Initial value representation for the classical propagator and S-matrix with the help of coherent states

10 July 1998

Chemical Physics Letters 291 Ž1998. 143–152

Initial value representation for the classical propagator and S-matrix with the help of coherent states Dmitrii V. Shalashilin 1, Bret Jackson

)

Department of Chemistry, UniÕersity of Massachusetts, Amherst, MA 01003, USA Received 31 July 1997; in final form 28 April 1998

Abstract This Letter demonstrates how coherent states can be used to construct an initial value representation for the propagator and classical S-matrix. Advantage is taken of the fact that the coherent state ‘coordinate’ describes trajectories in phase space, resulting in an unusual structure for the propagator. The method, which avoids the usual stability analysis, is successfully applied to nonreactive atom–diatom scattering. q 1998. Published by Elsevier Science B.V. All rights reserved.

1. Introduction Classical S-matrix theory w1–6x has made important contributions to the development of molecular dynamics by showing that classical trajectories can be used to obtain probability amplitudes for quantum transitions. This theory, along with numerous other semiclassical studies w7–21x is based on the Van Vleck approximation for the propagator w1x:

¦ ž

K Ž x f , x i , t . s x f exp y

i Ht "

/ ; Ýž xi s

traj

1

E2 S

2 p i " E xiE xf

1r2

/ ž exp

iS

i np y

"

2

/

,

Ž 1.

where S is the classical action along a classical trajectory from the initial coordinate x i at time t i to the final coordinate x f at time t f Ž t s t f y t i ., and n is the Maslov index. The pre-exponential factor in Eq. Ž1. is determined by the so-called stability analysis, i.e., by the integration of differential equations for the stability matrix along with Hamilton’s equations. In principle, the calculation of K Ž x f , x i , t . requires a Žroot. search for all trajectories satisfying these boundary conditions. In practice, the root search is not necessarily a problem, and several methods for calculating semiclassical wave functions or S-matrix elements avoid it. Among them are Miller’s initial value representation w5x, Pechukas’s and Child’s integral representations of the S-matrix w6x,

)

Corresponding author. E-mail: [email protected] Permanent address: Institute of Chemical Physics Russian Academy of Science, Kosygin St. 4, Moscow, 117977, Russia. E-mail: [email protected] 1

0009-2614r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 5 8 1 - 8

D.V. Shalashilin, B. Jacksonr Chemical Physics Letters 291 (1998) 143–152

144

Heller’s Gaussian wave packets and cellular dynamics method w7–15x, Kay’s smoothed Van Vleck propagator w16,17x, exact calculations of the Green function w18,19x, and others w20,21x. The initial value representation ŽIVR. takes advantage of the fact that the final coordinate is a function of the initial coordinate and momentum. For example, in the following transition matrix element, integration over the initial and final coordinates can be replaced by integration over the initial coordinate and momentum w22,23x:

¦ ž

n f exp y

i Ht "

/;

¦ ž

n f s HHd x i d x f² n f < x f : x f exp y

i Ht "

/ ;² xi

x i < ni :

s HHd x i d x f cn f ) Ž x f . K Ž x f , x i , t . cn iŽ x i . s HHd x i d p i

d xf d pi

cn f ) Ž x f Ž x i , p i . . K Ž x f , x i , t . cn iŽ x i . .

Ž 2.

In this Letter we explore another IVR, taking advantage of the special properties of the coherent state representation of oscillators. In quantum mechanics, coherent states ŽCS. are often the eigenfunctions of the creation and annihilation operators. In classical mechanics, they correspond to the complex coordinates z, z ) which describe phase space points and their trajectories. For example, for a harmonic oscillator of mass m and frequency v , the variables z, z ) are defined w24,25x zs

1r2

I

ž / ž / "

z)s

I

"

1r2

exp Ž yi

mv

1r2

1r2

1

ž / ž v/ v w. ž / ž v/

exp Ž i w . s

qqi

2"

m

1r2

qyi

s

2"

p,

2 m"

1

2 m"

Ž 3.

