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Glories in inelastic collisions: non-monotonic temperature dependence of collision rates
Volume
14, number
i
CHEMICAL
NON-MONOTONIC
PHYSKS
LETTERS
GLORIES IN INELASTIC COLLISIONS: TEMPERATURE DEPENDENCE OF COLLISION R.D. LEVINE?: Department
and B.R. JOHNSON
Citenzisrv, i%e Ohio Srute Universiv, Columbus, Ohio 432 IO, USA
manuscript
10 Jnnuary received
1972 25 February
It is suggested that non-monotonic temperature dependence of state-to-state occurrences of glories in the velocity dependence of the inelz&ic cross section. to the esistence of B minimum (i.e., a well) in the intermolecular potential.
rate constant represents an average distribution for the initial states of the reactants and a sum over all states of the products 11, 2j . It is therefore not a simple matter to extract &failed information on the energy dependence of the cross section from the temperature dependence of the rate. However, when one is able to measure a specific rate constant, i.e., a rate for transitions out of a given internal state of the reactants, the necessary averaging is considerably reduced. If intErnal states of both reactants and products are specified, the only averaging required is over the distribution of the initial kinetic enera. In this case one can obtain useful information about the energy dependence of the cross section from the knowledge of the temperature dependence of the rate constant [3-5 1. In particular, when the average over energy is performed using the saddle point procedure (3-61 it is seen that, to the leading order, the temperature dependence is determined by the value of the cross section at a particular energy. A non-monotonic energy dependence of the cross section is thus reflected in a non-monotonic temperature dependence of the specific rate constant. This dependence was previously considered by .4ndreev [7]. The thermal
over 5 canonical
? Alfred P. Sloan FclIow.
132
Jerusalem,
1972
collision rates may result from the As in the elastic case, these are due
2. Aims and claims
1. Introduction
University,
RATES
of
Received Revised
z$Also at the Depzrtment
I hlay 1972
of Physical
Israel.
Chemistry,
The Hebrew
The aim of the present note is to reiterate the general point made above and to illustrate one possibole source of non-monotonic example
chosen
energy
is an inelastic
dependence.
(energy
The
transfer)
transition, which is allowed by first order perturbation theory. For such a transition, it is possible to obtain an explicit approximation for the cross section [S, 91, which incorporates (to infinite order) parts of the contributions of higher orders in perturbation theory. This, “limited coupling” approximation improves as the impact parameter increases and hence provides a suitable approximation for the cross section (which gives greater weight to large impact parameters). We have previously noted [8-lo] that for realistic molecular potentials (which include a well) the limited coupling inelastic cross section will show an oscillatory energy dependence similar to the glory phenomena in elastic collisions [ II] , and in the total cross section [ 12- 141. The source of these oscillations is essentially the opposing contributions to the transition probability by the short- and long-
range forces, The well in the potential is another (related) manifestation of the opposing contributions, the short-range force being repulsive while the longrange force is attractive. As in the elastic case [ 111, the glory oscillations reported here are due to the existence of a stationary
Volume 14, number 1
CHEMICAL
point in
the impact parameter dependence of the transition probability. In summing over the partial waves, the region about the stationary point contributes a (non-random) oscillatory part which is superimposed on the smooth (energy-) averaged cross section. Minima in the impact parameter dependence of the transition probability in first order perturbation theory were noted by Cross and Go:don [ 131, Olson and Bernstein [ 141 and von Seggern and Toennies [IS]. Balint-Kurti and Levine [ 161 have shown that such minima are also present in both the exact transition probability and in the exponential approximation for it. In this note we reiterate our conclusion that such minima lead to non-random glory oscillations about the average energy dependence of the inelastic cross section. We present computational (close-coupling) results in support of this concept. The present results, indicating that the nonmonotonic temperature dependence is due to glory oscillations in the ineIastic cross section, should be compared to the recent discussion by Andreev [7] . There it was shown that in (distorted-wave) first crder perturbation theory, the transition probability for head-on collisions has a zero as a function of the energy. Using the modified wavenumber approximation Andreev then concludes that the cross section will have a minimum. Here we obtain a non-monotonic cross section due to collisions with impact parameter about the glory value, i.e., non-head-on collisions for which the glory condition is satisfied. In both cases, the effect is due to the well in the intermolecular potentiai. In the present approach however, the central role is played by “non-head-on” collisions. The low impact parameter collisions are expected to have a dominant role only in highly odiabotic collisions [9] , which necessarily have smaller cross sections.
