J. Phys. Chem. 1982, 86, 1087-1096
1087
The Utility of the CS and 10s Approximations for Calculating Generalized Phenomenologlcal Cross Sections in Atom-Diatom Systems D. E. Fltr, D. J. Kourl,’ Department of Chemisby, Universny of Houston, Houston, Texas 77004
WPK. LIu, F. R. McCOutt, Departments of Chemistry and Applied Methematics, Unh’ersky of Waterloo, Waterbo, Ontarlo, Canada N2L 301
D. Evans, and D. K. Hoffman Department of Chemisby, Iowa State Universify, Ames. Iowa 5001 1 (Received: June 22, 1981; In Final Form: August 7, 1987)
The calculation of shear viscosity and thermal conductivitycoefficienta in the presence of a magnetic field requires the accurate calculation of several types of generalized phenomenological cross sections in which velocity and angular momentum tensors are coupled with the orbital and rotational motion of the system. These cross sections are then averaged over energy in a fashion appropriate for the phenomenon of interest. The coupled states (CS) and/or infinite order sudden (10s)approximationshave been used to calculate several such cross sections for systems such as He-HC1, He-CO, He-H2, HD-Ne, Ar-N2, and Ne-H2. Excellent results are obtained compared with close-coupledmethods for cross sections which are symmetric in tensor index, especially in the CS approximation, and these results are not very sensitive to the choice of orbital wave parameter. On the other hand, the cross sections which are asymmetric in tensor index are much more sensitive to interference effects and are unsatisfactory in many cases,
Introduction The advent of experimental techniques for measuring Senftleben-Beenakker (SBE)effects’ has recently led to the possibility of probing the nature of potential anisotropies of diatomic molecules interacting with noble gases at infinite dilution. Recently, expressions have been developed which relate the experimentallymeasurable effects to the highly averaged generalized phenomenological cross sections. These cross sections were expressed in terms of scattering matrices by using either the total J coupling scheme2 or the translational-internal coupling ~ c h e m e . ~ For most atom-diatom systems of interest, an exact close-coupled calculation of the S matrices at the required energies is either formidable or unfeasible. Therefore, expressions have been developed for the appropriate cross sections by using various approximate quantal methods such as the coupled states (CS)4and infinite order sudden (IOS)5approximations. The utility and accuracy of the approximate quantum methods to calculate the necessary effects is currently untested. It is thus the object of this paper to test the accuracy of these methods a t the most detailed level of application, namely, in the calculation of the specific phenomenological cross sections necessary to calculate any
number of experimentally measurable effects. In particular, we shall give comparisons between the CC and CS methods for several generalized phenomenological cross sections for He-H2, HD-Ne, He-HCl, and He-CO. Furthermore, we shall make a comparison between the CS and 10s methods for Ar-N2 and between the CC and 10s methods for N2-He.
Formalism The details of the formalism relating any given effect to the transport cross sections (bracket integrals), which are appropriate averages over specific types of phenomenological cross sections, are well documented elsewhere2 and will not be repeated here. Instead, we shall begin by noting that the expression for the generalized phenomenological cross section relevant to transport properties can be defined in terms of a generalized opacity by
where
J f
(1)J. J. M. Beenakker,G. Scolee,H.F. P. Knaap, and R. M. Jonkman, Phys. Lett., 2, 5 (1962). (2)W.-K. Liu, F. R. McCourt, D. E.Fitz, and D. J. Kouri, J. Chem. Phys., 71,415(1979);W.-K. Liu, F. R. McCourt, and W. E.Kiihler, ibid., 71,2566 (1979);W.-K. Liu and F. R. McCourt, ibid., 71,3750 (1979);D. E.Fitz, D. J. Kouri,D. Evans, and D. K. Hoffman,ibid., 74,5022(1981); W.-K. Liu, F. R. McCourt, D. E.Fitz, and D. J. Kouri, ibid., in press. (3)D. A. Coombe and R. F. Snider, J. Chem. Phys., 71,4284(1979); 72,2445(1980);R.F. Snider, D. A. Coombe, and M. G. Parvatiyar, ibid., 74,5572 (1981). (4)P. McGuire and D. J. Kouri, J. Chem. Phys., 60,2488(1974);R. T Pack, ibid., 60, 633 (1974). See also the review by D. J. Kouri, “Rotational Excitation II. ApproximationMethods” in “Atom-Molecule Collision Theory”, R. B. Bernstain, Ed., Plenum, NY,1979. (5)T. P. Tsien and R. T Pack, Chem. Phys. Lett., 6,54 (1970);8,579 (1971);T. P. Tsien, G. A. Parker, and R. T Pack, J. Chem.Phys., 59,5373 (1973);C. F. Curties, ibid., 48,1725 (1968);D.Secrest, ibid., 62,710 (1975). 0022-3654/82/2086-7087~07.2510
[ ( 2 4 + 1)(22j + 1)(21i’ + 1)(21/
+
l)]”’ (If.0
KI’ 0
0 x 1;)
In this expression, j , 1, J , and E are respectively the rotational, orbital, and total angular momentum quantum numbers and the total energy of the atom-diatom system; kj is the precollision wavenumber corresponding to the initial translational energy; unprimed and primed quantities are pre- and post-collision quantities; Kj, Kl, and K 0 1982 American Chemical Society
The Journal of Physical Chemistry, Vol. 86, No. 7, 1982
1088
Fitz et al.
Z
are the rotational, orbital (from velocity), and total tensor indicies; and S is the Arthurs-Dalgarno S matrixU6 The S matrix for the CC approach must be calculated by solving the close-coupled equations subject to the appropriate boundary conditions.6 For the CS method, this S matrix is related to the CS S matrix by the relationship2
S Jj ti q j l i -
X[(21j + 1)(21j'
jtj+lj$-*Tj
+
1 ) ] 1 / 2
x
Ai
where ti is some function of li and li' to be specified. Furthermore, the 10s S matrix is related to the CS S matrix2 by l.,hj
s ftj
Y
X
Flgure 1. Spacafixed axis system for atom-diatom collisions with arbitrary incident and scattered directions.
