J. Phys. Chem. 1984, 88, 2031-2036
2031
Pressure-Broadenlng Cross Sections of Multiplet-Z Molecules: 0,-Noble Gas Mixtures G. C. Corey,? F. R. McCourt,*st and W.-K. Lid Guelph- Waterloo Centre for Graduate Work in Chemistry and Department of Chemistry, and Guelph- Waterloo Program f o r Graduate Work in Physics and Department of Physics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada (Received: May 2, 1983)
Expressions are obtained for the effective collision cross sections governing the pressure broadening of spectroscopic lines of paramagnetic linear molecules in a bath of closed-shell atoms. Using the Hund’s case (b) coupling scheme for the description of the paramagnetic species, and assuming that the impact approximation applies, we derived exact close-coupled,centrifugal sudden, and infinite-order sudden expressions, emphasizing the usefulness of transforming to a spin-free total4 basis for the description of the scattering S matrix. The pressure-broadening cross sections contain all electronic spin information in six-j symbols which multiply the spin-independentS-matrix elements, provided, of course, that the intermolecular interaction potential surface is itself spin independent. Upon setting the electron spin quantum number to zero, the expressions for the effective collision cross sections reduce to the well-known expressions for the cross sections governing the pressure broadening of the spectroscopic lines for diamagnetic molecules in a bath of structureless atoms.
1. Introduction
The calculation of binary collision cross sections for linear molecule-noble gas systems is by now almost a routine exercise when the linear molecule and the atom each possess closed electronic shells. Exact close-coupled (CC) expressions for state-to-state and total differential and integral cross sections’ are readily available and have been employed for calculations in hydrogenic binary mixtures.’ For non-hydrogenic systems, expressions obtained by using the centrifugal sudden (CS) and infinite-order sudden (10s) dynamical approximations are commonly Further, exact and approximate expressions for the effective cross sections governing the shear viscosity and diffusion phenomena,“ field-effect p h e n ~ m e n aand , ~ spectroscopic pressure broadening6 have been systematically developed and utilized for such closed-shell molecule-atom systems. For open-shell molecules possessing a net electronic angular momentum, the situation is not yet so well developed. A number of early studies have been made for open-shell “molecules” of astronomical importance, such as CH,8 CN,l0 and “,I0 and for Na,(B”I,)’’ which has been employed extensively in laser-induced fluorescence experiments. More recently, Alexander has obtained exact C C expressions as well as the CS and 10s approximations for state-to-state differential and integral cross sections for the species CN(22+)12and NO(211)13colliding with closed-shell atoms. Both papers deal with the Hund’s case (a) representation of the molecular species although, in the second paper, the 10s formulation is extended to the Hund’s case (b) limit. Calculations have been made on gaseous oxygen and its binary mixtures by utilizing the 10s approximation as well as classical and semiclassical trajectory expressions for the binary collision cross sections. In particular, classical’4a and s e m i ~ l a s s i c a l ’ ~ ~ trajectory calculations have been performed initially to describe the collisional broadening of the 0, fine-structure microwave lines and have been utilized more recently to calculate the pressure broadening of the Raman lines of o ~ y g e n . ’ ~10s ~ , ’calculations ~ have been reported for the differential and integral collision cross sections for O,-rare gas system^.^^^^^ In the 10s studies, it appears that diamagnetic diatom-noble gas atom IOSA routines have been utilized without detailed consideration being given to the presence of the multiplet structure in the 0, energy levels. Recently,18two of the present authors have examined in detail, using the Hund’s case (b) representation, the conditions under which such assumptions are justifiable. Both exact and approximate expressions have been given for the differential and integral, state-to-state and total collision cross sections, explicitly taking into account t Guelph-Waterloo Centre for Graduate Work in Chemistry and Department of Chemistry. Guelph-Waterloo Centre for Graduate Work in Physics and Department of Physics.
