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Comparison of quantum mechanical and semiclassical cross sections and rate constants for vibrational relaxation of N2 and CO colliding with 4He
39
Chemical Physics 107 (1986) 39-46 North-Holland, Amsterdam
COMPARISON OF QUANTUM MECHANICAL AND RATE CONSTANTS FOR VIBRATIONAL COLLIDING WITH 4He
AND SEMICLASSICAL CROSS SECTIONS RELAXATION OF N, AND CO
Gert Due BILLING Department
of Chemistry, Panum Institute,
University of Copenhagen, Copenhagen, 2200 N, Denmark
Received 3 February 1986
Semiclassical cross sections and rate constants for vibrational/rotational relaxation of CO and N, colliding with 4He are compared with quantum coupled-states calculations. If the same analytical potential energy surface is used good agreement is obtained. It is also shown that the small cross sections are very sentitive to the representation of the surface.
1. Introduction Exact quantum mechanical treatment of vibrational relaxation processes is at present virtually impossible due to the large number of vib/rot states which have to be included in the expansion of the total wavefunction. Another complicating feature is the small magnitude of the cross section at chemically interesting energies. Only for hydrogen containing diatomic molecules can one hope to carry out an exact converged calculation with present techniques. For these reasons there has in the past decade been a considerable interest in developing accurate approximate methods. Within the quantum mechanical frame work the most accurate of these methods is the coupled-states (CS) method in which the rotational projection states are decoupled. For a system as He + H, it has been shown [l] that the CS cross sections for the vib/rot transitions 10 --, Oj were underestimated with just 5-15% when compared to the exact close coupling (CC) values. Since one in the CS approach still solves a considerable number of coupled equations two decoupling methods, in which also the rotational states are decoupled, have been suggested. These methods are the infinite order sudden (10s) [2] and the breathing sphere (BS) [3] methods. Various recent investigations [4,5] have shown that the BS method yields cross sections which are typically an order of 0301-0104/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
magnitude smaller than the CS cross sections. Thus there is no reason to consider this method as an alternative for “better than order of magnitude” estimates of the relaxation rates. However it has also turned out that the 10s approach may underestimate the vib/rot cross sections significantly [4-61. Especially if the cross sections are small it was found [5,6] that the rates predicted by the VCC 10s (vibrational close coupling + 10s rotation) were too small by a factor of 2-4. However it appears that the actual correction factor depends not only upon the magnitude of the cross section but also upon the potential energy surface [4,7], the initial rotational state [5] or whether Coriolis (centrifugal stretch) coupling is included or not [8]. An alternative approach, for reducing the quantum dimensionality of the problem, is obtained when a classical mechanical description of some degrees of freedom is introduced. This is the basic idea behind the methods suggested a decade ago by Billing [8,9]. Comparison with quantum mechanical calculations and experimental vibrational relaxation data on He + H, showed [9] that the socalled V,R,T, method [lo] in which both rotation and translation are treated classically was accurate only at higher energies/temperatures. This is a natural consequence of the quantum mechnical nature of the H, rotational motion. However the alternative V,RgTc method [lo] which B.V.
40
G. D. Billing / Vibrational relaxation of CO and NJ cokding
treats also the rotation quantum mechnically yielded accurate cross sections at low energies [8]. Thus for H, or hydrogen containing molecules the V,R,T, method should be used to calculate rate constants only at temperatures higher than 300400 K. For heavier molecules as CO or N, we expect the V,R,T, method to be applicable in a larger energy range and hence rates down to about 50-100 K were believed to be accurately determined by this method (see, e.g., ref. [ll]). However no direct test of this approach was carried out for heavy molecules. The reason being that accurate CC or CS calculations only recently have become available. Thus large scale CS calculations on He + N, have been carried out by Banks et al. [5] and some CS cross sections for the He + CO have been calculated in ref. [7]. It is therefore relevant once again to discuss not only the accuracy of the 10s method as it was done in ref. [5] but also of the V,R,T, method and the problems involved in the calculations of accurate cross sections for vib/rot transitions. As mentioned above the cross sections are small, 10-‘“-10-4 A2, in the relevant energy range. This requires also an extreme accuracy of the potential energy surface and as mentioned in ref. [7] it appears that present ab initio methods are not sufficiently accurate to be able to predict the potential energy surfaces necessary for scattering calculations of rot/vib transitions. Therefore one most often uses semi-empirical surfaces in which parameters may be adjusted so as to obtain agreement for the 1 --) 0 rate constant. Then this surface can be used to generate other n --, m rates necessary for simulation of many bulk phenomena [12]. In order to investigate the sensitivity of vib/rot cross sections to the potential energy surface (PES), we calculate cross sections and rates obtained using an analytical PES and compare them with the values obtained using a so-called “accurate spline fit” to the surface. The result is quite surprising.
only the vibration is quantized and the equations of motion for the classical variables are integrated together with the equations for the time evolution of the quantum amplitudes [9,10]. Cross sections are obtained by averaging over a number of trajectories: u.,_.#)
= e-(2/+
1)(25+1)-’
IJ+jl
x
c
N-‘~‘~ay12,
treats the rotational and motion classically. Thus
0)
I
/=(J-Jl
where k$=2p(E-E,&C2,
(2)
E is the total energy, J$j the vibrational/rotational energy of the initial state, ~1 the reduced mass, J the total, j the rotational and 1 the initial orbital angular momentum. N is the number of trajectories, a$’ the amplitude for the quantum transition n to n’ associated with the ith trajectory and the prime indicates that the summation runs over trajectories having j(fina1) = lim ,_,j(t) between j’-p and j’+p where p =f for hetero- and p = 1 for homonuclear diatomics. If no resolution in final rotational quantum number, i.e. anj _ ,,, is needed then the prime is simply removed. Rate constants are now obtained by averaging over Boltzmann distributions over kinetic or kinetic and rotational energy, i.e. k,i,Ht(7’)
= ( E)1’2&Wd(BE,,n)BE,in
X exd -
PE,in>on,_ ,,’ ( E )
9
(3)
where ELin = E - E,,j and
XexP(-PE,in)QZ
5 (2j + 1)
j-0
Xexp(-PE,,,)a,,,.,(E),
2. The VqR,T, method The V,R,T, method the relative translational
with 4He
(4)
where /? = l/kT and Q,,, the rotational partition function. Introducing an integral instead of a summation over j and Enj = En + E,,, we obtain the
G. D. Billing / Vibrational relaxation
following expression valid in the range where a classical description of the rotation motion holds, i.e. T x==8_,,: k,_,(T)
= (y)‘”
x / omd(BE) exp[ -B(E - En)]
x(T,/T)“‘(u,,.~(E(U),T,h
of CO and N2 colliding with ‘He
correction to the usual arithmetic symmetrization may be neglected. In the present calculations we use for He-CO the analytical dumbbell potential determined in ref. [ll]. The reason for this choice is that previous VCC 10s [6] and even some CS calculations [7] are available on this surface. The PES is given by the following expression: V(R, r, y) =
(5)
= [ ntr2/2pI(
c i-o,
K(R, y)(G);/?‘,
(9)
1,2
where
where (un _ “,) is an average cross section: (0, -+.’ (E(U),
41
T,)) kTO)3’2]
(i=O, 1).