= Z ( - l ) A j [ ( 2 j + 1)(2j' + 1 ) ] ' / 2 X L;
v'-
where Liis restricted by j l I Li Ij' + j. Using these basic formulas we can calculate a variety of generalized phenomenological crms sections. The details of calculating CS and 10s S matrices are well documented elsewhere.' Before presenting the results of the calculations it is useful to describe the physical significance of eq 1 and 2. Although a clear picture of the general expression is not easy to grasp, there are certain special cases which may readily be visualized. The most straightforward case is when K j = K/ = Kl = K,' = K = 0. Then URj(0) = [(2j
+ 1)(2j'+
1)]-1/2"E(2J kj2iiu
In order to see physically to what this cross section corresponds, it is necussary to introduce a general form for the m-dependent differential cross section in which j and j ' are _quantizedwith respect to an arbitrary space fixed axis, r specififs the angles of the incident relative momentum, and R specifies the angles of the outgoing relative momentum (see Figure 1): darG'm '+jmlfi)
dR
kj'k = - r(j'mt+jmlfi)lz
S
(9)
The general scattering amplitude is defined within the Arthurs-Dalgarno6 phase convention by fr(.Ym'+jmIR) =
+ l)[S,jS,l, - ISJTtil(2](5)
If we note that the inelastic integral crms section is defined by where m, p, and M are the projections of j , I , and J on an arbitrary space-fixed z axis. If we define an integral by
QFj
then it is obvious that
(11) appropriate manipulation of 3 - j symbols and spherical harmonics leads directly to
and
A type of cross section which plays a dominant role in diffusion and viscosity phenomena results when we take Kl = Kl. = K and K j = Ky = 0. This case results from the existence of velocity gradienta in the bulk mixture. In this case 1 0KOKO .t.!
11 : j j
(K)=
(2K
+
X
l ) [ ( 2 j+ 1)(2j' + 1 ) ] 1 t 2
JiJf
(6) A. M. Arthurs and A. Dalgamo, R o c . R. SOC.London, Ser. A , 256, 540 (1960).
(7) D. E.Fitz, V. W e , and D. J. Kouri, Chem. Phys., 56,267 (1981); V. Khare, D.E. Fitz, and D. J. Kouri, J. Chem. Phys., 73, 2802 (1980).
Qf!j(K)= 4*(2K
+ 1)[(2j + 1)(2j'+
1)]1/2~$$0(K) j # j' (12)
where it is emphasized that this equality holds only for inelastic transitions. Physically, # O K 0 ( K ) is simply the weighted Kth moment of the degeneracy averaged differential cross section. Another type of cross section which is fundamental to NMR spin-pin and spin-rotation relaxation results when we take Kj = Ky = K and Kl = K1 = 0. In this case the generalized phenomenological cross section can be shown to reduce to
In order to see what this cross section represents, it is necessary to consider the general differential cross section given in eq 9. If instead of quantizing j and j ' along an arbitrary space-fmed z axis, we quantize j along some other
The Journal of Physical Chemistry, Vol. 86, No. 7, 1982 1089
Cross Sections in Atom-Diatom Systems
TABLE I: Cross Sections with Symmetric Rotational and Orbital Tensor Indicies for He-H, at E = 1520 cm-’ U.t?,’?.(l)
j ,j‘
191 1, 3 3, 3
cc
cs
4.18 0.134 (- 1) 4.83 u
..(3)
,3030 I . ,
0.???9.(2) I J :JJ
J J ;IJ
J I :II
cc
4.18 0.149 (- 1) 4.83
cs
3.27 0.819 (- 3) 3.66
?!?1..(1)
Uj??jj(
j J :JJ
j, j’
cc
cs
cc
1,1 1, 3 3,3
0.672 -0.466 (-1) 0.124
0.690 -0.419 (-1) 0.120
0.265 -0.177 (-1) 0.128
3.27 0.557 (-3) 3.66
cc
cs
2.62 0.816 (-4) 2.90 u
2)
2.62 0.411 (-4) 2.90
030?.(3)
j’j‘;~~
cs 0.270 -0.161 (-1) . . 0.128
cc
cs
o.oa
o*oa o.oa
0.oa 0.134
0.135
These are zero due to symmetry.
Z axis while quantizing j’ along yet another 2’ axis, the resulting differential cross section is given by
M’q’-j4@ %ifi(a,&y)12 (14) The angles (a,p,y) and (a’,P,y’) represent the Euler angles in the convention of Edmondss required to rotate from the x,y,z axis system to the X,Y,Z and X’,Y’,Z’axis systems, respectively. Let a,/3,i. represent the Euler angles necessary to rotate between the X,Y,Z and X’,Y’,Z’axis systems. It then follows that PK(C0S
B) = %(&B,i.)
=
c8 HQ(a,B,r)qo(.’,8’,Y’)
(15)
If we define the following integral cross section where j and j‘ are quantized as in eq 14:
then we can define another integral of the type 9Fj(IO = (87r2)-2Jd91d9’ uG’fi-jfi)
PK(cos
a)
(17)
where 9 = (cu,j3,y) y d 9’ = (cu’,P’,r’).After some manipulation of Wigner 3j symbols and rotation matrices, eq 16 can be shown to reduce to
for inelastic cross sections. If we take the inverse of eq 18 using the orthogonality properties of the Wigner 3j symbols,it becomes clear that the aOKOK(K)-type cross sections are particular momenta of eq 17 which is the Kth moment of the integral Am-dependent cross section averaged over all possible orientations of the j and j’quantization axes.
Calculations The potentials used here are well documented elsewhere in the literature and will not be repeated. Instead, a brief description of the potentials will be given with appropriate references. (8)A. R. Edmonds, *Angular Momentum in Quantum Mechanics”, Princeton University Press, Princeton, NJ, 1960. (9)U. Buck, F. Huisken, J. Schleusener, and J. Schifer, J. Chem. Phys., 72,1512 (1980). (10)5.Green and P. Thaddeus, Astophys. J., 206,766 (1976). (11)M. D. Pattengill, R. A. LaBudde, R. B. Bernstein, and C. F. Curtiss, J. Chem. Phys., 66,5517 (1971).
