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Opacity analysis of inelastic molecular collisions, computational studies of the exponential born approximation for rotational excitation
CHEMICAL PHYSICS LETTERS
Volume 7, number 1
I. October
19ig
.. OPACITP
ANALYSIS
COMPUTATIONAL
OF
STUDIES FOR
INELASTIC
OF
THE
MOLECULAR
EXPONENTIAL
ROTATIONAL
CCLLtSEONS. BORN
APPRCXIMATEOK
EXCITATION
G. G. BALINT-KURT1 and R. D. LEVINE* 2?epanlmenf of Physical Chetnistvy , The Hetweu~ University.
Jerasalenz
Received 3 Au@st
. Isvae.!
1970
The exponential Born approximation is used to compute S matrices and opacilies for rotational Eransitions in a model system corresponding to a heavy atom-heavy diatomic coilision. The rcsufts are in good agreement with the exact catculations in the limited coupling region. The use of the exponential Born approximation, together with a statistical npprosimation for regions of low angular momentum, make possible the practical computation of cross sections for inelastic rot+
tional tmnsitions.
1. E?TRODUCTXON The exponential Born approximation provides a fast procedure for the evaluation of opkcities and cross sections for rotational excitation_ The approximation consists of using the lowest order (Born) approximation for the generator G df the wave operator [ 11, and is equivalent to the approximations considered by Seaton [2 J, The resultant S-matrix is unitary, .and in the region of validity of the approximation has a magnitude and phase in good agreement with the exact results. it is thus possible to use the approximation to calculate not only opacities, and hence total cross sections, but also differential cross sections for rotational excitation. The present approach provides both a quantitative approximation and also a qualitative interpretation of the dependence of the S-matrix and the opacity on the reduced parameters (and interaction’ potentials) that characterise the collision. It thus combines the-advantage of the Born app~xin~a~on, in that the essential parts are analytic, with those features which are normally only found for exact numerical solutions, i.e. conservation of probability, accurate account of first order forbidden transitions (zero in the Born approximation) and the appearance of the ~imited’coupli~ and dominant coupling regimes. $ Department of Chemistry,
The Ohio State University, Columbus, Ohio 43210. USA.
The present
study examines
the actual
evalua-
tion of opacities via the exponential Born approximation for a reaRstic molecular interaction in
an energy regime where many channels are open. Comparison is also made with exact computations. A novel feature of the results. present in both the approximate and exact computations for the present model, is a dip in the opacities at the impact parameters corresponding to the vanishing of the Born integrals.
2. SCATTERING TMEORY The scattering tion is of the form SB = 1 -
matrix [3]
in the Born approxima-
is,
(1)
where S is a symmetric matrix, In terms of the usual notation for reduced variables #.
# For a potential of depth E and range (r. B= Zgizc?f?. Z = R/v is the reduced relative separation. A,), is the reduced wave vector in the channel ??L. A, = k,o and 1’ and 1 are the orbitnl nigular moliienb in cbanriels n and m respectivety. Ifr;. m &) is the in-
terchannel coupling, in units
of E.
107
CHEMICAI, PHYSICS LETTER!3
Volume 7, number 1
The diagonal elements of lljrare the Born approximations to the efastic phaees due to the distortion potentials V, #). It is sometimek convenient to define a modified B&n approximation that combines the advantage of (11, in that the radial integrals (2) can be done analytically, and the advantage of the distorted-wave approximation of exhibiting explicitly the distortion phases. This is the Borndistorted-wave (B.D. W.) approximation. SBJJW.
=
eXXp(idg)
fl-
is’)
+ZXpfidJ$.
(3)
Here 8’ is the off-diagonal part of B and 243 iS the diagonal part of B. The B.D.W. scattering matrix differs from the more usual D. W. B. scattering matrix [l, 3 ] in that in the D. W.B. the distortion phases are evaluated exactly and the radial integrals are between distorted waves. For high t values the two approximations should however be in close agreement. For lower 8 values, one is often in the dominant coupling regime where neither approximation is valid. The only region where B.D.W.