(6)
(10)
For i = 2 only the short-range term is included in eq. (10). The atom-atom distances are approximated by
In eq. (6) I is the moment of inertia and TO a reference temperature set to 300 K. The advantage of this expression is that the Monte Carlo approach yields a good estimate of the multiple integral (6) with rather few trajectories N = 50-100. In eqs. (5) and (6) the quantity U is the initial kinetic + rotational energy sometimes called the “classical energy”. In order to obtain the cross section (l), (6) at a given value of the “quantum energy” E - E,, we have to map the classical energy U onto the E energy scale. This “mapping” is done by extending the 1D symmetrization procedure to the 3D situation [lo]. Thus we use
where R is the distance from He to the center of mass of CO, r the CO distance and y the angle between R and r. The potential parameters are C, = 345.72 2, C, = 685.93 EI, (Y,= 3.442 A-‘, a2 = 3.751 A-‘, a, = 3.444 r^ A6”, a2 = 4.838 E^A6”, 2, = 0.7232 and c2 = 0.3631 (1 2 = 100 kJ/mol). If also the rotational motion is quantized it is convenient to expand y( R, y) in Legendre polynomials
E - E, = +AE + u+
Y(R,
AE~/I~u,
(7)
where AE = En, - E,. The expression (7) is obtained from the arithmetic mean velocity approach. This mapping may be shown [13] to be equivalent to imposing a common turning point for the upwards and downwards transition if AE is replaced by AE - AEc,, where AE,, is the energy transferred from the classical to the quantum system, i.e. AE,,=CIa,~(m)12E,~-E,, n’
R, = R - c,r cos y + $rfr2R -I sin2y, R, = R + c2r cos y + $:r’R
Y) =
c
J’,(cos
-’ sin2y,
Y)&(R).
I=0
.
01)
(12)
(13)
This expansion simplifies the calculation of coupling matrix elements between the rot/vib eigenstates. If the rotation is treated classically this expansion, which is sometimes slowly converging, is unnecessary and the expression (9) is used. Here we wish to investigate upon the performance of the expansion (13) using the V,R,T, method. Thus the expansion coefficients y,(R) are evaluated as:
(8)
where n is the initial vibrational state. However, in the present calculations AE,, & AE and this
Y,(R) = +-/-tdx
P,(x)v;.( R, x),
(14)
where x = cos y. The integral (14) is evaluated by
G. D. Billing / Vibrational relaxation of CO and N? coNiding with 4He
42
Table 1 Comparison of analytical and spline representation of expansion coefficients Vk (A, R
&CR> Y) (?I
6,
y=o
V,(R. y=T
analytical 2.0 2.5 3.0 4.0 5.0 6.0 8.0 10.0
4.676 0.9033 0.1549 2.092( - 6.415( - 2.410( - 3.872( -9.541(-6)
fit
3) =) 4) 4) 4)
-
4.722 0.9044 0.1549 2.092( 6.419( 2.410( 3.872( 9.543( -
3) 4) 4) 4) 6)
Y) (es. (9)) as a function of R and y
Y) (;)
y=o
y=n
analytical
fit
analytical
fit
analytical
fit
1.475 0.2130 2.490( -1.541(-3) - 5.826( - 1.923( - 3.263( - 8.369( -
1.476 0.2129 2.490( -1.542(-3) - 5.815( - 1.923( - 3.264( - 8.370( -
11.126 2.496 0.4555 1.209( - 2) - 1.247( - 3) - 1.235( -4) - 1.305( - 5) - 2.160( - 6)
11.661 2.502 0.4555 1.209( - 1.262( - 1.234( - 1.305( - 2.162( -
2.193 0.3393 4.784( - 2) 1.657(-4) - 1.688( -4) - 4.424( - 5) -4.166(-6) - 6.197( - 7)
2.198 0.3394 4.778( 1.647( - 1.689( - 4.423( -4.166(-6) -6.199(-7)
2) 3) 4) 5) 6)
2) 3) 4) 5) 6)
2) 3) 4) 5) 6)
2) 4) 4) 5)
s) Means 2.092 x 10F3. Table 2 Deviation A,(y)
(eq. (15)) for 17 distances R = Z.O(O.5)lO.O
Y
0 0.62832 1.25664 1.88491 2.51327 3.14159
A,(Y)(W) j=O
j=l
0.24 0.05 0.06 0.03 0.03 0.06
1.01 0.16 1.05 3.37 0.12 0.18
spline such a I’i( R, squares
approximation to comparison at two y). Table 2 shows deviation defined
V;,(R). Table 1 shows angles for I$( R, y ) and the average root-meanas:
A,(y)
=
Y> -
;;gl
numerical quadrature at 110 R-distances in the interval R E [1.5-12 A]. The elements K,(R) (i = 0, 1, 2 and I= 0, 1,. . ., 9) are spline fitted using the VC03AD Harwell subroutine which automatically evaluates the best knot positions. The quality of this approach can be estimated by comparing the analytical value of v( R, y) (eq. (10)) with the value obtained using the expansion (13) plus the
4ooo 3000 2000 1500 1000 750
2.47( 0.92( 1.77( 1.02( 5.13( 3.53( -
T-‘(
‘) Means 2.47 X lo-*.
R;, y))’
I
x 100%.