This potentials has an exponential repulsive wall with R-8, R-*, and R-’O long-range terms and Pl annd P2 anisotropies. Its isotropic well depth is about 25 cm-l. He-CO This potential’O has anisotropies through P5and has an isotropic well of about 23 cm-’. This potential” is a standard 6-12 LennardAr-N2 Jones type with a P2 anisotropy. The isotropic well depth is about 84 cm-’. He-H2 The form of this potential which we used differs slightly from ita original form12in that the P, R4, and R-l0 terms are not cut off as R 0. The anisotropic well depth is about 10 cm-’. He-HC1 This potential13has anisotropies through P.and has an isotropic depth of about 48 cm-’. N2-He This potential“ has an exponential dependence at close range fitted by a cubic spline to R4 long-range dependence and has a P2 anisotropy. Ita isotropic well depth is about 20 cm-‘. All CS calculations and the CC calculations except for He-HC1 and He-CO were done with the noniterative Volterra equation method.I6 The CC S matrices for He-HC1 and H e 4 0 had been calculated earlier by Green and Thaddeus’OJ6 and Green and Monchick,13J6respectively. The 10s S matrices were calculated by using Gauss-Mehler quadrature to calculate the WKB phase shift.4 The calculations for HD-Ne, He-CO, He-HC1, He-H2, Ar-N2, and N2-He were done at total energies of E = 254, 80, 200, 1520, 124.7, and 50 cm-’, respectively. Most of the S matrices were converged to at least three significant figures. The rotational constants which were used for H2, HD, N2, CO, and HC1 were 59.3,45.655,2.01, 1.93, and 10.59 cm-’, respectively. The maximum values of J necessary to obtain converged cross sections for all but the elastic imaginary cross sections for H2-He, HD-Ne, Ar-N2, He-CO, He-HC1, and N2-He were 54,40,150,30, 38, and 32, respectively. The CS and 10s calculations were done by using the 1 average, i.e. 2 = 1 / 2 ( 1 + 19, labeling scheme only. The 10scalculations for Ar-N2 were done by using h = ko and cros-s sections were calculated by using k? in eq 1replaced by k 2 in eq 1. On the other hand, the 10s calculations for N2-He were done by using the symHD-Ne
-
(12)M. Jacobs and J. Reuss, Chem. Phys., 61,427 (1980). (13)S.Green and L. Monchick, J. Chem. Phys., 63,4198 (1975). (14)W.-K. Liu, F. R. McCourt, D. E. Fitz, and D. J. Kouri, J. Chem. Phys., in press. (15)W. N. Sams and D. J. Kouri, J. Chem. Phys., 61, 4809, 4815 (1970);62,4144(1970);63,496(1970);D.Secrest in “Methods of Computational Physics”, Vol. 10,B. Alder, S.Fernbach, and M. Rotenberg, Ed., Academic Press, NY, 1971;R. A. White and E. F. Hayes, Chem. Phys. Lett., 14,98(1972);R. A. White and E. F. Hayes, J. Chem. Phys., 67,2985 (1972);E.R. Smith and R. J. W. Henry, Phys. Reu. A, 7,1585 (1973). (16)S. Green, private communication.
I090
The Journal of Physical Chemistty, Vol. 86, No. 7, 1982
Fitz et al.
TABLE 11: Cross Sections with Symmetric Rotational and Orbital Tensor Indicies for He-H, a t E = 1520 cm-' u ,f!f!l..(l)
u aooo ( 0 )
uj?;;j(
I I .I1
i'j';jj
2)
j,j'
cc
cs
cc
cs
cc
cs
1,1 1, 3 3, 3
0.284 -0.186 0.211
0.263 -0.169 0.196
1.26 0.633 (-2) 1.57
1.28 0.667 (-2) 1.59
0.641 -0.320 (-3) 0.729
0.641 -0.780 (-4) 0.729
u
j ,j'
1,1 1,3 373
u
!2??.(1)
1*1?.(
j'j';ii
j j ;JI
cc
0.790 0.466 (- 3) 0.974
UjJf'ij(3)
cc
cs
0.865 0.235 (- 2) 0.940
0.860 0.230 (- 2) 0.947
cs
0.762 0.402 (-3) 0.981
2)
cc
cs
0.825 0.817 (-3) 0.967
0.815 0.112 (-2) 0.965
TABLE 111: Cross Sections Not Symmetric in Tensor Indicies for He-H, a t E = 1520 cm-' ioI.!I!:.? 8 1 1.(1)
uj '0*99.(2) j';11
j,j' 191
1,3 3, 3
cc
cc
cs
cc
cs
0.396 0.344 (- 2) 0.137
0.406 0.417 (- 2) 0.133
-0.181 - 0.326 (- 2) -0.592 (-1)
- 0.182 - 0.187 (- 2) -0.605 (-1)
cs
-0.351 (-2) 0.903 (-2) -0.620 (-2)
-0.386 (-2) 0.634 (- 2) -0.678 (-2)
iUj!??.(2) I I :I1
TABLE IV: Cross Sections with Pure Orbital Coupling for He-CO at E = 80 cm-I ujJ;;;j(l)
u3???.(3)
uj'j9;;j(2)
I I 21
cs
cc
i,j '
cc
cs
cc
cs
0, 0 0, 1 0, 2 0, 3 0, 4 0, 5
12.88 -0.678 -0.798 0.222 (- 1) -0.915 (-2) 0.106 (- 1)
13.48 -0.997 - 0.709 0.349 (-1) -0.440 (-1) 0.241 (-1)
9.20 -0.262 -0.131 0.436 (- 2) -0.187 (-2) -0.396 (-1)
9.55 -0.411 -0.981 (-1) 0.145 (-1) -0.144 (-1) -0.508 (-2)
7.74 - 0.1 25 -0.159 0.289 -0.231 -0.107
(-1) (- 2) (-2) (-2)
7.95 -0.196 -0.847 0.475 0.908 -0.367
1,1 1, 2 1,3 194 1, 5
12.40 -0.553 -0.727 0.598 (-2) -0.110 (-1)
12.88 -0.757 - 0.666 0.577 (-1) -0.394 (-1)
9.32 -0.213 -0.105 0.519 (-3) -0.159 (-1)
9.59 - 0.288 -0.105 -0.666 (-2) -0.369 (-1)
7.99 -0.947 -0.126 -0.777 0.698
(-1) (-1) (-3) (-3)
8.14 -0.135 -0.370 (-1) 0.575 (-2) -0.799 (-3)
2, 2 2, 3 2, 4 2, 5
13.06 -0.537 -0.635 -0.417 (-2)
13.29 -0.519 -0.822 0.414 (-1)
9.97 - 0.200 -0.158 -0.892 (-2)
10.02 -0.180 -0.223 -0.122 (-1)
8.61 - 0.731 (- 1) -0.477 (-1) -0.146 (-1)
8.59 -0.584 (- 1) -0.107 (-1) -0.122 (-1)
3, 3 3,4 3, 5
14.12 -0.502 -0.579
13.91 - 0.287 -0.789
11.12 -0.113 -0.216
11.00 -0.122 - 0.348
9.78 - 0.257 (- 1) -0.616 (-1)
9.67 -0.479 (- 1) -0.128
4,4 4, 5
16.50 -0.326
15.27 -0.907 (- 1)
13.59 -0.236 (- 1)
12.80 0.259 (-1)
12.20 -0.534 (-1)
11.84 0.166 (- 2)
5. 5
26.46
27.18
20.58
20.69
16.99
16.96
(-2) (- 2) (- 2) (-3)
TABLE V : Cross Sections with Pure Rotational Coupling . -for He-CO a t E = 80 cm" 0101
uJ?go?.( 2) J :JJ
. . ..
j , i'
u ??to:.(
. . ..