3. Limited
1 May 1972
PHYSICS LETTERS
laP=
c
Pg;.
(2)
mfn
While (1) is invalid at low impact parameters when first order forbidden transitions are possible (i.e., where PnI,I f0 even if p(moil= 0) it does provide a useful approximation for intermediate impact parameters and is trivially exact at high impact parameters (where cr< 1). The cross section for the II + m transition is = (n/$) (3111 n
q
W
I)P,,,,
” ~d(n@)P,,,,(6), (3)
where we used the correspondence the limited coupling approximation
Consider, for simplicity, is of the form
I + f +? k,,6. In
the case where the potential
(5)
vmn tR ) = l’)&XR ),
so that the impact parameter dependence of tin,jz and of q is about the same (particularly so, in the sudden limit). Hence %tz
= Pm,l(nlk~)
T
W+l)si+
= Pt,l,rO,,.
(6)
where Pnzn = lvlnn WC ,,I 1v,, I2 and u,~ is the total inelastic cross section out of the state II. Clearly, u,,,, and a,, will show a glory behavior whenever the glory condition obtains
coupling
In the limited coupling approximation [8- 10, 161 the n + m transition probability (m f rz), is given by P m,t = (sina/cr)2P(o) mn’ Here PEA is the n + m transition probability as obtained by first order perturbation theory and
(1)
In fact, the expression for g,l is completely analogous to the result for pure e!astic scattering, if one replaces q by the elastic phase shift. Here [9] q is the difference in action (in units of h) between the two trajectories connecting the initial and final states. ‘SMe our discussion was based on the exponential approximation [9] a similar argument can be made wiithin the semi-classical theory of Miller [ 171 or Marcus [IS] . 133
Volume 14, number
Combining
1
CHEMICAL
(2) and (7) one sees the general
PHYSICS LE-ITERS
result
that the glory behaviour is essentially related to the minima in the impact parameter dependence of the transition probabilities (as functions of the impact parameter). These are a direct reflection of the opposing contributions of the short- and long-range contributions to f(R) [cf: ($1. Whenever f(R) has a minimum, one should expect that a solution to (7) can be obtained. The situation here is not disGmilar to that considered by Olson [ 191 for curve crossing. In the sudden limit c~is just the phase shift due to RR), and when an explicit form for flfi) is available, one can determine the velocities corresponding to the minima and maxima in the cross sectian. Combining (1). (3), (6) and (7) one can write (using a stationary phase approximation) Ornn = Fmn kr,> .
+
x C@S[2@()) + n/4]:. ,
hiay
1972
Also, to conform to the example studied by Andreev [7] , the coupling potential was taken as VIZ = (e/&,){Xexp[-12(X-l)]
-2exp[-6(X-l)]). (11)
sections (in units of nRi) were computed using the (reaction) amplitude density method 1201 as functions of E/E,. h was varied to examine the results, as was the reduced mass parameter
Cross
C= 2&,R;Itz*,
(12)
where g is the reduced mass. In terms of the velocity v the adiabaticity parameter, t, is { = (,Q,R,,/fiv) I? = E/E,,
[,3~%30(d2a/dB~)-‘~2]
1
= $(C/E*)‘i2,
(13)
and the glory parameter D, is [ 1 i]
(8)
1
where the brackets denote an average value obtained
6
by a “random-phase” (or SLL) [ 1 l] summation over I, and B, is the glory impact parameter, [al OCR] _ The second term in (8) is the glory contribution. As 3, varies with energy, this term oscillates leading to the glory maxima and minima.
