J. Phys. Chem. 1983, 87,2723-2730
2723
Inelastic Differential and Integral Cross Sections.for *’+‘E Linear Molecule-’S Atom Scattering: The Use of Hund’s Case (b) Representation 0. C. Corey and F. R. McCourt‘ Guelph-Waterlw Centre for G-aduate Work in Chemistry and Department of Chemisty, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada (Received: December 28, 1982)
Exact close-coupled scattering equations are established for multiplet-Z:molecule-structureless atom collisions by using a Hund’s case (b) coupling scheme for the description of the paramagnetic molecule. Coupled-states and infinite-order sudden approximation expressions are presented for T matrices and for differential and integral cross sections. The importance of transforming to a spin-freetotal-d basis is emphasized, especially its eventual usefulness in reducing the computing time that will be required for the solution of paramagnetic scattering problems. The result of a lack of j-state resolution in beam-scattering experiments is discussed and it is shown that, in such a case, formulas originally obtained for diamagnetic molecule-atom scattering apply.
1. Introduction
During the past several years, there has been considerable progress in the development of analytical approximations and efficient numerical codes for the evaluation of collision dynamics. Of particular interest has been the introduction of the centrifugal-sudden1 (CS) and infinite-order-sudden2 (10s) dynamical approximations. These two procedures have allowed a considerable reduction in computational effort while retaining a high degree of accuracy in the calculation of many collision cross sections. Much of the initial analytical development of these dynamical approximation procedures was quite naturally focussed upon atom-diamagnetic linear molecule collision processes and the consequent generalization to (diamagnetic linear) molecule-(diamagnetic linear) moled e 3 and atom-symmetric top4 collision processes. Indeed, until relatively recently, little attention has been paid to the development of dynamical approximation procedures for binary collisions in which one or both of the collision partners is paramagnetic. The development of exact close-coupled (CC) equations for the scattering process for collisions involving paramagnetic species was initially focussed upon the astronomically important ‘II, 211, and zZ+ molecular species found in interstellar cloud^.^-^ Much of this work has already been adequately reviewed in two very recent papers by Alexande@tgin which CS and 10s expressions have (1)P. McGuire, Chem. Phys., Lett., 23,55 (1973); P. McGuire and D. J. Kouri, J. Chem. Phys., 60,2488 (1974); D. J. Kouri, T. G. Heil, and Y. Shimoni, ibid., 65, 226, 1462 (1976); Y. Shimoni and D. J. Kouri, ibid., 66,2841 (1977); 67,86 (1977); G. A. Parker and R. T. Pack, ibid., 66,2850
(1977). (2) (a) C. F. Curtiss, J. Chem. Phys., 48, 1725 (1968); 49, 1952 (1968); T. P. Tsien and R. T. Pack, Chem. Phys. Lett., 6,54 (1970); T. P. Tsien, G. A. Parker, and R. T. Pack, J. Chem. Phys., 59,5373 (1973);R. T. Pack, ibid., 60,633 (1974); 66,1557 (1977); D. Secrest, ibid., 62,710 (1975). (b) See also the review by D. J. Kouri which appears in “Atom-Molecule Collision Theory: A Guide for the Experimentalist”, R. B. Bernstein, Ed., Plenum, New York, 1979, Chapter 9, and which contains an extensive bibliography of work on the CS and 10s approximations. (3) R. Goldflam and D. J. Kouri, J. Chem. Phys., 70,5076 (1979); T. G. Heil, and D. J. Kouri, Chem. Phys. Lett., 40,375 (1976);A. E. de Pristo and M. H. Alexander, J. Chem. Phys., 66, 1334 (1977); T. G. Heil, S. Green, and D. J. Kouri, ibid., 68, 2562 (1978). (4) S. Green, J . Chem. Phys., 64,3463 (1976); 70,816 (1979); 73,2740 (1980); B. J. Garrison and W. A. Lester, ibid.,66, 531 (1977). (5) H. Klar, J.Phys. E , 6,2139 (1973); S. Green and R. N. Zare, Chem. Phys., 7, 62 (1975). (6) D. P. Dewangen and D. R. Flower, J . Phys. B, 14, 2179 (1981). (7) %-I. Chu, Astrophys. J., 206,640 (1976); R. N. Dixon and D. Field, Proc. R. SOC.London, Ser. A, 366, 225 (1979); 368, 99 (1979). (8) M. H. Alexander, J. Chem. Phys., 76, 3637 (1982). (9) M. H. Alexander, J. Chem. Phys., 76, 5974 (1982).