cc
cs
cc
cs
11.66 -0.947 -1.30 - 0.239 -0.271
11.83 -0.996 -1.38 -0.226 -0.261
6.30 -0.505 -0.642 - 1.05 -0.112
6.35 -0.553 -0.613 -1.01 -0.111
10.44 -1.25 - 1.90 -0.402
10.33 -1.23 -2.10 -0.352
6.61 -0.604 - 0.840 -0.172
10.01 -1.50 -2.13
9.72 -1.24 -2.41 8.01
9.07
- 1.95
-1.12
11.01
10.82
3)
J. J. : J_. I
cc
cs
6.64 -0.586 -0.987 -0.163
5.05 -0.305 -0.394 -0.783 (-1)
5.03 -0.305 -0.479 -0.735 (- 1)
6.48 -0.736 -1.04
6.21 -0.632 -1.22
4.92 -0.414 -0.573
4.74 -0.363 - 0.694
6.02 -1.03
- 0.580
5.18
4.69 -0.621
3.98 -0.350
6.96
5.38
5.32
7.05
Cross Sections in Atom-Diatom Systems
The Journal of phvsical Chemistry, Vol. 86, No. 7, 1982 1091
TABLE VI: Cross Sections with Symmetric Orbital and Rotational Coupling for He-CO at E u!t?/2..(
I I :II
j,j'
cc
cs
1, 1 1, 2 1, 3 1,4 1, 5
2.38 -0.804 (-1) -0.217 -0.194 (-1) -0.288 (-1)
2.42 -0.109 -0.174 -0.102 (-1) -0.266 (-1)
2.89 -0.467 0.810 0.190 0.261
2, 2, 2, 2,
2 3 4 5
2.85 -0.108 -0.171 -0.176 (-1)
2.86 -0.111 -0.200 -0.159 (-1)
393 3, 4 3, 5
3.11 -0.936 (-1) -0.138
494 4, 5
3.59 -0.898 (-1)
5, 5
5.86
= 80 cm-' 0 .! 7,'?
2)
cc
,( 3)
I I .II
cs
cc
__
cs
(-1) (-1) (-1) (-1)
2.52 -0.781 (-1) -0.114 -0.499 ( - 2 ) -0.850 (-2)
2.61 -0.964 - 0.966 0.124 -0.101
2.71 -0.722 (-1) -0.179 (-1) 0.115 (-1)
2.66 -0.731 (-1) -0.409 (-1) 0.201 (- 1)
2.72 -0.105 -0.109 - 0.353 (- 2 )
2.79 -0.974 (-1) -0.143 - 0.303 (- 2)
3.03 -0.827 (-1) -0.204
2.73 -0.875 (-1) -0.531 (-1)
2.64 -0.382 (-1) -0.655 (-1)
2.91 -0.903 (-1) -0.104
2.89 -0.668 (-1) -0.155
3.38 -0.438 (-1)
3.09 -0.460 (-1)
2.83 -0.526 (-2)
3.47 -0.485 (-1)
3.19 -0.300 (-1)
5.90
4.84
(-1) (-1) (-1) (- 1)
2.96 -0.730 0.422 0.187 0.198
4.97
TABLE VII: Cross Sections with Asymmetric Orbital and Rotational Coupling for He-CO at E = 80 cm-' u .?f??.( 2 ) 0 ? ? ? Z . ( 2) II I J ;II
5.48
(-1) (- 1)
(-3) (-1)
5.57
= Kl?= 0; and (e) asymmetric in rotational coupling but symmetric in orbital coupling, Kj # K,,and K1 = K1, # 0. Of these classifications, cross sections of types (a)-(d) are purely real while type (e) is purely imaginary. j , j' cc cs cc cs Our discussion begins with cross sections of type (a) which are shown in Tables I, IV, IX, XI, XVI, and XIX. 0, 0 0, 1 -0.458 (-1) -0.114 A careful examination of the data reveals that the CS 0, 2 0.307 (-1) -0.246 (-2) elastic results are in excellent agreement with the CC re0, 3 0.617 (-1) 0.368 (-1) sults for He-H2, HD-Ne, and He-HC1 with very good 0, 4 0.660 (-1) 0.211 (-1) agreement for He-CO. The inelastic cross sections are not 0, 5 0.200 (-1) 0.104 (-1) quite so good as the elastic ones for these systems but are 1, 1 0.129 0.237 (-1) 0.129 0.239 (- 1) none-the-less quite reasonable for most transitions. There 1, 2 0.545 (-1) -0.623 (-2) 0.643 (-1) 0.587 (-1) are a few cases where the agreement is not good such as 1, 3 0.879 (-1) 0.631 (-2) 0.467 (-2) -0.529 (-1) the 0 4 transition of a3030(3)for He-CO in Table IV and 1 , 4 0.666 (-1) 0.374 (-1) 0.825 (-3) -0.279 (-2) the 0 3 transition of u3030(3)for He-HC1 in Table XI. 1, 5 0.700 (-1) 0.180 (-1) -0.111 (-2) -0.194 (-1) There are two possible explanations for this disagreement. 2, 2 0.128 -0.467 (-2) 0.128 -0.467 (-2) First, it is possible that there may be some orbiting col2, 3 0.692 (-1) 0.804 ( - 2 ) 0.276 (-1) 0.166 (-1) lisions for higher rotational states of HC1 and CO at the 2 , 4 0.945 (-1) -0.154 (-1) -0.338 (-1) -0.629 (-1) energies studied. The cross sections in question are quite 2, 5 0.653 (-1) 0.306 (-1) -0.109 (-2) -0.777 ( - 2 ) small to begin with and are subject to interference effects, 3, 3 0.468 (-1) -0.549 (-1) 0.468 (-1) -0.549 (-1) and the CS approximation is not valid when there is or3, 4 0.822 (-1) 0.142 (-1) 0.185 (-1) -0.413 (-2) biting. The second possible reason for disagreement could 3, 5 0.119 -0.166 (-1) -0.101 (-1) -0.642 (-1) involve the fact that the S matrices calculated were not 4, 4 -0.330 (-1) -0.998 (-1) -0.330 (-1) -0.998 (-1) of sufficient accuracy to correctly predict cross sections 4, 5 0.840 (-1) 0.893 (-2) -0.363 (-1) -0.219 (-1) where there may be considerable interference effects. This point will be discussed more later. The comparison be5, 5 -0.358 (-1) -0.371 -0.357 (-1) -0.371 tween CS and 10scross sections of type (a) for Ar-N2 in Table XVI and between the CC and 10s cross sections of metric choice of k = [(k? + k/2)/2]1/2 and by using the type (a) for N2-He is also reasonably good. correct k? in eq 1. Cross sections of type (b) are shown in Tables I, V, IX, Results and Discussion XII, and XM. The CS and CC results for HeH,, HeCO, The results have been divided into three basic groups. and H e H C l compare quite favorably for almost all of the In Tables I-XV results are presented where CC and CS(lnv) transitions. The results for HD-Ne in Table IX are cross sections are compared for He-H,, He-CO, He-HC1, somewhat disappointing, though. This is probably due to and HD-Ne. In Tables XVI-XVIII a comparison between the low energy of that system. In Table XIX good agreement is seen between the CC and 10s results for CS(1,) and IOS(1,) results for several types of cross sections is given for Ar-N2. In Table XIX some limited reN2-He. sults comparing the CC and IOS(lav)methods for N2-He In Tables 11, VI, IX, XIII, and XVII cross sections of are given. type (c) are given. The comparison between the CC and Since the results are rather voluminous it is convenient CS results for He-H,, He-CO, and He-HC1 is excellent for all elastic transitions and quite good for most of the to divide the discussions of them into several groups. inelastic transitions. The results for HD-Ne are similar These groups will be for cross sections which are (a) symmetric in purely orbital coupling, Kl = Kl, # 0 and K . = to the type (b) cross sections. The CS and 10s cross sections for Ar-N2 in Table XVII are quite good except KY = 0; (b) symmetric in purely rotational coupling, = KY # 0 and Kl= Kl, = 0; (c) symmetric in both rotational for the higher transitions where the energy sudden apand orbital coupling, Kj = KY # 0 and K I = Klt # 0; (d) proximation breaks down. asymmetric in rotational and orbital coupling, in particular The results for cross sections of type (d) are given in Tables 111, VII, X, XIV, XVI, and XIX. Only the CC and Kj = K1, = 2 while KY = K1 = 0 or KY = KI = 2 while Kj
--
4.
