Semiclassical approximations for scattering by nonlocal potentials. II Simpleapplications to atomic collisionsR. B. Gerber Citation: The Journal of Chemical Physics 58, 4949 (1973); doi: 10.1063/1.1679082 View online: http://dx.doi.org/10.1063/1.1679082 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/58/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Application of the semiclassical perturbation (SCP) approximation to diffraction and rotationally inelasticscattering of atoms and molecules from surfaces J. Chem. Phys. 78, 1801 (1983); 10.1063/1.444976 Semiclassical collision theory. Application of multidimensional uniform approximations to the atom–rigidrotor system J. Chem. Phys. 62, 913 (1975); 10.1063/1.430543 Semiclassical approximations for scatterings by nonlocal potentials. I The effectivemass method J. Chem. Phys. 58, 4936 (1973); 10.1063/1.1679081 On a semiclassical study of molecular collisions. II. Application to HClargon J. Chem. Phys. 58, 4149 (1973); 10.1063/1.1678972 Collinear Collisions of an Atom and a Morse Oscillator: An Approximate Semiclassical Approach J. Chem. Phys. 55, 1522 (1971); 10.1063/1.1676274

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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 58, NUMBER 11 1 JUNE 1973

Semiclassical approximations for scattering by nonlocal potentials. II Simple applications to atomic collisions

R. B. Gerber Department of Chemical Physics. The Weizmann Institute of Science. Rehovot. Israel

(Received 25 October 1971)

The effective-mass method for treating scattering by nonlocal potentials described in a previous publication. is applied to some problems in atomic collision theory. The applications are based on an effective equation for scattering in the elastic channel when coupling with other channels is included, giving rise to a nonlocal effective potential. We use the effective-mass theory to analyze the influence of coupling with closed channels on the elastic scattering in atom-atom collisions and point out that this may lead to a new type of rainbow effect. We show that strong interchannel coupling results in the appearance of poles and zeros in the effective mass and that the latter may lead to compound-state resonances and orbiting effects. It is observed that the occurences of Fesbach resonances in molecular encounters is always reflected in the presence of zeroes in the corresponding effective mass. We also employ the effective-mass method to deal briefly with the familiar problem of curve crossing and nonadiabatic transitions in atomic collisions.

I. INTRODUCTION

In the preceding paper D. Chem. Phys. 58, 4936 (1973)-hereafter referred to as Part I] a method for treating scattering by nonlocal potentials was devel­oped. This method is based on replacing the SchrOdinger equation with the nonlocal potential by an approximate equation, containing a local effective potential and a coordinate-dependent effective mass.

The effective mass pertaining to any scattering proc­ess by a nonlocal potential can be computed at once if the nonlocal potential is known. In the problems that are of interest to us, atomic and molecular collisions, the true physical interparticle forces are, of course, local. Nonlocal interactions arise, however, as effective potentials in the equation describing the scattering in a given channel, when one takes into account the cou­pling with the other channels. A general approach to many-channel collisions, based on the introduction of effective Hamiltonians involving nonlocal potentials, was formulated by Feshbach and by numerous other authors.l This formalism shows, in fact, that there are many possible definitions in general for each of the one­channel effective Hamiltonians. Each such definition corresponds to a particular choice of a certain projec­tion operator and this choice determines in effect the mathematical scheme by which the coupling between the channels will be calculated. Clearly all the possible effective potentials will yield the same scattering ampli­tude for a given process if fully exact calculations are carried out. This will not, however, be necessarily the case if one is working at an approximate level only. For each specific application, a particular scheme for coupling the channels may prove more advantageous than others and, accordingly, the definition of the

ical nature of the system under consideration. For our purposes, we shall be interested in the nonlocal oper­ators that pertain to two interchannel coupling schemes: the close-coupling expansion and the adiabatic-basis expansion.2

In Sec. II we derive and analyze the form of the effective mass that pertains to the adiabatic-basis cou­pling scheme for the case of atom-atom scattering. A two-channel approximation is employed. Results are obtained on the influence of a closed channel on the elastic scattering in atom-atom collisions. We also de­rive in this framework a familiar expression for the probability of nonadiabatic transitions in atom-atom scattering when there is a crossing of electronic energy curves. In Sec. III we similarly treat the effective mass corresponding to the close-coupling scheme for the colli­sion of a molecule with a structureless particle. Whereas it is possible in principle to carry out an ab initio calculation of the effective mass, we expect that in most practical applications it would be desirable to determine it from a limited set of scattering data (and then to use this quantity to calculate the values of other scattering observables). Section IV briefly de­scribes how this can be done. Finally, in Sec. V we employ the properties of the effective mass established in Secs. II and III to make some simple qualitative predictions on atomic scattering effects. Thus we sug­gest that interchannel coupling may lead to a new type of rainbow effect. Also we point out the relation be­tween the occurrence of Feshbach resonance in mo­lecular scattering and the existence of zeros of the effective mass.

II. THE EFFECTIVE MASS IN THE ADIABATIC­BASIS COUPLING SCHEME

effective, nonlocal potential that we shall choose to As an example for a problem involving the use of employ in any given problem will depend on the phys- the adiabatic-basis scheme, we consider a collision be-

4949

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4950 R. B. GERBER

tween two atoms. The Hamiltonian of this system is

H=-(ft2/2M)lh2+HE (R, r). (1)

He~e R is the internuclear distance vector and R= I R I, M IS the reduced mass of the two nuclei, r stands as a collective symbol for all the electronic coordinates. HE, which depends parameterically upon R, is the elec­tronic Hamiltonian, defined to include all the electronic kinetic energy terms as well as the entire potential energy function of the whole system (we assume that the center of mass motion was already separated out at the outset of the treatment and is not included in H). To solve the Schrodinger equation for the system

Hif;(R, r) = Eif;(R, r) (2)

one can consider first the electronic eigenvalue problem:

with

for all R. (3)

Since the f/Jn(R, r) form a complete set with respect to the coordinates r, if;(R, r) can be expanded as follows (expansion in an adiabatic basis set):

if;(R, r) = L f/Jn(R, rhn(R). (4) n

Substituting this into Eq. (2), one can obtain after some simple manipulations, a very well-known system of coupled equations for the nuclear wavefunctions3 ;

we give this result only for the special case that only two terms, n= 1,2, contribute significantly to Eq. (4):

[ - (ft2/2M) V'R2+Wn (R) +Cnn(R) JXn(R)

+Cnmxm(R) = EXn(R) (5)

with n~m, n, m= 1, 2. Cnm is defined by

Cnm= - W/2M) [Llnm(R) • VR+L2nm(R) J,

Llnm(R) = 2f f/Jn *(R, r) VRf/Jm(R, r)dr,

f CPn *(R, r) VR2f/Jm(R, r)dr= L2nm(R). (6)

The above definition for Cnm covers also the case n=m, but then, provided CPn(R, r) is real valued, we have the simplifying relation LInn = 0, which follows from the normalization of f/Jn(R, r). For m~n, with the elec­tronic wavefunctions assumed to be real valued, we have Llnm(R) = -Llmn(R). The quantity Cnn is known to be negligibly small compared with Wn 4 and we shall ignore it accordingly from now on. The two coupled equations (5) can readily be uncoupled. One obtains

[- W/2M)V'R2+Wn(R) JXn

+Cnm[E+ (ft2/2M) V'RL W m+itJ-ICmnXn= EXn (7)

for n, m= 1, 2, and with m~n. The significance of the infinitesimal quantity t in [E+ (ft2/2M) V'R2-

Wm(R)+itJ-l is familiar from the theory of Green's functions. While the operator Cnm[E+ (ft2/2M) V'R2-