0022-3654/83/2087-2723$0 1.50/0
been developed for T matrices and differential and integral cross sections for the cases of structureless atoms colliding with 2Z+molecules8 and with 211m o l e c ~ l e s .Both ~ treatments are based upon a Hund’s case (a) representation of the paramagnetic species: In the second paper on 2rI molecules, the 10s formulation was also extended to the case (b) limit. Finally, it should also be mentioned that Fitz and KourilO and Aquilanti et have obtained expressions for these cross sections for P state atoms colliding with ‘S state atoms and with H2. A paramagnetic molecule of atmospheric importance that has not been included in the studies referred to above ground electronic is the oxygen molecule, 02,with a 9,state. While no exact calculations have been carried out to our knowledge, semiclassical trajectory calculations have been made in order to determine the collisional broadening12J3of O2 fine structure lines and 10s studies of the differential and integral collision cross sections for the scattering of O2 against rare gases have been reported.14J5 In the 10s studies, in particular, it appears to have been assumed that diamagnetic diatomic molecule-atom IOSA routines could be utilized without further consideration being given to the multiplet structure of the O2 molecule. The conditions under which such an assumption is justifiable will be examined in section 5. In the following, CSA and IOSA expressions are obtained for T matrices, and differential and integral cross sections for multiplet-Z molecule-structureless atom interactions. Section 2 briefly reviews the Hund’s case (b) coupling scheme which is conventionally considered to be the “natural” basis for the description of multiplet-Z molecules16and then considers the structure of the exact CC equations for this representation. Exact CC expressions for the differential and integral cross sections are (10) D. E. Fitz and D. J. Kouri, J . Chem. Phys., 73, 5115 (1980); 74, 3933 (1981). (11) V. Aquilanti and G. Grossi, J . Chem. Phys., 73, 1165 (1980); V. Aquilanti, P. Casavecchia, G. Grossi, and A. Laganl, ibid., 73, 1173 (1980). (12) K. S. Lam, J . Quant. Spectrosc. Radiat. Transfer, 17,351 (1977). (13) E. W. Smith and M. Giraud, J . Chem. Phys., 71,4209 (1979);E. W. Smith, ibid., 74, 6658 (1981); M. BBrard, P. Lallemand, J. P. Cabe, and M. Giraud, ibid., 78, 672 (1983). (14) F. Battaglia, F. A. Gianturco, P. Casavecchia, F. Pirani, and F. Vecchiocattivi, Faraday Discuss., 73, 257 (1982); M. Faubel, K. H. Kohl, J. P. Toennies, and F. A. Gianturco, J . Chem. Phys., 78, 5629 (1983); F. A. Gianturco and A. Palma, “Intramolecular Dynamics”, J. Jortner and B. Pallman, Ed., Reidel, Dardricht, 1982, p 63. (15) M. Keil, J. T. Slankas, and A. Kupperman, J. Chem. Phys., 70, 541 (1979). (16) G. Herzberg, “Spectra of Diatomic Molecules”, 2nd ed, Van Nostrand, Princeton, 1950.
0 1983 American Chemical Society
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The Journal of Physical Chemistry, Vol. 87, No. 15, 1983
given explicitly since they are not only useful for illustrating the retrieval of results previously obtained for ‘Z and 22+molecules but also because we are interested primarily in the development of CS and 10s approximations to such exact expressions. Sections 3 and 4 present the CSA and IOSA expressions and factorization properties for the Hund’s case (b) coupling scheme. Section 5 focusses upon the experimental case in which there is a lack of j-state resolution: Observations initially made by Alexanders for the case of 2Z+ molecules are confirmed and extended. Finally, the paper concludes with a discussion of the current results and a summary of the salient features of the present work. 2. Close-Coupling Formulation of Collision
Dynamics in Hund’s Case (b) Molecules with nonzero electronic spin and/or orbital angular momentum are often represented in the limit of pure Hund’s case (a) or case (b) coupling. In particular, for multiplet-Z molecules, Hund’s case (b) coupling is conventionally considered to give an accurate description of the molecular wave functions. The molecular basis vector in pure Hund’s case (b) coupling is obtained in the general case by vectorially adding a rotational vector IKmKA), given in coordinate representation byl’Js
Corey and McCourt
(2.4)
+