5
1
4
4. Example 3
As an illustration of the flory phenomena for inelastic cross sections we show the results for a twostate problemt . The chaIlnei potentials (in units of the threshold energy, Eth) were VI, = (~/E~){exp[-12(X-1)]-2exp[-6(X-I)]),
!
0
(9)
and (10)
Here X = R/e, is the reduced collision coordinate with E the well depth and R,,, the equilibrium position. 7 Hex the limited coupling form is, in fact, exact, provided 2~ is inteqreted as the difference between the two eigen-
134
30
50
70
90
110
130
150E
I.0
Vziz= VII f i.0.
phases of the S matrix. In the exponential 191 this reduces to (1).
10
3pproxirnation
0.6
0.2
> Fig. 1. Computational results for C= 50, &!!‘a = 10 and A = 0.5. Shown is the energy dependence of the elastic (011) and inelastic (012) cross sections and of the head-on (I=01 transition probability.
Volume
14, number 1
CHEMICAL
PHYSICS
LETTERS
1 May 1972
y,,,(E) = 7r-4+_ La
I
1
.
.!I51
1
so that the rate constant k,,, (13 = jr--l
1
is
Y,, Qexp(-PE)dE/Q,
(16)
0
where Q is the partition function for the reactants (per unit volume) E= E, + #i2ki/2p and 0 is the inverse “temperature”. Performing the integration (16) by a saddle point method [4,6]
+.21
n v
+.I -
0 -.I
-
(a)
-
(b)
-r
I
D
= exp(-~E~)Y,,(Ep)(2d’lnY/ndE~)-1~2/hQ (17)
with Ep defined /3 = (dlnY(E)ldE&.
1
-_2 5
k,,n(T)
I
I
IO
!5
Fig. 2. The deviation function Au/(o), Au = u - co), versus the glory parameter D = 25(2/E * ) “’ for the elastic (a) and inelastic (IJ) cross sections shown in fig. 1.
The reduced well depth, c/Eth, was also varied. Fig. 1 shows a composite plot of the energy dependence of the elastic (aI 1), inelastic (u12) and head-on transition probability (PI2 for I=O). It is clear that the glory oscillations of the e!astic and inelastic cross sections interlace and that they bear no direct relation to the zeros of P12, apart from the very lowest maxima where the collision is nearly adiabatic (c e 26). These results are typical of those obtained in the range D > 1. Fig. 2 shows the deviation function, (a-GJ>)/(~> versus the glory parameter D,( 14) DcY(E*)-~/~, drawn by analogy to the elastic case [I I] _
in
by (18)
The non-monotonic energy dependence of Y(E) [due to the second term in (S)] may thus be reflected in k(T). It should however be noted that considerable smearing of the detailed energy dependence of Y(E) can result after energy averaging [21,22] . The integra! transform relation (16) between k(T) and YQ implies [cf. (IS)] that onIy such variations that have a “range” of the order of f3 can be detected. In general, one can regard this as an “uncertainty principle” between the energy and temperature characterization of the cross section. A more thorough discussion of this point will be given elsewhere.
6. Summary Arguments for the non-monotonic (glory-type) energy dependence of the state-to-state cross section for inelastic transitions have been presented and illustrated by a computational example. These oscillations may te the origin af the non-monotonic temperature dependence of the state-to-state rate constant, previously discussed by Andreev [7].
Acknowledgement 5. Rate constant It is convenient to introduce the n + m transition [2.4]
the yield function
for
We wouId like to thank the Computing Center of the Ohio State University for providing computing time for this work. 135
Volume
14, number
1
CHEhlICAL
References 11) M.A. El&on and J.O. Hirschfclder, J. Chem. Phys. 30 (1959) 1426. [ 21 R.D. Levine, Quantum mechanics of molccuIar rate processes (Cktrcndon Press, Oxford, 1959). [3] R.A. Marcus, J. Chem. Phys. 45 (1966) 2138. 2630. 141 R.D. Levine and R.B. Bernstein, Repcrt WS-k1-45~; J. Chem. Phys., to be published. (5i K. ,Morokuma, B.C. Eu and BI. Karplus, J. Chem. Phys. 51 (1969) 5193. la1 K.F. Herzfeld and T.A. Litovitz, Absorption and dispersion of ultrasonic waves (Academic Press, New York, 1959).