Fitr et ai.
1092 The Journal of Physical Chemistry, Vol. 86, No. 7, 1982
TABLE VIII: Purely Imaginary Sections with Asymmetric Orbital Coupling for He-CO at E = 80 cm-' io f !>?.(1) iu .t !??.(1) i o .! !:2. .( 2 ) I I ;I1 I I :I1 J I ill
_ _ I _ _ _ _ _
CC"
j , j'
0, 0 0,1 0, 2 0,3 0, 4 0, 5
0.0 0.0 0.0 0.0
1, 1 1, 2 1,3 1,4 1, 5 2, 2 2, 3 2,4 2, 5
-0.235' -0.487 -0.459 -0.432 -0.177
csa
0.0 0.0
csa
CC"
cc
cs
0.0" 0.393 (- 1) 0.783 (-1) 0.393 (-1) 0.482 (- 1) 0.990 (- 1)
0.0"
0.0
0.0
0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.459 (-1)' 0.518 (-1) 0.161 0.465 (- 1) 0.391 (-1)
- 0.169' 0.675 (- 1) 0.122 0.786 (-1) 0.161 (-1)
- 0.018
0.0 0.0 0.0 0.0
0.0
'
(-1) (-1) (-2) (-2)
-0.332 -0.232 -0.721 -0.208 -0.175
(- 1)' (- 1)
(-1) (-1) (-1)
0.356' 0.115 0.146 0.398 (- 1) 0.118 (-1)
-0.229' -0.109 (-1) 0.110 (-1) -0.254 (-2)
-0.484 -0.258 -0.549 -0.130
(-1)' (-1) (-1) (-1)
0.204' 0.950 (- 1) 0.972 (-1) 0.192 (-1)
0.654 (- 1)' 0.576 (-1) 0.123 0.291 (-1)
-0.267' 0.145 0.218 0.324 (-1)
-0.029' 0.0 0.0 0.0
3, 3 3, 4 3, 5
-0.198' 0.340 (-1) -0.344 (-2)
-0.328 (-1)' -0.270 (-1) -0.383 (-1)
0.125' 0.819 (- 1) 0.409 (- 1)
0.216 (-1)' 0.604 (-1) 0.856 (-1)
-0.261' 0.221 0.582 (-1)
-0.035' 0.0 0.0
4,4 4,5
0.421 (- 1)' 0.916 (- 2)
-0.350 (-1)' -0.293 (-1)
0.221' 0.608 (- 1)
0.229 (- 1)' 0.655 (- 1)
0.273' 0.901 (-1)
-0.039' 0.0
5, 5
-0.469 (- 2)'
0.163 (-1)'
0.127'
- 0.068'
-0.149'
0.165'
0.0 0.0 0.0 0.0
~ ( the 1 )j,f~ = 0 vanishes due to symmetry. The cross sections with j = 0 vanish due to symmetry. For ~ ~ ~ ' only ' These cross sections have not converged but are becoming more positive with increasing partial waves. a
TABLE IX: Cross Sections with Symmetric Rotational and Orbital Coupling for HD-Ne at E = 254 cm-' u .????.(2) 0 ?40° ( 0 ) u j y ; j ( 1) j , j'
cc
cs
CC"
CS"
CC"
CS"
0, 0
6.38 -3.68 3.27
6.25 -3.60 3.21
0.0 0.0 2.00
0.0 0.0 2.43
0.0 0.0 1.15
0.0
0, 1 1.1
uj'p';gj(2)
U.tvo.(l)
--___1 I
j ,j'
:JJ
CC
CS
CC
CS
cc
cs
6.69 0.102 7.86
5.08 0.013 5.83
5.09 0.008 5.79
4.05 -0.0053 4.58
4.05 -0.0050 4.63
u,!?::.(l)
cc
j ,j'
>
u f,' 2. .( 3)
ul'i'.;.( 2) 3
J J :I1 _____
J
I I :I1
cs
cc
cs
1.44
1.48
1.69
cc
cs
1.85
1.53
HD-Ne
i
a
0
-0 5-
ujyt3;j(3)
6.74 0.097 7.72
_____.
0, 0 0, 1 1, 1
0.0 1.16
-cc
0.5-
I J :I1
I j :jj
- 1 01
0
\
1
CJ1110111
/
I
I
1
I
I
1
5
10
15
20
25
30
35
J
-
+
Flgure 2. Comparison of CC and CS(I,,) ( 2 4 1)-weighted opacities for the 1 1 transition of the C T ' " ~ ( ~cross ) section for HD-Ne at E = 254 cm-'.
l__l
1, 1
0.844
0 ??lo: .(2 ) I I :I1
u ?!?! .(1 ) I I :JJ
j ,j'
1,1 a
cc
cs
cc
cs
1.76
1.37
1.25
0.75
The cross sections for j = 0 vanish due to symmetry.
TABLE X : Cross Sections Not Symmetric in Tensor Indicies for HD-Ne at E = 254 cm-' io j 1''!.(2) ' j ' : ~ ~
CC
j,j'
CS
CC"
CSa
cca
csa
0, 1
0.28 0.14 0.0 0.0 0.0 0.0 1, 1 -0.20 -0.09 0.219 0.136 0.666 (-1) 0.631 (-1) a
The cross section for j = 0 vanishes due to symmetry
CS results for He-H2 in Table 111 and for He-HC1 in Table XIV agree favorably. The remaining systems give agreement which is mediocre at best and poor in the comparison between CC and 1 0 s for N2-He. In these last results, however, one should bear in mind that the total energy is sufficiently low that one should not expect reliable results for the 10s approximation.