Here, [xl] represents 2x, 1 (and [x1x2-.x,] similarly will represent the appropriate product of factors 2 x + 1)and ml is the projection of 1 onto a space-fixed axis of quantization. The present coupling scheme may be represented by the vector equations N + S = j; j+l= J (2.5) In the total J representation, the scattering wave function labeled by the entrance channel NSjl may be expanded as
Y&!jl(f,R) = C R-’ UJ (N’S’j’I’,NSjIJR)(i,RI(N’S9j 4 ’;J M ) (2.6) “S‘j‘l‘
where i gives the orientation of the figure axis of the linear molecule and R = RR is the position vector of the structureless atom relative to the center of mass of the linear molecule. Substitution of this expansion into the Schroedinger equation leads to the usual set of coupled equations for the radial functions UJ(N’S’j’l’+-NSjl(R), namely
(ffPyIKmKh)E [(2K + 1 )/8T2]112(-l)mK-.4DK* fnK.4(ffPr 1 (2.1) to a vibronic basis vector luASms) as ,u2\KSjm) =
UJ(N”S”j”l”,NSjl(R)(2.7)
mKmS
(2.2)
where (:: :) represents a Wigner 3 - j symbol,lsK is the total molecular angular momentum apart from electronic and nuclear spins, and A is the projection of the electronic orbital angular momentum upon the molecular axis. The quantities oKmKa(aPy) are matrix elements of the rotation operator in coordinate representation: The Euler angles a, p, and y are defined in the EdmondV8 convention and describe the orientation of the molecule-fixed reference frame relative to a space-fixed frame, mK,ms, and m are projection quantum numbers in a space-fixed frame, and j is the total molecular angular momentum apart from the nuclear spin. For the present case of multiplet-2 molecules, the molecular angular momentum K reduces to the nuclear rotational angular momentum N (since A is zero). If we ignore for the present discussion the dependence of the basis vectors on the vibrational quantum number u , the basis vectors for multiplet-2 molecules reduce to INSjm). These basis vectors have definite parity under inversion of the space-fixed axis system, ~ i 2 . l ~
isFINSjm)= (-1)””INSjm) (2.3) where isF is the inversion operator for the space-fixed reference frame and w has the value unity for 2- states and zero for 8+ states. As in the initial formulation of Arthurs and Dalgarnolg for a ‘ 2 molecule interacting with a ‘S atom, the total J representation is defined by vectorially adding the molecular basis element INSjm) to Ilml),an element of the basis set for the orbital motion of the colliding pair: (17) R. N. k e , A. L. Schmeltekopf, W. J. Harrop, and D. L. Albritton,
J. Mol. Spectrosc., 46, 37 (1973). (18)A. R. Edmonds, ‘Angular Momentum in Quantum Mechanics”, Princeton University Press, Princeton, 1957. (19) A. M. Arthurs and A. Dalgamo, R o c . R. SOC.London, Ser. A , 256, 540 (1960).
in which the wavenumber k’ is defined via (2.8)
where p is the atom-molecule reduced mass, E is the total energy, and 4N’j’) is the energy of the molecule in the N‘S‘j’state. Since the potential V is independent of the orientation of the space-fixed frame of reference (rotational invariance) with respect to which the vectors in (2.4) are defined, the matrix elements of V are diagonal in J , M and independent of M. Matrix elements of the scattering operator are defined through the boundary condition UJ(N’S’j’l’,NSjllR)
-
where G(x1’x2/”~3c~~x1xZ~~~x,) denotes a product of Kronecker deltas hXJx. In Hund’s case (b) coupling, the spin and rotational angular momenta are weakly coupled. The assumed spin independence of the intermolecular potential requires that the binary collision does not directly affect the magnitude or orientation of the spin angular momentum S. Rather, the effect of the collision is to cause a sudden change in the rotational angular momentum N. Because S and N are weakly coupled, the spin cannot follow this change in N and the only effect of the collision on S is to cause a change in its axis of quantization. The collision indirectly affects the molecular angular momentum j through the recoupling of S to N after the target recedes. Because of this collisional mechanism, matrix elements of V can be cast into a convenient form by transforming to a total-J
Cross Sections for Paramagnetic Molecule-Atom Scattering
basis represented by the coupling scheme N+l=d; d+S=J
The Journal of Physical Chemistry, Vol. 87,
No. 15, 1983 2725
where (2.10)
This basis is defined through
0
= arccos (R-P)