Sphne fit
5.51( - 3) 1.91(-2) 1.09( - 3) 6.40( - 3) 7.96( - 3) 2.61( - 3)
(15)
obtained using the analytical and the spline fitted
(%,(U Tl)) 2) s) 3) 5) 6) 9) 11)
Y)~~‘~“~]
The tables show that the spline fit represents the potential better than a few percent. Typical deviations are as small as 0.2%. The question is now how important such a deviation is in scattering calculations? Normally one would not hesitate to use a potential which represents the data points with this accuracy. In order to investigate the
Table 3 Average cross sections (o,e(U, Te)) and (AE,,,) as a function of U for He+CO surface. 50 trajectories were used at each energy, b,,, = 2 A and R, = 10 A Analytical
~(4,
l/-J
X
U (cm-‘)
([ q(R;,
2.48( - 2) 1.03( - 3) 3.48( - 5) 3.78( - 6) 1.51(-7) 6.13( - 8)
(A*)
(Wet) Cc) 5.51( - 3) 1.91( - 2) 1.08( - 3) 6.39(-3) 7.96( - 3) 2.61( - 3)
43
G. D. Billing / Vibrational relaxation of CO and N, colliding with ‘He
problem we calculated the average cross sections (a(E(U), T,)) defined by eq. (6) using ,both surfaces. The. trajectories were started and stopped at R = 10 A, the total angular momentum is selected randomly between 0 and J,,,,,, rotational angular momentum between 0 and j,,,,, and I between 1J-j 1 and J + j. The angles conjugate to j and I were selected between 0 and 2n [lo]. J max was set equal to A-‘~,,,,,(2~U)‘/* with b,,,,, = 2 A and j,,,,, = tl-‘(21U)“*. This defines the initial kinetic energy Ekin = U - E,,. Since we wish to test small differences in the potential we used the same initial conditions in the two calculations. In order to test whether the classical trajectories are the same the classical quantity = N- ’ 5 ( &fz;; - @;:;‘t$’) i=l
(A&,)
w
point
C
--10
-15i-
-2c)-
J
10
I 20
I 30
I LO
-10 -
point
73 lic
--15
-
-to-
/
t Ilo-‘kc Fig. 2. Same as fig. 1 but with 11=1250
_E
tuying
turning &
(16)
was recorded. The number of trajectories N was set to 50 giving a standard deviation of about lo-20% upon the cross sections. Table 3 shows that the cross sections deviate more and more the
^o ci
I
I 50 t/lo-t4
set
Fig. 1. The probability for vibrational deexcitation In(P,,) of CO(u =l) colliding with 4He as a function of time. Dashed curve obtained using the analytical potential energy surface and the solid curve using a spline representation. The energy is U= 2000 cm-‘.
cm-‘.
smaller the cross section or probability. Thus transition probabilities of the order lo-’ deviate by a factor of four but probabilities of the order lo-‘* deviate by orders of magnitude! In order to understand this we follow the probability P, _0 as a function of time during the collision at two energies U = 2000 cm-’ (fig. 1) and U = 1250 cm-’ (fig. 2). We see that the probability increases to a rather large value at the turning point in both cases and that the two surfaces yield almost identical values at the turning points, namely P,, = 1.541 X 10e4 versus 1.527 X low4 at 2000 cm-’ and P,O = 5.831 x 10e5 versus 5.889 X lo-’ at U = 1250 cm-‘. However when the particles move away from each other we see that the probability is quenched but that this quenching is not nearly as effective on the spline fitted surface. This is most likely due to slight difference in asymmetry around the turning point on the two surfaces with the analytical surface being the most “symmetric”. This asymmetry of the spline surface prevents an effective quenching of the probability. The smaller the probability the more accurate must the potential be in order to obtain the correct asymptotic value. The difference in the average cross sections shown in table would give more than an order of magnitude difference for rates around lOPi9 cm3/s decreasing to a factor of two for rates around
G. D. Billing / Vibrational relaxation
44
ofCO
and N, colliding with 4He
Table 4 Cross sections qi _ a( E) (K) for vibrational relaxation of CO colliding with 4He at Eki, = 50 meV as a function of initial CO rotational state. N = 100 trajectories were used to obtain the V,R,T, values. The uncertainty is about 10% but the same trajectories were used in the runs with and without Coriolis coupling, i.e. the change in c is independent of the smd on the individual numbers j(h)
0 4 6 8 Fig. 3. Rate constants for vibrational deexcitation of CO( u = 1) colliding with 4He as a function of temperature. The upper curve obtained using the spline representation of the surface the lower using the analytical surface.
lo-” cm3/s as shown in fig. 3. Only for rates around or above lo-l6 cm3/s would we get better than 10% agreement. This result has some bearing upon recent comparison between semiclassical, VCC 10s and CS quantum calculations on CO relaxation by collisions with H, carried out in ref. [4]. In ref. [4] it was found that the CO vibrational rate obtained by the two methods deviated by about a factor of three. Apparently the same potential was used [14] but contrary to what was done in the semiclassical calculations carried out in ref. [15] Bacic et al. [4] expanded the CO anisotropy in Legendre polynomials. We have just shown that unless this expansion is done either
Table 5 Cross sections for vibrational relaxation (a,,,( U, T,)),
u (cm-‘)
(ero(U, (K)
T,))
872 1137 1672 1915 2650 3483 4313
9.7q-11)*’ 7.21( - 9) 1.39( - 6) 5.62(-6) 1.43( - 4) 1.4q-3) 1.25( - 2)
E b, (ev)
eic-o(E) (A’)
ALi (9
0.0119 0.0336 0.0881 0.115 0.200 0.300 0.400
2.84(- 10) a) 0.0135 2.33( - 9) 0.0115 1.18( - 7) 0.0218 3.48(-7) 0.0265 5.41( - 6) 0.0315 3.3q - 5) 0.0346
e,j_.,,(E)
eis+s(E) (AZ) 9.02( 4.91( 2.17( 3.62( 6.16( 5.17(
Semiclassical V,R,T,
Quantum CS
with Coriolis coupling
without Coriolis coupling
(ref. [71)
1.6(-8) a) 4.2( - 8) 4.2( - 8) 9.2( - 8)
3.6( - 8) 4.q - 8) 7.1( - 8)
2.6( 2.7( 3.q 3.4(
-
8) 8) 8) 8)
=) 1.6(-8)=1.6x10-‘.
analytically or if one uses spline fits more accurate than indicated by table 1 one cannot expect to obtain accurate cross sections at low energies. We therefore think that the deviation of a factor of three on the Hz-CO vibrational relaxation rate could be due to slight differencies in the potentials used. Such a deviation would also contradict previous comparisons between quantum and semiclassical theory (for a summary see ref. [16]) and recent derivations [13,17] of the semiclassical equations which show that the approach in the weak coupling limit with turning point dominating coupling is essentially exact. Despite the fact that the PES is represented in different ways we have shown in table 4 a comparison between quantum CS calculations from ref. [7] which uses an expanded version of the PES
and AE,,,(j=O, A&t (f)
- 10) ‘) 0.0125 - 9) 0.0126 -7) 0.0276 - 7) 0.0272 - 6) 0.0413 -5) 0.0591
5, 9) for “He-N,
crs+o(E) (AZ)
AE,i
Quantum(ref.