1r2

p,

where I, w are the action-angle variables, and p, q are the Cartesian momentum and position. Although coherent states can be used not only for the harmonic oscillator, we consider only the harmonic case in this Letter. For the harmonic oscillator the quantum coherent state is related to the Gaussian wave packet. Special properties of the propagator in the coherent state representation simplify the integrals in Eq. Ž2., generating a simple IVR form and a convenient expression the for the propagator and S-matrix. The coherent state representation was used by Ovchinnikova w26x for analytical calculation of the S-matrix, and by others later w27–29x. However, much more is known about coherent states w24,30,31x in field theory and solid state physics. We hope to apply these ideas to molecular dynamics more comprehensively. In this Letter we apply our approach to the non-reactive atom–diatom scattering problem, which often serves as a simple benchmark for theories of this sort.

2. The properties of harmonic oscillator coherent states We summarize briefly some properties of coherent states needed in this Letter. More details can be found in the literature w24,30,31x. The classical equations of motion for z and z ), for a generic classical Hamiltonian H Ž z ), z ., are i EH

dz sy dt

" Ez)

,

i EH

d z) s dt

" Ez

.

Ž 4.

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145

Thus, z ) and z Žstrictly speaking z )ri and z . form a canonical pair. Given the transformation of Eq. Ž3., the classical Hamiltonian of a harmonic oscillator becomes p2

cl Hvib s

mv 2q 2

s " v zz ) . 2m 2 which has the trivial solution: z Ž t . s z Ž 0 . exp Ž yi v t . , z ) Ž t . s z ) Ž 0 . exp Ž i v t . . The variables z ) and z are the classical analogs of the quantum creation and annihilation operators aˆq a, ˆ and Eq. Ž5. is the classical analog of the quantum Hamiltonian 1 Hˆvib s " v aˆq aˆ q . 2 The quantum coherent state is defined to be the eigenstate of the aˆ operator, aˆ < z : s z < z : , and can be written w24,25x as a superposition of harmonic oscillator eigenfunctions, < n ) : q

ž

/

< z : s Ý < n:² n < z : s ²0 < z : Ý n

n

zn

Ž n! .

< z<2

ž /Ý

< n: s exp y 1r2

2

n

zn

Ž n! .

1r2

< n: .

Ž 5. Ž 6. and Ž 7. Ž 8.

Ž 9.

The function

cn Ž z . s

² z < n: p 1r2

s

< z<2

ž /

Ž z ).

p 1r2

Ž n! .

exp y

2

n

1r2

Ž 10 .

can be viewed as a normalized harmonic oscillator eigenfunction < n ) in the coherent state representation ŽCSR.. This wave function is plotted in Fig. 1. It has a maximum around the classical trajectory in phase space, thus giving the appearance of an orbit. These wave functions are orthogonal, i.e. Hcn ) Ž z . cm Ž z . d z 2 s dn m Ž 11 . This is a consequence of the identity operator, which is expressed in the CSR as follows: 1 Iˆs Ý < n:² n < s H < z :² z < d z 2 . Ž 12 . p n The integration in Eqs. Ž11. and Ž12. is over the whole complex plane Ž q, i p ., i.e., d z 2 s d ReŽ z .d ImŽ z . s d pd qr2". A convenience of the CSR is that both the Cartesian coordinate and the momentum of an oscillator

Fig. 1. Plot of
D.V. Shalashilin, B. Jacksonr Chemical Physics Letters 291 (1998) 143–152

146

are represented. However, this does not contradict the uncertainty principle, and it can be easily shown from Eq. Ž9. that unlike in the standard coordinate representation, where ² q1 < q2 : s d Ž q1 y q2 ., the coherent states are not orthogonal Ži.e., they overlap., and ² z 1 < z 2 : s exp Ž z 1 ) z 2 . y

< z1 < 2

y

2

< z2 < 2 2

.

Ž 13 .

The right-hand side of Eq. Ž13. is not a d-function. However, there is a relationship 1 p

² z1 < z 2 : l d Ž z1 y z 2 . ,

Ž 14 .

in the sense that 1 H

p

² z 1 < z 2 :cn Ž z 2 . d z 22 s cn Ž z 1 . .

Ž 15 .

This can be written for any function which is a linear combination of the cnŽ z ..

3. The propagator in the coherent state representation To compute Eq. Ž2. in the CSR we divide the propagator into N short steps D t s trN, and insert the identity N q 1 times to find

¦ ž

n f exp y 1

s

i "

/;

ˆ ni Ht

Nq 1

¦ ž

Hd z N2 PPP Hd z 02 ² n f < z N : z N exp y

ž / p

i "

/ ; ¦ ž

ˆ t z Ny1 PPP z1 exp y HD

i "

/;

ˆ t z0 ² z0 < ni : . HD Ž 16 .