~ n:(R) +itJ-ICmn is nonlocal (an integral operator), It IS nevertheless of a somewhat more complicated nature than the nonlocal operators considered in Part I, because it includes also Cmn, which is a differential operator. In what follows we shall assume that the term - (ft2/2M)L2nm(R) of Cnm can be neglected com­pared with - (ft2/2M) Llnm(R) • VR. This postulate is not essential and comes only to simplify the treatment. One can see from the definitions of L1nm and L2nm that it is justified to make the above conjecture when the variation of the electronic states with the internuclear distance is slow. The assumption can also be easily validated when the collision wavenumber k is suffi­ciently large (to see this in the most trivial manner it suffices to consider the respective action of each of the two terms of Cmn on the Born approximation for the nuclear wavefunction eik.,). With approximations simi­lar to those described in Eq. (49) of Part I, one finds that

3

~Uln L (aXn/aRj)+U2nV'R2Xn, (8) j=l

where the Rj are the Cartesian components of R· U1n " , , U2

n are defined by

The Llinm(R) denote the Cartesian components of the vector function Llmn(R). We stress that (hn, U2n de­fined above are nonlocal operators that act, respectively, on the functions

and

We can apply to these potentials the moment expan­sion given in Eq. (48) of Part I. However, in the present case we shall obtain contributions to the effec­tive mass and, consequently, to the effects that follow from it, already from the zeroth-order approximation

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SCATTERING BY NONLOCAL POTENTIALS. II 4951

for U1n, U2n. This is so because U2n acts on V'R2Xn in (8) and not on Xn itself. Thus we shall use the approxi­mation

A 33 Ii} a Uln~Uln(R) =t L L -2 Llinm(R) -

i=l 1=1 4M aRi

x f (;m(R, R')Lllmn(R')dR', (11)

x f Gm(R, R') Lllmn(R') dR'. (12)

Gm(R, R') is defined as the kernel of the integral operator [E+ (1i,2/2M) V'RL W meR) +iEJ-I. Uln(R) , U2n(R) do not depend upon the direction of R be­cause this kernel is rotationally invariant. Equation (7) can now be approximated by

[ - (~ - U2n) V'R2+u1n(R) t.!....- +Wn(R)] xn(R) 2M j=l aRj

=EXn(R) , (13)

which, of course, can be transformed into

[(1i,2/2JLn(R) V'R2+Wn(R) l~n(R) = EXn(R) , (14)

where xn(R) = g,,(R)xn(R) and gn(R) is a solution of

- - +U2n L _n _!U1ngn=O. (

li,2 ) 3 ag

2M j=l aRj

The effective mass JLn(R) is given by

li,2

JLn(R) = 2[(1i,2/2M) _ U2nJ . (15)

Equation (15), combined with Eq. (12), constitutes an explicit expression for the effective mass and we can thus compute the latter quantity if we know Wm(R) , the second electronic energy curve, and also the elec­tronic matrix elements Llkmn. However, before dwelling on the conclusions that can be drawn from the above formulas, we shall first pose to deal with a particular question that may arise in examining the results ob­tained for U1n, U2n. In Part I it was stated that in keeping with the spirit of semiclassical approximation, we were not going to retain terms of order higher than fi2 in our effective Hamiltonian. Seemingly U1n and U2n are a violation of this principle, as they involve factors of fi'. However, we can assure ourselves of the consistency of our treatment in this respect, if we take the order in Ii, of G(R, R') into account. Consider the integral operator [E+ (fi2/2M) V'R2- W m+iEJ-I and re-

call that E= (fi2km2/2M)+Wmo, where km is the wave­number in the channel m and

Wmo= lim Wm(R)

is the dissociation limit of the electronic curve W m(R). Now in a crude sense E and _(fi2/2M)V'R2+Wm(R) are of the same order of magnitude as operators acting on functions relevant to our discussion. For instance, in the limit of zero coupling Cnm=O we have [E+ (1i2/2M) V'RL W mJXm= O. It follows from the above considerations that fi2km2/2M is of the same order of magnitude, in the above sense, as (fi2/2M) V'R2+ W m­W mO. This implies that the operator [E+ (fi2/2M) V'R2-W m+iEJ-l is of the order of 1i,-2. We conclude that by Eqs. (11) and (12), U1n, U2n are of the order of fi2.

We can now proceed to examine our results and to establish the conclusions that follow from them. We note that:

(1) All the influence of the channel m on the scattering in the channel n is contained in the effective mass. This is true for both the absorption and the distortion effects. In fact, the effective local potential in the channel n is Wn(R), the same as in the pure adiabatic (zero cou­pling) limit. It is especially interesting that within the approximate frame'Xork Aused (e.g., only the lowest­order moments of U1n, U2n were taken into account) a local optical potential term does not arise and the imaginary part of the Hamiltonian of Eq. (14) is entirely due to JLn(R). We may conclude from this fact that for problems that are best treated by an adiabatic­basis-type of interchannel coupling scheme, it is most probably a much superior approach to use a real­valued potential combined with an effective (complex) mass than to employ an ordinary optical potential. This conclusion is in fact quite general and is not re­stricted to the simple system treated above, as long as we are concerned with problems for which the optimal coupling scheme is the adiabatic one. This criterion for using the effective mass formalism should prove a useful one, since in many cases physical intuition sug­gests which interchannel coupling scheme should "most naturally" be associated with the system under con­sideration.

(2) Clearly JLn(R) depends in general on the channel n that we consider. However JLl(R)~JL2(R) when W2(R)~Wl(R) in those regions of R at which LI12(R) attains relatively large magnitudes. Note that, as we pointed out earlier, LI12(R) = -LI21(R).

(3) The ratio JLn(R)/M differs significantly from 1 especially for those values of R at which LI12(R) is very large [see Eqs. (12) J. This, of course, is com­pletely obvious from qualitative physical considerations.

(4) Let WI(R) be the ground state electronic curve. Consider now the case where W2(R) > E for all values of R. Recalling that the kinetic energy operator is posi­tive definite, it is obvious that [E+ (fi2/2M) V'R2- W 2]-1

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4952 R. B. GERBER

is a negative-definite operator in the present case. One can thus set E= 0 directly in the nonlocal potential of Eq. (7). The kernel G2(R, R') of the above operator will therefore be real valued and negative; and by the Hermicity of -- (li,2/2M) V'R2+ W2 it will also be sym­metric in R', R. Suppose first that the LIP(R) have all the same sign and that the latter does not change as R is varied. Then from Eq. (12), together with the relation LIP (R) = --LIP (R), it follows immediately that U2

1(R) >0. Consider yet another case in which we refrain from making the above assumption on LIP, but instead we shall suppose that these quantities as­sume significant magnitudes only in a bounded region of extension a in the coordinate R. (LiP always tends rapidly to zero as R-H~ in physically realistic prob­lems hence the above postulate is most likely to be gene~ally justified). Let us now write down the spatial average of U21(R) , using Eq. (12):

(U21)= _1_ f U21(R)dR 4/31ra3

3

XL LllI2(R')dR'dR>0. (16) 1=1

The inequality sign in Eq. (16) is obtained from the fact that G2(R, R') is the kernel of a Hermitian nega­tive-definite operator. In deriving Eq. (16) we also used the antisymmetry of LItm with respect to the indices m, n. Thus under the condition that all the LIP be of the same (constant) sign, we found that U2I(R) >0. When this restriction was removed, ,:e could still show that U21 is positive at least in a certam average sense. From this, using (15), we conclude that (}J.I»M provided (U21)<li,2/2M [(}J.I) is the spatial average of }J.I(R)]. However, when (U21» li,2/2M, we have (p.I)<O. The latter possibility is likely to be of physical relevance only in very few cases, as the matrix elements Llj12 are usually very small. As­suming that (U21) <li,2/2M, we see that one of the consequences of coupling with closed channels (when the coupling is not very large) is an effective increase in the reduced mass of the nuclei for scattering in the open channel. We demonstrated this above in a rather special case, but the result is fairly general. In fact, we shall comment later on the extension of this discussion and its conclusions to more complex systems. If we assume, in the general spirit of the adiabatic approx.i­mation, that the magnitude of the nuclear kinetic energy is small compared with that of the total elec­tronic energy of the molecule (for both the ground state and the excited state of the electronic Hamilton­ian) we can obtain from Eqs. (12) and (15) a rough, order of magnitude estimate of the effective increase in

the nuclear reduced mass owing to the coupling be­tween the channels. We find that

M (17)

where a is the length of the domain of the distance in R for which Llj12(R) are appreciable in magnitude and 'Y is a typical value which these matrix elements attain in the mentioned region.