(f)
CSA
0.0081 0.0101 0.0184 0.0356 0.0440
8.29( 7.06( 1.92( 6.02( 7.25( 4.76( 1.89(
closed 8.03( 1.06( 3.57( 5.05( 3.47(
-
9) 7) 7) 6) 5)
‘) N - 50 trajectories and R, = 15 A. For the other energies N = 65 and R, = 10 A. (1 Z = 100 kJ/mol.) b, E = Eki, + E,,
151) e,,_,,(K) vcc
-
10) 9) 7) 7) 6) 5) 4)
10s
l.OO(- 10) 1.23( - 9) 4.84( - 8) 1.62( - 7) 2.40(-6) 1.88( - 5) 8.15( -5)
45
G.D. Billing / Vibrational relaxation of CO and N2 colliding with 4He
‘He+N2
Table 6 Vibrational relaxation rates k,,(T) for 4He-N, obtained by the semiclassical VgR,T, method, the quantum CS method and experiment
TW
ho(T)
(cm3/s)
semiclassical
quantum. (ref.
100 132 149 156 175 210 262 291
/
/
5.9( - 20) a) 1.8( - 19) 2.9( - 19) 3.1( - 19) 5.8( - 19) l.l(-18) 3.2(-18) 5.7(-l@
‘) Means 5.9
x
lO_“,
1511
5.6( 1.5( 2.q 2.9( 4.6( 9.9( 2.7( 4.3( uncertainty
20) 19) 19) 19) 19) 19) 18) 18)
experiment (ref. [ZO]) 3.2+1.5(-20) 1.6+0.15(-19) 2.7 &0.2( - 19) 3.3 f 0.3( - 19) 5.0+0.3( - 19) 1.2*0.05(-18) 3.2+0.1(-18) S.l+O.l(-18)
5-108.
L-
1o-zoI’
1o-21o Fig. 4. Rate constants for vibrational deexcitation of Nr( u = 1) colliding with 4He as a function of temperature: k,,(T) (solid line), k,,,,(T) (lower dashed line), k,,_,(T) (dotted line) and k ,9_o(T) (upper dashed line).
(see ref. [18]) and the V,R,T, semiclassical cross sections obtained using the analytical expressions (9) and (10). Since the quantum calculations neglected Coriolis coupling we also report the semiclassical cross sections obtained neglecting the Coriolis coupling (centrifugal distortion) term [9]. The numbers show that u,,,_,, is underestimated semiclassically but that aii_,, increase somewhat faster with j than the quantum CS values. The difference being a factor of 2 at j = 8. Since exactly the same PES surface was probably not used the comparison of such small cross section may however not be meaningful. In the order to support the above findings further we should perform a comparison between a quantum calculation - preferable a CS calculation and the V,R,T, calculation on the same analyticalpotential energy surface. CS calculations have recently been carried out by Banks et al. [5]. It was found that the CS cross sections were typically a factor of 5 larger than the VCC 10s values and
that uli _ e( E) increased with j. This latter result is due to the change in energy gap for the lj + 0 j + n transition and to the effect of Coriolis coupling. Similar results have previously been found for He-H, [S]. The strong j-dependence of the cross sections shows that the VCC 10s method cannot be used but also for j = 0 the 10s cross sections are underestimated (see table 5). We also expect the semiclassical V,R,T, method to be less accurate for j = 0 since rotation becomes more classical as j increases. Calculations using the program ADIAV [19] and the PES surface determined in ref. [5] were carried out and the results are in table 5 compared with CS and VCC 10s values for qo_,,(E). We see that the semiclassical values are somewhat smaller than the quantum CS but larger than the VCC 10s values. The table also shows that the cross section u,~_~(E) increases strongly with j although the initial kinetic energy decreases. This is in agreement with the CS calculations and table 6 shows that for the rotational averaged rate constant k , _ o(T) we obtain good agreement with the CS values in the whole temperature range. The reason being that the Boltzmann rotational distribution weights higher-j states for which the agreement is better than for j = 0. The semiclassical values lie lo-20% above the CS values and might if one can generalize from the He-H, system [l] even be in better agreement with the exact CC values.
46
G. D. Billing / VIbrational relaxation of CO and N2 colliding with ‘He
Acknowledgement This research was supported by The Danish Natural Science Research Council and EEC (grant no. STI-062-J-C(CD)). Drs. D.C. Clary and A.J. Banks are acknowledged for supplying the programs for the He-N, potential.
[7] [S] [9] [lo] [ll] [12]
References [l] P. McGuire and J.P. Toennies, J. Chem. Phys. 62 (1975) 4623. [2] R. Schinke and P. McGuire, Chem. Phys. 31 (1978) 391. [3] F.A. Gianturco, U.T. Lamanna and G. Petrella, Chem. Phys. 64 (1982) 299. (41 Z. Bacic, R. Schinke and G.H.F. Diercksen, J. Chem. Phys. 82 (1985) 236; 82 (1985) 245. [5] A.J. Banks, D.C. Clary and H.-J. Werner, J. Chem. Phys. 84 (1986) 3788. [6] G.D. Billing and D.C. Clary, Chem. Phys. Letters 90 (1982) 27;
1131 (141 [15] (161
[17] (181 [19] 1201
R.J. Price, DC. Clary and G.D. Billing, Chem. Phys. Letters 101 (1983) 269. R. Schinke and G.H.F. Diercksen, J. Chem. Phys. 83 (1985) 4516. G.D. Billing, Chem. Phys. 30 (1978) 387. G.D. Billing, Chem. Phys. 9 (1975) 359. G.D. Billing, Computer Phys. Rept. 1 (1984) 237. G.D. Billing and M. Cacciatore, Chem. Phys. Letters 86 (1982) 20. M. Cacciatore, M. Capitelh and G.D. Billing, Chem. Phys. 82 (1983) 1. G.D. Billing, J. Chem. Phys., to be published L.L. Paulsen, Chem. Phys. 68 (1982) 29. G.D. Billing and L.L. Poulsen, Chem. Phys. 70 (1982) 119. G.D. Billing, Molecular astrophysics, eds. G.H.F. Diercksen, W.F. Huebner and P.W. Langhoff (Reidel, Dordrecht, 1985) p. 517. J.T. Muckermann, unpublished results. R. Schinke, M. Meyer, U. Buck and G.H.F. Diercksen, J. Chem. Phys 80 (1984) 5518. G.D. Billing, Computer Phys. Commun. 32 (1984) 45. M.-M. Maricq, E.A. Gregory, C.T. Wickham-Jones, DC. Cartwright and C.J.S.M. Simpson, Chem. Phys. 75 (1983) 347.