(2.18)
and PAis a Legendre polynominal. Equation 2.17 can also be written as a contraction of spherical harmonics by means of the spherical harmonic addition theorem as
I (NI)dSilM, =
V(P,R)= wherein INld&) is similarly defined through
EA-2 X4T+ 1 VA(R)Yx(P)*Yx(R) (2.19)
The potential matrix elements V'(N'l',NL) are then given byz1 (2.12)
The vector INld&) is an element of the spin-free total-d basis appropriate for linear molecules in a lZ electronic state interacting with a lS atom. The total-J basis elements defined by (2.4) and (2.11) are related through6J8,20
V'(N'l',NI) =CVh(R)(-1)NtN~+B[N'NI'i]'/2 X h
N' (0
h
0
N I'
o)(o
h 1 d I' 0 O){h N
N' ( 2 . 2 0 ) I
}
Substitution of (2.20) into (2.16) allows the summation over d to be performed via the Racah-Elliott relationz0
(2.13)
where (:::)denotes a 6 - j symbol.l8 The dependence on spin has been extracted from the molecular basis through the recoupling of the total-J basis and it can thus be factored out of matrix elements of spin-independent operators in the total-J representation. If we use the fact that V depends neither upon spin nor upon the orientation of the space-fixed frame of reference, its matrix element in the uncoupled basis simplifies to (("1 ?d'Jn',S'mS'l VI(Nl)d&,Sms) = 6 (d '& 'S 'ms'ld &Sms) Vd( N'1',Nl) (2.14) where Vd(N'l',N1)is the matrix element
Vd(N'l',N1) (N'l'd&lVlNld&)
(2.15)
From (2.11) and (2.13), the matrix element of V in the total-J representation becomes ( ( N ' S )' ' I ' ;JM'I V l ( N Su l $ M ) = 6
( S ' J M ' I S J M ) [dl ~ [ j , j l 1 , 2 ( - 1 ) - ~ ' + 1 x+ ~ ' - ~
with the result that the potential matrix element (2.16) is reduced to ( ( N ' S ) j ' I;JMI ' Vl(NS)jl$M) = C V h ( R)(-I ) S - h - x h
In principle, the close-coupled equations (2.7) together with the potential matrix elements in (2.22) can be solved numerically and the S matrix elements can be extracted through the boundary condition in (2.9). However, because the potential is independent of spin, the coupled equations can be significantly simplified by defining a spin-independent radial function Ud("l',NIIR) through the relation V ( N ' S ' j ' / ' , N S j i l R=) 6 s ' s C [ ~ ~ [2 j( - ~I )jJ 'l- "l- J + '
The summation over d is finite and restricted by the triangle inequalities A(Nld),A(N'l'd), and A(SJd). Because the interaction potential is spin independent, the matrix elements factorize into a spin-free dynamical part Vd(N'l',N1) (which is identical with the potential matrix element of a 'Z state rigid rotor interacting with a 'S state atom) times a spin-dependent spectroscopic factor originating from the orthogonal transformation relating the two total-J bases. The spectroscopic factor is independent of the intermolecular interaction and depends only upon the angular momentum quantum numbers of the atom-molecule scattering system and the quantum numbers specifying the spectroscopic state of the molecule. All dependence on the intermolecular potential is thus contained in the dynamical factor. Since the atom-molecule collisions are dynamically independent of spin, the intermolecular potential can be expanded as
W , R ) = C V J R ) pA(cOse) x
(2.17)
(20) A. Messiah, 'Quantum Mechanics", Vol. 2, Wiley, New York, 1962, Appendix C.
x
d
d
Substituting (2.23) into (2.7) and using (2.16), one can readily see that U8(N'1',N1IR)satisfies the usual closecoupled equations for scattering of a 'Z state molecule by a 'S state atom
2/1.
2 V' ( N'1 ',N "I h 2N',l"
") U' ( N"1 ",N1IR ) ( 2.24)
Scattering matrix elements S d N ' I ~can f l l then be defined from the asymptotic behavior of U*(N'l',NlIR) which is similar to the boundary condition in (2.9). These SdNtIrfl1 are related to the S*N,~~rs?isjl by exactly the same relation as in (2.23). However, it wll be more convenient to employ transition or T matrices, defined in terms of S matrices by PWsfrflsjl = 8(N'S'jr1lNSj1)- SJNat~lrflsjl (2.25) (21) I. C. Percival and M. J . Seaton, Proc. Cambridge Phil. SOC.,53, 654 (1957); D. M. Brink and G. R. Satchler, 'Angular Momentum", 2nd ed, Clarendon Press, Oxford, 1968, see especially eq 5.14.
2726
Corey and McCourt
The Journal of Physical Chemistry, Vol. 87, No. 15, 1983
(2.26)
TBy,/!-V[ 3 6(Nr11N1)- SaLvpsy/
rather than the S matrices themselves in the following. Of course, the T matrices are also related in precisely the same fashion, viz.