*
0022-3654/84/2088-203 1$01.50/0
the presence of the nonzero electronic spin of multiplet-I: state linear molecules colliding with %-state atoms. The present work is a logical extension of our previous work to generalized relaxation cross sections. Calculations of these cross sections for the 0,-He and 02-Ar systems will be reported in a later paper. Section 2 presents the general line-shape expressions, valid within the impact approximation for the collisional broadening cross sections, both for the magnetic dipole-allowed and for the Raman spectral lines of 0, infinitely dilute in a noble gas. Exact CC expressions for the calculation of the generalized relaxation cross sections are presented in section 3. The appropriate CS and 10s expressions, obtained from the exact C C expressions, are presented in sections 4 and 5, respectively. The paper concludes with a discussion of the present results in section 6. 2. General Line-Shape Expression
Under the impact approximation in which it is assumed that (1) See, e.g., S. Stoke and J. Reuss (Chapter 5) and D. Secrest (Chapter 8) in “Atom-Molecule Collision Theory: A Guide for the Experimentalist”, R. B. Bernstein, Ed., Plenum Press, New York, 1979. (2) L. D. Thomas, M. H. Alexander, B. R. Johnson, W. A. Lester, Jr., J. C. Light, K. D. McLenithan, G. A. Parker, M. J. Redmon, T. G. Schmalz, D. Secrest, and R. B. Walker, J . Comput. Phys., 41, 407 (1981). (3) See, e.g., D. J. Kouri, ref 1, Chapter 9. (4) G. A. Parker and R. T. Pack, J . Chem. Phys., 68, 1585 (1978); A. S. Dickinson and D. Richards, J . Phys. B, 16, 2801 (1983). (5) W.-K. Liu, F. R. McCourt, D. E. Fitz, and D. J. Kouri, J. Chem. Phys., 71, 415 (1979); 75, 1496 (1981); 76, 5112 (1982); D. E. Fitz, D. J. Kouri, D. Evans, and D. K. Hoffman, ibid., 74, 5022 (1981); D. E. Fitz, D. J. Kouri, W.-K. Liu, F. R. McCourt, D. Evans, and D. K. Hoffman, J . Phys. Chem., 86, 1087 (1982); F. R. W. McCourt and W.-K. Liu, Discuss. Faraday. SOC., 73, 241 (1982). (6) (a) S. Green, L. Monchick, R. Goldflam, and D. J. Kouri, J . Chem. Phys., 66, 1409 (1977); (b) R. Goldflam and D. J. Kouri, ibid., 67, 4149 (1977); (c) R. T. Pack, ibid., 70, 3424 (1979). (7) S. I. Chu, Astrophys. J., 206, 640 (1976). (8) R. N. Dixon and D. Field, Proc. R . SOC(London),Ser. A , 368, 99 (1979). (9) D. P. Dewangen and D. R. Flower, J . Phys. B, 14, 2179 (1981). (10) R. N. Dixon and D. Field, Proc. R . SOC.(London), Sec. A , 366, 225 (1979). (11) H. Klar, J . Phys. B, 6, 2139 (1973); H. Klar and M. Klar, ibid., 8, 129 (1975). (12) M. H. Alexander, J . Chem. Phys., 76, 3637 (1982). (13) M. H. Alexander, J . Chem. Phys., 76, 5974 (1982). (14) (a) R. G. Gordon, J. Chem. Phys., 46,448 (1967); U. Mingelgrin and R. G. Gordon, ibid., 70, 3828 (1979); (b) K. S. Lam, J . Quant. Spectrosc. Radiat. Transfer, 17, 351 (1977); E. W. Smith and M. Giraud, J. Chem. Phys., 71, 4209 (1979); E. W. Smith, ibid., 74, 6658 (1981). (15) (a) J. P. Cebe, M. Giraud, and E. W. Smith, Chem. Phys. Lett., 81, 37 (1981); (b) M. Berard, P. Lallemand, J. P. Cebe, and M. Giraud, J. Chem. Phys., 78, 672 (1983). (16) F. Battaglia, F. A. Gianturco, P. Casavecchia, F. Pirani, and F. Vecchiocattivi, Faraday. Discuss. 73, 257 (1982); F. A. Gianturco and A. Palma, “Intramolecular Dynamics”, J. Jortner and B. Pullman, Ed., Reidel, Dordrecht, 1982, p 63. (17) M. Faubel, K. H. Kohl, J. P Toennies, and F. A. Gianturco, J. Chem. Phys., 78, 5629 (1983). (18) G. C. Corey and F. R. McCourt, J . Phys. Chem., 87,2723 (1983).
0 1984 American Chemical Society
Corey et al.
