ~O~Q» e »))» 0» dQ)-277 0

(j +2i}'8'(r) = v(»') . + (4.e3) e i(a»+((/2)~ -(Z )p i ~ (4.71)

The relation between x arid the time t for the un-perturbed oscillator is

2 -Z/2

t(~)= J dr ]z-v, (~)]r 0 ~V

(4;64)

d'v E —U~ & (4.66)

For the oscillator in a potential well, the period is

The additional phase ei i enters because of thesinQ, terms in the Fourier expansion of A.

E

The Fourier expansion is a general techniquewhich can be applied to molecules of considerablecomplexity. Through this examination of the an-harmonic oscillator, me have shown that the Four-ier expansion leads to the same sum-and-productof Bessel functions for the S-matrix expression,modified only by an additional phase factor.

SEMICLASSICAL PERTURBATION THEORY OF. . . 951

V. CONCLUSIONS

We have given general expressions and severalillustrative examples showing how the sum-and-

/

product Bessel function representation can be de-rived in a systematic way for a very wide class ofcollision problems involving electrons, ions,atoms, and molecules. Exactly the same tech-niques can be applied to interactions with morecomplicated couplings of angular momenta includ-ing higher multipoles and other more complicatedpotentials, and higher anharmonic terms in vibra-tional coordinates. The principal limitation is therequirement that there be no change in chemicalstructure, such as dissociation or ionization ofmolecule or exchange of an atom, electron, orother structural unit.

Ordinarily, when the potentials are analyticallywell behaved over the entire range of the dynami-cal variables, it appears to be a very generalfeature of this approach that the arguments of theBessel functions, the functions g, of our expres-sions, can be explicitly evaluated through the useof various combinations of the modified Besselfuncti. ons Ko and K, whose arguments depend onlyon the dynamical variables such as collision energyand on the unperturbed quantum numbers. Whenthe interactions involve less well behaved poten-tials, such as step functions, or functions ex-pressed as a numerical tabulation, the argumentsP, must be evaluated by numerical, integration overthe unperturbed trajectories. At the expense of anacceptable increase in numerical manipulations,the procedure can then be adapted to these lesswell behaved interactions, and the S matrix willstill take the form of a sum and product of Besselfunctions.

The multiple Bessel function form of our Smatrix is most directly suited to evaluation whenone of the collisions partners is an electron, be-cause the coupling coefficients P,. are small enoughso that the expressions will converge with the useof Bessel functions of comparatively low order.For heavy-particle collisions, the same expres-sions are formally valid but the actual evaluationof cross sections will require greater use of fur-ther devices such as the introduction of asymptoticexpansions for the Bessel functions. In such cases,our procedure will lead to forms similar to .thoseobtained by Percival, Richards, Clark, and Dick-inson."

The expressions we have obtained have a numberof desirable properties. They correctly preservethe symmetry of the S matrix, and to a good ap-proximation its unitarity (except when some of thequantum numbers are extremely small). They in-clude the Born approximation as a special limiting

ACKNOW( LEDGMENTS

This work was supported by the Air Force Officeof Scientific Hesearch under Contract No. F44620-'?5-C-0073 and the Air Force Geophysics Labor-atory under Contract No. F19628-75-C-0050.

APPENDIX

A. Unitary Bessel functions

We define the following for integer order p

through the standard Bessel functions J,(z):

g, (z) =z J,(z), (A&)

and we use the notation g(Z) = g*,(z) because ofthe connection of J„with the S matrix, such as

„(Z)=S, , A number of symmetry prop-2 1 2 1

erties follow:

(A2)

(AS)

(A4)

case in the region where it is valid, essentially aregion of quantum tunneling where one-quantumtransitions predominate and transition probabilitiesare not very large. They behave in a quantitativelycorrect way through the important transition re-gion from quantum tunneling to classically allowedmotion, and they describe the quantum. oscjllations(interference effects) that are often characteristicin the classically allowed region. When properaveraging over the oscillations is carried out, thepurely classical result should be obtained. Thusthey build a correct bridge between the Born andclassical types of motion and describe both of them

.with a comparatively simple and flexible functionalform adapted to a great variety of pro'blems.

With these advantages go certain limitations.These reflect the limitations of any perturbationtheory in that they cannot correctly describe themotion in regions where the interaction is verystrong and the limitations of the semiclassical the-ory, which does not fully represent all the detailsof a c-omplete quantum-mechanical description,particularly when angular momentum quantum num-bers are very small. It is significant to note thatthis breakdown of the semiclassical theory is like-ly to be much more severe for angular momentumquantum numbers than it is for vibrational quantumnumbers. The reason for this is that the assumedequivalence j"-(j+-,') 8 between classical variablesand quantum numbers is imprecise for small angu-lar momenta when j is small, but the correspond-ing one for vibrations v"-(v+ —,'}8 is correct evenfor v=0.

H. MILLER AXD F. T. SMiTH

The Neumann addition theorem becomes

g„(u —v) = Q g„,(u) 8,*(v)k.=-~

= Q g„,(u) g*,(v)

where n, /1n are as in Eq. (A10), and

4n' =n3-nl, hn" =n2 -n3 =An —An' .