nificant reduction in the dimensionality of the coupled equations is achieved by block diagonalization of the potential matrix. In a Hund's case (b) coupling scheme, this is achieved by replacing as usual the centrifugal terms in the CC equations by an average value t(t + 1)/ R2 and then transforming to a new spin-free total-d basis defined by
'(L
I.\'Ldsrli = x ( - l ) L [ l ] ' -;)I.VLMd
Td.y
i
/ ,.VI ,
(2.27)
The scattering amplitude for scattering from channel NSjm to N'Sy'm' is given by
1
(3.1)
This transformation is equivalent to defining a new total-J basis element I(NL)dSJM),which can be related to the basis element I(NS)jLJM)through (3-L ) dSJM, =
n
f ( X 'S' j ' vi ' +-)VSjm1 R ) =:
(3 2 )
by inverting the transformation between the two total-J bases in (2.13). Upon evaluation of the summation over J with the Racah-Elliott relation (2.21), the potential matrix elements in the new basis simplify to
((A-L )a s Jil.rIVl(NL)dSJM, = 5s ~ h 6d x V \ ( R ) ( - l ) d ' L + L ' " [ l ' r ] [ N IV]'
A
/ /\
(; Notice that the scattering amplitude cannot be written as a linear combination of the spin-free scattering amplitudes f(N'm,'+-NmdR) because it is the total molecular angular momentum j and not the nuclear rotational angular momentum N that enters into the 3 - j symbol in (2.28). From the scattering amplitude, the state-testate differential and integral cross sections summed over final and averaged over initial degeneracies are given by du -((N'S'j'+-NSj) = dn k' ( 2 j l)-l If(N'S'j'm'+-NSjmlR)12 (2.29)
+
m 'm
3. Coupled-States Approximation for Hund's Case (b) Coupling In principle, exact close-coupled calculations can be performed starting from a given intermolecular potential. However, CC calculations are not feasible for many systems of interest due to the large number of coupled equations that arise from the degeneracy of the rotational states.2b,22Moreover, for molecules in multiplet-Z electronic states, the number of channels is further compounded by the (2s+ 1)-fold multiplicity of each rotational state arising from the nonvanishing electronic spin. An approximation procedure that has proven extremely useful and accurate for systems in which the interaction is dominated by short-range forces is the centrifugal-sudden or coupled-states (CS) approximation in which a sig(22) H. Rabitz in 'Dynamics of Molecular Collisions", W. H. Miller, Ed., Plenum Press, New York, 1976, Part A, p 33.
hJ(l
\
l)(l
o o / o o o
Ai O L
"(1
-L
N
d))( d 1
O L - L
h n ' llv
1
(3.3)
The summations over 1 and 1' can now be evaluated through the
Substitution of (3.4) into (3.3) and simplification of the summations over 1 and I' through the orthogonality properties of the 3 - j symbols gives for the matrix elements of V the result
This matrix element is equivalent to (NZI VlNL),which is the CS approximation to the matrix element of the intermolecular potential for a molecule interacting with a 'S state atom. Verification of this result can be obtained by applying the transformation of the spin-free total-6 basis element in (3.1) to the potential matrix elements in (2.20). Hence, within the CS approximation, the CC equations in Hund's case (b) coupling that govern rotationally inelastic collisions between a molecule in a multiplet-Z electronic state and a structureless target are replaced by a set of equations equivalent to those arising in the CS approximation to the diamagnetic problem, viz.
This set of equations is solved subject to the boundary conditions
The Journal of Physical Chemistry, Vol. 87, No. 75, 1983 2727
Cross Sections for Paramagnetic Molecule-Atom Scattering
l', the summations over 1 and J can be simplified through the orthogonality properties of the 3 - j symbols to give f F L ( N ' S ' j ' m ' - N S j mI f f ) =
The scattering matrix STL(N1N) defined through the boundary condition (3.7) is equivalent to ((NZ)dSJM(SI(NL)dSJM). This matrix is diagonal in L and independent of d, S, J, and M. The relation between the CC and CS total-J T matrices may be obtained by inverting the transformation in (3.2), thereby yielding
-L-m
"L Jm )
x
Finally, within the CSA, the degeneracy-averaged integral cross section is given by o(N'S'j'-NSj) =
TJN'S'j'l',NSjl =
(3.8) T i L ( N ' I N ) T * j L ( N ' I N )( 3 . 1 4 )
where
TlL(N'ln3
bN"
- SlLW'IN)
(3.9)
An overall phase factor corresponding to the "correct phase" convention of Khare et al.23has been incorporated into (3.8). Within the CS approximation, the spin-free total-d T matrix can be expressed asz4
o(N'S'j'-NSj) =
T6N'l',~l
(3.10)
In section 2 the paramagnetic scattering problem was essentially reduced to the usual coupled equations for scattering of a l2 molecule by a IS atom. Hence, the relation between the CC and CS total-JT matrices in (3.8) may alternatively be obtained by substituting (3.10) into (2.27). Substitution of (3.4) into (3.8) and subsequent simplification gives the total-J T matrix within the CS approximation as T J N , s , j , l , , N s j l=
Within the CSA, the state-to-state scattering amplitude is given by f ( N ' S ' j ' m ' + N S j mlkj =
J
In the initial4 (IL) labeling scheme, 1 is chosen to be 1. Within both the IL and FL labeling schemes, the summations over the orbital angular momentum and over J can be simplified and the integral cross section can be reduced to
I'
) T T ~ ( N ' I N ) Y ~ ~ (,3~. 1' 2( )~ ~ )
In the final-1 (FL) labeling scheme where 1 is chosen to be (23) V. J. Khare. D. E. Fitz, and D. J. Kouri, J. Chem. Phys., 73, 2802 (1980). (24)W.-K. Liu, F. R. McCourt, D. E. Fitz, and D. J. Kouri, J . Chem. Phys., 71,415 (1979);R.Goldflam, S.Green, and D. J. Kouri, ibid., 67, 4149 (1967).