2032 The Journal of Physical Chemistry, Vol. 88, No. 10, 1984 the time scale of interest is much greater than the duration of a collision, the line-shape function has been given by Fano'' and Ben-Reuven20 in terms of Liouville vectors as
m
where p is the spectral transition operator for the radiatively active molecule coupled to the external electromagnetic field, p is the equilibrium density matrix for the active molecule, Lo is the unperturbed Liouville operator for the active molecule, expressed in terms of the Hamiltonian (using units of h = 1 ) as
Figure 1. Schematic representation of a spin multiplet level of molecular oxygen: The ordering of j = N - 1 and j = N 1 is inverted for N >
Lo = [Ho, 1 (2.2) and A is a relaxation matrix whose elements are given below. In the Liouville space formalism, an operator A in ordinary Hilbert space becomes a vector IA)) in Liouville space, in which inner products are defined by taking the trace ( ( A I B ) )= T r AtB
(2.3)
The trace implicit in eq 2.1 is most conveniently evaluated in terms of a basis of eigenfunctions of Ho.For diatomic molecules, the eigenvectors can be denoted by lajmj) satisfying Holdmj) = E ( a , j )l&j)
(2.4)
where j denotes the total angular momentum of the diatom apart from nuclear spin, mj denotes its projection along the space-fixed z axis, and a represents the remaining quantum numbers. When this basis is used, the line-shape expression of (2.1) becomes2'
(2.5) wheref represents a f j f ,etc. In this equation pJ is the reduced matrix element ( afjhlpllaij i ) obtained via the Wigner-Eckart theorem22 ( a f j f r n j f I ~ Q l a i j i m j i (-1y'f"jf )=
(
lji)
if -mjf
( a fjfllgllaiji)
Q (2.6)
being the tensorial rank of ~ 1 .For electric or magnetic dipole transitions, K = 1 , while for Raman or electric quadrupole transitions, K = 2. Further, coo is a diagonal matrix whose elements are eigenvalues of Lo, i.e. (2.7) [ w o l f t i 'fi = 6 f f j S i i rwfi K
where w f i = E ( a f j f )- E(aiji);n is the density of the perturber, ( u ) = ( 8 k B T / r p 1 ) 1is / 2the average relative velocity (pIis the reduced mass of the colliding pair), and u(")is called the relaxation cross section matrix. The elements of u(") are given by (ufiiSi) =
where
Jdx xe-Xu)J,fi(Ek);
x = Ek/kBT
(2.8)
is given by
i=N
+
5.
pair, and SJ(a'j'l',ajl) is the S-matrix element defined in section 3. Equation 2.5 gives the spectral function for an entire band. A first-principles calculation involves the computation of all the relaxation cross sections and the inversion of the complex matrix appearing in eq 2.5. Such an exact calculation is extremely time consuming and has not been attempted. However, at low pressure individual spectral transitions are well resolved, Le., the linecoupling relaxation frequencies n( u ) u$)t,fi(f # f', i # i 3 are much smaller than the transition frequencies wfi and only the diagonal cross sections ux!,, are important. In such a case the spectrum appears as a sum of Lorentzian lines with the i f line having line width and line shift given by n ( u ) Re (u$h) and - n ( u ) Im (u$h,) respectively. For the collisional broadening of the microwave absorption by O2 molecules in the 3Z.9electronic ground state, the molecular state can be accurately approximated by the pure Hund's case (b) basis INSjmj) corresponding to the coupling scheme N = j - S where N and S are the nuclear rotational and electronic spin angular momenta, respectively, and S = 1. For a given value of N , a triplet of states corresponding to j = N + 1, N , and N - 1 is formed, represented schematically in Figure 1. Since O2 is homonuclear, it has zero permanent electric dipole moment and electric dipole transitions are thereby forbidden. However, the total spin is nonzero and magnetic dipole transitions are allowed. The spectral transition operator for magnetic dipole transitions is23
-.
b =
pB[gls + (gf-g,)S(t
+ gNw
PBgLS ( 2 . 1 0 )
where pg is the Bohr magneton, g,, gf, and gN are magnetic g factors defined in ref 23 and f is a unit vector along the axis of the O2 molecule. The allowed transitions are given by AN = 0 and Aj = 0, f 1 . (In an exact treatment of the O2 system by Tinkham and Strandberg,23 AN = 2 transitions are weakly allowed; such transitions would have much higher frequencies and weaker intensities, and will not be discussed here.) The Aj = f l transitions correspond to resonant transitions, while Aj = 0 gives rise to nonresonant transitions. The reduced matrix elements are ( N , j = NlIplIN, j = N 1) = & B [ N ( ~ Ni3 ) / ( N -k 1)I"2g,
+
(N,j=AIlpIlN,j=N-l) = - 2 p g [ ( 2 N - 1 ) ( N + l ) / w ' 1 2 g l (2.11) ( N , j = AIIPIIN,j = N ) = ~ P B [ ( +~ 1N) / N ( N -I-l)11i2gl
[6(af'jf'ai'ji'l'lafjfai j i 1) - SJf(a~'jf'l',olfjfl)SJi*(ai'ji'l', aiji l ) ]
(2.9)
In eq 2.9 6(x'y'-.z'lxpz) denotes the product of Kronecker deltas 6 ~ x 6 y t y . . . 6 z ~Ek z , = h2k2/2p1is the precollisional kinetic energy, J and I are the total and orbital angular momenta of the colliding (19) U. Fano, Phys. Rev., 131, 259 (1963). (20) A. Ben-Reuven, Phys. Reu., 141, 34 (1966); 145, 7 (1966). (21) R. Shafer and R. G . Gordon, J. Chem. Phys., 58,5422 (1973); W.-K. Liu and R. A. Marcus, ibid., 63, 272 (1975). (22) A. R. Edmonds, "Angular Momentum in Quantum Mechanics", Princeton University Press, Princeton, 1957.
