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The role of memory effects in scholastics approach to collinear diatom—diatom collisions
Chemical Physics 68 (1982) 317-321 North-Holland Publishing Company
THE ROLE OF MEMORY TO COLLINEAR
EFFECTS
DIATOM-DIATOM
IN STOCHASTIC
APPROACH
COLLISIONS
A J. MAKOWSKI Instituteofphysics, Nicholas Coper)licus Unfversity, ul. GN&+zLz
5.87-100
TONER.PoIand
Received 11 December 1981
Theroleofmemoryeffects in the stochastic approach to vibrational-vibrational transitions in collinear diatom-diatom collisions is studied. It is shown that with the help of a new recurrence relation for Clebsch-Gordan coefficients we were able to solve exactly a non-mzkovian master equationfor a model hamiltoniar~ Tne derived solution for probabilities of V-Y transitions is compared with markovian as xvell as exact semiclassical results over a wide rang,e of velocities of the colliding molecules. We have found a substantial improvement of markovian results, both qualitative and quantitative, when the non-markovian effects have been included in the stochastic theory. This is in contrast with a recent study by King and Schata who got for a restricted V-T model more accurate probabilities from the markovian approach than from the nonmarkovian one. The reasons for this are also discussed.
I. Introduction In recent years the socalled stochastic method of the collision theory has been intensively developed. Roughly speaking it consists in deiiving and then applying in collisional dynamics equations of the Fokker-Planck type [ 1,2,8--IO] and Pauli-like master equations [l-7]. The detailed conditions for such a type of description to be justified have already been discussed (1,6,8,9]. In applying the master equations it is most frequently assumed that the collisional system can be described by a Markov process. In this way we deal with approximated master equations without memory [l-4] (cf. an interesting exception in ref. [S]). In other cases we get “exact” non-markovian master equatiors [6,7] _These latter equations are, of course, much more difficult to solve than the markovian ones [4,1 l-131. Thus it is important to determine on what conditions it is appropriate to approximate the non-markovian master equation by its markovian counterpart. For the first time the problem has been examined in ref. [ia] in the case of the system having two degrees of freedom, i.e., for the V-T transitions in atom-diatomic molecule collisions. A surprising result of that paper was that the marko\tian probabili-
0301-0104/82/ClOOO-OOoo/S
02.75 0 1982 North-Holland
ties were generally more accurate than the nonmarkovian ones. The authors of the paper regarded it as an accident connected with a special model problem under consideration_ The aim of our paper is also an examination of non-markovian effects which we shall study in the case of collinear diatom-diatom collisions, i.e., for three degrees of freedom. To this end, we will adopt a model used in our previous paper [7] and show that even in the non-markovian case its exact solution can be found analytically_
2. Master equations
and their solutions
Following our previous paper [7] in the lowest order of a Born series expansion [19] we get the nonmarkovian master equation
where P&)
is a probabtity state Ik) 2t time f, and
L&l
= E-1 f t V,@)ljj$$
distribution
- Iv,m;&I
in a quantum
.
(2)
Al
318
Ma.bvskijCMhe~
As in ref. [7]) we shall apply eq. (1) to the co&near collisions between two homonucIear diatomic molecules treated as harmonic osciIIators. Their interaction is descdbed by [IS] V-I@) = G(t) (A-
,
i-o-b”)
(3)
where the boson operators LI+and b* act in the subspaces of vibrational asympiotic eigenstates of both molecules, respectively. The shape of the function G(t) depends on an assumed form for a classical trajectory. Here we take the function first introduced in ref. f 161 G(t) = (u%jSLZo)sech
(Irvtj4L)
_
(8)
Now, we want to solve exactly eq. (5) for our restricted problem with the perturbation given by eq. (3). It can be performed if the following recurrence relation for the Clebsch-Gordan coefficients [es_ (7)] is appIied
+ (2np - 2,2 +p)c@, -t(rQ-
722+rz)C@?,n-
i-(t~P-n~in)P,_~CO,r~)]dt,,
ior a more rigorous classification we refer the reader to ref. [ 181 or [20]. Besides, it is worth to point out
The above identity can easily be proved by applying the-results of our previous paper [7] and can aIso be verified by using explicit formula for the vdues of the Clebsch-Gordan coefficients (7) (see e.g. eq. (27.9.1) in ref. [21]). We Iook for the exact solution of eq. (5) in the form
correiations
where an unknown function h (r, s) is to be detennined. Inserting eq. (10) into eq. (5) and then ta!!ng advantage of relation (9), we get
syat +(2/fi*j s(s f I)G(T)
am
thst eq. (l), and hence eq. (5) too, follow from neg,!ecting translational-vibrational
and
iron1 replacing a quantum ensemble average of an interaction potential by a classical average as discussed in ref. [6]_ it has been shown in ref. [7] that by the approximation P,(rl) = Pl{r) (no memory) eq. (5) reduces to
s m~rkovhn equation the exact solution ofwhich
has
the form
X
JrG(r@(s,
The integrodifferential
&tat2
equation can be replaced by
+ (Irvf4L) th (7i7~t/4L) ah/at
(12)
sech”(nutj4L)h(t,s)=O,
if eq. (11) is differentiated with respect to time and eq. (4) is used. Now, defining a new function
e-T(r~~i)C(P,,,,)C,~~~~
s=o
03)
??(f,s>=h&s),
S) in eq. {66; denotes the Clebsch-
Cordan coefficient C@,j, s) = W&P/2,
(11)
tl) dtl = 0 .