The propagator above can be written in an operator form:

ž

exp y

i "

ˆ s HHd z N2 d z 02 < z N : K Ž z N , z 0 , t . ² z 0 < . Ht

/

Ž 17 .

Following Ref. w24x, for a sufficiently short time interval we can write

¦ ž

z n exp y

i "

/ ;

ˆ t z ny1 f ² z n < z ny1 : y HD

iD t "

² z n < Hˆ < z ny1 : f ² z n < z ny1 :exp y

ž

i "

H˜ Ž z n ), z ny1 . D t ,

/

Ž 18 . where H˜ Ž z n ), z ny1 . s

² z n < Hˆ < z ny1 : ² z n < z ny1 :

.

Ž 19 .

Inserting this into our expression for K leads to w24,32x i

K Ž z N , z 0 , t . s H D w path x e

"

S˜

.

Ž 20 .

D.V. Shalashilin, B. Jacksonr Chemical Physics Letters 291 (1998) 143–152

147

The action S˜ is: S˜N 0 s

t

H0

i"

½ ž 2

dz

z)

dt

yz

d z) dt

/

5

y H˜ Ž z ), z . dt s SNcl0 q w ,

Ž 21 .

where z Ž0. s z 0 and z Ž t . s z N . Note, that H˜ and S˜ are not necessarily the classical Hamiltonian and the cl classical action. For Hˆ s Hˆvib ŽEq. Ž7.. H˜ s ² z < Hˆ < z : s " v Ž zz ) q 12 . s Hvib q 12 " v . S˜ can thus be written as a sum of the classical action and a term resulting from the zero-point energy, which plays the same role as the Maslov index in the propagator, Eq. Ž1.. The matrix elements of the propagator, Eq. Ž17. with the kernel K Ž zX , z, t . have been examined by Kuratsuji and Suzuki, Klauder, and Weissman w32–34x. Clearly, in quantum mechanics z, zX , and t are all independent. In the classical limit of Eq. Ž20., z Žt . obeys Hamilton’s equations ŽEq. Ž4.., and z Ž0. and t uniquely define an endpoint z Ž t .. The double ended boundary conditions in K can be satisfied by the use of Žunphysical. complex trajectories w34x or double trajectories w33x; one starting at z and the other ending at zX at time t. For classically allowed processes, however, as pointed out by Weissman, there is a single real trajectory, and zX s z Ž t .. Ref. w32x has also suggested that K is zero unless zX s z Ž t .. We thus propose the following form for K : K Ž z , z, t. s

i

1

X

X

pBŽ z, t.

d Ž z yzŽ t. .e

"

Ž S cl Ž z , t .q w Ž z , t ..

,

Ž 22 .

Note that S˜s S cl q w is a function of initial z and time t. Note also that for t s 0, our semiclassical propagator ˆ in the CSR, U CSR f eyi H t r " , defined by Eqs. Ž17. and Ž22. must equal the identity Eq. Ž12.. Thus X X K Ž z , z, 0. s py1d Ž z y z ., and the delta function form satisfies the t ™ 0 limit for B Ž z, 0. s 1. We present here only the simplest way to evaluate the normalizing factor B. If B and w are smooth functions i.e. for z 1 f z, B Ž z 1 . f B Ž z . and w Ž z 1 . f w Ž z ., the condition of unitarity ŽU CSR .q U CSR s I,ˆ or y

Hd z 12 Hd z 2 < z 1 :

e

i S zclX z X

i S zcl< z 1 "

p B ) Ž z1 .

² zX1 < zX :

e

"

pBŽ z.

² z
1 p

Hd z 2 < z :² z <

Ž 23 .

leads to the simple equation < BŽ z. <2 s

1 p

H² zX1 < zX :e i

S zclX zyS zcl< z X "

² z < z 1 : d z 12 .

Ž 24 .