The molecular energies E(a,j ) = E ( N ,j ) can be found in ref 23, from which the transition frequencies wfi = E ( N ,jJ) - E ( N ,j , ) can be evaluated. Alternatively, the experimental values of the resonant frequencies can be used in the evaluation of the line-shape expression. 3. The Close-Coupled Equations In order to calculate the relaxation cross sections of eq 2.9, the scattering problem between the radiatively active diatomic molecule and its structureless perturber must be solved. The (23) (a) M. Mizushima, J. S. Wells, K. M. Everson, and W. M. Welch, Phys. Reu. Lett., 29,831 (1972); (b) M. Tinkham and M. W. P. Strandberg, Phys. Reu., 91, 931, 951 (1955).
The Journal of Physical Chemistry, Vol. 88, No. 10, I984 2033
Pressure Broadening of Multiplet-Z Molecules S-matrix elements required can be obtained from the asymptotic behavior of the wave function of the colliding system. Following Arthurs and D a l g a r n ~ the , ~ ~eigenvectors of the total angular momentum for the colliding pair can be defined by
Cm1
1
lO]l,JM) =
(-l)-l+l-M(2J t
mp1
1)1/2
J
ml -M) ~ l m . T ) l h ) (3.1)
where Ilml) is the eigenvector describing the orbital motion
(Lj Iml J-Id) is a Wigner 3-j coefficient for the vector coupling scheme j+I=J (j=N+S) (3.2) and ml, mj, and M are the projections of the orbital angular momentum I , the total angular momentum of the diatom (apart from nuclear spin) j, and the total angular momentum of the colliding pair Jrespectively, on the space-futed z axis. For diatomic molecules in multiplet-t: electronic states described by the Hund’s case (b) basis, a represents the pair of quantum numbers (NS). The wave function is then expanded in terms of laj1,Jm) as
X P ; ~ ( ~ R=) ~ , p a ’ j ~ i f , a j i(l i~~) l a j i , ~ (3.3) ~) where i describes the orientatioc of the internuclear axis of the diatomic molecule and R = R R is the position vector of the perturber relative to the center of mass of the diatom. The radial functions UJ(a‘jfl‘,ajl~R) satisfy the close-coupled equations
where the ((N1)dJn)basis elements, given by I(N1) d A )=
c
(-1)-N+’-”(2dt 1 y
mNml
cNLl
-:>.-NHlrni)
(3.9)
are identical with the total-J angular momentum eigenvector of a spin-free system. The orthogonal transformation between the two total-J coupling schemes defines a Wigner 6 - j symbol and the basis elements in (3.1) and (3.8) are therefore related by1* I ( N S )jl,J M ) =
The matrix elements of the spin-independent intermolecular potential in the basis of (3.8) then simplify to ( ( N ?’) d’N,S’mil Vl (Nl)dM,Sm,) = G(d’AZ’S’m~~bMSm,) Vd(Nfl’,NI)(3.1 1) where Vd(Nfl’,NI)= ((N’l’)dJnlJI(Nl)dM).If eq 3.10 is used, the matrix elements of Vappearing in the close-coupled equations (3.4) become ( ( N’ S ’ ) j ‘1 ’, JIM I VI (NS)jlJ M ) = 6 sfs(-1 y’ ‘-j-”+’ [ (2j
I‘ N ’ d L’ d ( 2 d + 111 S J j’
11
+ 1) ( 2 j + 1) ]
112
X
1 N d
S J
/ V ‘ ( N ’ l ’ , N l ) (3.12)
It is often convenient to express V in a Legendre expansion as V = C V ~ ( R ) P ~ ( C6) OS
(3.13)
x
where V is the potential for the nuclear motion of the colliding particles within the Born-Oppenheimer approximation, and k’ is the wave number of the (a’j?’) channel given by h 2 k f 2 / 2 p r= Etot- E(a’j’) (3.5) in which E,,, is the total energy of the colliding system. Since the potential Vis independent of the orientation of the space-fixed frame of referepce, the matrix elements of V are diagonal in J and independent of M . The S-matrix element is defined by imposing the following boundary condition on the radial function: UJ(a’j?’,ajIIR) Rzm d(a’j?’lajl)exp[-i(kR - 1 ~ / 2 ) ](k ’/ k)’/2SJ(a’jfl’,ajl) exp [i(k’R - I ’7r/2)] (3.6) It will be assumed that the