LL'P
(3.15)
4. Infinite-Order Sudden Approximation and Factorization Properties In atom-molecule systems in which the interaction is dominated by short-range forces, the collision energy is large compared with the level spacing, and closed channels do not play a significant role, the close-coupled equations may be further simplified through the infinite-order-sudden approximation (IOSA). In section 3, it has been demonstrated that, in Hund's case (b) coupling, the CSA leads to a set of equations equivalent to those replacing the CC equations in the diamagnetic problem. Further, the CS approximation to the total-J T matrix can be obtained through the CS approximation to the total-d T matrix. Consequently, it is evident that the IOSA leads to a set of uncoupled equations equivalent to those occurring in the spin-free problem and that the 10s approximation to the total-J T matrix may be obtained directly from the 10s approximation to the spin-free T matrix. The 1 0 s approximationto the spin-free total-d T matrix is given by (3.10), but with the coefficients TiL(N1N)given byz4
TrL(NIN)= ~ Y ~ ~ Y * ~ Z ( B , O ) T I ( Bd8) Y(4.1) ~~(~,O) where 8 is the angle between the figure axis of the molecule and the vector joining its center of mass with the atom. The 10s T matrix Tl(8) is obtained by solving the uncoupled one-dimensional 1 0 s radial equations25at fixed orientation 8. If we expand the angle dependence of the (25)G.A. Parker and R. T. Pack, J. Chem. Phys., 68,1585 (1978).
2728
The Journal of Physical Chemistty, Vol. 87,No. 15, 1983
1 0 s T matrix in terms of the same set of functions used to expand the interaction potential, viz.
T*'(O)= ~TT',P,(cose) x
Corey and McCourt
-(N'S'j'+LVSj) do = dn
(4.2)
the 10s approximation to the spin-free T matrix in the total-d representation becomes24
(4.9)
where du dQ
-(Xto)
The coefficients Tixare obtained by inverting the expansion in (4.2). Now, substituting (4.3) into (2.27) and summing over d with the Racah-Elliott relation (2.21) yields the 1 0 s approximation to the total-J T matrix element in the Hund's case (b) coupling scheme as
=
kx -Ifxl2
(4.10)
k0
is the differential cross section for scattering out of the N = S = j = 0 state. Following the substitution of (4.4) into (2.30),the summation over J can be evaluated by using the orthogonality properties of the 6 - j symbols. Hence, within the IOSA, the degeneracy-averaged integral cross section is given by u (N'S'j'+NSj) =
Upon substitution of (4.4) into (2.28) and algebraic simplification, the 10s approximation to the state-to-state scattering amplitude becomes f(N'S'j'm'+-.VSjmff ) =
where the cross section u(X+-O), defined as
represents inelastic transitions out of the N = S = j = 0 level. For both the IL and FL labeling schemes, the summation over the orbital angular momentum can be simplified and, in these schemes, the cross section for transitions out of the N = S = j = 0 level becomes (4.13)
In the FL labeling scheme, the summation over I can be simplified to give
If the final j state cannot be resolved, the cross section for scattering out of an initial state NSj into the N'spin multiplet is obtained by summing (4.11) over j', giving
(4.14)
Following Goldflam et al.,24the scattering amplitude can be factorized as fFL(Ar'S'j'm'-NSjmif )
Further, the cross section for scattering out of an initial NSj state without regard to final state is obtained by summing (4.14) over N', giving
= k02
Cu(N'S'j'+-NSj) = 6sts -Eu(X+-O) N'j' k2 x
where
and the wavenumbers ko and k x correspond to the states N = S = j = m = 0 and N = A, S = 0, j = X,m = 0, respectively. Setting N = S = 0 in (4.7),f x can be identified as the 10s differential scattering amplitude for the transition N = S = j = m = 0 - N = X, S = 0, j = A, m = 0. Within the 10s approximation, the scattering amplitude for the transition NSjm NS'j'm'can be expressed in terms of the scattering out of the N = S = j = m = 0 state and degeneracy-averaged paramagnetic differential cross sections can readily be obtained from (4.7). The factorized form of the degeneracy-averaged differential cross section is
-
(4.15)
Identical arguments lead to analogous expressions for the differential cross sections. As discussed by several aut h o r ~in~the ~ context * ~ ~ of~ the ~ ~diamagnetic problem, within the IOSA the total differential and integral cross sections are independent of the initial state of the diatom except for the factor ko2/k2. It is a well-known result of the 10s approximation that general state-to-state inelastic cross sections can be obtained as in eq 4.11 in terms of the cross sections for scattering out of the lowest rotational level (with rotational quantum number 0). In the case of molecular oxygen, for which the most common species is 1602,such a result may appear at first sight contradictory since occupation of the N = 0 level is nuclear spin symmetry forbidden. The intermolecular potential surface, however, does not depend upon the nuclear spin statistics so that even in the case of 1 6 0 2 , where only the odd-numbered rotational levels are occupied, general rotationally inelastic cross sections can nonetheless be calculated within the IOSA from a
The Journal of Physical Chemistry, Vol. 87, No. 15, 1983
Cross Sections for Paramagnetic Molecule-Atom Scattering