A comment is due now on the situation in which LI12(R) is very large and the coupling strong enough for the relation (U21)<li,2/2M to hold. This is probably a feasible possibility only in cases where the electronic states rln(R, r), c/>2(R, r) become degenerate or nearly degenerate at some configuration Ro, since then the coupling matrix elements LIP(R) are known to be extremely large in magnitude.5 In this event }J.I(R) may be singular at the vicinity of Ro and become a negative quantity after passing through the singularity. As a function of R, }J.I(R) is likely to have then the following behavior: far from the degenerate configura­tion LI12(R) is small and we have }J.I(R) >M. As we approach the regime of strong coupling where U21(R)~ li,2/2M, }J.I(R) will become singular and if as we approach Ro the strength of the coupling still increases, the effective mass will be negative. It is possible that sufficiently near to or at Ro the extremely strong coupling limit will be reached, by which we mean that U21(R)>> li,2/2M. Then }J.I(R)~O and the effective mass ha~ a zero. We shall postpone the discussion of the dynamical consequences of the zero and the singularity in I;'I(R) to Sec. V. Here we only point out that these conSidera­tions are purely academic as long as the con~ition E<Wm(R) is maintained: one can hardly conceive of two electronic energy curves of reasonable shape (e.g., Morse functions, Lennard-Jones potentials, etc.) that cross at a classically accessible configuration and that, at the same time are consistent with the conditions E<W2(R) for ali R; and E> WI(R) for sufficien.tly large values of R. The requireme?t that. the crossmg shall occur at a classically accessible reglOn and not, say, at the highly repulsive domain a~ R~, is essen~ial if we want the crossing to have slgmficant phYSical effects. Actually the consideration of poles and zeros in }J.I(R) due to curve crossing becomes relev~nt only as we remove the condition E<Wm(R), as will be done below.

(5) Suppose that

lim W2(R) = W2o> E,

but that for some region of R, W2(R) <E. In this case the second channel is still closed and the nonlocal op­erator in Eq. (7) is again real valued (one can then set E=O in the Green's operator). However, now [E+ (li,2/2M) V'R2- W2]-1 is not a negative-definite op­erator. As we already stated earlier, there will usually

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SCATTERING BY NONLOCAL POTENTIALS. II 4953

exist a finite region R<a, such that for R>a the L1P(R) practically vanish. Assume now that for R<a we have WI(R) <E. Since nuclear kinetic energy is usually, in molecular collisions, much smaller than the electronic energy, then heuristic types of arguments suggest that LI/2(R) [E+ (Ji2/2M) V'R2 - W2]-ILI.j12 is positive definite, at least when acting on the set of functions relevant to scattering at low and intermedi­ate energies. One can show then that (U2

1 )<0. If the coupling is weak, I (U21) I <Ji2/2M, then it follows that (/-II)<M. Clearly, if the coupling is strong, I U21(R) I ~ Ji2/2M, /-Il(R) will be singular, and when the coupling is extremely large, I U21(R) I »Ji2/2M, the effective mass may vanish. In the present case, /-II(R) is, how­ever, always positive. Another obvious case to consider is when W2(R) >E for R<a. It is easy to argue that in this case (U2

1) and, under certain conditions, even the function U21(R) itself, is positive. In this case the be­havior of /-Il(R) will be the same as in the description given in item (4) above [for the situation when W2 (R) > E for all R]. Note that the strong coupling case is of practical interest here since the removal of the condi­tion W2 (R) > E for all R, renders feasible the occurrence of a crossing of the electronic curves W 1(R), W2 (R) in classically allowed regions.

(6) We consider now the case where

E> lim W2 (R) = W2o, R-+oo

i.e., the second channel is also open. In this case the nonlocal potential involved, hence also the effective mass, is complex valued. By a well-known result in dis­tribution theory we have6 :

[E+ (Ii2/2M) V'R2- W2+iEJI

= P[E+ (Ji2/2M) V'RL W2J 1

-i1l'·o[E+ (Ji2/2M) V'R2- W2], (18)

where P[E+ (Ji2/2M) V'RL W2Jl denotes the principal part of the Green's operator involved. By Eq. (12) we have

ReU21(R) = - (Ji'/4M2) L12 (R)

XP[E+ (Ii2/2M) V'R2 - W2]-IL12, (19a)

ImU21(R) = (Ji~/4M2)Lu(R)

are therefore given, respectively, by

( )-!.Ji2 (Ji2/2M)-ReU21(R)

Re/-ll R - 2 [(Ji2/2M) -ReU21J2+ (ImU21)2' (21a)

(R) - !.Ji2 ImU21 (R) Im/-ll -2 [(Ii2/2M)-ReU21J2+(ImU21)2' (21b)

The imaginary part of the effective mass is responsible for "absorption" of particles from channel 1 into chan­nel 2, that is to say, for nonadiabatic transitions in the system under consideration. We shall investigate in some detail the behavior of the imaginary part of the mass in the case of curve crossing. Thus, we suppose that there exists a configuration Re such that W1(Re) = W2(Re). We recall that the coupling coefficients that determine the strength of the nonadiabatic transitions are generally large only in a small vicinity of the cross­ing point.6 A consideration of expression (19b) for ImU21(R) reveals that only the behavior for R~Re of the operator o[E+ (fi2/2M) V'R2 - W2(R)] is of interest to us here, since, for values of R that differ consideraby from Re, ImU21(R) will practically equal zero. We shall use here classical arguments of the kind fre­quently employed in the context of nonadiabatic transi­tions.7 We shall also assume for convenience that while the coupling at Re is large enough to produce significant physical effects, it is nevertheless considerably smaller than the other local contributions to the Hamiltonian in Eq. (7). Classically, the following condition must be satisfied at the crossing7 : Tl(Re) = T2(Re) , where Ti(R) is the classical kinetic energy at the configuration R, for motion in the potential field Wi(R). In the close vicinity of the crossing, R~Re, we will therefore have T2(R)~Tl(R)~E-W 1(R) (we disregarded coupling contributions to the total energy of the system). With the above crude approximations we have

for R~Rc.

(22)

The above relation should not be taken as being valid in general, for one can only hope that it will be a reasonable approximation to the quantum-mechanical operator §[E+(Ji2/2M)V'R2-W2] when acting on the junction k(R). Using a well-known property of the o function,8 we find that

o[W2(R) - W 1(R)]

= I (aWl/dR)-(aWdaR) IR-R.-1·o(R-Rc). (23)

where

Xo[E+(Ii2/2M)V'R2-W2]L12, (19b) Equations (19b) and (23) readily give h~

ImU21(R)~-4M2

(20)

The real and the imaginary parts of the effective mass Recall now the assumption that we made earlier, that

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4954 R. B. GERBER

the nonadiabatic coupling term in Eq. (7) is consider­ably smaller in magnitude than both the potential and the kinetic energy terms. Under this condition we have (h2/2M)>> 1 ReU2

1 I; 1 ImU21 I. Using this together with Eq. (24), we obtain from Eqs. (21b)

7rh2 1 £12(R) [2 Im}.l1(R) = 21 (awt/aR)-(aW2/aR) [R=R, ·o(R-Re).