-m
+ [s(s+l)u4/32w2L4]
The symbol C&j.
1,S)=S(S+l)C@,n_S)_(9)
(5)
where @= n +iV, while tz andlV denote oscillatory quantum numbers of the two molecuIes. We shall refer to eq. (5) as non-markoiian or master equation -aith ~.lemcry. _kithough this is the usual interpretation
wnck 5 P?” J.--R (B ’ r) = (-1,
n, s)
2n2+n.P,(P, t1>
fl! - (2m/3-
and the function
7(w) = (Vj2WL)2 .
(4)
In eq. (4) u is an initial relative velocity of the coIIiding molecules, L is a potential parameter and w the frequency of the molecule under consideration. Inserting eq_ (3) into eq. (2) and the result into eq. (l), we have [7]
x Pn+ilj3.
dinrom-diatom co~isioIs
where
H-j7
5-(t) reads
+E+jlsO)
(7)
[Fli’*
jsech(%J -m
dtl , (14)
A.J. MabwsFcifColiinea.?diarom-diatom wllisions we get a very simple differential
a+/ag
equation
+ s(g) = 0 .
(15)
319
.. . . .. . Es NM ---
M
The solutions of eq. (15) together with eqs. (14) and (13) and the obvious initial condition h (s, t = -) = 1 lead us to the final result
$:$%W
= (-1r+”
go cos{[2s(s+1)1-(t)]~~2)
X C@,n, 4 C@,k, 4 .
(16)
In the next sectjon we compare transition probabilities obtained from both the exact markovian 31) [eq. (6)] and the exact non-markovian (NM) [eq. (16)], as well as the exact semiclassicaI (ES) solutions. ‘For simplicity we restrict ourselves to transitions involving one of the molecules initially in the ground state [i.e., k = 0 ineqs_ (6) and (1611. In such a case the ES formula is given by [15,17]
qzy’(9 =
I!
?Pt! (m
-
I)!
SirFX(f) cos2(m-+(f), (17)
where
and ‘rhe function
3. Numerical
FC_ 1. Comparison of probabilityPi=: for exact semiclassical (ES) (dot-dot), nonmarkovian (NM) (solid line), and markovian Ofl (dash-dash) versus the initial relative velocity U.
G(f) is defined in eq. (4).
examples
and discussion
For the sake of comparison we choose the oscillator frequencies: .qHZ = 8294 x 10’4 s-1 ) w = 4.393 x 1014 s-1, WC1 = 1.087 X 1014 s-l:Ae 2 steepness parameter L = 0.2 A and the relative velocitiesofthecolli.Sion1X104cm/s
fects have been examined by King and Schatz in the energy range 1 Q E,, <4 for the V-T problem in the systems: H + 12, H + Ha and He i HBr. They found a
Fig. 2. As in fg. 1 but farG$!.
A.J. Makowski/CoiBxar diatom-d~tom coiItions
320
satisfactory agreement of the *M,NM and ES probabilities for ~~ 5 1. At higher values of CQ there appeared some discrepancies among those three levels of appro.ximations with, in general, better results from the M than NM theory. As can be seen in our figures, the stochastic approach when applied to the resonance V-V transitions in the systems HZ + Hzr N2 f N2 asid Cl2 + Cl, works excellently over a much wider energy range than in the V-T case. This is connected with the fact that the hamiltonian in our model [cf. eq. (3)] does not contain any oscillating terms contrarytotheV-TmodeL[14].Hence,thevibrationaltranslational correlations are much weaker in our case than in the former one and approximations leading to an equation like (I) [6] are much better satisfied in the resonance V-V calculations. To compare the N and Ml results let us take the series expansions [cf. eqs. (6) and (16)] : exp [-(u/~w~)~s(s
f I)] = 1 - s(s +I) (v/2wLj2
+; [s(s i- l)]+Jj2wL)4-
... )
Wa)
cos {(u/hdL) 12s(s + I)] l/2) = 1 - s(s +1) (v/2wL)2 +~[s(s+I)]~(u/2cL)4-
. ..)