S2 expfi&B),
s2 = exp(-iBY form)
3, ROTATIONAL EXCITATION The cross section a(j’,jf for the j 4 j’ rotational trnnsition can be written in terms of the appropriate opacity function as [4]
where pzcj’,jl
(51 or alternatively
.s2 = expf-2itn-l(El’/2)]
(6)
(the H.B.D.W. form), Eqs. (5) and (6) should be compared with eq. (12) of ref. [l], which gives the corresponding exponential approximations for the distorted-wave-Born scheme. Within the framework of the ordinary Born approximation, the formulae corresponding to (5) and (6) are S = exp(:iB)
and the symbols have their usual meaning, total inelastic opacity Pin(j) is given by
The
(4)
where S2 is obtained from B’ by using some exponential unitarization procedure. In particular, following our previous study [lJ of the exponential approximation for the distorted-wave-Born matrix, we take
(the E.B.D.W.
imations can, be derived from a variational principle and hence&l are equivalent to order (B)2, The approximations based on the B.D.W.. form are superior to: the approximations based on the .ordinary Born form when C& has large elements comparei to the. elements of B’ (say,, for low anis&ropies).
will almost always fail
while D. W.B. may work, is just above threshold. The exponential approximation used in the present computational study corresponds to replacing (3) by S = i?xp(i$$
1 October 1970
and the total opacity P,(j) isf
In the present computational study the potential used $ corresponds to a structureless atom interacting with $ homonuciear rotor. The reduced parameters we’re similar # to those used by Fenstermaker and Bernstein f5f. Calculations
were performed at the reduced energy @Z/E)= 10.15 an3 the radial Born integrals required in (2) are of a standard form [6, ?]. Exact calcula-
tions were performed using the (reaction) density
amplitude method [S]. Fig. I shows the opacities obtained for: (a) the j = 0 -+I’ = 2 first order allowed transition, (b) t&j =O+j'= 4 first order forbidden transition
02 $ V= 4c[ (u/R)~~l+ObP~(cos8))-(u/R)~lcO.~~(co~8j)l
and S= expf-2i tn-‘(B/Z)]
(6)
referred to as the exponential-Born (E.B.) and the Heitler-Born (H.B.) respectively. When dg and B’ commute the E.B. and the.E.B.D.W. formulas are equivalent. The tLZ3.D.W. and H.B. procedures.are never identical. All four appxwc-
8 Is the angle between the roLor and the directionoLreIativemoMon. S B= 403Zl?,(E&/C) = 0.0145 witxeE& is the lowest threshDIdenergy (j= 0-r 2) transition. This choice was msant to simulate an Ar + TIF collision. Note howeverthat the potential usedcontains nocdd Legendrepolynomfai~ andhence is moreappmpriate for a homonuclear rotor. where
CHEMICAL PHYSICS LETTERS
Volume 7, number 1
(b)
v--L.150
200
250
I
1”
350
*
Fig. 1. Results of 9-state exact (dots) and exponential approximations for inelastic opacities for rotational excitation. (a) For the j = 0 + 2 transition, @) for the j - 0 -r4 transition and (c) for the total inelastic transitions our of the j = 0 stab. For I 3 200 the computation has converged, as increasing the basis size to a 49-&b computation does not significanLly alter the results (cf. table 1). For lower Zvaluee, dominant coupling sets in. and (c) the total inelastic opacity out of thej = 0 state, aa a function of 2. The plots show (22+1) times the opacities, so that the area under the curves is proportional to the appropriate cFos6 section. The E.B.D.W. is In good agreement with the exact results down to about Z = 200 while the ordinaxy Born approximation is valid only down to
L October 1970
about 1 = 300 and fails badIy ir?.the limited coupling regime. The H.B.D.W. is ‘51reasotile agreement with the exact results down to about Z = 200 but becomes unreasonable at Lower Z values (where the predicted total inekstic opacity goes to zero). The simple KB. approximation was also examined and found to yietd opacities which were generaPly several orders of magnitude too smaI1. The failure of the Heitler Qpe exponential approximation was also noticed in other computational studies [9]. This is not unexpected since one can show [lo] that the Reitler form is exact in the limit where energy is conserved in all intermediate transitions. This is the opposite extreme to the limit where the reiative motion is nearly classical and which is more appropriate to the present study. On the other hand, one should-recall that the Heitler scheme can be derived from a variational procedure and thus the H.B. scheme should work well when the norm of f3 is small whiIe the B.B.D. W. should provide accurate results when the norm of 8’ is small (irrespective of the ‘magnitude of the diagonai elements of B). The latter condition defines the limited coupling regime and, as is evident from fig. 1, the H.B.D.W. does provide satisfactory results in that region. Table I lists some nunerical results for the j = 0 4 j’ = 2 opacity. Besides the calculations shown in fig. IA, some 49 state E.B.D.W. calculations are aIso presented. These show that the 9 state caicuIations have essentially converged for Z 5 180. Both the exact and the exponentiat approximations opacities show dips around those values of the angular momenta that correspond to the vanishing of the radial Born integrals [Ill due to the cancellation of the contributions from fhe long range and short range parts of the potential. This feature occurs in the exact results ody when the energy is high enough to ensme that the radiaL Born integrals provide a good approximation to the distorted-wave radial inte&rats z. At Lower Z values the exponential opacities oscillate erratically. The oscillations are, however, approximately centred about the correct statistical values. A study of the energy dependence of these oscillations suggests that they would be largely smoothed out, if an energy average were taken.