Chemical Physics 107 (1986) 39-46 North-Holland, Amsterdam
COMPARISON OF QUANTUM MECHANICAL AND RATE CONSTANTS FOR VIBRATIONAL COLLIDING WITH 4He
AND SEMICLASSICAL CROSS SECTIONS RELAXATION OF N, AND CO
Gert Due BILLING Department
of Chemistry, Panum Institute,
University of Copenhagen, Copenhagen, 2200 N, Denmark
Received 3 February 1986
Semiclassical cross sections and rate constants for vibrational/rotational relaxation of CO and N, colliding with 4He are compared with quantum coupled-states calculations. If the same analytical potential energy surface is used good agreement is obtained. It is also shown that the small cross sections are very sentitive to the representation of the surface.
1. Introduction Exact quantum mechanical treatment of vibrational relaxation processes is at present virtually impossible due to the large number of vib/rot states which have to be included in the expansion of the total wavefunction. Another complicating feature is the small magnitude of the cross section at chemically interesting energies. Only for hydrogen containing diatomic molecules can one hope to carry out an exact converged calculation with present techniques. For these reasons there has in the past decade been a considerable interest in developing accurate approximate methods. Within the quantum mechanical frame work the most accurate of these methods is the coupled-states (CS) method in which the rotational projection states are decoupled. For a system as He + H, it has been shown [l] that the CS cross sections for the vib/rot transitions 10 --, Oj were underestimated with just 5-15% when compared to the exact close coupling (CC) values. Since one in the CS approach still solves a considerable number of coupled equations two decoupling methods, in which also the rotational states are decoupled, have been suggested. These methods are the infinite order sudden (10s) [2] and the breathing sphere (BS) [3] methods. Various recent investigations [4,5] have shown that the BS method yields cross sections which are typically an order of 0301-0104/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
magnitude smaller than the CS cross sections. Thus there is no reason to consider this method as an alternative for “better than order of magnitude” estimates of the relaxation rates. However it has also turned out that the 10s approach may underestimate the vib/rot cross sections significantly [4-61. Especially if the cross sections are small it was found [5,6] that the rates predicted by the VCC 10s (vibrational close coupling + 10s rotation) were too small by a factor of 2-4. However it appears that the actual correction factor depends not only upon the magnitude of the cross section but also upon the potential energy surface [4,7], the initial rotational state [5] or whether Coriolis (centrifugal stretch) coupling is included or not [8]. An alternative approach, for reducing the quantum dimensionality of the problem, is obtained when a classical mechanical description of some degrees of freedom is introduced. This is the basic idea behind the methods suggested a decade ago by Billing [8,9]. Comparison with quantum mechanical calculations and experimental vibrational relaxation data on He + H, showed [9] that the socalled V,R,T, method [lo] in which both rotation and translation are treated classically was accurate only at higher energies/temperatures. This is a natural consequence of the quantum mechnical nature of the H, rotational motion. However the alternative V,RgTc method [lo] which B.V.
40
G. D. Billing / Vibrational relaxation of CO and NJ cokding
treats also the rotation quantum mechnically yielded accurate cross sections at low energies [8]. Thus for H, or hydrogen containing molecules the V,R,T, method should be used to calculate rate constants only at temperatures higher than 300400 K. For heavier molecules as CO or N, we expect the V,R,T, method to be applicable in a larger energy range and hence rates down to about 50-100 K were believed to be accurately determined by this method (see, e.g., ref. [ll]). However no direct test of this approach was carried out for heavy molecules. The reason being that accurate CC or CS calculations only recently have become available. Thus large scale CS calculations on He + N, have been carried out by Banks et al. [5] and some CS cross sections for the He + CO have been calculated in ref. [7]. It is therefore relevant once again to discuss not only the accuracy of the 10s method as it was done in ref. [5] but also of the V,R,T, method and the problems involved in the calculations of accurate cross sections for vib/rot transitions. As mentioned above the cross sections are small, 10-‘“-10-4 A2, in the relevant energy range. This requires also an extreme accuracy of the potential energy surface and as mentioned in ref. [7] it appears that present ab initio methods are not sufficiently accurate to be able to predict the potential energy surfaces necessary for scattering calculations of rot/vib transitions. Therefore one most often uses semi-empirical surfaces in which parameters may be adjusted so as to obtain agreement for the 1 --) 0 rate constant. Then this surface can be used to generate other n --, m rates necessary for simulation of many bulk phenomena [12]. In order to investigate the sensitivity of vib/rot cross sections to the potential energy surface (PES), we calculate cross sections and rates obtained using an analytical PES and compare them with the values obtained using a so-called “accurate spline fit” to the surface. The result is quite surprising.
only the vibration is quantized and the equations of motion for the classical variables are integrated together with the equations for the time evolution of the quantum amplitudes [9,10]. Cross sections are obtained by averaging over a number of trajectories: u.,_.#)
= e-(2/+
1)(25+1)-’
IJ+jl
x
c
N-‘~‘~ay12,
treats the rotational and motion classically. Thus
0)
I
/=(J-Jl
where k$=2p(E-E,&C2,
(2)
E is the total energy, J$j the vibrational/rotational energy of the initial state, ~1 the reduced mass, J the total, j the rotational and 1 the initial orbital angular momentum. N is the number of trajectories, a$’ the amplitude for the quantum transition n to n’ associated with the ith trajectory and the prime indicates that the summation runs over trajectories having j(fina1) = lim ,_,j(t) between j’-p and j’+p where p =f for hetero- and p = 1 for homonuclear diatomics. If no resolution in final rotational quantum number, i.e. anj _ ,,, is needed then the prime is simply removed. Rate constants are now obtained by averaging over Boltzmann distributions over kinetic or kinetic and rotational energy, i.e. k,i,Ht(7’)
= ( E)1’2&Wd(BE,,n)BE,in
X exd -
PE,in>on,_ ,,’ ( E )
9
(3)
where ELin = E - E,,j and
XexP(-PE,in)QZ
5 (2j + 1)
j-0
Xexp(-PE,,,)a,,,.,(E),
2. The VqR,T, method The V,R,T, method the relative translational
with 4He
(4)
where /? = l/kT and Q,,, the rotational partition function. Introducing an integral instead of a summation over j and Enj = En + E,,, we obtain the
G. D. Billing / Vibrational relaxation
following expression valid in the range where a classical description of the rotation motion holds, i.e. T x==8_,,: k,_,(T)
= (y)‘”
x / omd(BE) exp[ -B(E - En)]
x(T,/T)“‘(u,,.~(E(U),T,h
of CO and N2 colliding with ‘He
correction to the usual arithmetic symmetrization may be neglected. In the present calculations we use for He-CO the analytical dumbbell potential determined in ref. [ll]. The reason for this choice is that previous VCC 10s [6] and even some CS calculations [7] are available on this surface. The PES is given by the following expression: V(R, r, y) =
(5)
= [ ntr2/2pI(
c i-o,
K(R, y)(G);/?‘,
(9)
1,2
where
where (un _ “,) is an average cross section: (0, -+.’ (E(U),
41
T,)) kTO)3’2]
(i=O, 1).