V’b)
which are valid when u -%2wi
or in the scale of energicsE
As we can see, the function T(t) increases monotonically during the collision for all three systems and exhibits a similar behaviour in alI the cases wi’& r(a)
[=O] < T(O) [=(zJ/4wL)2]<7(-)
[=(lJ/%JLyq -
This is one more reason why the stochastic theory including memory effects better works in the resonance V-V problem than in the V-T case. As it has been pobted out in ref. [ 141, if the energy transfer parameter is oscillating in time then the vibrational-translational correlations can play an important role and the exact conditions for the markovian approximation to be valid are much diffl,cuit to determine. In our model the parameter ?r,t> would be an oscillating function of time in the case of the V-V-T energy transfer problem because then the perturbation like that of eq. (3) would include oscillating factors [15]. However, the roIe of non-markovian effects for the above type of collisions and for collisions with more realistic potentials has to be studied in future. To complete our considerations, it is worth to note that by accident Po-1 ‘*@JM) = Pky(ES) as can be easily verified by inspection of eqs. (16) and (17). Besides, some of the NM and M multiple transition probabilities were found to be negative_ This deficiency of the stochastic method hzs already been known for some time [22,14].
(W
and is similar to a condition found in ref. [ 141. With the inequality (20) satisfied a non-markovian master equation may be approsimated by its markovian counterpart. Typical solutions for the two kinds of master equations are plotted in i&s. 1 and 2. There we see that at moderate and high velocities the NM r2sults are more accurate than those obtained from the markovian solutions. As it should be, those former show an oscillatory behaviour and the latter tend to their fixer3 values. For B given energy, say eO = 4, the differences between N?J and i\l results are smaller for N, than, say for Hz, since (mu)& < (mw)c12 < (mw)& [cf. (2O)J. Some conclusions can also be deduced by examining the the dependence of the energy transfer parameter T(t) defined by eqs. (8) and (4). After simple integrations we get
Acknowledgement This work has been supported by rhe Polish 1Ministry of Science and Higher Education (Project M.R.I.5).
References
[I] S.D. Augstin and H. Rabitz, J. Chem. Phys. 64 (1976) 1223. [2] SJ). Aupstin and H. Rabitz, J. Chem. Phys. 66 (1977) 269. [3] S.D. Augusti and H. Rabitz, J. Chem. Phys. 67 (1977)
2082. 141:G.C. Schatz, J. Chem. Phys. 66 (1977) 5220. [S] Y. Alhassid and R.D. Levine, Chem. Phys. Letters 72
(1980) 401.
A.J. Afa~wskijCaIlinear drhtom-diatom collislbns [6] G.C. Schatz, F-T. McLafferty Phys. 66 (1977) 3609.
and J. Ross, I. Cbem.
[7] AJ. Makowski and T. Orlikowski, Chem. Phys. 64 (1982) 231. [8] B.C. Eu, Chem. Phys. Letters 47 (1977) 555. [9] B.C. Eu,Cbem. Phys. 27 (1978) 301. [lo] G.D. Bill&, B.C. Eu and C. Nyeland, Chem. Phys. 29 (1978) 427. [ll] S.D. Augustiu and H. Rabitz, J. Chem. Phys. 70 (1979) 1286. 1121 C. Nyeland and G.D. Billing, Chem. Phys. 40 (1979) 103. 1131 G.C. Schatz, Chem. Phys. 24 (1977) 263. [14] F.W. King and G.C. Schatz, Mol. Phys. 38 (1979) 257.