potential Vis independent of the electronic spin of the diatom. Therefore, the binary collision between the radiatively active diatom and the structureless perturber does not 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 nuclear rotational angular momentum N. Because S and N are weakly coupled the spin cannot follow this change in N. The collision indirectly affects the total molecular angular momentum j through the recoupling of N to S after the target recedes. Because of this collisional mechanism it is convenient to transform to a total-J representation defined through the coupling scheme N+I=d d+S=J (3.7) The basis elements in this coupling scheme are defined by
where 6 is the angle between the axis of the diatom and the line joining its center of mass to the colliding atom. The matrix element of )’in the basis of (3.9) is then given by the well-known formula2* d
V ( N ’ I ’ ,N 1) = N’ h N
~Vh(R)(-l),Y”’~d[N‘NE’l]l’Z h (0
0 O)(O
I’ h 1
d 1’ N ’
0 O)[k N 1
1
(3.14)
where [xIxz...x,] denotes (2x1 + 1)(2x2 + 1)...(2x, + 1). Substituting eq 3.14 into eq 3.12 and evaluating the sum over d yields ((N’S‘)j’l’, JJIIVI(NS)jI,JM)= 6sisc Vh(R)(-l)S-h-JIN’Ni’iI’ll ‘ I 2 x
x
In principle, the close-coupled equations (3.4) together with eq 3.15 can be solved numerically and the S-matrix elements can be extracted from the boundary condition in (3.6). However, because of t’he result (3.12), the dimension of the set of coupled equations cam be significantly reduced by defining a spin-independent radial function Ud(Nfl‘,NIIR)by UJ(N’S’j‘l’,NSjlIR) = 6s1Sz(2dt l)[i’j]”2(-1Y”-”’’ x d
I (NI)d S,JM) =
Substituting eq 3.16 into eq 3.4 and using eq 3.12, one can show that Ud(N7‘,NIIR)satisfies the usual close-coupled equations for the scattering of a ‘%state diatom by a ‘S-atom, viz. (24) A. M. Arthurs and A. Dalgarno, Proc. R . SOC.(London),Ser. A , 256, 540 (1960).
(25) I. C.Percival and M. J. Seaton, Proc. Cambridge Philos. Soc., 53, 654 (1957); D. M. Brink and G. R. Satchler, “Angular Momentum”,2nd ed, Clarendon Press, Oxford, 1968, see especially eq 5.14.
2034
Corey et al.
The Journal of Physical Chemistry, Vol. 88, No. 10, 1984
2 h ",," c Vd(N'i:N"i'?Ud(N'ri':Ni(R) (3.17) 2
The S-matrix elements Sd(Nrl',NI) can then be defined by the asymptotic behavior of Ud(N'l',NIIR) which is similar to that given in (3.6). They are related to the SJ(NS'jrl',NSjl) elements by a relation analogous to eq 3.12:
(J
M -mZ3' jl3' jm,' l f ) ( i 2 , mz,' t - m 3 mz' i,
y ( . V S ' j 'I' ,NSjl) =
(3.18)
If we substitute eq 3.18 into eq 2.9, the relaxation cross section becomes
"
I
)
(4.2)
Because of the selection rule on the m's in the Wigner 3 - j symbols for given m l , m2, m,', and m i , the sum on the left-hand side of (4.2) is only over j12 and that on the right-hand side is only over one of M , m23, m2{, and m3. This result can easily be derived by first expanding the product of a 6 - j symbol and a 3 - j symbol into a product of three 3 - j symbols by eq 6.2.8 of ref 22, and utilizing the orthogonality relation then performing the sum over j12 eq 3.7.7 of ref 22. Substituting eq 4.1 into eq 3.18 and using eq 4.2 to carry out the sum over 6, we obtained the following expression for the CS approximation to the total-J S-matrix:
S d f ( N , ' l ' , N f l ) S dI " ( N i ' l ' , N , l ) ]( 3 . 1 9 )
The finite summations over 6 ~ a n 6, d in eq 3.19 are restricted by the triangle inequalities A(SJfdf) and A(SJ,d,). Because the intermolecular potential is independent of the electronic spin of the diatom, the dependence of the pressure-broadening cross section on S enters only through the 6 - j symbols relating the two total-/ coupling schemes.