(25)

The imaginary part of the effective mass is thus a sharply peaked function, attaining its maximum value, which is proportional to

1 £12(Re) 12

at R=Re. Since it is the complex part of the effective mass that gives rise to the nonadiabatic transitions, then one would expect that the probability for such a transition to occur should increase with the quantity

1 £12(Re) 12

[ (aWt/aR) - (aW2/aR) IR=R, •

There is, in fact, no difficulty in calculating the transi­tion probability in a quantitative way, using Eq. (23) and the semiclassical approximation to the wavefunc­tion. The result that obtains is essentially the same as the celebrated Landau-Zener formula.7 The details of the calculation will not be given here. We shall return now to Eq. (21a) and examine the real part of }.I1(R). The deviation of Re}.l1(R) from the value M, which is attained only in the limit of zero coupling, is a measure of the nonabsorptive distortion effects of the channel 2 on the scattering in channel 1. To gain some insight into the physical consequences of the distortion effects, we shall adopt a very simple model which retains nevertheless the essential realistic gross featurel" We shall suppose that the coupling coefficient k(R) assumes large magnitudes only in the region R<d and that it is practically a constant throughout that region. We shall further assume that also W2(R) varies very slowly with R for R<d. Finally we shall consider only cases for which E-W2(R»0 in the interval men­tioned (Le., the potential W2 is either attractive or weakly repulsive in that region). For R', R such that 1 R' I, 1 R 1 <d, the kernel G(R', R) of the nonlocal operator

is given by

G(R', R)

= - (M/27rh2) Icos[k2 1 R-R' [ J/ [R-R' [j, (26)

where k2= (l/h) [2M(E- W2) J1/2, W2 denotes the (con­stant) value of W2(R) in the region R<d. Using the

fact that £12(R) is practically a constant for R<d, and vanishes by our assumption for R>d, we find from Eq. (19), after a calculation,

ReU21(R) ex: M-11 £12 121 [sink2R sink2d/Rk22J

X (k2- 1+d) -k2- 21 for R<d. (27)

Clearly, our model is far too trivial and simple for attaching any real meaning to the particular function form that was obtained for ReU21(R). The qualitative implications of Eq. (27) are likely, however, to be generally reliable. The main conclusion that we can draw from this expression is that for cases where k2 is large (e.g., when W2 is strongly attractive) there will be regions of R in which ReU21(R) , hence also }.I1(R) , will exhibit a strong oscillatory behavior. Below we outline a number of results that can be learned from the above model, but that can be similarly established also for any other problem involving adiabatic-basis coupling, not only for atom-atom collisions in the two­state approximation:

(i) There will be an oscillatory contribution to ReU21(R) , }.I1(R) from coupling with any channel i for which the adiabatic potential is algebraically smaller than the energy, E-WieR) >0 in some region of R.

(ii) The oscillations will be of physically significant magnitude only when the coupling coefficients are large in the interval in which the inequality of (i) is satis­fied, and if ki= (l/h) [2M(E-W i )J1/2 is also large in that domain (the latter condition can be expressed by kidi»l, where di is the length of the interval of strong coupling between the channels 1 and i). The oscillations will be strongest for a highly attractive potential WieR) .

Dynamical consequences of strong undulations III

the effective mass will be discussed in Sec. V.

III. THE EFFECTIVE MASS IN THE CLOSE-COUPLING SCHEME

We now turn to consider another collision problem, which will be treated by means of the close-coupling scheme for calculating interchannel coupling.2 We shall be concerned with the scattering of a structureless particle, which will be referred to as the "projectile," by a molecule. The relevant Hamiltonian for this problem is

H=TR+Ho(r)+V(r, R), (28)

where r stands as a collective symbol for all the in­ternal degrees of freedom of the molecule, R is the distance vector between the projectile and the mole­cule's center of mass, and TR, Ho(r), VCr, R) denote, respectively, the kinetic energy of relative motion of the collision partners, the internal Hamiltonian of the molecule, and the interaction potential between the projectile and the molecule. Let <I>;(r) denote the ith

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SCATTERING BY NONLOCAL POTENTIALS. II 4955

eigenstate of the molecular internal Hamiltonian Hoipi(r) =Eiipi(r). We can then expand the total wavefunction of the collision system 1/I(r, R) in terms of the <I>;'s:

00

1/I(R, r) = 1:: ipi(rhi(R) (29) i-I

with the 7J/s being interpreted as wavefunctions of the projectile. Substituting this expansion into the eigenvalue problem H1/I=E1/I for the Hamiltonian (28), one obtains after some manipulations, the close-cou­pling equations.2 Again, for the sake of simplicity, we shall state the result in the case that only two terms in the expansion (29) contribute significantly to the wave­function:

[TR+ Vii(R) J7Ji(R) + Vij(Rhj(R) = (E-Ei)7Ji(R)

(30)

for i,j= 1,2 and with i~j. Vij for i,j= 1,2 is given by

Vij(R) = Jcpi*(r) VCr, R)cpj(r)dr.

Now the above system of equations can be readily uncoupled to yield

{TR+ Vii(R)

+ V.;[1/(E-Ej-TR- Vjj+iE) JVjihi(R)

= (E-Eihi(R) (31)

with i~j and for i, j= 1, 2. Since the Vj;'s are just multiplicative operators [unlike the em" of (6)J, the nonlocal operator of Eq. (31) is of simpler nature than that which appears in Eq. (7). In fact, for Eq. (31) the treatment given in Part I [for Eq. (46) thereJ can be repeated without alterations. This leads to the fol­lowing result, which is nothing but Eq. (51) of Part I, with the notation adopted to accord with that used in Eq. (31) above:

{[fi2/2Ili(R) JVR2+ Vii(R) }7i(R) = (E- Ei )7i(R) , (32)

where Vii(R) = Vii(R)+Uoi(R). (33)

The functions U"i(R) are, of course, moments of the nonlocal potential of Eq. (31) and are defined by

Uoi(R) = Vi;[1/ (E-Ej- TR- Vjj+iE) J[Vji(R) J = Vij(R) JGj(R, R') Vji(R') dR', (34a)

where Gi(R, R') denotes the kernel of the integral op­erator [E-Ej- TR- Vjj+iEJ-I,

3

Uli(R) =j 1:: Vii(R)fGi(R, R') Vji(R') (R/-Rz)dR', I-I

(34b) 3

U2i(R) = l 1:: Vij(R) JGi(R, R') Vji(R') (Rz' - RI)2dR'. I-I

(34c)

In the derivation of the above results we made use of the approximations outlined in Eq. (49) of Part 1. The effective mass !Ji(R) is, of course, given by

ft,2

lli(R) = 2[(fi2/2M) _ U2iJ ' (35)

where M is the reduced mass of the projectile and the molecule. The relation between 7ii(R) of Eq. (32) and 7Ji(R) of Eq. (30) is again of the form 7Ji(R) = gi(R)7ii(R), where gi is a function satisfying

ft2 3 ago .(R) 1:: !lR' +U1(R)gi(R) =0

Il, 1=1 U I

[see Eq. (55) of Part IJ. We have now obtained explicit expressions for the

effective mass Ili and the effective potential Vii that pertain to the channel i. Note the difference between Eqs. (32) and (14), the corresponding result in the adiabatic-basis coupling scheme: The effective-mass Schrodinger equation that arises in the close-coupling framework involves an energy-dependent optical po­tential as a correction to the channel "static" potential Vii(R) (which governs the scattering completely when coupling does not exist). Unlike the situation in the adiabatic-basis scheme, the effective potential in any channel represents at least part of the absorption and distortion effects that are caused by the coupling. It is clear that at the level of approximation of our approach, the absorption and distortion effects of JoLi(R) are not negligible, but nevertheless they are 'probably less sig­nificant than the contributions of Vii(R). This is a simple consequence of the fact that Vii(R) contains Voi(R) as a term, whereas lli(R) is dependent only upon the second-order moment U2i(R) of the nonlocal potential of Eq. (31). Thus in those cases for which the two-state a'pproximations (30) and (5) are strictly equivalent, the effective local potential in the close coupling scheme will produce most of the effects that are induced by lli(R) in the adiabatic scheme and the few remaining effects will be due to the close-coupling effective mass.