321
[15] I.D. KeIley, J. Chem. Phys. 56 (1972) 6108. 1151 D. Rapp and T.E. Sharp, J. Chem. Phys. 38 (1963) 2641. 1171 AJ. Makowski, Chem. Phis. 18 (1976) 477. [I81 N-G. van Kampen,Phys. Rep. C24 (1976) 171. [19] R_ Zwanzig, in: Lectures in theoretical physics, Vol. 3, eds. WE. B&tin, B.W. Downsand J. Downs (Interscience, New York, 1961) p. 106. [ZO] I. Oppenheixu and K.E. Shuler. Phys. Rev. B138 (1965) 1007. [21] XI. Abramowitz and LA. Stwn, Handbook of matbematical functions, Applied hlathematics Series 55 (National Bureau of Standards, Washington; 1964). [22] G.C. Schatz, Chem. Phys. Letters 58 (1978) 368.
THE ROLE OF MEMORY TO COLLINEAR
EFFECTS
DIATOM-DIATOM
IN STOCHASTIC
APPROACH
COLLISIONS
A J. MAKOWSKI Instituteofphysics, Nicholas Coper)licus Unfversity, ul. GN&+zLz
5.87-100
TONER.PoIand
Received 11 December 1981
Theroleofmemoryeffects in the stochastic approach to vibrational-vibrational transitions in collinear diatom-diatom collisions is studied. It is shown that with the help of a new recurrence relation for Clebsch-Gordan coefficients we were able to solve exactly a non-mzkovian master equationfor a model hamiltoniar~ Tne derived solution for probabilities of V-Y transitions is compared with markovian as xvell as exact semiclassical results over a wide rang,e of velocities of the colliding molecules. We have found a substantial improvement of markovian results, both qualitative and quantitative, when the non-markovian effects have been included in the stochastic theory. This is in contrast with a recent study by King and Schata who got for a restricted V-T model more accurate probabilities from the markovian approach than from the nonmarkovian one. The reasons for this are also discussed.
I. Introduction In recent years the socalled stochastic method of the collision theory has been intensively developed. Roughly speaking it consists in deiiving and then applying in collisional dynamics equations of the Fokker-Planck type [ 1,2,8--IO] and Pauli-like master equations [l-7]. The detailed conditions for such a type of description to be justified have already been discussed (1,6,8,9]. In applying the master equations it is most frequently assumed that the collisional system can be described by a Markov process. In this way we deal with approximated master equations without memory [l-4] (cf. an interesting exception in ref. [S]). In other cases we get “exact” non-markovian master equatiors [6,7] _These latter equations are, of course, much more difficult to solve than the markovian ones [4,1 l-131. Thus it is important to determine on what conditions it is appropriate to approximate the non-markovian master equation by its markovian counterpart. For the first time the problem has been examined in ref. [ia] in the case of the system having two degrees of freedom, i.e., for the V-T transitions in atom-diatomic molecule collisions. A surprising result of that paper was that the marko\tian probabili-
0301-0104/82/ClOOO-OOoo/S
02.75 0 1982 North-Holland
ties were generally more accurate than the nonmarkovian ones. The authors of the paper regarded it as an accident connected with a special model problem under consideration_ The aim of our paper is also an examination of non-markovian effects which we shall study in the case of collinear diatom-diatom collisions, i.e., for three degrees of freedom. To this end, we will adopt a model used in our previous paper [7] and show that even in the non-markovian case its exact solution can be found analytically_
2. Master equations
and their solutions
Following our previous paper [7] in the lowest order of a Born series expansion [19] we get the nonmarkovian master equation
where P&)
is a probabtity state Ik) 2t time f, and
L&l
= E-1 f t V,@)ljj$$
distribution
- Iv,m;&I
in a quantum
.
