Semiclassical theory of rotational transitions in collisions of diatomic moleculesC. F. Curtiss Citation: The Journal of Chemical Physics 67, 5770 (1977); doi: 10.1063/1.434836 View online: http://dx.doi.org/10.1063/1.434836 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/67/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Semiclassical picture of collision-induced Λ-doublet transitions in diatomic molecules J. Chem. Phys. 107, 5473 (1997); 10.1063/1.474252 On the semiclassical theory of collisioninduced vibrational–rotational transitions in molecules J. Chem. Phys. 72, 3805 (1980); 10.1063/1.439595 Semiclassical theory of rotational excitation in collisions of diatomic molecules J. Chem. Phys. 63, 2738 (1975); 10.1063/1.431625 Semiclassical Theory of Vibrational Excitation for Diatomic Molecules with Conserved Collision Energy J. Chem. Phys. 54, 1539 (1971); 10.1063/1.1675052 Semiclassical Theory of Rotational Excitation of a Diatomic Molecule by an Atom J. Chem. Phys. 43, 2930 (1965); 10.1063/1.1697252

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Semiclassical theory of rotational transitions in collisions of diatomic moJecules8 )

C. F. Curtiss

Theoretical Chemistry Institute. University of Wisconsin. Madison. Wisconsin 53706 (Received 2 September 1977)

In earlier papers. the theory of rotational transitions in collisions of diatomic molecules has been developed in terms of a generalized phase shift. In the present paper. the classical limit of this function is discussed and limiting expressions are developed. These expressions may be used to obtained expressions for the classical limit of various quantities associated with the collisions. in particular. an expression is developed for the moments of the transition probability with respect to the rotational inelasticity.

In a series of papers, we have discussed the theory of rotational transitions in collisions of diatomic mole­cules. The theory has been developed through the intro­duction of a generalized phase shift. This is accom­plished by first transforming the usual coupled equa­tions to a new set based on a coupling scheme involving directly the vector changes in the angular momentum of the problem. The resulting set of coupled second-order equations is then transformed to a set of coupled first­order equations. The functions determined by these equations may be interpreted as complex amplitudes and generalized phase shifts are defined in terms of the asymptotic forms of the solutions.

The usual coupled scattering equations may be writ­ten! in the form

1/!J(jajbjl;j!j~j'l') = 13(jajbjl;j~j~j'l')j/(kr) + G~9)(k')

x

where ja and jb are the angular momentum quantum num­bers of the rotating diatomic molecules and l is the an­gular momentum quantum number associated with the relative motion (the impact parameter); j is the result of the vector coupling of ja and jb and J describes the total angular momentum of the system. The unprimed values are the values before the collision and the primed are those after the collision. The function on the left is a function of the separation distance r between the two molecules and is the expansion coefficient in the partial wave analysis. The first factor on the right is a product of four Kronecker deltas and j/(kr) is a spherical Bessel function; GjO) (k) is the one-dimensional Green's opera­tor associated with the noninteracting molecules and V J(j~j;j' zt Ij~'j;'j"lll) is a matrix element of the inter­action potential.

Through a unitary transformation, the coupled equa­tions, Eqs. (1), have been rewritten2 as a completely equivalent set

X(AA') = 13(.\; .\')13(L:L;L'; OOO)j,(kr)

+ G~9)(k') L: W(AA'A")X(AA") , (2) A"

where .\ is a collective index for the tripletjajbl, and A is symbolic for the six indices jajblLaLbL. The trans-

alThis research was carried out under Grant CHE-74-17494 from the National Science Foundation.

formation replaces the three indices jj' J by three others, L~L:L', which describe the vector changes in the three basic angular momenta of the problem: the angular mo­mentum of each of the molecules and that associated with the relative motion. The essential point of the trans­formation arises from the rather surprising properties of the transformed coupling matrix W(AA' A").