The discussion of the properties of the effective mass in the close-coupling framework does not differ basically from that which we gave in the adiabatic scheme. Thus:

(1) Consider the scattering in the channel i= 1. When E- V22(R)-E2<0, below the threshold of the first Feshbach resonancel we have ~I)<M, where ~1) is the spatial average of IlI(R), provided the coupling is not large [Le., fi2/2M> U21(R)]. If the coupling is large, P-I(R) may be singular and may also become negative for some values of R. As in the adiabatic case, extremely strong coupling, U21(R)>>fi2/2M may pro­duce a zero in the effective mass.

(2) When E- V22 (R) -&>0 in the region of R for which the coupling is strong, IlI(R) will exhibit an undulatory behavior in this domain of R.

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4956 R. B. GERBER

(3) If E>E2, above the inelastic threshold, JLl(R) will have an imaginary part.

We will make only a few trivial comments on the properties of the effective potential Vu (R), although this subject deserves a thorough investigation.

(1) Vu(R) is real valued below the inelastic thresh­old, but becomes complex valued when E> E 2•

(2) If E-V22(R)-&>O in the regions where the coupling is strong, then there will be undulations in Vn(R) as a function of R.

(3) Vii(R) may have singularities when E-Ei-Vii(R) >0, even if the coupling function V12(R) varies very smoothly with R.

The question now arises as to how the treatment given above for two very simple model problems can be extended to more general and more realistic situ­ations. For this to be achieved one needs a procedure that will enable one to reduce an arbitrary collision problem to a set of uncoupled one-channel equations involving nonlocal potentials. Fortunately such a method is avilable, due to contributions by Feshbach and others.l •9 We shall illustrate this method for a system governed by the Hamiltonian (28). Let Pi be the projection operator on the eigenstate ~i(r) of Ho• lts action on a general function VCR, r) of the con­figuration space pertaining to the system under consid­eration, is given by

PiVeR, r) =tPi(r) ItPi(r') VCR, r')dr'. (36)

Let Qi be the orthogonal-complement projection of P i ,l,9 then

(37)

and in addition, of course, P i2=Pi , Qi2=Qi. We can now apply these projection operators to the SchrOdinger equation Hy,(R, r) = Ey,(R, r), and follow a procedure known as the partitioning technique.9 The following result, a very well-known one, is obtainedl,9:

Pi! TR+ Vii(R) + P iVQi[l/Qi(E- H) QiJQi V Pi} PiXi

= (E-Ei)PiXi, (38)

where Xi= Piy,(R, r) is the wavefunction for the scatter­ing of the projectile in the channel i. Equation (38) is of the same form as Eq. (46) of Part I and can be reduced to an effective-mass equation by the same procedure as given there. Clearly, this is a generaliza­tion of the two-states close-coupling model treated before. We shall not analyze it here, as it leads to the same conclusions as reached in the special model with respect to the properties of the effective mass.

In a similar way, one can also generalize our two­channel model treatment of the adiabatic coupling scheme. All we have to do to this effect is use the parti­tioning technique with an operator Pi that corresponds to the adiabatic interchannel coupling scheme. The

form of the appropriate projection operator is quoted in the literature9 •JO and so is the effective Schrodinger equation with the nonlocal potential, which describes the scattering in the subspace Pi. Using this approach, one obtains similar qualitative statements on the prop­erties of JLi(R) as in the two-channel model.

IV. THE DETERMINATION OF THE EFFECTIVE MASS FROM SCATTERING DATA

An obvious conclusion from the examples of the pre­vious section is that the effective mass and effective local potential that govern the scattering in any channel are not uniquely defined, but depend on the scheme that is adopted for calculating the interchannel cou­pling. This, of course, poses no problems if we want to calculate the effective mass theoretically from the underlying Hamiltonian: We decide upon a particular coupling scheme, a choice that will be motivated in each particular case by physical considerations as to what should be the most efficient coupling scheme for the system under study, and then the effective mass and effective potential that pertain to each channel are completely well defined and can be computed ac­cording to the outline and the illustrations given in the foregoing section. However, in general we want to avoid carrying out a theoretical ab initio computation of the quantities JLi(R) , Vi(R). Rather, we prefer to approach them phenomenologically, i.e., to determine them from a particular set of experimental data, and once these quantities are found, use them to calculate theoretically some other type of data. Such kinds of approach are familiar, for instance, in the optical po­tential method.n In this case, if we wish to obtain and to employ unique and well-defined values for JLi(R) , Vi(R), we have to impose suitable conditions with regard to the procedure of determining these quanti­ties, in order to assure the uniqueness mentioned. One possible application of this method is the prediction of inelastic or reactive cross section from data on the concurrent elastic scattering, a rather typical use of phenomenological theories. ll The most practical proce­dure to follow in general for the determination of JLi(R) , VieR) is probably numerical data fitting: One will choose, for any given channel i, trial functions for the effective potential and the effective mass, Vi (a., R), JJ.i(b., R), respectively, where a v , b. stand for sets of free parameters that are included in these functions. One can then proceed to evaluate the elastic cross sec­tion in terms of these parameters and thereafter deter­mine the "correct" values of the latter by looking for the best fit with the experimental data. Once the quan­tities Vi, JJ.i were determined in this way, we can use them in calculating inelastic cross sections, which are determined by the imaginary parts of the effective mass and potential. lt is clear that by choosing the forms of Vi(a., R) and JJ.i(b., R) we also associate

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SCATTERING BY NONLOCAL POTENTIALS. II 4957

implicitly with our treatment a particular coupling scheme. For instance, if we take Vi (a., R) to be of the form of a typical electronic energy curve (e.g., a Morse function) and the only !-Li(b., R) to be a complex valued and possibly oscillatory function of R, then the fit to the elastic data will essentially yield an approxi­mation to the effective mass and potential in the adiabatic interchannel coupling scheme.