Fig. 2 shows the phase of SJ
$ The dips occur around a reduced impact parameter of I. As E increases the dips occnr at higher Zvalues and hence the Born radial integrak provide a better approximation due to tie increasing importance of the centrifugal repulsion.
109
Volume
7, number
1
CHEMICAL PHYSICS LETTE-?tS
Opacitias 1
-__
Exact a)
for
the
Table 1
first order allowed j = 0 + 2 transition
E.B.D.W.
180
0.524
0.231
a)
E.B.D.W.,
b)
1 October 1970
(E/E) = 10.15 H.B.D.W’.
0.281
0.150
a)
Born 116.8
200
0.313
0,547
0.572
0.552
220
0.261
0.494
0.498
0.445
0.827
240
0.557
0.553
0.562
0.524
1.24
260
0.547
0.506
0.510
0.460
0.900
0.386
0.387
0.350
0.542
0.256
0.256
0.236
280 300
0.287
320
340
0.102
.
1.67
:
0.310
0.158
0.151
0.176
0.095
0.093
0.101
a) g-state cakulations. b) 49-&&e calculations.
ic phase) as a function of- Z (Z = J). The E.B.D.W. provides a good approximation for.the phase. In the present model, for Z > 200, the matrices dB and S’ nearly commute and hence the E.B.D.W. results are in agreement .withthe E.B. results. In other studies. this is not the case and the E.B.D.W. often provides the best approximation in the limited coupling regime.
4. N_JhlMARY The exponential Born approximation permits a fast, accurate, calculation of .opacities for systems of the type considered here (high B, high
E/Eth). Due to its speed, S matrices of very large dimensions may be evaluated. The use of such large S matrices is essential in order to join smoothly the limited coupling and dominant coupling regimes and thus provide.a complete prescription for the efficient calculations of cross sections. Work in progress includes an extension of the present, analytic, approximation to energies nearer threshold where diitortion effects due to channel potentials are important.
ACKNOWLEDGEMENT This work was supported by the Israel Academy of Sciences and Rumanities. We would also like to thank Dr. B. R, Johnson for the use of his amplitnde density program and Dr. M. Shapiro for many useful discussions. REFERENCES
-.
[I] R.‘b. Levine and G. G. Balint-Kurti, Letters 6 (1970) 101.
Chem. Phys.
Proo. Phys. Sot. (London) 89 (l966) 469; 77 0961) 174; [Sl R. D. Levine. Quant’k m&!hanics of molecular [2] M.J.Seaton,
rate processes (Oxford Univ. Press, Oxford,
196!)). [4] R. D. Levine and R. B.Bernstein. J. Chem. Pbys.. .to be published. [5]-R.W. Fenstermaker and II. B. Bernstein, J. Chem.
Phys. 47 (I967) 4417.
“-~-YkO~~ 200
240
A
Fig..2. Exact (dots) and E.B.D;W. rzx~lt~ for the phase of the diagonal element SJ(Or.OJ) versus
110
2 (I = a).
[S] Ft. J. Cro& and It. G. Gordon, J. Chem. Phys. 45. (1966) 3571, [7] I. S, Gradshteyn and-I. M. Ryzhik, Tables of Integrals series and products [Academic Press, New York, 1965) p. 692. -_ r
Volume 7, number 1
CHEMICAL PHYSICS LETTERS
[8j B. R. Johnson and f). Secrest, J. Ckem. Phys. 48 (1968) 4682. [S] B.R. Johnson and R. D. Levine, J. Chem. Phye, SO (1969) 1992,
3. October 1970
[lo] R. D. Levine. Proc. Phys.Soc. B. (Atom. Mol. Phys.} 2 (1969) 639. Ill] R. E.Olson and R. B. Bernstein. J. Chem. P&\-s. 50 (I969j 246; M. von Seggern and J. P. Toennies. preprint (L970).