(2)
Al
318
Ma.bvskijCMhe~
As in ref. [7]) we shall apply eq. (1) to the co&near collisions between two homonucIear diatomic molecules treated as harmonic osciIIators. Their interaction is descdbed by [IS] V-I@) = G(t) (A-
,
i-o-b”)
(3)
where the boson operators LI+and b* act in the subspaces of vibrational asympiotic eigenstates of both molecules, respectively. The shape of the function G(t) depends on an assumed form for a classical trajectory. Here we take the function first introduced in ref. f 161 G(t) = (u%jSLZo)sech
(Irvtj4L)
_
(8)
Now, we want to solve exactly eq. (5) for our restricted problem with the perturbation given by eq. (3). It can be performed if the following recurrence relation for the Clebsch-Gordan coefficients [es_ (7)] is appIied
+ (2np - 2,2 +p)c@, -t(rQ-
722+rz)C@?,n-
i-(t~P-n~in)P,_~CO,r~)]dt,,
ior a more rigorous classification we refer the reader to ref. [ 181 or [20]. Besides, it is worth to point out
The above identity can easily be proved by applying the-results of our previous paper [7] and can aIso be verified by using explicit formula for the vdues of the Clebsch-Gordan coefficients (7) (see e.g. eq. (27.9.1) in ref. [21]). We Iook for the exact solution of eq. (5) in the form
correiations
where an unknown function h (r, s) is to be detennined. Inserting eq. (10) into eq. (5) and then ta!!ng advantage of relation (9), we get
syat +(2/fi*j s(s f I)G(T)
am
thst eq. (l), and hence eq. (5) too, follow from neg,!ecting translational-vibrational
and
iron1 replacing a quantum ensemble average of an interaction potential by a classical average as discussed in ref. [6]_ it has been shown in ref. [7] that by the approximation P,(rl) = Pl{r) (no memory) eq. (5) reduces to
s m~rkovhn equation the exact solution ofwhich
has
the form
X
JrG(r@(s,
The integrodifferential
&tat2
equation can be replaced by
+ (Irvf4L) th (7i7~t/4L) ah/at
(12)
sech”(nutj4L)h(t,s)=O,
if eq. (11) is differentiated with respect to time and eq. (4) is used. Now, defining a new function
e-T(r~~i)C(P,,,,)C,~~~~
s=o
03)
??(f,s>=h&s),
S) in eq. {66; denotes the Clebsch-
Cordan coefficient C@,j, s) = W&P/2,
(11)
tl) dtl = 0 .
-m
+ [s(s+l)u4/32w2L4]
The symbol C&j.
1,S)=S(S+l)C@,n_S)_(9)
(5)
where @= n +iV, while tz andlV denote oscillatory quantum numbers of the two molecuIes. We shall refer to eq. (5) as non-markoiian or master equation -aith ~.lemcry. _kithough this is the usual interpretation
wnck 5 P?” J.--R (B ’ r) = (-1,
n, s)
2n2+n.P,(P, t1>
fl! - (2m/3-
and the function
7(w) = (Vj2WL)2 .
(4)
In eq. (4) u is an initial relative velocity of the coIIiding molecules, L is a potential parameter and w the frequency of the molecule under consideration. Inserting eq_ (3) into eq. (2) and the result into eq. (l), we have [7]
x Pn+ilj3.
dinrom-diatom co~isioIs
where
H-j7
5-(t) reads
+E+jlsO)
(7)
[Fli’*
jsech(%J -m
dtl , (14)
A.J. MabwsFcifColiinea.?diarom-diatom wllisions we get a very simple differential
a+/ag
equation
+ s(g) = 0 .
(15)
319
.. . . .. . Es NM ---
M
The solutions of eq. (15) together with eqs. (14) and (13) and the obvious initial condition h (s, t = -) = 1 lead us to the final result
$:$%W
= (-1r+”
go cos{[2s(s+1)1-(t)]~~2)
X C@,n, 4 C@,k, 4 .
(16)
In the next sectjon we compare transition probabilities obtained from both the exact markovian 31) [eq. (6)] and the exact non-markovian (NM) [eq. (16)], as well as the exact semiclassicaI (ES) solutions. ‘For simplicity we restrict ourselves to transitions involving one of the molecules initially in the ground state [i.e., k = 0 ineqs_ (6) and (1611. In such a case the ES formula is given by [15,17]
qzy’(9 =
I!
?Pt! (m
-
I)!
SirFX(f) cos2(m-+(f), (17)
where
and ‘rhe function
3. Numerical
FC_ 1. Comparison of probabilityPi=: for exact semiclassical (ES) (dot-dot), nonmarkovian (NM) (solid line), and markovian Ofl (dash-dash) versus the initial relative velocity U.
G(f) is defined in eq. (4).
examples
and discussion
For the sake of comparison we choose the oscillator frequencies: .qHZ = 8294 x 10’4 s-1 ) w = 4.393 x 1014 s-1, WC1 = 1.087 X 1014 s-l:Ae 2 steepness parameter L = 0.2 A and the relative velocitiesofthecolli.Sion1X104cm/s
fects have been examined by King and Schatz in the energy range 1 Q E,, <4 for the V-T problem in the systems: H + 12, H + Ha and He i HBr. They found a
Fig. 2. As in fg. 1 but farG$!.