In certain specific applications, the above procedure for the determination of the effective mass can be im­proved or replaced by a more systematic method that does not involve the use of arbitrary parametrized functions. Consider the problem of atom-atom colli­sions. It is natural to employ the adiabatic coupling in this case, as it takes advantage of the generally good accuracy of the Born-Oppenheimer approximation.12

This choice of the coupling scheme will be reflected in the effective potential and mass having a relatively smooth behavior, except when there is a breakdown of the Born-Oppenheimer approximation (due to curve crossing or pseudocrossing, say), but even in such a case the coupling effects will be pronounced only in a small region of the internuclear configuration space, outside which the smoothness of !-L(R) will prevail. We shall suppose that the initial channel pertains to the electronic ground state curve Wl(R) and that the latter is known from a calculation done on the eigen­value problem of the electronic Hamiltonian. From the experimental elastic cross section, one can determine the scattering amplitude f(k, (}) .13 Knowledge of this quantity enables one to compute the phase shift X as a function of the impact parameter b, by means of a Bessel transforml4 :

(2ik )-I[e2ix(bL1J= ~ /1 d( cos(})f(k, (})Jo(2bk sint{}), 2 0

(39)

where we assumed that 2bk»1 (in which case the angular momentum quantum number I can be regarded as a practically continuous variable, related to b by l=bk-t). Now that we know x(k, b), suppose that the Eikonal approximation is valid, so that we can use Eq. (61) of Part I. In the present case we identify the potential VCR) of Eq. (61) in Part I with the func­tion WI(R) = Wl(R) - WIO, where

W1o= lim Wl(R) .. R-",

It is well known that in the high-energy approximation the potential and the phase shift are simple transforms of one another. IS Thus, Eq. (61) of Part I can be in­verted to yield

!-Ll(k, R)W1(R)+[M-Ml(k, R)JE

4M d ( ('" kx(k, b) db ) = -:; dR R J

R b(b2-R2)1/2 =F(k, R). (40)

In (40), E=E'-W1o, where E' is the total energy of the system, hence Iik= (2ME)1/2. The effective mass is in general k dependent and we chose to indicate this explicitly in Eq. (40). Since Wl(R), the electronic ground state curve is assumed known and, since the function F(k, R) can be calculated from the experi­mental phase shifts, then the effective mass can readily be obtained from Eq. (40). Suppose now that we are below the first inelastic threshold, so that !-Ll(k, R) is real valued. The crude estimate (17) for Ml(k, R) strongly suggests the following approximate ansatz for Ml(k, R) :

!-LI(k, R) =M/(l-q(R) [E-U(R)J-I}, (41)

where q(R) and U(R) are real-valued functions. Now suppose that we have determined Ml(k, R) from experi­mental data for two values, EI and E2 of the energy. From Eqs. (40) and (41) we get

!-Ll(k i , R)[WI(R)-EiJ

=M[Wl(R) -EiJ/{ 1-q(R)[Ei- U(R)J-Il

=F(ki, R) -MEi (42)

for i=l, 2. We have obtained now two equations from which the functions g(R), U(R) can readily be found. Substituting the values obtained for these quantities into (41), allows us to calculate the effective mass for all values of the energy below the first inelastic thresh­old. Since the results obtained in this method do take into account the effects of the closed channels on the elastic scattering, we should be able to find in this way effects such as nonadiabatic resonances. For energies lying above the first inelastic threshold E t , the mass that will be determined from the experimental data will be complex valued; thus, as it stands, the ansatz (41) cannot be valid in this case. However, instead of the real-valued function g(R), we can use the func­tion ql(R) +iq2(R){}(E-E t ) where {}(E- E t ) = 0 for E<Et , {}(E-E t ) = 1 for E>Et (ql, q2 are real valued). In this case too !-Ll(k, R) can be determined for all k, if the results of measurements at two energies (above threshold) are available. While there is a variety of possibilities to determine the effective mass from a limited set of data and for its subsequent use to com­pute other results, these methods clearly depend on the context of the particular application considered. Thus, having provided one simple illustration, we shall leave this subject here.

V. POSSIBLE APPLICATIONS OF THE EFFECTIVE-MASS METHOD

The purpose of the present section is to offer a very sketchy outline, on a purely heuristic and qualitative level, of various conclusions that can be reached through the effective mass approach. Clearly these must be thought of as trivial odd comments and hints

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4958 R. B. GERBER

that do not reflect what we believe is the full power of the method. Other, quantitative applications are left to future publications. Most of what is stated below follows simply out of the properties of the effec­tive mass, as analyzed in Secs. II and III. We shall make very extensive use of the results of these sections.

A. General Conclusions about the Scattering System from the Shape of the Experimentally

Determined Effective Mass

Suppose that we have detennined Jl.(R) , the effec­tive mass pertaining to the channel labeled by 1, from experimental data, according to the outline of Sec. IV. Then some crude qualitative information on the struc­ture of the potential surfaces involved and on the nature of the channels that influence the scattering may be obtained from the functional fonn of Jl.I(R). In what follows we shall refer to the specific example of an atom-atom collision and use the notation of the corresponding discussion in Sec. II, but our statements should be valid for many other types of collision proc­esses, provided the experimentally determined effective mass can be identified with Jl.I(R) of the adiabatic­basis coupling scheme (in Sec. IV it was pointed out when such an identification can be made). We shall also choose to restrict ourselves here to collisions below the first inelastic threshold, i.e., we assume that the experimentally determined effective mass was found real valued. It is obvious that the quantity E(R) = [fl1(R) -M]/M

provides us with a very good quantitative measure for the strength of the interchannel coupling at the dis­tance R. The following regimes of coupling strength are of special interest for purposes of interpretation: weak coupling, I feR) I «1; strong coupling, I E(R) I ~1; extremely strong coupling, I E(R) I »1. If the experi­mentally determined )J.I(R) obeys Jl.I(R) >M for all values of R, then this can serve as evidence that the collision energy lies below the first nonadiabatic reso­nance, or at least that the channel that couples most strongly with channel 1 pertains to an electronic en­ergy curve WieR) such that E-Wi(R) <0 in all the region in which coupling is significant (i.e., nonadi­abatic resonances due to channel 1 are forbidden). If O<J.LI(R)«M for some interval of R, then we can be sure that E lies above the lowest nonadiabatic reso­nance threshold and, moreover, this indicates that E-WieR) >0 in the above region of R, where i is the channel that contributes most to the coupling with 1. If Jl.I(R) >M for R>Ro and Jl.I(R) <0 for R<Ro then this indicates again that E-WieR) <0 in the d~main of significant coupling. In fact, in this case we must always have

Jl.I(R) )± 00 (R-Ro)-+O(±)

and the coupling must be strong in the neighborhood of

Ro. Another possibility is that Jl.I(R) found from the scattering data, will have the behavior Jl.I(R) >M for R>Ro and O<Jl.I(R) <M for R<Ro. In this case it is obvious that the potential curves WieR) in the channel i intersects with the energy E at R= Ro. If E(R) changes its sign more than once, then we can conclude that a corresponding number of points, RI , R2, "', etc., exists for which W(Rn) =E. Between any two such adjacent points, WieR) must have either a minimum or a maximum. We will know that WieR) has a mini­mum between RI and R2 if 0 < Jl.I (R) < M for RI < R < R2. Similarly, if J.LI(R) >M or Jl.I(R) <0 for RI<R<R2, then WieR) attains a maximum in that interval. Finally, if the effective mass Jl.I(R) exhibits strong oscillations in some domain of R, then WieR) must be strongly attractive in that region. One could elaborate further on these points, but we hope that the above considerations can serve to illustrate the method of qualitative analysis, although they are the simplest possible ones of their kind. A similar discussion, albeit a somewhat more complicated one, can be given also when ImJl.I~O, above the inelastic threshold.