111
Volume 7, number 1
I. October
19ig
.. OPACITP
ANALYSIS
COMPUTATIONAL
OF
STUDIES FOR
INELASTIC
OF
THE
MOLECULAR
EXPONENTIAL
ROTATIONAL
CCLLtSEONS. BORN
APPRCXIMATEOK
EXCITATION
G. G. BALINT-KURT1 and R. D. LEVINE* 2?epanlmenf of Physical Chetnistvy , The Hetweu~ University.
Jerasalenz
Received 3 Au@st
. Isvae.!
1970
The exponential Born approximation is used to compute S matrices and opacilies for rotational Eransitions in a model system corresponding to a heavy atom-heavy diatomic coilision. The rcsufts are in good agreement with the exact catculations in the limited coupling region. The use of the exponential Born approximation, together with a statistical npprosimation for regions of low angular momentum, make possible the practical computation of cross sections for inelastic rot+
tional tmnsitions.
1. E?TRODUCTXON The exponential Born approximation provides a fast procedure for the evaluation of opkcities and cross sections for rotational excitation_ The approximation consists of using the lowest order (Born) approximation for the generator G df the wave operator [ 11, and is equivalent to the approximations considered by Seaton [2 J, The resultant S-matrix is unitary, .and in the region of validity of the approximation has a magnitude and phase in good agreement with the exact results. it is thus possible to use the approximation to calculate not only opacities, and hence total cross sections, but also differential cross sections for rotational excitation. The present approach provides both a quantitative approximation and also a qualitative interpretation of the dependence of the S-matrix and the opacity on the reduced parameters (and interaction’ potentials) that characterise the collision. It thus combines the-advantage of the Born app~xin~a~on, in that the essential parts are analytic, with those features which are normally only found for exact numerical solutions, i.e. conservation of probability, accurate account of first order forbidden transitions (zero in the Born approximation) and the appearance of the ~imited’coupli~ and dominant coupling regimes. $ Department of Chemistry,
The Ohio State University, Columbus, Ohio 43210. USA.
The present
study examines
the actual
evalua-
tion of opacities via the exponential Born approximation for a reaRstic molecular interaction in
an energy regime where many channels are open. Comparison is also made with exact computations. A novel feature of the results. present in both the approximate and exact computations for the present model, is a dip in the opacities at the impact parameters corresponding to the vanishing of the Born integrals.
2. SCATTERING TMEORY The scattering tion is of the form SB = 1 -
matrix [3]
in the Born approxima-
is,
(1)
where S is a symmetric matrix, In terms of the usual notation for reduced variables #.
# For a potential of depth E and range (r. B= Zgizc?f?. Z = R/v is the reduced relative separation. A,), is the reduced wave vector in the channel ??L. A, = k,o and 1’ and 1 are the orbitnl nigular moliienb in cbanriels n and m respectivety. Ifr;. m &) is the in-
terchannel coupling, in units
of E.
107
CHEMICAI, PHYSICS LETTER!3
Volume 7, number 1
The diagonal elements of lljrare the Born approximations to the efastic phaees due to the distortion potentials V, #). It is sometimek convenient to define a modified B&n approximation that combines the advantage of (11, in that the radial integrals (2) can be done analytically, and the advantage of the distorted-wave approximation of exhibiting explicitly the distortion phases. This is the Borndistorted-wave (B.D. W.) approximation. SBJJW.
=
eXXp(idg)
fl-
is’)
+ZXpfidJ$.
(3)
Here 8’ is the off-diagonal part of B and 243 iS the diagonal part of B. The B.D.W. scattering matrix differs from the more usual D. W. B. scattering matrix [l, 3 ] in that in the D. W.B. the distortion phases are evaluated exactly and the radial integrals are between distorted waves. For high t values the two approximations should however be in close agreement. For lower 8 values, one is often in the dominant coupling regime where neither approximation is valid. The only region where B.D.W.