knowledge of the scattering out of the ground rotational level. 5. Lack of j-State Resolution Alexander8has recently demonstrated for 2Z+diatomatom interactions that, within the IOSA, when parity doublets cannot be resolved the collision dynamics becomes equivalent to that of a lZ+diatom scattering against an atom. It is important to note that this interesting result is in fact independent of any dynamical approximation and is valid at the level of the CC formalism. If the final j state cannot be resolved, the integral cross section for scattering out of an initial NSj state into the N’spin multiplet is obtained by summing eq 2.30 over j’. The summation over j’and J can be evaluated through the orthogonality properties of the 6 - j symbols. In particular, the summation over j’ gives add, and the summation over J gives a factor 1/[m. The integral cross section for scattering out of a well-defined j‘ state into the N‘spin multiplet is then given by NCtS
The total integral cross section can now be obtained by summing (5.1) over N’. When the nuclear rotational states are resolved but the spin multiplets are not, the integral cross section for a transition between N states is obtained by averaging over the unresolved j j‘ transitions. The N N’degeneracy-averaged integral cross section is then given by
-
-
N+S
u(N’+-N) =
N’+S
j=N-S w j j’=NCS
u(N’Sj’+-NSj)
(5.2)
scattering amplitude for scattering from channel NmN into in the spin-free molecule. The total differential cross section can now be obtained by summing eq 5.6 over N’. The unresolved j-state N N’ degeneracy-averaged differential cross section is given by
N’“’
(5.8) N+S
= GI/
c
j=N-S
lil
= Gl/[NSI
(5.3)
denotes the statistical weight associated with each j state. In eq 5.1, only the wavenumber depends on j. Consequently, the unresolved j-state N N’ degeneracy-averaged integral cross section is given by
-
where
-
The only difference between eq 5.4 and 5.7 and the degeneracy-averaged state-bstate differential and integral cross sections for a ‘Z molecule-colliding with a structureless target is the factor ( k N / k ) ’ . Similarly, the total differential and integral cross sections for the paramagnetic molecule are ( k N / k j ) ’times the total cross section in the corresponding spin-free molecule. The reduction to the spin-free problem is valid for a linear molecule in any multiplet-Z electronic state and is an immediate consequence of the fact that the total-J and total-d T matrices are related through the orthogonal transformation in eq 2.27. Cross sections for transitions between N states have often been calculated1*J5by employing formulas originally derived for diamagnetic molecules. The present discussion appears to be the first rigorous justification of such a procedure. An interesting corollary obtained from the reduction to the spin-free problem at the level of the CC formalism is that the application of any dynamical approximation to the total-6 T matrices must lead to an analogous reduction valid within that approximation. In particular, within the IOSA
where w j , defined by wj
2729
R2, defined by (5.5)
is the statistically averaged wavenumber for the N multiplet. Of course, similar expressions can be obtained for the differential cross section by averaging eq 2.29 over the unresolved j j‘transitions. Upon substitution of eq 2.28 into eq 2.29 each of the four pairs of 3 - j and 6 - j symbols can be replaced by the identity in eq 3.4 and the resultant expression can then be simplified through the orthogonality properties of the 3 - j symbols. The differential cross section for scattering out of a well-defined j state into the N’ spin multiplet is thus given by
-
Here, kN is the wavenumber for the spin-free mqlecule in the Nth rotational state and f(N’mN’+-NmNIR) is the
where u ( X - 4 ) is the cross section for scattering out of the N = 0 level. Further, when it is possible experimentally to resolve only transitions between the nuclear rotational states, j-dependent state-to-state cross sections can be predicted within the IOSA from the experimental N = 0 X cross sections through eq 4.9 and 4.11. An analogous result was first obtained for ‘W diatomic molecules in Hund’s case (a) coupling scheme by Alexander.8
-
6. Summary and Discussion The CC formalism for rotationally inelastic collisions developed initially for diamagneticmolecules colliding with structureless atoms has been extended to paramagnetic molecules in multiplet-Z electronic states. The extension was found to be most conveniently carried out in terms of a Hund’s case (b) coupling scheme for the molecular angular momenta. By exploiting the fact that the intermolecular potential energy surface governing such collisions is independent of spin, the scattering problem has been rigorously reduced to a much simpler spin-free problem through the orthogonal transformation (2.13). Computationally, the expense of solving the paramagnetic scattering problem is of the order of n:, where n is the number of coupled channels that must be include: in the expansion of the scattering wave function. If there had been no spin angular momentum, the corresponding scattering problem would have had nd coupled channels, where np and nd are related by np = (2s+ l)n& Since the summation over 6 in eq 2.27 is restricted by the triangle inequality A(S,J,d), an exact solution to the paramagnetic scattering problem requires the corresponding spin-free scattering problem to be solved at most 2 s + 1 times. Consequently, upon transforming to the spin-free basis,