(6)
(10)
For i = 2 only the short-range term is included in eq. (10). The atom-atom distances are approximated by
In eq. (6) I is the moment of inertia and TO a reference temperature set to 300 K. The advantage of this expression is that the Monte Carlo approach yields a good estimate of the multiple integral (6) with rather few trajectories N = 50-100. In eqs. (5) and (6) the quantity U is the initial kinetic + rotational energy sometimes called the “classical energy”. In order to obtain the cross section (l), (6) at a given value of the “quantum energy” E - E,, we have to map the classical energy U onto the E energy scale. This “mapping” is done by extending the 1D symmetrization procedure to the 3D situation [lo]. Thus we use
where R is the distance from He to the center of mass of CO, r the CO distance and y the angle between R and r. The potential parameters are C, = 345.72 2, C, = 685.93 EI, (Y,= 3.442 A-‘, a2 = 3.751 A-‘, a, = 3.444 r^ A6”, a2 = 4.838 E^A6”, 2, = 0.7232 and c2 = 0.3631 (1 2 = 100 kJ/mol). If also the rotational motion is quantized it is convenient to expand y( R, y) in Legendre polynomials
E - E, = +AE + u+
Y(R,
AE~/I~u,
(7)
where AE = En, - E,. The expression (7) is obtained from the arithmetic mean velocity approach. This mapping may be shown [13] to be equivalent to imposing a common turning point for the upwards and downwards transition if AE is replaced by AE - AEc,, where AE,, is the energy transferred from the classical to the quantum system, i.e. AE,,=CIa,~(m)12E,~-E,, n’
R, = R - c,r cos y + $rfr2R -I sin2y, R, = R + c2r cos y + $:r’R
Y) =
c
J’,(cos
-’ sin2y,
Y)&(R).
I=0
.
01)
(12)
(13)
This expansion simplifies the calculation of coupling matrix elements between the rot/vib eigenstates. If the rotation is treated classically this expansion, which is sometimes slowly converging, is unnecessary and the expression (9) is used. Here we wish to investigate upon the performance of the expansion (13) using the V,R,T, method. Thus the expansion coefficients y,(R) are evaluated as:
(8)
where n is the initial vibrational state. However, in the present calculations AE,, & AE and this
Y,(R) = +-/-tdx
P,(x)v;.( R, x),
(14)
where x = cos y. The integral (14) is evaluated by
G. D. Billing / Vibrational relaxation of CO and N? coNiding with 4He
42
Table 1 Comparison of analytical and spline representation of expansion coefficients Vk (A, R
&CR> Y) (?I
6,
y=o
V,(R. y=T
analytical 2.0 2.5 3.0 4.0 5.0 6.0 8.0 10.0
4.676 0.9033 0.1549 2.092( - 6.415( - 2.410( - 3.872( -9.541(-6)
fit
3) =) 4) 4) 4)
-
4.722 0.9044 0.1549 2.092( 6.419( 2.410( 3.872( 9.543( -
3) 4) 4) 4) 6)
Y) (es. (9)) as a function of R and y
Y) (;)
y=o
y=n
analytical
fit
analytical
fit
analytical
fit
1.475 0.2130 2.490( -1.541(-3) - 5.826( - 1.923( - 3.263( - 8.369( -
1.476 0.2129 2.490( -1.542(-3) - 5.815( - 1.923( - 3.264( - 8.370( -
11.126 2.496 0.4555 1.209( - 2) - 1.247( - 3) - 1.235( -4) - 1.305( - 5) - 2.160( - 6)
11.661 2.502 0.4555 1.209( - 1.262( - 1.234( - 1.305( - 2.162( -
2.193 0.3393 4.784( - 2) 1.657(-4) - 1.688( -4) - 4.424( - 5) -4.166(-6) - 6.197( - 7)
2.198 0.3394 4.778( 1.647( - 1.689( - 4.423( -4.166(-6) -6.199(-7)
2) 3) 4) 5) 6)
2) 3) 4) 5) 6)
2) 3) 4) 5) 6)
2) 4) 4) 5)
s) Means 2.092 x 10F3. Table 2 Deviation A,(y)
(eq. (15)) for 17 distances R = Z.O(O.5)lO.O
Y
0 0.62832 1.25664 1.88491 2.51327 3.14159
A,(Y)(W) j=O
j=l
0.24 0.05 0.06 0.03 0.03 0.06
1.01 0.16 1.05 3.37 0.12 0.18
spline such a I’i( R, squares
approximation to comparison at two y). Table 2 shows deviation defined
V;,(R). Table 1 shows angles for I$( R, y ) and the average root-meanas:
A,(y)
=
Y> -
;;gl
numerical quadrature at 110 R-distances in the interval R E [1.5-12 A]. The elements K,(R) (i = 0, 1, 2 and I= 0, 1,. . ., 9) are spline fitted using the VC03AD Harwell subroutine which automatically evaluates the best knot positions. The quality of this approach can be estimated by comparing the analytical value of v( R, y) (eq. (10)) with the value obtained using the expansion (13) plus the
4ooo 3000 2000 1500 1000 750
2.47( 0.92( 1.77( 1.02( 5.13( 3.53( -
T-‘(
‘) Means 2.47 X lo-*.
R;, y))’
I
x 100%.