C. Reduction of matrix products

(A13)

= Q g„„(u)g„(v), (A5)

Matrix products such as Eq. (A12) can be evalu-ated by substituting a Bessel form for the S matrix.With the further definitions

8,(u+v) = Q 9„,(u) 8, (v)oo

I

= Q g„,(u)g „(v); (A6)

n' = —2'(s1, + ns) =n —,' b,n",—

we can write

n" =-,'(res+ n2) =n + ,'an', -

The unitarity of the g functions (at constant argu-ment) is a special case of Eq. (A5):

U~„= g g~„,(Z (n')) g~t„„(Z(n")), (A15)nn' ~ -n+,'a, n

and rearrange the sum into two terms:

g;(Z) =-,' [g„,(Z) +g„„(Z)],vg, (Z) =-,'iz[g, ,(Z) -g,„(Z)].

(A8)

(A9)

B. Perturbation notation for the S matrix

=8.(0) = &(v) ~ (A7)

Thus, if $„,,„=g, „,(Z), and Z is assumed to be a,

constant independent of n, and n» the unitarity and

symmetry of $ follow from Eqs. (A7) and (A2), re-spectively. The recurrence relations of greatestimportance are

w.here

(f/,"„)'= g 8,„,(Z(s1')) g„„(Z(21")),

-no

gv = Q g „,(z(n')) g „„(z(n")),h,n'=-~

with n' an integer such that

n —2gn —1&n1 0 1

(A18)

(A17)

(A18)

(A19)

It is convenient to label the S matrix separatelywith the unperturbed variables (quantum numbers)and with the changes induced by the collision, and

sometimes only with the latter. We use the defin-itions

(nip u2 ) n 2(u1 n2)p +n n2 +19)

(m„m, )-m =-,'(m, +m, ), Am =m, -m, , etc.

(A10)

Then we write

/

Finally, we can expand the $'s about Z =—Z (I).Utilizing the identities (A2) to (A9), we find thefirst-order correction terms in (A17) vanish, and

the expression up to second order reduces to

8 Z(ir" )'=5(an) —18

x [5(bn —2) +25(hn) +5(b,n+2)] ~

(A20)

Except when n' is very small, it suffices to takethe leading term in (A18):

(A11)/1 tr/, „=g„,(Z) g„*,~„(Z) . — (A21)

When matrix products are formed, it is impor-tant to note that the upper as well as the lower in-dices' in these expressions will vary with the dum-

my index of the summation. Consider, for exam-ple, the product involved in a test of unitarity:

U =Un~1 2 &2 +& &1&3 ~3&2

n3= o

$n-t1n/2+an'/2$tn+6n'/2 (A12)An' hn-hn'

Equation (A21) gives a measure:of the deviationfrom unitarity caused by the existence of a finitelower bound in the summation of Eq. (A12) or(A15). Equation (A20) measures the deviation fromunitarity [i.e., from the leading term 6(an)] causedby the variation of Z with n.

Similar reductions can be carried out when the$ matrix is represented by a product of P's. Inthat case, it must be remembered that each Z& isgenerally a function of all the unperturbed quan-tum numbers n, : Z; =Z,.(n, , n2, . . .).

SKMICI ASSICAI PERTURBATION THKOR Y OF. . .

W. H. Miller, J. Chem. Phys. 53, 1949 (1970); for re- '

views, see W. H. Miller, Adv. Chem. Phys. 25, 69(1974); 30, 77 (1975).

D. L. Bunker, Methods Comput. Phys. 10, 287 (1971).R. J. Cross, Jr., J. Chem. Phys. 49, 1753 {1968).; J.Chem. Phys. 58, 5178 (1973).I. L. Beigman, L. A. Vainshtein, and I.. I. Sobel'man,Zh. Eksp. Teor. Fiz. 57, 1465 (1969)[Sov. Phys. JETP30, 920 (1970)].

R. D. Levine and B. R. Johnson, Chem. Phys. Lett. 7,404 (1970).I. C. Percival and D. Richards, J. Phys. B 3, 1035(1970).

A. P. Clark, A. S. Dickinson and D. Richards, Adv.Chem. Phys. 36, 63 (1977).

W. H. Miller, J. Chem. Phys. 53, 3578 (1970).~See, for example, W. H. Miller, J. Chem. Phys. 49,

2373 (1968).~OSee, for example, (a) R. J. Cross, J. Chem. Phys. 47,

3724 (1967); 48, 4838 (1968); and (b) R. D. Levine,Chem. Phys. Lett. 2, 76 (1968); R. D. Levine andG. G. Balint-Kurti, ibig. 7, 101 (1970).For the eikonal approximation in potential scattering,see. L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968), p. 339.F. T. Smith, D. L. Huestis, D. Mukherjee, and W. H.Miller, Phys. Rev. Lett. 35, 1073 (1975).J.R. Stine and R. A. Marcus, J. Chem. Phys. 59,5145 (1973); see also, W. I-I. MQler, Adv. Chem. Phys.30, 77 (1975).

~~With regard to the anharmonic oscillator, see also thediscussion pertaining to Eqs. (3.94)-(3.100) in W. H.Miller, Adv. Chem. Phys. 25, 69 (1974).