Note that for t s 0, B s 1, and the zero-time limit is correct. In Eqs. Ž22. – Ž24. the prime refers to the final time t. The phase w in Eq. Ž22. is an analog of the usual Maslov index, and like the latter it can be found by applying the semiclassical propagator to a simple problem which has an exact solution. As described above, for the harmonic oscillator w s y 12 i v t s yi E0 tr"; the phase simply accounts for the zero-point energy. For the harmonic oscillator, one can show that the matrix elements

² zf
CSR <

z i s HHd zX22 d z 12

:

1 pBŽ z.

i

² z f < zX2 :d

Ž

zX2 y z 1

Ž t . .e

"

Ž S cl q w .

² z1 < z i :

Ž 25 .

are the same as the exact result w33x. A more rigorous test is the free particle case, since the pre-exponential factor must describe spreading of the wave packet. We can show that Eqs. Ž17., Ž22. and Ž25. give the exact result w33x for coherent states matrix elements, except for the phase. The way to obtain w is somewhat ambiguous, although w is related to the commutator for creation and annihilation operators. We feel slightly uncomfortable with this but find it interesting.

D.V. Shalashilin, B. Jacksonr Chemical Physics Letters 291 (1998) 143–152

148

Eqs. Ž17., Ž22. and Ž24. are the first result of this Letter. Their first convenience is that we do not need to do the usual stability analysis while we propagate. We need only the information obtained from running a swarm of trajectories. Namely, the arrays of initial z, final zX s z Ž t ., and action S. Secondly, the IVR comes naturally from the fact that the coherent state defines both classical position and momentum. The third convenience is that equations similar to Eqs. Ž17., Ž22. and Ž24. can be used for other unitary operators such as the S-matrix Žsee below.. From Eq. Ž25. one can see that our propagator Eq. Ž22. projects the initial coherent state z i on the whole manifold of coherent states. After that, the whole manifold is propagated classically and projected back onto the final coherent state z f . If we are interested in matrix elements between n i and n f rather then between individual coherent states we have to project on those eigenstates, i.e., weight the propagator by initial and final wave functions, Eq. Ž10..

4. The S-matrix in the CSR In this Letter we apply the CSR to the case of a non-reactive collinear atom–diatom collision. The Hamiltonian, which often serves as a simple benchmark, is Hˆ s

Pˆ 2 2M

q Hˆvib q " v eya Ž Ry b q . ,

Ž 26 .

where P is the momentum operator of the incident atom with coordinate R, Hvib is the Hamiltonian of the oscillator with coordinate q Žgiven by Eq. Ž7.., and the last term is the potential energy of interaction. Our brief derivation follows Miller w3,4x. The difference is that we use the CSR for the oscillator instead of action-angle variables. Similar to Ref. w3x we use the standard coordinate representation for the translation of the atom Ž R, P .. The S-matrix Žnot to be confused with the action S . can be expressed through the Green function, the Fourier transform of the propagator, as follows w3x Sn f n iŽ E . s y

lim Ri , R f ™ `

ž

"2 k i k f M2

1r2

/

< : exp Ž yi k i R i y i k f R f . ² n f R f < Gq E ni R i ,

Ž 27 .

where `

² n f R f < Gq < : E ni R i s

H0 d t exp

i Et

ž /¦ "

ˆ Ht

ž / ;

n f R f exp yi

"

ni R i .

Ž 28 .

The initial and final states of the oscillator are labeled by n i and n f , respectively, and the initial and final atomic moment are Pi s y"k i and Pf s "k f , respectively. We can introduce the scattering operator and write for the S-matrix Sn f n iŽ E . s ² n f < Aˆ< n i : .

Ž 29 .

Similar to Eqs. Ž17. and Ž22. we introduce a scattering operator which is nonvanishing only if zX s z Ž R f ., where R f is the end of classical trajectory starting at z, R i : Aˆs Hd zX 2Hd z 2

1 pB

< zX :d Ž zX y z Ž R f . . A zX z² z < .

Ž 30 .

D.V. Shalashilin, B. Jacksonr Chemical Physics Letters 291 (1998) 143–152

149

Similar to eq. Ž3.13. in Ref. w3x we can write the scattering amplitude between two coherent states z and zX in a standard double ended boundary conditions form as follows A zX z s y

ž

lim Ri , R f ™ `

"2 k f k i M

S cl

ž

=exp i

y "

2

/

`

exp Ž yi k i R i y i k f R f .

i " v Ž tf y ti .

i np y

2

2

/

H0 d t exp

i Et

ž /ž "

1

E 2 S cl

2 p i " ER f ER i

1

/

2

.