B. Rainbow Effects in Elastic Scattering Due to Interchannel Coupling

Even crude knowledge of the properties of J.LI(R) may suffice to predict or interpret certain qualitative effects caused by interchannel coupling. Consider again, as a simple example atom-atom scattering by the elec­tronic ground state curve WI (R) and suppose that WI(R) is either purely repulsive or that it has only a very weak attractive well. Now a monotonic potential does not produce rainbow effects and one would expect the latter to be practically unobservable also in the case of a potential curve with a very shallow deep. Suppose, however, that above WI(R) there is another electronic curve W2 (R) [pertaining to a state of the same symmetry as epl(R, r)], which attains a minimum at the distance Rm. Consider now a collision at the en­ergy E', such that E-W2(Rm) = ~<O, where ~ is ex­tremely small. If the coupling coefficients L 12 (R) [de­fined in Eq. (20) of Sec. II] do not vanish or nearly vanish, then an inspection of Eq. (12), or more clearly of Eq. (17), shows that the effect of nonadiabatic coupling, reflected in the magnitude of Jl.I(R) , is likely to be very large. In Part I, Eq. (57b) we defined a function V that plays the same role in determining the Eikonal phase shifts as does the potential in the con­stant-mass case:

VCR) = [fl1(R)/M]Tl'I(R) + {1-[flI(R) /M]IE,

where

li\(R) = WI(R) - lim WI(R), E'=E- lim WI(R). R_oo

If i 12 (R) is not large, then for R»Rm we will have Jl.I(R)~M. Thus for R»Rm , VCR) = WI(R) >0 [the last inequality follows from our assumption that WI(R)

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is a repulsive curve]. However, in the neighborhood of Rm the coupling will be very strong and we will have either fJ.l(R)>>M or fJ.l(R) <0. [These are the only possibilities under the condition E-W2(Rm) <Q-see Sec. II, Conclusion (4).J When fJ.l(R)>>M then VCR) <0 for &:::::!-Rm provided that E> W 1(R) in that neighborhood. If E<W1(R), then W 1(R) and W2(R) have an intersection or a near intersection at Rm [recall that E-W2(Rm) =.6, where .6 is smallJ. Conse­quently the coupling in the vicinity of Rm will be very strong and it is likely that fJ.l(R) <0 there (see the relevant discussion in Sec. II). Now it is easy to note that also when E<W1(Rm ) together with fJ.l(R) <0, I fJ.l(R) I »M we have VCR) <0. Finally, when O<R« Rm , we reach again a weak -coupling regime where fJ.l(R)~M and V(R)~Wl(R»O. We thus conclude that the effective potential V that determines the elastic phase shifts will have, most probably, an attractive as well as a repulsive part in this case. Strictly speaking, we have reached a conclusion that applied directly to the Eikonal phase shifts only. However, it is easy to show that an equivalent result holds for the JWKB phase shifts The JWKB phase shifts are determined by the quantity P I2(R) = 2fJ.l(R)[E-W 1(R) J-[l(l+1)/W]' The corresponding quantity for zero potential and constant mass is °PI2(R) = 2ME-[l(l+1) /R2J Con­sider the situation first when fJ.l(R) = M If the potential is repulsive, say, for R«Rm and R»Rm but attractive for R~Rm then P8R) will have the property PI2(R) < °PI2(R) for R»Rm and R«Rm, PI2(R) >OPI2(R) for R~Rm. Now in our Case the potential is purely repulsive but fJ.l(R)~M for R»Rm and R«Rm, fJ.l(R)>>M for R~Rm. [For simplicity we shall consider here only the case where fJ.l(Rm) >0, E-W(Rm) >O.J Therefore, we shall also have here P I2(R) >OPI2(R) for R~Rm and P I2(R) <OPI2(R) for R»Rm. We can thus conclude also in the framework of the JWKB approximation that the phase shifts will behave in this case as though they were determined by a nonmonotonic potential, although W 1(R) is purely repulsive. We would therefore expect the scattering amplitude to exhibit the rainbow effects, which are typical of scattering by nonmonotonic po­tentials.16 (We will not give the derivation here-as it is essentially the same as in the constant-mass case.) However, the rainbow effect we predict will differ in one important detail from the usual pattern of this phenomenon: It will be extremely sensitive to the en­ergy, since only when E<W(Rm) will the effective potential V have the desired properties: At lower ener­gies the coupling will be too weak, at higher energies, the inequality E-W(Rm) >0 will tend to decrease the effective mass (see Sec. II).

C. Orbiting and Barrier Phenomena in s-Wave Scattering

It is well known that heavy particle scattering in­volves contributions from any partial waves when the

energy lies in the intermediate region of thermal ener­gies. However, at extremely low energies (a few degrees Kelvin), even atom-atom collisions are dominated by s-wave scattering.17 There has been a great deal of activity recently in the subject of very low energy collisions between atoms, and we may mention the very interesting work by Miller17 and the calculations by Dondi et al. 18 as examples. We shall make here one simple comment on this topic. In most cases the elec­tronic energy curve that provides the interaction be­tween two atoms does not have an intrinsic barrier. The question arises whether it would be correct to ex­pect very low energy scattering (which is predominantly s-wave scattering for which there is no centrifugal barrier) to be free from orbiting and tunneling effects.19

The answer to this is trivial, and is best divided into two parts:

If E-WieR) <0 for all R, i~ 1, for every electronic state <i>i(R, r) which is of the same type of symmetry as the state <i>1(R, r), then s-wave orbiting is indeed impossible. As we have previously shown, the influence of interchannel coupling is in this case to increase the mass or, equivalently, to add an attractive contribu­tion to the adiabatic potential. This clearly cannot have the effect of a barrier.

The situation may, however, be altogether different if, say, -E+W2 (R) <0 in some region of R in which the channels 1 and 2 are moderately or strongly cou­pled. Note that this may be relevant even to scattering at zero incident velocity. It may well be the case that

lim W 1(R) > W 2(Rm ), R-oo

where Rm is the distance for which W2(R) attains a minimum. We saw in Sec. II that if E-W2(R) >0 for the domain in R for which coupling is significant then fJ.l(R) <M. An inspection of the effective potential VCR) shows that this has the effect of adding a repulsive contribution to the potential: The term Cf..Ll(R) /MJW1(R) , which represents an attractive force in the region of interest, decreases as fJ.l(R) decreases, whereas 11-Cf..Ll(R)/MJ}E increases as the effective mass decreases and is positive for fJ.l(R) <M. A similar conclusion that the decrease in the effective mass is equivalent to adding a repulsive potential term to the interaction, can be reached by studying the function P I2(R) , which determines the JWKB phase shifts. The effect considered will be of interest especially when it is strong enough to have the equivalent influence to adding a large hump to the potential Wl(R). This is likely to happen when Cf..Ll(R)/MJ«1. Suppose that the last inequality is satisfied in the strongest possible manner: that fJ.l(R) has a zero, at the energy E and the configuration Ro. Since Pt2(Ro) =0, then Ro has the significance of a "turning point" for the ~cattering of all the partial waves. For R«Ro, R»Ro, VeE, R) will remain attractive. Thus at the energy E, VeE, R)

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4960 R. B. GERBER

behaves as a potential curve with two attractive regions (it may be also that only one such region exists, as for the exterior region V(E, R)"'O, if Ro is far enough from the origin) separated by a barrier the top of which is ex­actly at height E, V (E, Ro) = E. In this case, orbiting will occur.19 [Note that similar conclusions can be reached by studying the function P02(R) rather than the potential V (E, R).] If at an energy E the effective mass has two distinct neighboring zeros, Rl, R2 say, then the effect on the scattering will be the same encountered in ordinary barrier phenomena [when 1L1(R) is constant] for energies below the top of the barrier. We shall not analyze here the dynamics of orbiting and barrier penetration in this case, but only point out that they are similar to the "ordinary" cases of barrier effects and that this is to a large extent due to the importance of connection formulas in this con­text,20 combined with the fact that the connection laws are the same for a zero of the effective mass as for a turning point of the type E-W(Ra)=O (see Part I).