I71 EA. Andreev, Chem. Whys. Letters 11 (197 1) 429. ISI R.D. Levine, J. Chem. Phys. 54 (197 1) 997. 191 R.D. Levine, Mol. Phys. 22 (1971)497. 1101 RD. Levine, Atomic and molecular collisions (NorthHolland, Amsterdam, 1971) p. 3 12.
136
PHYSICS LETTERS
1 May 1972
[ 1 l] R.B. Bernstein, Advan. Chem. Phys. 10 (1966) 76. [lZ] L. Biolsi, J. C&m. Phys. 53 (1970) 390i. [ 131 R.J. Cross Jr. and R.G. Gordon, J. Chem. Phys. 45 (1966; 3571. __ ._ 1141 R.E. Olson ‘and R.B. Bernstein, I. Chem. Phys. 50 (1969) 246. [I51 hf. van Seggern snd J.P. Toennies, Chem. Phys. Letters S (1970) 613. 1161 G.G. Baiint-Kurti and R.D. Levine, Chem. Phys. Letters 7 (1970) 107. 1171 WM. hfiiler. Accounts Chem. Res. 4 (1971) i61. !l81 R.A. Marcus, J. Chem. Phys. 54 (1971) 3965. I191 RX. Olson, Phys. Rev. A2 (1970) 121. 1201 B.R. Johnson and D. Secrest, I. Chem. Phys. 48 (1968). 4682. [211 H. Pauly and J.P. Toennics, Advan. At. Mol. Phys. 1 (1965) 201. 1221 GgH?4iche and E.A. hIsson, J. Chem. Phys. 53 (1970)
14, number
i
CHEMICAL
NON-MONOTONIC
PHYSKS
LETTERS
GLORIES IN INELASTIC COLLISIONS: TEMPERATURE DEPENDENCE OF COLLISION R.D. LEVINE?: Department
and B.R. JOHNSON
Citenzisrv, i%e Ohio Srute Universiv, Columbus, Ohio 432 IO, USA
manuscript
10 Jnnuary received
1972 25 February
It is suggested that non-monotonic temperature dependence of state-to-state occurrences of glories in the velocity dependence of the inelz&ic cross section. to the esistence of B minimum (i.e., a well) in the intermolecular potential.
rate constant represents an average distribution for the initial states of the reactants and a sum over all states of the products 11, 2j . It is therefore not a simple matter to extract &failed information on the energy dependence of the cross section from the temperature dependence of the rate. However, when one is able to measure a specific rate constant, i.e., a rate for transitions out of a given internal state of the reactants, the necessary averaging is considerably reduced. If intErnal states of both reactants and products are specified, the only averaging required is over the distribution of the initial kinetic enera. In this case one can obtain useful information about the energy dependence of the cross section from the knowledge of the temperature dependence of the rate constant [3-5 1. In particular, when the average over energy is performed using the saddle point procedure (3-61 it is seen that, to the leading order, the temperature dependence is determined by the value of the cross section at a particular energy. A non-monotonic energy dependence of the cross section is thus reflected in a non-monotonic temperature dependence of the specific rate constant. This dependence was previously considered by .4ndreev [7]. The thermal
over 5 canonical
? Alfred P. Sloan FclIow.
132
Jerusalem,
1972
collision rates may result from the As in the elastic case, these are due
2. Aims and claims
1. Introduction
University,
RATES
of
Received Revised
z$Also at the Depzrtment
I hlay 1972
of Physical
Israel.
Chemistry,
The Hebrew
The aim of the present note is to reiterate the general point made above and to illustrate one possibole source of non-monotonic example
chosen
energy
is an inelastic
dependence.