Sphne fit
5.51( - 3) 1.91(-2) 1.09( - 3) 6.40( - 3) 7.96( - 3) 2.61( - 3)
(15)
obtained using the analytical and the spline fitted
(%,(U Tl)) 2) s) 3) 5) 6) 9) 11)
Y)~~‘~“~]
The tables show that the spline fit represents the potential better than a few percent. Typical deviations are as small as 0.2%. The question is now how important such a deviation is in scattering calculations? Normally one would not hesitate to use a potential which represents the data points with this accuracy. In order to investigate the
Table 3 Average cross sections (o,e(U, Te)) and (AE,,,) as a function of U for He+CO surface. 50 trajectories were used at each energy, b,,, = 2 A and R, = 10 A Analytical
~(4,
l/-J
X
U (cm-‘)
([ q(R;,
2.48( - 2) 1.03( - 3) 3.48( - 5) 3.78( - 6) 1.51(-7) 6.13( - 8)
(A*)
(Wet) Cc) 5.51( - 3) 1.91( - 2) 1.08( - 3) 6.39(-3) 7.96( - 3) 2.61( - 3)
43
G. D. Billing / Vibrational relaxation of CO and N, colliding with ‘He
problem we calculated the average cross sections (a(E(U), T,)) defined by eq. (6) using ,both surfaces. The. trajectories were started and stopped at R = 10 A, the total angular momentum is selected randomly between 0 and J,,,,,, rotational angular momentum between 0 and j,,,,, and I between 1J-j 1 and J + j. The angles conjugate to j and I were selected between 0 and 2n [lo]. J max was set equal to A-‘~,,,,,(2~U)‘/* with b,,,,, = 2 A and j,,,,, = tl-‘(21U)“*. This defines the initial kinetic energy Ekin = U - E,,. Since we wish to test small differences in the potential we used the same initial conditions in the two calculations. In order to test whether the classical trajectories are the same the classical quantity = N- ’ 5 ( &fz;; - @;:;‘t$’) i=l
(A&,)
w
point
C
--10
-15i-
-2c)-
J
10
I 20
I 30
I LO
-10 -
point
73 lic
--15
-
-to-
/
t Ilo-‘kc Fig. 2. Same as fig. 1 but with 11=1250
_E
tuying
turning &
(16)
was recorded. The number of trajectories N was set to 50 giving a standard deviation of about lo-20% upon the cross sections. Table 3 shows that the cross sections deviate more and more the
^o ci
I
I 50 t/lo-t4
set
Fig. 1. The probability for vibrational deexcitation In(P,,) of CO(u =l) colliding with 4He as a function of time. Dashed curve obtained using the analytical potential energy surface and the solid curve using a spline representation. The energy is U= 2000 cm-‘.
cm-‘.
smaller the cross section or probability. Thus transition probabilities of the order lo-’ deviate by a factor of four but probabilities of the order lo-‘* deviate by orders of magnitude! In order to understand this we follow the probability P, _0 as a function of time during the collision at two energies U = 2000 cm-’ (fig. 1) and U = 1250 cm-’ (fig. 2). We see that the probability increases to a rather large value at the turning point in both cases and that the two surfaces yield almost identical values at the turning points, namely P,, = 1.541 X 10e4 versus 1.527 X low4 at 2000 cm-’ and P,O = 5.831 x 10e5 versus 5.889 X lo-’ at U = 1250 cm-‘. However when the particles move away from each other we see that the probability is quenched but that this quenching is not nearly as effective on the spline fitted surface. This is most likely due to slight difference in asymmetry around the turning point on the two surfaces with the analytical surface being the most “symmetric”. This asymmetry of the spline surface prevents an effective quenching of the probability. The smaller the probability the more accurate must the potential be in order to obtain the correct asymptotic value. The difference in the average cross sections shown in table would give more than an order of magnitude difference for rates around lOPi9 cm3/s decreasing to a factor of two for rates around
G. D. Billing / Vibrational relaxation
44
ofCO
and N, colliding with 4He
Table 4 Cross sections qi _ a( E) (K) for vibrational relaxation of CO colliding with 4He at Eki, = 50 meV as a function of initial CO rotational state. N = 100 trajectories were used to obtain the V,R,T, values. The uncertainty is about 10% but the same trajectories were used in the runs with and without Coriolis coupling, i.e. the change in c is independent of the smd on the individual numbers j(h)
0 4 6 8 Fig. 3. Rate constants for vibrational deexcitation of CO( u = 1) colliding with 4He as a function of temperature. The upper curve obtained using the spline representation of the surface the lower using the analytical surface.
lo-” cm3/s as shown in fig. 3. Only for rates around or above lo-l6 cm3/s would we get better than 10% agreement. This result has some bearing upon recent comparison between semiclassical, VCC 10s and CS quantum calculations on CO relaxation by collisions with H, carried out in ref. [4]. In ref. [4] it was found that the CO vibrational rate obtained by the two methods deviated by about a factor of three. Apparently the same potential was used [14] but contrary to what was done in the semiclassical calculations carried out in ref. [15] Bacic et al. [4] expanded the CO anisotropy in Legendre polynomials. We have just shown that unless this expansion is done either
Table 5 Cross sections for vibrational relaxation (a,,,( U, T,)),
u (cm-‘)
(ero(U, (K)
T,))
872 1137 1672 1915 2650 3483 4313
9.7q-11)*’ 7.21( - 9) 1.39( - 6) 5.62(-6) 1.43( - 4) 1.4q-3) 1.25( - 2)
E b, (ev)
eic-o(E) (A’)
ALi (9
0.0119 0.0336 0.0881 0.115 0.200 0.300 0.400
2.84(- 10) a) 0.0135 2.33( - 9) 0.0115 1.18( - 7) 0.0218 3.48(-7) 0.0265 5.41( - 6) 0.0315 3.3q - 5) 0.0346
e,j_.,,(E)
eis+s(E) (AZ) 9.02( 4.91( 2.17( 3.62( 6.16( 5.17(
Semiclassical V,R,T,
Quantum CS
with Coriolis coupling
without Coriolis coupling
(ref. [71)
1.6(-8) a) 4.2( - 8) 4.2( - 8) 9.2( - 8)
3.6( - 8) 4.q - 8) 7.1( - 8)
2.6( 2.7( 3.q 3.4(
-
8) 8) 8) 8)
=) 1.6(-8)=1.6x10-‘.
analytically or if one uses spline fits more accurate than indicated by table 1 one cannot expect to obtain accurate cross sections at low energies. We therefore think that the deviation of a factor of three on the Hz-CO vibrational relaxation rate could be due to slight differencies in the potentials used. Such a deviation would also contradict previous comparisons between quantum and semiclassical theory (for a summary see ref. [16]) and recent derivations [13,17] of the semiclassical equations which show that the approach in the weak coupling limit with turning point dominating coupling is essentially exact. Despite the fact that the PES is represented in different ways we have shown in table 4 a comparison between quantum CS calculations from ref. [7] which uses an expanded version of the PES
and AE,,,(j=O, A&t (f)
- 10) ‘) 0.0125 - 9) 0.0126 -7) 0.0276 - 7) 0.0272 - 6) 0.0413 -5) 0.0591
5, 9) for “He-N,
crs+o(E) (AZ)
AE,i
Quantum(ref.