Purely imaginary cross sections of type (e) are shown in Tables 111, VIII, X, XV, and XVIII. The agreement between CC and CS cross sections for He-H2 in Table 111, HD-Ne in Table X, and He-HC1 in Table XV is generally excellent for P 2 ( 1 )and ul1l2(2)type cross sections. H e CO gives only good agreement in scattered cases. The comparison between CS and 1 0 s results for Ar-N2 in Table XVIII is also quite reasonable. For cross sections of the type d1l0(l), the CS approximation fails since the results when converged with respect to J go to zero. For the CS(I) approximation one can rigorously show that crow sections of this type vanish. The vanishing of the u"lo(l) cross sections for the CS(1,J approximation is apparently a truly dynamical effect. To illustrate this point the (Wi 1)-weightedopacity for the 1 1 transition of HD-Ne at E = 254 cm-' is shown in Figure 2. Although the CC results do not exactly cancel, the CS results do if carried to large enough values of Ji. Further insight can be gained concerning why certain types of cross sections can be more reliably calculated by the CS approximation than others by examining the behavior of the (Wi + l)-weighted opacity as a function of Ji for various types of cross sections. In Figures 3-8 are
-
+
Cross Sections in Atom-Diatom Systems
The Journal of Physlcal Chemistry, Vol. 86, No. 7, 7982 1093
TABLE XI: Cross Sections with Purely Orbital Coupling for He-HCl at E = 200 cm-' ..
. . ...
j , j'
cc
cs
cc
01 0 0, 1 0, 2 0, 3
9.00 -0.809 (-1) 0.322 (-1) -0.377 (-1)
8.95 -0.586 (-1) 0.183 (-1) -0.394 (-1)
6.91 0.499 (-2) -0.300 (-2) 0 4 2 6 (-1)
1 9 1 1, 2
1,3
9.42 -0.121 0.233 (-1)
9.38 -0.110 0.117 (-2)
7.31 -0.785 (-2) -0.165 (-1)
2, 2 2, 3
10.56 -0.133
10.58 -0.252
3, 3
14.87
14.65
.
cs
...
cs
5.84 0.251 (-2) -0.404 (-2) -0.359 (-2)
5.84 -0.514 (-2) -0.489 (-2) 0.115 (-2)
7.28 0.990 (-2) -0.106 (-1)
6.22 -0.572 (-2) -0.318 (-2)
6.20 -0.113 (-1) 0.620 (-2)
8.48 -0.812 (-1)
8.46 -0.575 (-1)
7.31 -0.171 (-1)
7.28 0.968 (-2)
12.40
12.39
10.75
6.87 0.664 (- 2) 0.102 (-1) 0.231 (-2)
1 1 .
10.78
U.J
cc --- cs -
0.4 -
He-CO
5
-u
0.2-
L
6
8
10 12
1L 16 18 20 22
2020 (21
261
V
2
- cc
E = 80crn-'
0.3-
0
~
cc
0
L
2
6
10 12 1L
8
J
-
+
Figure 3. Comparison of CC and CS(l,) (2JI lbweighted opacities for the 1 1 transition of the uoZo2(2) cross section for HD-Ne at E = 254 cm-I.
E = 80 cm-'
2-0.L-
-u
20 22
+
Figure 6. Comparison of CC and CS(l,) (2JI lbweighted opacities for the 1 2 transition of the aZom(2)cross section for HD-Ne at E = 80 cm-'. +
- cc cs
He-CO 0.5
16 18
J
I'
0202 121
',
0.3
- cc - - - cs
HD-Ne
---
E = 25L cm-'
- 0 0220(21
0.2-0.1 0
2
L
6
8
12 1L 16 18 20 22
10
'C
+
Figure 4. Comparison of CC and CS(,l) (24 lbweighted opacities for the 1 2 transition of the uom2(2)cross section for He-CO at E = 80 cm-'.
6
0
J
L
2
6
J
-
16 18 20 22
+
Figure 7. Comparison of CC and CS(I,) (2J, lbweighted opacities for the 1 1 transition of the uozZo cross section for HD-Ne at E = 254 cm-I. 0.16 -
i
10 12 I C
8
,' 1
He-CO
f
,
I
,
A
0.08 -
>r
.-
-u
c
0220 12)
V
0
Q
0
a
0
0-
-0.08I
0
-
2
L
6
8
10 12
1L 16 18 20 22
J
+
Figure 5. Comparison of CC and CS(,l) (2Jl lkweighted opacities for the 1 1 transition of the umm(2)cross section for HD-Ne at E = 254 cm-'.
0
-
2
,
I
L
,
,
6
I
I
8
,
I
I
I
,
I
10 12 14
,
I
16
I
I
I
I
L
18 20 22
J
+
Figure 8. Comparison of CC and CS(I,) ( 2 4 1)-welghted opacities for the 1 2 transition of the uoZ2O(2) cross section for HD-Ne at E = 80 cm-'.
1094
The Journal of Physical Chemistry, Vol. 86, No. 7, 1982
Fitz et al.
TABLE XII: Cross Sections with Pure Rotational Coudine for He-HC1 at E = 200 cm-'
cs
cc
cs
cca
CS"
1,1 1,2 1,3
~2.85 -0.498 -0.952 (-1)
2.80 -0.474 -0.842 (-1)
1.58 -0.286 -0.748 (-1)
1.55 - 0.282 -0.692 (-1)
0.00 0.00 0.00
0.00 0.00 0.00
2, 2 2, 3
3.09 -0.826
3.02 -0.774
1.88 --0.390
1.84 -0.368
1.43 -0.195
1.39 -0.182
3, 3
3.03
3.12
1.99
1.96
1.53
1.53
cc
j , j'
a
The cross sections with j = 1 vanish due t o symmetry.