Finally, we must ask the question under what kind of physical conditions can the effective mass attain a zero. This can happen only when U1

2 is singular. An examination of Eq. (12) shows that there are two obvious situations in which this may occur: (i) when the coupling coefficient L12 (R) has a singularity (this may be the case when there is a curve crossing); (ii) when the operator (E-TN-W2)-1 is singular, i.e., when we are at a nonadiabatic (Feshbach) resonance.

D. Form of the Differential Cross Section When the Effective Mass is a Strongly Oscillating Function

We pointed out in Sec. II that if there is strong cou­pling between a given channel 1 and many other chan­nels for each of which E-Wl(R) >0 (in the domain of strong coupling), and if the Wi(R) are strongly attrac­tive potentials, then the effective mass will have a pronounced oscillatory behavior. Let us investigate the consequences of such a behavior of 1L1(R) for atom­atom collisions, disregarding for the moment the ques­tion of how realistic are the above assumptions likely to be in this context. If 1L1(R) oscillates very strongly, there will also be pronounced undulations in the magni­tude of P I2(R) as a function of R. Consider the expres­sion given for the JWKB phase shift 01 in Eq. (45) of Part I. It is clear that because of the oscillatory inte­grand in that expression, 01 will depend very strongly on the end point of the integration R", which varies, of course, with l. Thus, we would expect 01 to be an extremely rapidly varying function of l. In the extreme limit of infinitely rapid oscillations in the magnitude of 1L1(R) , sinol and cos 01 will become essentially random functions (with magnitude bounded by unity) of .the variable l. We have at our hands a situation that resembles the random phase approximation,16 although it arises from a completely different physical reason.

The characteristic pattern of the differential cross­section form in the random phase approximation is well known: it involves strong peaks in the forward and backward directions and a relatively diffuse and uniform structure for wide angles.16 The same pattern must hold in our case. The above discussion is of no interest for atom-atom collisions because: (i) even without contributions from the effective mass, elastic atom-atom scattering is known to exhibit the above pattern. (ii) It is unlikely that the effective forces due to the inelastic (closed) channels will be strong enough at thermal energies to produce pronounced oscillations of the effective mass. However, the above qualifitative discussion we gave can be repeated also for other sys­tems that can be suitably treated in the adiabatic coupling scheme. In the case of reactive molecular col­lisions, strong interchannel coupling frequently occurs, and there should be systems with strong attractive potentials governing the nonelastic channels. For this situation our earlier conclusions on the structure of the cross section may be of relevance and interest.

E. Feshbach Resonances and the Properties of the Effective Mass

It is important to observe that in principle, the ap­pearance of Feshbach resonances will be exhibited in the effective mass model whenever they occur for the corresponding actual system. It is not difficult to see in which way the presence of a Feshbach resonance is reflected in the behavior of the effective mass. Con­sider first the adiabatic two-state model of atom-atom collisions brought in Sec. IV. From Feshbach's theory1 it follows that these resonances will occur for energies E that make the operator [E+(h2/2M)V'RL W 2]-1

singular. But for such energies, U21(R) of Eq. (12) is a singular function, and consequently 1L1(R) will have one or a number of zeros as a function of R at that energy. [This is most easily illustrated in the crude estimate (17) of the effective mass.] It is a trivial matter to show that this conclusion is always valid when we employ the adiabatic coupling scheme: At any Feshbach resonance En, the effective mass 1L1(En, R) has a zero or a number of zeros [p.l(En, R) may, in principle, even vanish for the entire region in which the coupling coefficients are not zero or "almost" zeroJ.

In the close-coupling scheme the appearance of a Feshbach resonance will be exhibited in the two follow­ing features: (1) The local potential Vll (R) becomes a singular function at Feshbach resonance [this follows from the behavior of U 01 of Eq. ( 34a), which is the zeroth-order moment of the nonlocal potential involved and which is one of the terms that contribute to Vll (R) J. (2) 1L1(R) will have a zero, or a number of zeros, or will vanish in a whole interval of R. This can be deduced at once from Eq. (34c) for U2

1•

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SCATTERING BY NONLOCAL POTENTIALS. II 4961

VI. DISCUSSION

In the present article we gave some simple examples at an essentially qualitative level to illustrate the pos­sible applications of the effective-mass method to atomic collisions. We obtained and studied expressions for the effective mass corresponding to atom-atom and to structureless particle-molecule scattering and from these we were able to draw some conclusions on the dynamics of the collision processes.

The simplicity of applying the effective-mass method is due to the fact that this approach retains many of the simple features of potential scattering theory. The effective-mass method appears to be a natural general­ization of the optical potential model: Both methods

IH. Feshbach, Ann. Phys. (N.Y.) 5, 237 (1958)/; Ann. Phys. (N.Y.) 19, 287 (1962).

2R. D. Levine, Quantum Mechanics of Molecular Rate Processes (Oxford, U. P., London, 1969).

3M. Born, Nachr. Akad. Wiss. GO'ttingen, Math. Physika. K1. IIa 1951,1.

'A. Dalgarno and R. McCarroll, Proc. R. Soc. A 237, 383 (1956).

sw. D. Hobey, J. Chern. Phys. 43, 2187 (1965). OM. L. Goldberger and K. M. Watson, Collision Theory

(Wiley, New York, 1964), p. 72. 7L. D. Landau and E. M. Lifshiftz, Quantum Mechanics

(Pergamon, London, 1965), 2nd Ed., Sec. 90. IA. Messiah, Quantum Mechanics (North-Holland,

Amsterdam, 1964), Vol. I, p. 469. 9Reference 2, Chaps. 3.l., 3.2, 3.3. lOR. B. Gerber and M. Karplus, J. Chem. Phys. 51, 2726

(1969). "For example, see: (a) J. L. J. Rosenfeld and J. Ross, J. Chern.

Phys.44, 188 (1966); (b) C. Nyeland and J. Ross, J. Chern. Phys. 49, 843 (1968); (c) R. D. Levine and R. B. Bernstein,

describe effects of coupling with other channels on the scattering in the channel under consideration, but the effective-mass approach takes into account properly also nonabsorptive effects due to interchannel coupling (e.g., Feshbach resonances). The effective-mass method is of the same order of complexity as the optical poten­tial model and is therefore as easily applicable as the latter to phenomenological analysis of collisions and to making qualitative predictions on various effects in scattering.

ACKNOWLEDGMENT

I wish to thank Professor M. Karplus for many help­ful discussions and valuable comments.

Isr. J. Chern. 7, 315 (1969). 12R. B. Gerber, Proc. R. Soc. A 309, 221 (1969). 13R. G. Newton, J. Math. Phys. 9, 2050 (1968). A. Martin,

Nuovo Cimento 59, 131 (1969). R. B. Gerber and M. Karplus, Phys. Rev. D 1, 998 (1970). In general, the elastic scattering amplitude can be determined from elastic cross section only when inelastic scattering is absent. However, as an approximation such a determination can be carried out if the inelastic differential cross section are much smaller than the elastic one.

I'T. Adachi and T. Kotani, Prog. Theor. Phys. 35, 485 (1966). ISE. Predazzi, Ann. Phys. (N.Y.) 36, 228 (1966). lOR. B. Bernstein in Advances in Chemical Physics, edited by

J. Ross, (Interscience, New York, 1966), Vol.X, p.75. 17W. H. Miller, Chern. Phys. Lett. 10, 7 (1971). 18M. G. Dondi, G. Scoles, F. Torello, and H. Pauly, J. Chern.

Phys. 51, 392 (1969). 19K. W. Ford and J. A. Wheeler, Ann. Phys. (N.Y.)

7,259,287 (l959). 2('Reference 8, Chap. VI.

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