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The Journal of Physical Chemistry, Vol. 87,No. 15, 1983
the overall computational expense is reduced from the order of np3 to the order of n,3(2S + 1)F2.The most commonly encountered multiplet-8 molecules have doublet or triplet electronic states, in which case the transformation to the spin-free basis will reduce the computational expense of the solution of the scattering problem by about 75% for the doublet case and about 88% for the triplet case from that required for the untransformed problem. This reduction in the number of coupled channels upon transformation to the spin-free basis appears to be unique to the case (b) representation. For example, Alexanders has recently presented the CC formalism in a Hund’s case (a) coupling scheme for rotationally inelastic collisions between 2X+ molecules and structureless atoms. When the interaction potential is independent of spin, the dynamical problem in a case (a) representation becomes formally equivalent to the spin-free symmetric top-atom scattering problem without any net reduction in the number of coupled channels. The CS and 10s approximation procedures were extended to include multiplet-8 molecule-atom interactions within Hund’s case (b) coupling. By utilizing the reduction to the spin-free scattering problem, we have demonstrated that the dynamical aspects of the CS and 10s approximations are equivalent to those previously obtained for the diamagnetic problem. The CS and 10s approximations to the total-J T matrices in eq 3.11 and 4.4,respectively, are only slightly more complicated than the corresponding CS and 10s approximations to the diamagnetic T matrices. Further, through the transformation to the spin-free basis, exact and approximate T matrices, differential and integral as well as generalized cross section^^^^^' for multiplet-2 molecule-atom collisions can be calculated almost as easily as for diamagnetic molecules by using computational codes originally developed for diamagnetic molecules. In principle, of course, the exact CC expressions and the CS and 10s approximations for rotationally inelastic collisions between multiplet-Z molecules and structureless atoms could be obtained in Hund’s case (a) coupling scheme from the present analysis through the orthogonal transformation connecting the bases. This would result in a generalization to molecules with arbitrary spin of the formulation for 2Pmolecules recently obtained by Alex(26) G. C. Corey, W.-K. Liu, and F. R. McCourt, in press. (27) G. C. Corey and F. R. McCourt, to be submitted for publication.
Corey and McCourt
ander.a Alternatively, the results reported here could also have been obtained through a simple generalization of Alexander’s case (a) results to molecules with arbitrary spin, followed by a transformation to the case (b) coupling scheme. It should be emphasized here that, for molecules with arbitrary electronic spin and/or orbital electronic angular momentum, a general expression for the orthogonal transformation between the Hund’s cases (a) and (b) coupling schemes has been obtained by Freed% using Van Vleck’s technique of reversed angular momentumz9and the well-known algebra of spherical tensors. Both procedures have the disadvantage, however, that they become as algebraically involved as a formulation from first principles developed consistently within a particular coupling scheme. Thus, the transformation between the two schemes will prove to be of greatest value in comparing specific details of the full CC, CSA, or IOSA formulations in the two coupling schemes. Recently, it has been demonstrateds for 5+moleculeatom interactions that, when parity doublets cannot be resolved, the collision dynamics becomes equivalent within the 10s approximation to that of the l2+molecule-atom case. Use of the Hund’s case (b) coupling scheme facilitates the elucidation of the meaning of this interesting result. In fact, within the case (b) representation, it is apparent that at the level of the CC formalism, when the nuclear rotational states are resolved but the spin multiplets are not, the collision dynamics for state-to-state transitions between the nuclear rotational states in paramagnetic molecules may be analyzed in terms of formulas originally derived for diamagnetic molecules. The present discussion in section 5 appears to be the first rigorous justification of such a procedure. Moreover, the reduction to the equivalent diamagnetic problem is independent of any dynamical approximation and can be traced to the orthogonal transformation connecting the two total-J basis elements which were defined in section 2 in the limit of Hund’s case (b) coupling.
Acknowledgment. We are grateful to Dr. Wing-Ki Liu for his interest in the present work and for numerous helpful discussions. This research was supported in part by a NSERC of Canada grant in aid of research. (28) K. F. Freed, J. Chem. Phys., 45,4214 (1966); see also, Y.-N. Chiu, ibid.,41, 3285 (1964). (29) J. H. Van Vleck, Reu. Mod. Phys., 23, 213 (1951).