(energy
The
transfer)
transition, which is allowed by first order perturbation theory. For such a transition, it is possible to obtain an explicit approximation for the cross section [S, 91, which incorporates (to infinite order) parts of the contributions of higher orders in perturbation theory. This, “limited coupling” approximation improves as the impact parameter increases and hence provides a suitable approximation for the cross section (which gives greater weight to large impact parameters). We have previously noted [8-lo] that for realistic molecular potentials (which include a well) the limited coupling inelastic cross section will show an oscillatory energy dependence similar to the glory phenomena in elastic collisions [ II] , and in the total cross section [ 12- 141. The source of these oscillations is essentially the opposing contributions to the transition probability by the short- and long-
range forces, The well in the potential is another (related) manifestation of the opposing contributions, the short-range force being repulsive while the longrange force is attractive. As in the elastic case [ 111, the glory oscillations reported here are due to the existence of a stationary
Volume 14, number 1
CHEMICAL
point in
the impact parameter dependence of the transition probability. In summing over the partial waves, the region about the stationary point contributes a (non-random) oscillatory part which is superimposed on the smooth (energy-) averaged cross section. Minima in the impact parameter dependence of the transition probability in first order perturbation theory were noted by Cross and Go:don [ 131, Olson and Bernstein [ 141 and von Seggern and Toennies [IS]. Balint-Kurti and Levine [ 161 have shown that such minima are also present in both the exact transition probability and in the exponential approximation for it. In this note we reiterate our conclusion that such minima lead to non-random glory oscillations about the average energy dependence of the inelastic cross section. We present computational (close-coupling) results in support of this concept. The present results, indicating that the nonmonotonic temperature dependence is due to glory oscillations in the ineIastic cross section, should be compared to the recent discussion by Andreev [7] . There it was shown that in (distorted-wave) first crder perturbation theory, the transition probability for head-on collisions has a zero as a function of the energy. Using the modified wavenumber approximation Andreev then concludes that the cross section will have a minimum. Here we obtain a non-monotonic cross section due to collisions with impact parameter about the glory value, i.e., non-head-on collisions for which the glory condition is satisfied. In both cases, the effect is due to the well in the intermolecular potentiai. In the present approach however, the central role is played by “non-head-on” collisions. The low impact parameter collisions are expected to have a dominant role only in highly odiabotic collisions [9] , which necessarily have smaller cross sections.
3. Limited
1 May 1972
PHYSICS LETTERS
laP=
c
Pg;.
(2)
mfn
While (1) is invalid at low impact parameters when first order forbidden transitions are possible (i.e., where PnI,I f0 even if p(moil= 0) it does provide a useful approximation for intermediate impact parameters and is trivially exact at high impact parameters (where cr< 1). The cross section for the II + m transition is = (n/$) (3111 n
q
W
I)P,,,,
” ~d(n@)P,,,,(6), (3)
where we used the correspondence the limited coupling approximation
Consider, for simplicity, is of the form
I + f +? k,,6. In
the case where the potential
(5)
vmn tR ) = l’)&XR ),
so that the impact parameter dependence of tin,jz and of q is about the same (particularly so, in the sudden limit). Hence %tz
= Pm,l(nlk~)
T
W+l)si+
= Pt,l,rO,,.