S2 expfi&B),
s2 = exp(-iBY form)
3, ROTATIONAL EXCITATION The cross section a(j’,jf for the j 4 j’ rotational trnnsition can be written in terms of the appropriate opacity function as [4]
where pzcj’,jl
(51 or alternatively
.s2 = expf-2itn-l(El’/2)]
(6)
(the H.B.D.W. form), Eqs. (5) and (6) should be compared with eq. (12) of ref. [l], which gives the corresponding exponential approximations for the distorted-wave-Born scheme. Within the framework of the ordinary Born approximation, the formulae corresponding to (5) and (6) are S = exp(:iB)
and the symbols have their usual meaning, total inelastic opacity Pin(j) is given by
The
(4)
where S2 is obtained from B’ by using some exponential unitarization procedure. In particular, following our previous study [lJ of the exponential approximation for the distorted-wave-Born matrix, we take
(the E.B.D.W.
imations can, be derived from a variational principle and hence&l are equivalent to order (B)2, The approximations based on the B.D.W.. form are superior to: the approximations based on the .ordinary Born form when C& has large elements comparei to the. elements of B’ (say,, for low anis&ropies).
will almost always fail
while D. W.B. may work, is just above threshold. The exponential approximation used in the present computational study corresponds to replacing (3) by S = i?xp(i$$
1 October 1970
and the total opacity P,(j) isf
In the present computational study the potential used $ corresponds to a structureless atom interacting with $ homonuciear rotor. The reduced parameters we’re similar # to those used by Fenstermaker and Bernstein f5f. Calculations
were performed at the reduced energy @Z/E)= 10.15 an3 the radial Born integrals required in (2) are of a standard form [6, ?]. Exact calcula-
tions were performed using the (reaction) density
amplitude method [S]. Fig. I shows the opacities obtained for: (a) the j = 0 -+I’ = 2 first order allowed transition, (b) t&j =O+j'= 4 first order forbidden transition
02 $ V= 4c[ (u/R)~~l+ObP~(cos8))-(u/R)~lcO.~~(co~8j)l
and S= expf-2i tn-‘(B/Z)]
(6)
referred to as the exponential-Born (E.B.) and the Heitler-Born (H.B.) respectively. When dg and B’ commute the E.B. and the.E.B.D.W. formulas are equivalent. The tLZ3.D.W. and H.B. procedures.are never identical. All four appxwc-
8 Is the angle between the roLor and the directionoLreIativemoMon. S B= 403Zl?,(E&/C) = 0.0145 witxeE& is the lowest threshDIdenergy (j= 0-r 2) transition. This choice was msant to simulate an Ar + TIF collision. Note howeverthat the potential usedcontains nocdd Legendrepolynomfai~ andhence is moreappmpriate for a homonuclear rotor. where
CHEMICAL PHYSICS LETTERS
Volume 7, number 1
(b)
v--L.150
200
250
I
1”
350
*
Fig. 1. Results of 9-state exact (dots) and exponential approximations for inelastic opacities for rotational excitation. (a) For the j = 0 + 2 transition, @) for the j - 0 -r4 transition and (c) for the total inelastic transitions our of the j = 0 stab. For I 3 200 the computation has converged, as increasing the basis size to a 49-&b computation does not significanLly alter the results (cf. table 1). For lower Zvaluee, dominant coupling sets in. and (c) the total inelastic opacity out of thej = 0 state, aa a function of 2. The plots show (22+1) times the opacities, so that the area under the curves is proportional to the appropriate cFos6 section. The E.B.D.W. is In good agreement with the exact results down to about Z = 200 while the ordinaxy Born approximation is valid only down to
L October 1970
about 1 = 300 and fails badIy ir?.the limited coupling regime. The H.B.D.W. is ‘51reasotile agreement with the exact results down to about Z = 200 but becomes unreasonable at Lower Z values (where the predicted total inekstic opacity goes to zero). The simple KB. approximation was also examined and found to yietd opacities which were generaPly several orders of magnitude too smaI1. The failure of the Heitler Qpe exponential approximation was also noticed in other computational studies [9]. This is not unexpected since one can show [lo] that the Reitler form is exact in the limit where energy is conserved in all intermediate transitions. This is the opposite extreme to the limit where the reiative motion is nearly classical and which is more appropriate to the present study. On the other hand, one should-recall that the Heitler scheme can be derived from a variational procedure and thus the H.B. scheme should work well when the norm of f3 is small whiIe the B.B.D. W. should provide accurate results when the norm of 8’ is small (irrespective of the ‘magnitude of the diagonai elements of B). The latter condition defines the limited coupling regime and, as is evident from fig. 1, the H.B.D.W. does provide satisfactory results in that region. Table I lists some nunerical results for the j = 0 4 j’ = 2 opacity. Besides the calculations shown in fig. IA, some 49 state E.B.D.W. calculations are aIso presented. These show that the 9 state caicuIations have essentially converged for Z 5 180. Both the exact and the exponentiat approximations opacities show dips around those values of the angular momenta that correspond to the vanishing of the radial Born integrals [Ill due to the cancellation of the contributions from fhe long range and short range parts of the potential. This feature occurs in the exact results ody when the energy is high enough to ensme that the radiaL Born integrals provide a good approximation to the distorted-wave radial inte&rats z. At Lower Z values the exponential opacities oscillate erratically. The oscillations are, however, approximately centred about the correct statistical values. A study of the energy dependence of these oscillations suggests that they would be largely smoothed out, if an energy average were taken.