TABLE XIII: Cross Sections with Symmetric Rotational and Orbital Coupling for He-HCl a t E = 200 cm-' I I ;I1
cc
cs
1,1 1,2 1,3
1.64 0.162 (-1) 0.121 (-1)
1.65 0.257 (-1) 0.105 (-1)
2, 2 2, 3
2.22 0.284 (-1)
2.23 0.338 (-1)
3, 3
3.19
3.10
j , j' -
cc
cs
2.11 -0.390 (-1) -0.800 (-2)
2.10 -0.434 (-1) -0.960 (-2)
1.82 -0.364 (-3) 0.461 (-2)
1.81 0.102 (-2) 0.330 (- 2)
1.98 --0.351 (-2)
2.00 -0.183 (-1)
2.14 -0.180 (-1)
2.15 -0,283 (-1)
2.76
2.72
2.97 u .7
cc
cs
cc
131 1, 2 1, 3
2.80 0.259 (-1) 0.102 (-1)
2.81 0.328 (-1) 0.106 (-1)
1.47 -0.131 (-1) -0.524 (-2)
1.46 -0.120 (-1) -0.550 (-2)
2, 2 2, 3
3.16 0.557 (-2)
3.21 0.106 (-1)
1.71 -0.154 (-1)
1.70 -0.133 (-1)
3, 3
4.51
4.47
2.49
.(3)
I I :I1
I I :I1
cs _-__-_____-
I _ _
3.03
u .???2..( 2)
I j :]I ___-__-_-
cc
I I ;I1
cs
cc____
u .!l t l ? .(1 )
j , j'
u .t?!I 2. .(3)
u .!???.(2)
u j ! y j (1)
______
cs
0.0 0.0 0.0
0.0 0.0 0.0
1.06 -0.101 (-2)
2.51
1.05 0.133 (-3)
1.54
1.54
" The cross sections with j = 1 vanish due t o symmetry TABLE XIV: Cross Sections with Asvmmetric Orbital and Rotational Coupling for He-HCl at E = 200 cm-'
j , j'
___
0, 0, 0, 0,
0.0 0.644 (-1) 0.474 (-1) 0.225 (-1)
1 2 3
CC" _
_
_
CSa
~
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
0.0
0.565 (- 1) 0.306 (- 1) 0.123 (- 2)
1,1 1, 2
- 0.161 0.624 (-1) 0.539 (-1)
- 0.180 0.437 (-1) 0.213 (-1)
- 0.161 0.197 (-1) 0.232 (-1)
- 0.180 0.335 (-1) 0.191 (-1)
2, 2 2, 3
-
0.132 0.914 (- 1)
- 0.142 0.130 (- 1)
- 0.132 0.320 (- 1)
- 0.142 0.209 (- 2)
1, 3
a
cs
cc
_______________.________-_
0
-0.976 (-- 1) - 0.210 3, 3 The cross section with j = 0 vanishes due t o symmetry.
-
- 0.210
0.976 (- 1)
TABLE X V : Purely Imaginary Cross Sections with Asymmetric Rotational Coupling for He-HCl at E = 200 cm-'
i ,i'
CC" 0.0 0.0 0.0 0.0
cca
CS" 0.0 0.0 0.0 0.0
CSa
cc o.oa
cs 0.0a
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
0.153 (-1) -0.362 (-2) -0.125 (-1)
0.0 0.0 0.0
-0.760 (-l)b -0.292 (-1) -0.118 (-1)
-0.626 ( - l ) b -0.228 (-1) -0.917 (-2)
0.101b 0.71 2 (- 1) 0.245 (-1)
0.126b 0.509 (-1) 0.205 (- 1)
-0.603 (-l)b 0.162 (-1) -0.501 (-2)
0.0 0.0 0.0
-0.760 ( - l ) b -0.229 (-1)
-0.504 ( - l ) b -0.217 (-1)
0.817 ( - l ) b 0.624 (- 1)
0.936 (-l)b 0.486 (- 1)
-0.656 ( - l ) b 0.122 (-1)
0.0 0 .o
-0.510 ( - l ) b
-0.676 ( - l ) b
0.16Zb
0.131b
0.419 (-l)b
0.0
The cross sections with j = 0 vanish due t o symmetry. For u " ' O ( l ) , only that for j = j' = 0 vanishes due to symmetry. These cross sections are not converged with respect t o the number of partial waves. a
Cross Sections in Atom-Diatom Systems
The Journal of Fhysical Chemistry, Vol. 86, No. 7, 1982 1095
TABLE XVI: Several Types of Cross Sections for Ar-N, at E = 124.7 em-' u .f ?! ? .( 1) I I ;I1
u ??1?.( 2) I J ;I1
uI.?9? J VI.(2)
cs
10s
cs
10s
-1.09 0.984 (-2)
23.4 -1.32 -0.177 -0.103 (-1)
23.6 -1.36 -0.329 -0.249 (-1)
-0.442 0.428 (-1) 0.936 (-1)
0.0' -0.456 0.115 (-2) 0.540 (-1)
29.7 -3.34 -0.541
29.8 -3.31 -0.240
22.4 -1.24 -0.157 (-1)
22.9 -1.20 -0.132 (-1)
0.182 -0.191 0.946 (-1)
0.161 -0.186 0.968 (-1)
4,4 4, 6
30.2 -3.22
35.0 -1.61
22.5 -1.21
23.6 -0.63
0.219 -0.217
0.885 (-1) -0.226 (-1)
6, 6
30.3
44.7
22.5
28.4
cs
i,j'
10s
0,o 0, 2 0,4 096
32.5 -3.44 -0.723 0.362 (-1)
2, 2 2,4 2, 6
" The cross section for j = j'
32.4
- 3.52
0.0"
0.177
-0.165
= 0 vanishes due to symmetry.
TABLE XVII: Cross Sections Symmetric in both Orbital and Rotational Coupling for Ar-N, at E = 124.7 em-' u :111.(0) j ~';IJ
u
j , i' 2, 2 234 2, 6
10s
cs
10s
cs
10s
cs
12.09 -0.758 -0.324
13.1 -0.673 -0.272
9.86 -1.04 -0.124
9.80 -0.934 0.135 (-1)
10.75 -0.925 -0.204
11.1 -0.830 -0.101
494 4,6
11.58 -0.996
14.5 -0.707
9.50 -1.05
10.7 -0.362
10.33 -1.03
12.2 -0.517
6, 6
11.44
18.3
13.6
10.28
15.5
9.50
TABLE XVIII: Purely Imaginary Cross Sections with Symmetric Orbital Coupling are for Ar-N, at E = 124.7 cm-' j , j'
10s
cs
2, 2 2, 4 2, 6
-0.218" 0.497 0.298
-0.95" 0.682 0.215
4, 4 4, 6
0.105' 0.364
6, 6
0.110"
cs
10s 0.834 (-1)' -0.222 -0.133
0.165 (-2)' -0.304 -0.962 (-1)
0.665' 0.472
-0.566 (-1)' -0.163
-0.348' -0.211
1.64"
-0.539
(-lp
-0.735a
TABLE XIX: Cross Sections Calculated for N,-He at E = 54 cm-'
:
u ?? 9 .( 1 ) I J ;I1
u *OZO (2) j'j';jj
j , j'
cc
10s
cc
10s
1, 1 1, 3 3, 3
13.5 -0.31 17.0
14.4 -0.25 14.6
11.2 -0.11 15.3
12.1 -0.79 (-1) 12.1 u ??? .(2) j j ;IJ
u 02?.(2) j'i';Jj
i'
1, 1 1, 3 3, 3
cc 0.78 (-1) 0.17 0.13
'
cc
10s
10s
-0.84 (-1) 0.78 (-1) 0.11 -0.34 (-1) 0.13 -0.24 uj?j;;j(
figures are quite typical of all the cross sections of this type, namely, the elastic opacities are always positive whereas the inelastic opacities are always negative. In Figures 5 and 6 the u2O2O(2)cross sections for the 1 1 transition of HD-Ne and the 1 2 transition of H e 4 0 are shown. These plots are also typical of all type (a) cross sections, namely, the elastic opacities are always positive while the inelastic opacities may be both positive and negative. This accounts for the excellent agreement between the CC and CS results for the elastic cross sections for most systems compared to the occasional disagreement for some of the inelastic cross sections. The disagreements are due to interference effects. In Figures 7 and 8 we show the ~ ~ ~cross ~ sections ~ ( 2 ) for the 1 1transition of HD-Ne and the 1 2 transition of He-CO. The behavior shown in these figures is typical of all cross sections of this type, namely, the opacities may change in sign with respect to Ji and the elastic and inelastic opacities are very similar in magnitude. This pronounced interference behavior clearly accounts for the difficulty in accurately predicting these cross sections. Although not shown, cross sections of type (c) behave very much like cross sections of type (a). The somewhat unnatural oscillations in Figures 4,6, and 8 bring up another difficulty in calculating cross sections which exhibit strong interference effects. It is likely that these oscillations are the result of failure to converge the CC and to a lesser extent the CS results to sufficiently accurate values of the S matrices. This sort of error is less severe for cross sections in which the opacities are all of the same sign. Up to this point we have examined the behavior at a very detailed level. We are now in a position to comment on the utility of approximate methods for calculating various phenomena. Most bulk effects which one can measure involve averaging over rotational states of the appropriate phenomenological cross sections as well as averaging over energy to obtain the desired observable. The fact the cross sections of type (b) give consistently good results for both the elastic and inelastic transitions
-
-
-
'These cross sections have not converged.