The CSA expression for dK) is then obtained by substituting eq 4.3 into eq 2.9 and evaluating the sum over Jfwith eq 4.2. The result is
4. The Centrifugal Sudden Approximation (CSA)
The expressions developed in the previous sections for the relaxation cross section, eq 2.9 and 3.19, imply an "exact" solution to the scattering dynamics (eq 3.4 or 3.17). Such close-coupled calculations can, in principle, be carried out by various numerical methods developed in the past decade2 but in practice these have only been performed for systems in which a small number of channels are coupled by the potential. For most molecular systems, dynamical approximations must be introduced to reduce the computational effort. One of the most successful of such approximation schemes is the centrifugal sudden approximation (CSA)3 in which the dimensionality of the close-coupled equations is reduced by replaci!g the orbital angular momentum operator by an effective value 1 and block diagonalizing the intermolecular potential matrix. Details of the CSA are well documented in the l i t e r a t ~ r eand , ~ only the C S approximation to the S-matrix will be discussed here. Within the CSA, the Sd(N'l',NI) appearing in (3.18) can be expressed as3
S i h f ( N f ' ,N f ) s i A i*(Nj',N j ) ] ( 4 . 4 )
The same result can of course be obtained by substituting eq 4.1 into eq 3.19 and using eq 4.2 repeatedly. If the AVL scheme is employed, no further simplication can be found since f depends on both 1 and I ! For the IL/FL schemes, the summation over /'/I can be performed explicitly and subsequently the sum over J1can be carried out to give the simpler result
S ' ( N I ~ ~ , N=I )
where ST~(N',N)is the CSA S-matrix element obtained by solving a set of close-coupled equations similar to eq 3,17 but with a lower dimensionality. In general, the parameter I depends on both I and 1' and- the accuracy of the CSA depends on the particujar choice of 1. Three common choices of 1 are the following: 1 = 1 (the initial-I [IL] labeling scheme), f = I' (the final-1 [FL] labeling scheme), and f = 1/2(1+ 1') (the average1 [AVL] labeling scheme). The AVL scheme is the most accurate, but the IL and FL schemes allow significant simplification of the expression for dX)and are sufficiently accurate in many applications. The following identity of the Wigner symbols is very useful for manipulations within the CS approximation:
[ S (Nl'Nf'INiNf)- sihl'(N,',Nl)Sihf(Nj,Nf)l (4 5 )
5. The Infinite-Order Sudden Approximation (IOSA)
Although the CSA affords a significant reduction in computational effort in comparison with the C C approach, solving the CS coupled equations to obtain the Sn(N',N)'s may still prove to be impractical for many molecular systems. A further sim-
Pressure Broadening of Multiplet-2 Molecules
The Journal of Physical Chemistry, Vol. 88, No. 10, 1984 2035
plification, known as the infinite-order sudden approximation (IOSA)? can then be made. This approximation invovles solving a one-dimensional Schrodinger equation for various fixed orientations 0 of the diatomic molecule with respect to its colliding partner for the S-matrix element
SL7(@)= exp[2ix7(6)1
(5.1)
factor u L ( f f ) is equivalent to the well-known Percival-Seaton coefficient2tf;(N)l”N,NflK). Preliminary calculations;* within the 10s approximation, of the pressure-broadening cross sections for 02-noble gas mixtures show that neglect of the nonzero electronic spin in molecular oxygen can introduce an error of the order of 10%. For energetically elastic transitions N N, eq 5.2 gives f = k where k is the wave number of the entrance channel. Thus, the diagonal relaxation cross sections u,!$fi whose real and imaginary parts give the width and shift, respectively, of an isolated spectral line, exhibit the factorization property such that once the dynamical quantity +
where qe7(6)_isthe phase shift corresponding to an effective channel labeled by Z and k . Note that 1 has the same interpretation as in the CSA, while f is an effective wave number which in general is a function of N’ and N. In order that the resulting energyaveraged cross sections satisfy detailed balance, a simple choice of f is obtained from the relation
+
h2f2/2/.lr = E,, - f/z[E(N’) E(N)]
(5.2)
The Sd-matrix element is then given as Sd(N’Z’,NI) = i‘“’-2 j ( N’l’dJnlSi 7(6) INldJn)
(5.3)
) be expanded in terms of the Legendre series used The S ~ j ( 0can to expand the intermolecular potential as
and (5.3) then gives S d b V l ’ . N l )=
has been evaluated, the cross section ufi);klfor any spectral line i ’fmay be obtained without any further dynamical calculations ) eq 5.7. The quantity by substituting uLoin place of o L ( f f , i lin uLo defined in eq 5.10 has the following simple interpretation obtained by considering the degeneracy-averaged state-to-state integral cross section u(N’+ NlEk) =
+ 1)-’(2d + 1)[6(N’lINI) - IS‘(N’l’,NI)IZ] (5.11) k2 di? for inelastic transitions from the N to ”levels of the corresponding spinless diatomic molecule. When the IOSA result of eq 5.5 is substituted into eq 5.1 1 uLo can be identified as U(L+OlEk) = ( k 2 / P ) U L O ( E k ) (5.12) ?I.