(6)
where Pnzn = lvlnn WC ,,I 1v,, I2 and u,~ is the total inelastic cross section out of the state II. Clearly, u,,,, and a,, will show a glory behavior whenever the glory condition obtains
coupling
In the limited coupling approximation [8- 10, 161 the n + m transition probability (m f rz), is given by P m,t = (sina/cr)2P(o) mn’ Here PEA is the n + m transition probability as obtained by first order perturbation theory and
(1)
In fact, the expression for g,l is completely analogous to the result for pure e!astic scattering, if one replaces q by the elastic phase shift. Here [9] q is the difference in action (in units of h) between the two trajectories connecting the initial and final states. ‘SMe our discussion was based on the exponential approximation [9] a similar argument can be made wiithin the semi-classical theory of Miller [ 171 or Marcus [IS] . 133
Volume 14, number
Combining
1
CHEMICAL
(2) and (7) one sees the general
PHYSICS LE-ITERS
result
that the glory behaviour is essentially related to the minima in the impact parameter dependence of the transition probabilities (as functions of the impact parameter). These are a direct reflection of the opposing contributions of the short- and long-range contributions to f(R) [cf: ($1. Whenever f(R) has a minimum, one should expect that a solution to (7) can be obtained. The situation here is not disGmilar to that considered by Olson [ 191 for curve crossing. In the sudden limit c~is just the phase shift due to RR), and when an explicit form for flfi) is available, one can determine the velocities corresponding to the minima and maxima in the cross sectian. Combining (1). (3), (6) and (7) one can write (using a stationary phase approximation) Ornn = Fmn kr,> .
+
x C@S[2@()) + n/4]:. ,
hiay
1972
Also, to conform to the example studied by Andreev [7] , the coupling potential was taken as VIZ = (e/&,){Xexp[-12(X-l)]
-2exp[-6(X-l)]). (11)
sections (in units of nRi) were computed using the (reaction) amplitude density method 1201 as functions of E/E,. h was varied to examine the results, as was the reduced mass parameter
Cross
C= 2&,R;Itz*,
(12)
where g is the reduced mass. In terms of the velocity v the adiabaticity parameter, t, is { = (,Q,R,,/fiv) I? = E/E,,
[,3~%30(d2a/dB~)-‘~2]
1
= $(C/E*)‘i2,
(13)
and the glory parameter D, is [ 1 i]
(8)
1
where the brackets denote an average value obtained
6
by a “random-phase” (or SLL) [ 1 l] summation over I, and B, is the glory impact parameter, [al OCR] _ The second term in (8) is the glory contribution. As 3, varies with energy, this term oscillates leading to the glory maxima and minima.
5
1
4
4. Example 3
As an illustration of the flory phenomena for inelastic cross sections we show the results for a twostate problemt . The chaIlnei potentials (in units of the threshold energy, Eth) were VI, = (~/E~){exp[-12(X-1)]-2exp[-6(X-I)]),
!
0
(9)
and (10)
Here X = R/e, is the reduced collision coordinate with E the well depth and R,,, the equilibrium position. 7 Hex the limited coupling form is, in fact, exact, provided 2~ is inteqreted as the difference between the two eigen-
134
30
50
70
90
110
130
150E
I.0
Vziz= VII f i.0.
phases of the S matrix. In the exponential 191 this reduces to (1).
10
3pproxirnation
0.6
0.2
> Fig. 1. Computational results for C= 50, &!!‘a = 10 and A = 0.5. Shown is the energy dependence of the elastic (011) and inelastic (012) cross sections and of the head-on (I=01 transition probability.
Volume
14, number 1
CHEMICAL
PHYSICS
LETTERS
1 May 1972
y,,,(E) = 7r-4+_ La
I
1
.
.!I51
1
so that the rate constant k,,, (13 = jr--l
1
is
Y,, Qexp(-PE)dE/Q,
(16)
0
where Q is the partition function for the reactants (per unit volume) E= E, + #i2ki/2p and 0 is the inverse “temperature”. Performing the integration (16) by a saddle point method [4,6]
+.21
n v
+.I -
0 -.I
-
(a)
-
(b)
-r
I
D
= exp(-~E~)Y,,(Ep)(2d’lnY/ndE~)-1~2/hQ (17)
with Ep defined /3 = (dlnY(E)ldE&.
1
-_2 5
k,,n(T)
I
I
IO
!5
Fig. 2. The deviation function Au/(o), Au = u - co), versus the glory parameter D = 25(2/E * ) “’ for the elastic (a) and inelastic (IJ) cross sections shown in fig. 1.