A.J. Makowski/CoiBxar diatom-d~tom coiItions
320
satisfactory agreement of the *M,NM and ES probabilities for ~~ 5 1. At higher values of CQ there appeared some discrepancies among those three levels of appro.ximations with, in general, better results from the M than NM theory. As can be seen in our figures, the stochastic approach when applied to the resonance V-V transitions in the systems HZ + Hzr N2 f N2 asid Cl2 + Cl, works excellently over a much wider energy range than in the V-T case. This is connected with the fact that the hamiltonian in our model [cf. eq. (3)] does not contain any oscillating terms contrarytotheV-TmodeL[14].Hence,thevibrationaltranslational correlations are much weaker in our case than in the former one and approximations leading to an equation like (I) [6] are much better satisfied in the resonance V-V calculations. To compare the N and Ml results let us take the series expansions [cf. eqs. (6) and (16)] : exp [-(u/~w~)~s(s
f I)] = 1 - s(s +I) (v/2wLj2
+; [s(s i- l)]+Jj2wL)4-
... )
Wa)
cos {(u/hdL) 12s(s + I)] l/2) = 1 - s(s +1) (v/2wL)2 +~[s(s+I)]~(u/2cL)4-
. ..)
V’b)
which are valid when u -%2wi
or in the scale of energicsE
As we can see, the function T(t) increases monotonically during the collision for all three systems and exhibits a similar behaviour in alI the cases wi’& r(a)
[=O] < T(O) [=(zJ/4wL)2]<7(-)
[=(lJ/%JLyq -
This is one more reason why the stochastic theory including memory effects better works in the resonance V-V problem than in the V-T case. As it has been pobted out in ref. [ 141, if the energy transfer parameter is oscillating in time then the vibrational-translational correlations can play an important role and the exact conditions for the markovian approximation to be valid are much diffl,cuit to determine. In our model the parameter ?r,t> would be an oscillating function of time in the case of the V-V-T energy transfer problem because then the perturbation like that of eq. (3) would include oscillating factors [15]. However, the roIe of non-markovian effects for the above type of collisions and for collisions with more realistic potentials has to be studied in future. To complete our considerations, it is worth to note that by accident Po-1 ‘*@JM) = Pky(ES) as can be easily verified by inspection of eqs. (16) and (17). Besides, some of the NM and M multiple transition probabilities were found to be negative_ This deficiency of the stochastic method hzs already been known for some time [22,14].
(W
and is similar to a condition found in ref. [ 141. With the inequality (20) satisfied a non-markovian master equation may be approsimated by its markovian counterpart. Typical solutions for the two kinds of master equations are plotted in i&s. 1 and 2. There we see that at moderate and high velocities the NM r2sults are more accurate than those obtained from the markovian solutions. As it should be, those former show an oscillatory behaviour and the latter tend to their fixer3 values. For B given energy, say eO = 4, the differences between N?J and i\l results are smaller for N, than, say for Hz, since (mu)& < (mw)c12 < (mw)& [cf. (2O)J. Some conclusions can also be deduced by examining the the dependence of the energy transfer parameter T(t) defined by eqs. (8) and (4). After simple integrations we get
Acknowledgement This work has been supported by rhe Polish 1Ministry of Science and Higher Education (Project M.R.I.5).
References
[I] S.D. Augstin and H. Rabitz, J. Chem. Phys. 64 (1976) 1223. [2] SJ). Aupstin and H. Rabitz, J. Chem. Phys. 66 (1977) 269. [3] S.D. Augusti and H. Rabitz, J. Chem. Phys. 67 (1977)
2082. 141:G.C. Schatz, J. Chem. Phys. 66 (1977) 5220. [S] Y. Alhassid and R.D. Levine, Chem. Phys. Letters 72
(1980) 401.
A.J. Afa~wskijCaIlinear drhtom-diatom collislbns [6] G.C. Schatz, F-T. McLafferty Phys. 66 (1977) 3609.
and J. Ross, I. Cbem.
[7] AJ. Makowski and T. Orlikowski, Chem. Phys. 64 (1982) 231. [8] B.C. Eu, Chem. Phys. Letters 47 (1977) 555. [9] B.C. Eu,Cbem. Phys. 27 (1978) 301. [lo] G.D. Bill&, B.C. Eu and C. Nyeland, Chem. Phys. 29 (1978) 427. [ll] S.D. Augustiu and H. Rabitz, J. Chem. Phys. 70 (1979) 1286. 1121 C. Nyeland and G.D. Billing, Chem. Phys. 40 (1979) 103. 1131 G.C. Schatz, Chem. Phys. 24 (1977) 263. [14] F.W. King and G.C. Schatz, Mol. Phys. 38 (1979) 257.
321
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