- Z(2N
where Bk = h2k2/2~.,.This factorization property is a generalization of a result first obtained by Goldflam et a1.6bfor diamagnetic molecules. As in the CSA, no further simplication of the relaxation cross section is obtained if the AVL scheme is employed. However, for the IL or FL schemes, the sums over l’or 1 in eq 5.8 can be carried out explicitly, yielding The sum over d has been performed by using eq 6.2.12 of Edrnonds.,, The IOSA expression for up&, can be obtained by substituting either eq 5.6 into eq 2.9 or eq 5.5 into eq 3.19 with the sum over Jfand Ji evaluated by using a Racah-Elliot relation (eq 6.2.12 of Edmonds22)and the orthogonality property (eq 6.2.9 of Edmonds) of the 6 - j symbols, respectively, to give finally (K)
‘ j ” i ’ ,f i
(Ek) = x(-1)L-ji’-ji-K [ N i ’N i j i ’ l i ~ ’ f i v f i f ’ j f ]x” 2
where
Upon setting S = 0, eq 5.7 can be reduced to (see, e.g., eq 6.3.2 of Edrnonds2,) the 10sapproximation to the effective cross section appropriate for linear diamagnetic molecules infinitely dilute in a bath of closed-shell atoms, namely6b f r i t , f i ( E k ) =T ( - 1 ) - K [ N i ’ N i N f ’ N f ] ”X2
o(K)
u L ( f f , f i= ) ?C(2L k2 7
+ 1)-’(2f + 1 ) ( ~ -5 S~f, ~ Si;)
(5.13)
in either labeling scheme.
6. Discussion The C C formalism for rotationally inelastic collisions between open-shell linear molecules and closed-shell atoms, as set out recently in ref 18, has been applied to the description of the effective cross sections governing the pressure broadening and shifting of spectroscopiclines. It has been assumed that the impact approximation, which lies at the heart of most spectroscopic line-broadening theories, remains valid for these systems so that the basic line-shape theory of Fanolg and Ben-Reuven20 may be applied. The description presented here has been given in terms of the Hund’s case (b) coupling scheme for the states of multiplet-2 linear molecules, keeping in mind the best-known stable species, 0, with its 3Z; electronic ground state. It has been shown in ref 18, for example, that transformation to an equivalent spin-free total-d basis for multiplet-8 molecule-closed-shell atom collisions significantly reduces the number of coupled channels that one needs to consider. This reduction in the dimensionatlity of the coupled channel scattering problem is independent of any dynamical approximation. This seems to be a unique advantage obtained by employing the case (b) representation. Of course, even 0, in its electronic ground state does not represent a molecule which is describable as a pure Hund’s case (b) molecule, although it is nearly so [see, e.g., ref 15b and
L
The collection of Racah coefficients multiplying the dynamical
(26) K. Altmann, G. Streg, J. G. Hochenbleicher, and J. Brandmuller, 2. Nuturforsch, A , 27, 56 (1972); M. Bdrard and P. Lallemand, Opt. Commun., 30, 175 (1979). (27) D. C. Clary, J. N. L. Connor, and H. Sun, Mol. Phys., 49, 1139 (1983). (28) G. C. Corey and F. R. McCourt, to be submitted for publication.