Ž 31 .

Note that in Eq. Ž31., k i and k f are functions of E, z and zX . The classical action along the trajectory connecting z and zX is dz d z) dR tf i " S cl s z) yz qP y H cl Ž z ), z , R , P . d t . Ž 32 . 2 dt dt dt ti

H

½ ž

5

/

The classical Hamiltonian H cl is defined by Eqs. Ž4. and Ž26.. In Eq. Ž31., n is the number of turning points for R-motion only, since the oscillator zero-point energy term takes care of the vibrational motion turning points. In the derivation of Eqs. Ž31. and Ž32. the computation of ² z < Hˆ < z : leads to an additional term in S cl Žfrom the coupling in Eq. Ž26.. which we have shown to be negligible. Similar to the derivation of eq. Ž3.16. in Ref. 3 we use the fact that Et

E 2 S cl

M2 s

EE ER f ER i

Pi Pf

,

Ž 33 .

and obtain after stationary phase integration over time A zX z s e iW zX z Ž E .r " . Thus, Eq. Ž30. becomes Aˆs HHd zX 2 d z 2

1 pB

i

< zX :d Ž zX y z Ž R f . . e "

W zX z Ž E .

² z< ,

Ž 34 .

where WzX z Ž E . s yk i R i y k f R f q

tf

Ht

i

i"

½ ž 2

z)

dz dt

yz

d z) dt

/

qP

dR dt

5

dty

i np

Ž 35 .

2

is the reduced action defined by the classical trajectory which starts at point R i , z before the collision, ends at point R f , zX after the collision, and has a total energy of E y 12 " v . The zero-point energy must be subtracted from the total energy because of the zero-point energy term in Eq. Ž31., which results from the expression for Hˆ given in the paragraph after Eq. Ž21.. The normalizing factor B in Eq. Ž34. can be approximated in a similar fashion as in Eq. Ž24.: Wy W 1 1 < B Ž z . < 2 s H² zX1 < zX :e i " ² z < z 1 : d z 12 . Ž 36 . p It is also convenient to introduce new variables z˜ s z exp Ž yi vt i . ,

z˜X s zX exp Ž i vt f . ,

Ž 37 . ya R

Žthe elastic where t i, f is the time for a trajectory to reach its turning point on the exponential potential " v e part of the potential in the Hamiltonian.. For large R t i, f f R i, f MrPi, f . This makes z˜ and z˜X independent of R i and R f , and is similar to the introduction of the angle a s a y v Ž RMrP . in Refs. w3,6x. Due to the d-function the integration in Eq. Ž34. is in fact over initial z only, and Eq. Ž29. becomes Sn f n i s H² n f < z˜X : A z˜X Ž z˜., z˜² z˜ < n i : d z 2 s Hcn f ) Ž z˜X Ž z˜ . .

ei

W zX t Ž E . "

< BŽ z. <

cn iŽ z˜ . d z 2 .

Ž 38 .

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150

This integral is trivial to evaluate via a trajectory simulation. Eq. Ž38., which is the second result of this Letter, gives the S-matrix in the CSR. In summary, to calculate the S-matrix SŽ E . for a given total energy E using the CSR: Ž1. Pick a point Ž q, p . in the phase space of the oscillator. The corresponding coherent state will be z s Ž q q i p .r '2 . Ž2. Calculate z˜ according to Eq. Ž37. and the wave function cn Ž z˜. of Eq. Ž10.. i Ž3. Run a trajectory with the initial conditions

(

q, p, R i s large, Pi s y

ž

2 M Ey

m2v 2 qi2 q p i2 2m

"v y 2

/

.

Ž4. When after the collision the trajectory reaches the point R f , stop the integration. Find the final coordinate q , momentum pX , coherent state zX , corrected coherent state z˜X , action along this trajectory WzX z Ž E ., and the wave function cn fŽ z˜X .. Ž5. Repeat steps Ž1. – Ž4. several times. After that for every z calculate the prefactor Eq. Ž36.. Ž6. Calculate X

F Ž z . s cn f ) Ž z˜X .

ei

W zX z Ž E . "