1,
Uj?j?,ij( 2)
111!.(1)
j'j';u
-0.84 (-1) -0.29 (-1) -0.24
2)
j , j'
cc
10s
1,1 1, 3 3,3
4.07 0.76 3.61
4.66 0.72 4.54
- -
illustrated some typical opacities for H e C O and HD-Ne. In Figures 3 and 4 specifically the 1 1transition for the uom2(2)cross section of HD-Ne and the 1 2 transition for the uOm(2) cross section of He-CO are shown. These
-
J. Phys. Chem. 1982, 86,1096-1098
1096
of most systems indicates that the CS approximation would give excellent results for spin-spin and spin-rotation NMR relaxation times. The fact that cross sections of type (a) give consistently good results for the elastic cross sections while the inelastic cross sections are much smaller but not so accurate as the elastic ones indicates that the CS approximation should give good results for diffusion and viscosity related phenomena in the absence of external fields. More sensitive phenomena such as the deviations of thermal conductivity or shear viscosity in the presence of a magnetic field (SBE) depend quite sensitively upon rotationally and energy averaged cross sections of type (d). Because the CS approximation often does not agree well with the CC results for cross sections of this type, the success of the CS (or 10s for that matter) is questionable in determining SBE effects. On the other hand, the CS method does properly predict the shape of the opacity which determines these cross sections. Because the elastic and inelastic cross of this type are small, often opposite in sign, and similar in magnitude, it is also clear that averages over them are going to be extremely sensitive to
the accuracy of even the exact CC values of the S matrices. Thus, the challenge presented in calculating cross sections which are extremely sensitive to interference effects extends beyond the approximate methods to include accurate CC approaches as well. Much work remains to be done before these cross sections can be fully utilized in obtaining detailed information about molecular potential anisotropies. Acknowledgment. D. K. Hoffman and D. J. Kouri both benefited from the wonderful experience of being graduate students at Professor J. 0. Hirschfelder’s Theoretical Chemistry Institute. In addition to very best wishes on the occasion of his 70th birthday, they wish to express their gratitude to Professor Hirschfelder for his great kindness both during and following their graduate studies and for providing a stimulating and friendly environment in which to learn. The authors also gratefully thank Sheldon Green for providing us his CC S matrices for the He CO and He + HC1 systems. This work was supported in part by the National Science Foundation Grant CHE 77-22911.
+
A New Algorithm for the Evaluation of Vector-Coupling Coefficients in an MCSCFXI Study 0. Das Chemistry Division, Argonne National Labomby, Argonne, Iiiinols 60439 (Received June 22, 1981; In Final Form: August 17, 1981)
A new method is presented for an efficient handling of the vector-coupling coefficients in an MCSCFICI calculation. The method is suited to those problems (such as transition metal clusters) where only specialized sets of angular momentum couplings are important.
Introduction Owing to the increasing use of ever larger sets of configurations in theoretical studies based on configuration interaction (CI) or many-body-perturbation theory (MBPT), the problem of efficient handling of the vector-coupling efficients has attracted the attention of both chemists and physicists. The methods1$that have emerged are ideally suited to those cases where all spin couplings are treated on the same footing. For many potential surface calculations this generality is desirable. There are, however, two classes of problems for which such generality is very seldom required: (i) studies aimed at computing spectroscopic properties of systems in static geometries and (ii) those that deal with systems such as transition metal clusters, many of the open shells being relatively compact. As an example of the latter case consider a system l i e Crz. A recent calculation3 on this system has revealed that bonding in the ground state of this system corresponds largely to an antiferromagnetic pairing of the spins on each Cr. This implies that the molecular wave function is very nearly valence-bond-like, consisting of ’S atomic ground states of Cr. For chromium multimers, therefore, it is (1) J. Paldus in ‘Theoretical Chemistry: Advancea and Perspectives”, Vol. 2, H. Eyring and D. J. Henderson, Ed.,Academic Press, New York, 1975. (2) I. Shavitt, Znt. J. Quantum Chem. Symp., 11, 131 (1977); 12, 5 (2978);Chem. Phys. Lett., 63, 421 (1979). (3) M. M. Goodgame and W. A. Goddard, III, J.Chem. Phys., in press.
0022-3654/82/2086-1096$01.25/0
conceivable that the molecular states are best expressed in terms of groups of atomic orbitals in a limited number of atomic spin states leading to a drastic simplification of the “vector-coupling” problem. The unitary group approach,l obviously, cannot reproduce such simplicity. Moreover, even for potential surface calculations based on GVB/CI or MCSCF/CI it is generally true that not all types of configurations are important such that what is considered the strength of the unitary group approach, viz., the full generality of the configuration space, may in fact be its weakness. For example consider a recent calculation on N02.4 Over the interactive region on the N + O2 side of the potential surface, the following four configurations turn out to be the most important: = { core)YO12XolX~Z~Yb2Zb2x02 91 = 9 2
(Yb2
= (zb2
- z;)@O Y;)@O
a3 = (YbZb y,za)@O where the orbitals (X,, Y,, Z t ) , E = N, 0 , O l are approximately the 2px, 2py, and 2pz orbitals centered on the centers (5) and Yb, Zb, Ya, and Z, are bonding and antibonding orbitals, Y’s for the u bond in NO and Z’s for the u bond in 02.Moreover, all important single excitations +
(4) G. Das and P. A. Benioff, Chem. Phys. Lett., 7 5 , 519 (1980).
@ 1982 American Chemical Society