J. Phys. Chem. 1984, 88, 2036-2045
2036
ref 261. The deviation of the state of O2 from a pure Hund’s case (b) description, while rather important from the point of view of spectroscopic line intensities, for example, would not be as important in the description or calculation of the pressure broadening of such spectroscopic lines. Such an assumption has already been made recently by B&rard et al.’5b Provided the intermolecular potential surface is spin independent (a very reasonable assumption for the interaction of multiplet-8 molecules with closed-shell atoms), the electron spin quantum number of the multiplet-2 molecule appears only in the six-j symbols multiplying the product of S-matrix elements evaluated in the spin-free t o t a l 4 basis. The basic C C expression for the effective cross section governing the pressure broadening (and shifting) of spectroscopic lines is given by eq 3.19: K has the value 0 for rotational relaxation, 1 for electric or magnetic dipole-allowed transitions, and 2 for electric or magnetic quadrupole-allowed transitions and for Raman and depolarized Rayleigh light scattering. The equivalent expression within the C S dynamical approximation is given by eq 4.4 and its simplified version (obtained
by using either initial-1 (IL) or final-1 (FL) labeling) is presented in eq 4.5. For molecules such as 0 2 , even the CSA expressions will be prohibitively expensive to use for numerical calculations. Thus, it will likely be necessary to employ the more approximate but nonetheless reasonably accurate2’ infinite-order sudden approximation. The appropriate generalization of the diamagnetic IOSA expressions6for the pressure-broadening cross sections have been given in eq 5.7. The effective cross sections required for a description of the “diffusional narrowing” (or Dicke narrowing) effect in low-density gases will be presented in a later publication. An application of the present theoretical description to the calculation of differential and integral cross sections as well as to the pressure-broadening cross sections for the magnetic dipoleallowed and pure rotational Raman lines of O2 in Ar and H e mixtures is presently underway and will be presented in due course. Acknowledgment. This research was supported in part by NSERC of Canada grants in aid of research. Registry No. 02,7782-44-7.
Two New Anisotropic Potential Energy Surfaces for N,-He: The Use of Hartree-Fock SCF Calculations and a Combining Rule for Anisotropic Long-Range Dispersion Coefficients R. R. Fuchs, F. R. W. McCourt,* A. J. Thakkar,* Guelph- Waterloo Centre for Graduate Work in Chemistry and Department of Chemistry, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
and F. Grein Department of Chemistry, University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A3 (Received: October 1 I , 1983)
Two new anisotropic potential energy surfaces of the Hartree-Fock plus damped dispersion (HFD) type have been obtained for the He-N2 van der Waals molecule. The SCF energies utilized in constructing the short-range part of the interaction were computed at 30 points on the surface by using a basis set of somewhat better,than dodble {plus polarization quality. An improved functional representation of the SCF energy, taking into account the presence of induction energy contributions at this level, has been utilized in order to obtain more accurate repulsion energies. Anisotropic C6and C8dispersion coefficients, utilized in constructing the long-range part of the interaction, have been calculated with, a new combination rule reported here and tested on the H2-He, -Ne, -Ar interations. Two novel means are introduced for estimating anisotropic damping factors which are used in the description of the correlation energy contributions in the HFD model. The essential differences between the two surfaces reported here arise from these estimates. Comparison is made with other available surfaces for the He-N2 system. A detailed study shows that the Drude correction included in the recent He-N2 surfqce of Habitz, Tang, and Toennies causes their surface to be significantly more repulsive than any other He-N,,surface. In a later paper, all these surfaces will be compared with experiment on the basis of their predictions of the temperature dependence of the second virial coefficient, shear viscosity, and diffusion, and of differential scattering cross sections at a fixed energy.
1. Introduction
Over the past 20 years it has become increasingly obvious that simple spherical models of intermolecular potentials,’ such as the Lennard-Jones model, are inadequate2 for the description of many physical properties. As a result, more elaborate potential models have been proposed initially for atomatom interactions2and, more (1) Hirschfelder, J. 0.; Curtiss, C. F.; Bird, R. B. “Molecular Theory of Gases and Liquids“; Wiley: New York, 1954. (2) Maitland, G. C.; Rigby, M.; Smith, E. B.; Wakeham, W. A. “Intermolecular Forces: Their Origin and Determination”; Oxford University Press: Oxford, 198 1,
0022-3654/84/2088-2036$01,50/0
recently, for anisotropic atom-diatom system^.^ In recent years, a number of models which contain at most a single parameter (and often none) have been developed.e’’ With (3) (a) Loesch, H. J. Adv. Chem. Phys. 1980,42,421. (b) LeRoy, R. J.; Carley, J. S. Ibid. 1980, 42, 353. (c) Pack, R. T.; Valentini, J. J.; Cross, J. B. J . Chem. Phys. 1982, 77, 5486. (4) Ahlrichs, R.; Penco, R.; Scoles, G. Chem. Phys. 1977, 19, 119. (5) Tang, K. T.; Toennies, J. P. J . Chem. Phys. 1977, 66, 1496. (6) Feltgen, R. J . Chem. Phys. 1981, 74, 1186. (7) Ng, K. C.; Meath, W. J.; Allnatt, A. R. Mol. Phys. 1979, 37, 237. (8) Douketis, C.; Scoles, G.; Marchetti, S.; Zen, M.; Thakkar, A. J. J . Chem. Phys. 1982, 76, 3057. (9) Varandas, A. J. C.; Brandao, J. Mol. Phys. 1982, 45, 857.
0 1984 American Chemical Society