Fig. 2 shows the phase of SJ
$ The dips occur around a reduced impact parameter of I. As E increases the dips occnr at higher Zvalues and hence the Born radial integrak provide a better approximation due to tie increasing importance of the centrifugal repulsion.
109
Volume
7, number
1
CHEMICAL PHYSICS LETTE-?tS
Opacitias 1
-__
Exact a)
for
the
Table 1
first order allowed j = 0 + 2 transition
E.B.D.W.
180
0.524
0.231
a)
E.B.D.W.,
b)
1 October 1970
(E/E) = 10.15 H.B.D.W’.
0.281
0.150
a)
Born 116.8
200
0.313
0,547
0.572
0.552
220
0.261
0.494
0.498
0.445
0.827
240
0.557
0.553
0.562
0.524
1.24
260
0.547
0.506
0.510
0.460
0.900
0.386
0.387
0.350
0.542
0.256
0.256
0.236
280 300
0.287
320
340
0.102
.
1.67
:
0.310
0.158
0.151
0.176
0.095
0.093
0.101
a) g-state cakulations. b) 49-&&e calculations.
ic phase) as a function of- Z (Z = J). The E.B.D.W. provides a good approximation for.the phase. In the present model, for Z > 200, the matrices dB and S’ nearly commute and hence the E.B.D.W. results are in agreement .withthe E.B. results. In other studies. this is not the case and the E.B.D.W. often provides the best approximation in the limited coupling regime.
4. N_JhlMARY The exponential Born approximation permits a fast, accurate, calculation of .opacities for systems of the type considered here (high B, high
E/Eth). Due to its speed, S matrices of very large dimensions may be evaluated. The use of such large S matrices is essential in order to join smoothly the limited coupling and dominant coupling regimes and thus provide.a complete prescription for the efficient calculations of cross sections. Work in progress includes an extension of the present, analytic, approximation to energies nearer threshold where diitortion effects due to channel potentials are important.
ACKNOWLEDGEMENT This work was supported by the Israel Academy of Sciences and Rumanities. We would also like to thank Dr. B. R, Johnson for the use of his amplitnde density program and Dr. M. Shapiro for many useful discussions. REFERENCES
-.
[I] R.‘b. Levine and G. G. Balint-Kurti, Letters 6 (1970) 101.
Chem. Phys.
Proo. Phys. Sot. (London) 89 (l966) 469; 77 0961) 174; [Sl R. D. Levine. Quant’k m&!hanics of molecular [2] M.J.Seaton,
rate processes (Oxford Univ. Press, Oxford,
196!)). [4] R. D. Levine and R. B.Bernstein. J. Chem. Pbys.. .to be published. [5]-R.W. Fenstermaker and II. B. Bernstein, J. Chem.
Phys. 47 (I967) 4417.
“-~-YkO~~ 200
240
A
Fig..2. Exact (dots) and E.B.D;W. rzx~lt~ for the phase of the diagonal element SJ(Or.OJ) versus
110
2 (I = a).
[S] Ft. J. Cro& and It. G. Gordon, J. Chem. Phys. 45. (1966) 3571, [7] I. S, Gradshteyn and-I. M. Ryzhik, Tables of Integrals series and products [Academic Press, New York, 1965) p. 692. -_ r
Volume 7, number 1
CHEMICAL PHYSICS LETTERS
[8j B. R. Johnson and f). Secrest, J. Ckem. Phys. 48 (1968) 4682. [S] B.R. Johnson and R. D. Levine, J. Chem. Phye, SO (1969) 1992,
3. October 1970
[lo] R. D. Levine. Proc. Phys.Soc. B. (Atom. Mol. Phys.} 2 (1969) 639. Ill] R. E.Olson and R. B. Bernstein. J. Chem. P&\-s. 50 (I969j 246; M. von Seggern and J. P. Toennies. preprint (L970).
111