< BŽ z. <

cn iŽ z˜ . ,

which is, in fact, a function of the initial z only. Integrate Eq. Ž38. and obtain the S-matrix. 5. Results and discussion We applied the above method to the calculation of the S-matrix for our model Hamiltonian. Fig. 2 shows the probabilities < Sn f n i < 2 for b s 1, a s 0.3, m s 2r3, and E s 10. The circles are our result Eq. Ž38.. The squares are the exact quantum calculation w35x. The diamonds show the result for < B < s 1, i.e. with no normalization of the propagator. The triangles show the probabilities obtained with no zero-point energy subtracted; i.e. Pi s y 2 M Ž E y Ž m2v 2 qi2 q pi2 . r2 m . . The normalized S-matrix satisfies detailed balance, it is unitary with a good degree of accuracy, and the agreement of Eq. Ž38. with the exact results is very good. There are two computational advantages of the CSR. First, the integration in Eq. Ž38. can be made by means of the Monte Carlo method. The biased random walk method, which is a standard technique for Monte Carlo estimations w36x, can be easily applied. Some of the points shown in Fig. 2 were in fact obtained by Monte Carlo, as well as by integration on a regular grid. The second advantage is the absence of the usual stability analysis required by all semiclassical methods based on trajectory propagation, with the exception of the frozen Gaussian approximation ŽFGA., which will be discussed below. For example, the pre-exponential term in Miller’s formula w3x includes the factor d Ifrd w i , which is the derivative of the final action over with the initial angle. It normalizes the S-matrix, making it unitary, and is obtained by the integration of a system of first order differential equations for the stability matrix. For our simple model Hamiltonian, comprised of two degrees of freedom, four Hamilton equations should be solved for the two generalized coordinates and momenta along with 16 coupled equations for the elements of a 4 = 4 stability matrix. Remarkably, in the CSR only four Hamilton equations should be solved. The same goal Ži.e., the normalization and unitarization of the propagator or S-matrix. is achieved simply by calculation of the integral in Eq. Ž24. or Eq. Ž36.. Of course, the values for B depend upon the behavior of neighboring trajectories, which is why Eqs. Ž24. and Ž36. can successfully replace the standard stability analysis. Note that this normalization scales with the number of trajectories and not the dimensionality. The coherent state of a harmonic oscillator in the coordinate representation is the Gaussian wave packet. Thus, there is an obvious relationship between the present approach and that of Heller w7–15,37x. The term ‘coherent state’ often refers to an arbitrary Gaussian wave packet in Refs. w7–15,37x. In this Letter it means only

(

D.V. Shalashilin, B. Jacksonr Chemical Physics Letters 291 (1998) 143–152

151

Fig. 2. Ža. Probabilities for the n i ™ n f quantum transitions, < Sn f n i < 2 , for n i s 1. Values are shown for the results of integration of Eq. Ž38. with B given by Eq. Ž38. Žcircles., B s 1 Ždiamonds., and B given by Eq. Ž36. but without zero-point energy subtracted Žtriangles.. The exact results from Ref. w35x are shown by squares. Žb. Same as Ža. but for n i s 2. Žc. Same as Ža. but for n i s 3.

the packet which is the eigenstate of the annihilation operator a. ˆ In a harmonic potential, the superposition in Eq. Ž9. results in a Gaussian wave packet whose width remains constant in time. Thus, if B s 1 our approach is similar to the FGA suggested in Ref. w37x. The FGA w37x does not require stability analysis at all. The stability analysis was brought back to the FGA by Herman and Kluk w38x. Our integral form of the pre-exponential factor, Eqs. Ž24. and Ž36., provides a reasonable alternative to their differential equations for the stability matrix. In summary, we have presented a simple IVR method for propagating wave functions and computing the S-matrix with the help of coherent states. The usual stability analyses is replaced by the calculation of a simple normalization integral. In conjunction with Monte Carlo techniques the method should prove to be useful for describing multidimensional systems, such as given by the Hamiltonian Hˆ s

P2 2M

q V Ž R ,  aˆq, aˆ4 . q Ý " v i Ž aˆq ˆ i q 12 . i a

Ž 39 .

i

and can possibly be generalized to the anharmonic case.

Acknowledgements This work was supported by the National Science Foundation under Grant No. CHE-9318853. DVS acknowledges M.Ya. Ovchinnikova, who initiated his interest in this problem, and D.L. Thompson for useful discussions.

D.V. Shalashilin, B. Jacksonr Chemical Physics Letters 291 (1998) 143–152

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