The reduced well depth, c/Eth, was also varied. Fig. 1 shows a composite plot of the energy dependence of the elastic (aI 1), inelastic (u12) and head-on transition probability (PI2 for I=O). It is clear that the glory oscillations of the e!astic and inelastic cross sections interlace and that they bear no direct relation to the zeros of P12, apart from the very lowest maxima where the collision is nearly adiabatic (c e 26). These results are typical of those obtained in the range D > 1. Fig. 2 shows the deviation function, (a-GJ>)/(~> versus the glory parameter D,( 14) DcY(E*)-~/~, drawn by analogy to the elastic case [I I] _
in
by (18)
The non-monotonic energy dependence of Y(E) [due to the second term in (S)] may thus be reflected in k(T). It should however be noted that considerable smearing of the detailed energy dependence of Y(E) can result after energy averaging [21,22] . The integra! transform relation (16) between k(T) and YQ implies [cf. (IS)] that onIy such variations that have a “range” of the order of f3 can be detected. In general, one can regard this as an “uncertainty principle” between the energy and temperature characterization of the cross section. A more thorough discussion of this point will be given elsewhere.
6. Summary Arguments for the non-monotonic (glory-type) energy dependence of the state-to-state cross section for inelastic transitions have been presented and illustrated by a computational example. These oscillations may te the origin af the non-monotonic temperature dependence of the state-to-state rate constant, previously discussed by Andreev [7].
Acknowledgement 5. Rate constant It is convenient to introduce the n + m transition [2.4]
the yield function
for
We wouId like to thank the Computing Center of the Ohio State University for providing computing time for this work. 135
Volume
14, number
1
CHEhlICAL
References 11) M.A. El&on and J.O. Hirschfclder, J. Chem. Phys. 30 (1959) 1426. [ 21 R.D. Levine, Quantum mechanics of molccuIar rate processes (Cktrcndon Press, Oxford, 1959). [3] R.A. Marcus, J. Chem. Phys. 45 (1966) 2138. 2630. 141 R.D. Levine and R.B. Bernstein, Repcrt WS-k1-45~; J. Chem. Phys., to be published. (5i K. ,Morokuma, B.C. Eu and BI. Karplus, J. Chem. Phys. 51 (1969) 5193. la1 K.F. Herzfeld and T.A. Litovitz, Absorption and dispersion of ultrasonic waves (Academic Press, New York, 1959).
I71 EA. Andreev, Chem. Whys. Letters 11 (197 1) 429. ISI R.D. Levine, J. Chem. Phys. 54 (197 1) 997. 191 R.D. Levine, Mol. Phys. 22 (1971)497. 1101 RD. Levine, Atomic and molecular collisions (NorthHolland, Amsterdam, 1971) p. 3 12.
136
PHYSICS LETTERS
1 May 1972
[ 1 l] R.B. Bernstein, Advan. Chem. Phys. 10 (1966) 76. [lZ] L. Biolsi, J. C&m. Phys. 53 (1970) 390i. [ 131 R.J. Cross Jr. and R.G. Gordon, J. Chem. Phys. 45 (1966; 3571. __ ._ 1141 R.E. Olson ‘and R.B. Bernstein, I. Chem. Phys. 50 (1969) 246. [I51 hf. van Seggern snd J.P. Toennies, Chem. Phys. Letters S (1970) 613. 1161 G.G. Baiint-Kurti and R.D. Levine, Chem. Phys. Letters 7 (1970) 107. 1171 WM. hfiiler. Accounts Chem. Res. 4 (1971) i61. !l81 R.A. Marcus, J. Chem. Phys. 54 (1971) 3965. I191 RX. Olson, Phys. Rev. A2 (1970) 121. 1201 B.R. Johnson and D. Secrest, I. Chem. Phys. 48 (1968). 4682. [211 H. Pauly and J.P. Toennics, Advan. At. Mol. Phys. 1 (1965) 201. 1221 GgH?4iche and E.A. hIsson, J. Chem. Phys. 53 (1970)