(f)
CSA
0.0081 0.0101 0.0184 0.0356 0.0440
8.29( 7.06( 1.92( 6.02( 7.25( 4.76( 1.89(
closed 8.03( 1.06( 3.57( 5.05( 3.47(
-
9) 7) 7) 6) 5)
‘) N - 50 trajectories and R, = 15 A. For the other energies N = 65 and R, = 10 A. (1 Z = 100 kJ/mol.) b, E = Eki, + E,,
151) e,,_,,(K) vcc
-
10) 9) 7) 7) 6) 5) 4)
10s
l.OO(- 10) 1.23( - 9) 4.84( - 8) 1.62( - 7) 2.40(-6) 1.88( - 5) 8.15( -5)
45
G.D. Billing / Vibrational relaxation of CO and N2 colliding with 4He
‘He+N2
Table 6 Vibrational relaxation rates k,,(T) for 4He-N, obtained by the semiclassical VgR,T, method, the quantum CS method and experiment
TW
ho(T)
(cm3/s)
semiclassical
quantum. (ref.
100 132 149 156 175 210 262 291
/
/
5.9( - 20) a) 1.8( - 19) 2.9( - 19) 3.1( - 19) 5.8( - 19) l.l(-18) 3.2(-18) 5.7(-l@
‘) Means 5.9
x
lO_“,
1511
5.6( 1.5( 2.q 2.9( 4.6( 9.9( 2.7( 4.3( uncertainty
20) 19) 19) 19) 19) 19) 18) 18)
experiment (ref. [ZO]) 3.2+1.5(-20) 1.6+0.15(-19) 2.7 &0.2( - 19) 3.3 f 0.3( - 19) 5.0+0.3( - 19) 1.2*0.05(-18) 3.2+0.1(-18) S.l+O.l(-18)
5-108.
L-
1o-zoI’
1o-21o Fig. 4. Rate constants for vibrational deexcitation of Nr( u = 1) colliding with 4He as a function of temperature: k,,(T) (solid line), k,,,,(T) (lower dashed line), k,,_,(T) (dotted line) and k ,9_o(T) (upper dashed line).
(see ref. [18]) and the V,R,T, semiclassical cross sections obtained using the analytical expressions (9) and (10). Since the quantum calculations neglected Coriolis coupling we also report the semiclassical cross sections obtained neglecting the Coriolis coupling (centrifugal distortion) term [9]. The numbers show that u,,,_,, is underestimated semiclassically but that aii_,, increase somewhat faster with j than the quantum CS values. The difference being a factor of 2 at j = 8. Since exactly the same PES surface was probably not used the comparison of such small cross section may however not be meaningful. In the order to support the above findings further we should perform a comparison between a quantum calculation - preferable a CS calculation and the V,R,T, calculation on the same analyticalpotential energy surface. CS calculations have recently been carried out by Banks et al. [5]. It was found that the CS cross sections were typically a factor of 5 larger than the VCC 10s values and
that uli _ e( E) increased with j. This latter result is due to the change in energy gap for the lj + 0 j + n transition and to the effect of Coriolis coupling. Similar results have previously been found for He-H, [S]. The strong j-dependence of the cross sections shows that the VCC 10s method cannot be used but also for j = 0 the 10s cross sections are underestimated (see table 5). We also expect the semiclassical V,R,T, method to be less accurate for j = 0 since rotation becomes more classical as j increases. Calculations using the program ADIAV [19] and the PES surface determined in ref. [5] were carried out and the results are in table 5 compared with CS and VCC 10s values for qo_,,(E). We see that the semiclassical values are somewhat smaller than the quantum CS but larger than the VCC 10s values. The table also shows that the cross section u,~_~(E) increases strongly with j although the initial kinetic energy decreases. This is in agreement with the CS calculations and table 6 shows that for the rotational averaged rate constant k , _ o(T) we obtain good agreement with the CS values in the whole temperature range. The reason being that the Boltzmann rotational distribution weights higher-j states for which the agreement is better than for j = 0. The semiclassical values lie lo-20% above the CS values and might if one can generalize from the He-H, system [l] even be in better agreement with the exact CC values.
46
G. D. Billing / VIbrational relaxation of CO and N2 colliding with ‘He
Acknowledgement This research was supported by The Danish Natural Science Research Council and EEC (grant no. STI-062-J-C(CD)). Drs. D.C. Clary and A.J. Banks are acknowledged for supplying the programs for the He-N, potential.
[7] [S] [9] [lo] [ll] [12]
References [l] P. McGuire and J.P. Toennies, J. Chem. Phys. 62 (1975) 4623. [2] R. Schinke and P. McGuire, Chem. Phys. 31 (1978) 391. [3] F.A. Gianturco, U.T. Lamanna and G. Petrella, Chem. Phys. 64 (1982) 299. (41 Z. Bacic, R. Schinke and G.H.F. Diercksen, J. Chem. Phys. 82 (1985) 236; 82 (1985) 245. [5] A.J. Banks, D.C. Clary and H.-J. Werner, J. Chem. Phys. 84 (1986) 3788. [6] G.D. Billing and D.C. Clary, Chem. Phys. Letters 90 (1982) 27;
1131 (141 [15] (161
[17] (181 [19] 1201
R.J. Price, DC. Clary and G.D. Billing, Chem. Phys. Letters 101 (1983) 269. R. Schinke and G.H.F. Diercksen, J. Chem. Phys. 83 (1985) 4516. G.D. Billing, Chem. Phys. 30 (1978) 387. G.D. Billing, Chem. Phys. 9 (1975) 359. G.D. Billing, Computer Phys. Rept. 1 (1984) 237. G.D. Billing and M. Cacciatore, Chem. Phys. Letters 86 (1982) 20. M. Cacciatore, M. Capitelh and G.D. Billing, Chem. Phys. 82 (1983) 1. G.D. Billing, J. Chem. Phys., to be published L.L. Paulsen, Chem. Phys. 68 (1982) 29. G.D. Billing and L.L. Poulsen, Chem. Phys. 70 (1982) 119. G.D. Billing, Molecular astrophysics, eds. G.H.F. Diercksen, W.F. Huebner and P.W. Langhoff (Reidel, Dordrecht, 1985) p. 517. J.T. Muckermann, unpublished results. R. Schinke, M. Meyer, U. Buck and G.H.F. Diercksen, J. Chem. Phys 80 (1984) 5518. G.D. Billing, Computer Phys. Commun. 32 (1984) 45. M.-M. Maricq, E.A. Gregory, C.T. Wickham-Jones, DC. Cartwright and C.J.S.M. Simpson, Chem. Phys. 75 (1983) 347.