The elements of the transformed coupling matrix de­pend on the separation distance between the two diatoms through the interaction potential. We have, however, obtained in a previous paper3 explicit expressions for the eigenvectors X('\)\ S) of this matrix. These eigen­vectors, surprisingly, do not depend on the separation distance or the form of the interaction potential. The eigenvectors are labeled by six indices which may be in­terpreted as the six Euler angles associated with two ro­tations Sa and Sb or more concisely S.

In a more recent paper, 4 the solutions of the trans­formed equation, Eqs. (2), are written in the form

(3) (j{+) (.\')* exp iQ{+) (AS) -j{-) (.\')* expiQ{-)(AS)] ,

wherej(+)(.\') andj<-)(.\') are independent solutions of the radial wave equation involving a spherically aver­aged "unperturbed" potential. The exponential functions exp iQ(~) (AS) may be interpreted as amplitudes of incom­ing and outgoing waves. The generalized phase shift H(AS) is defined by the asymptotic value of the amplitude of the outgoing wave,

H(AS)=tlimQ{+)(AS) • (4) r~~

In the limit that the interaction potential is spherical, this quantity becomes simply the usual phase shift TIJ ariSing in the solution of the spherical scattering prob­lem. As discussed later in this paper, all of the scat­tering information is contained in H(AS).

The equations for the complex amplitudes are cou­pled first order differential equations in the separation distance r between the diatoms. These equations may be written in the formS

(5)

5770 J. Chern. Phys. 67(12)' 15 Dec. 1977 0021-9606/77/6712·5770$01.00 © 1978 American Institute of Physics

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C. F. Curtiss: Rotational transitions in collisions 5771

where the ° Ii are operators in the space of two sets of Euler angles, indicated symbolically by S. The boundary conditions on these complex amplitudes are

r- oo ;

r-O; Q(+)-Q(-). (6)

To define the ° ii' we return to a pair of operators de­fined earlier6 by the relations

g:(z)y(S) = f as' F(±) (ASS')y(S') , (7)

where the F(z) (ASS') are matrix elements defined by Eq. (23) of Ref. 4. In the present discussion, it is conve­nient to introduce the Hermitian adjoint operator,

g:(z)ty(S) = f as' F(z) (AS'S)*Y(S') • (8)

With these definitions, it readily follows that

f as ytCS)*g:t Y2(S) = (Jas Y2(S)*g: Y1(S»)* (9)

In terms of the operators just defined and the aniso­tropic portion, v(1), of the interaction potential, the ° Ii operators are

(10)

Since the interaction potential is real, it follows from these definitions that

011 = ° 11 ,

012=0 a1 ,

ok = 0 a2 ,

i. e., the operators, On and 0aa are Hermitian.

(11)

The 01j are operators in the space of the double set of Euler angles S which depend parametrically on the co­ordinate r. Let us consider .the equations

(12)

for the operators (J>1 and (J>a in the same space (i. e., the differentiation with respect to r affects only the operator and does not "carry through") along with the conditions

(J>l(oo) =1

(J>a(OO) =1 • (13)

Clearly, solutions of these operator equations exist of the form

( i (1»)

!Pl =exp - ti '0 ,

(14)

if

i l -(It/h)'lJ(l) a '0(1) (Itlh)'lJ(l) 011 =- dte --e

o Or

= _ '0 (1)' + ;n- ['0(1), '0(1)1] _ ~(~y ['0(1) , ['0(1), '0(1)1]] + ••• ,

1 (2) (15)

I _(it/h)'lJ(2) av (/t/h)'lJ(2) ° 22 = dt e ---ar-e o

= '0 (2)' _ ;n-[v(a), '0(2)'] + ~G)\U(2), ['V (a) , '0(2)1]] + •••

where the [ ] indicates the commutator and the prime on the operator indicates the derivative with respect to r.

The last two series may be inverted by conSidering series expressions for '0 (1) and '0 (2) ordered according to the power of the 011 or 022 operators. Thus

'0 (1) - ~ '0(1) -L-J n , n=l

'0(2) - ~ 1; (2) -L-J n ,

(16)

n=l

where

1; (1), - .i. ['0(1) 0 ] 2 --2n- 1,11,

'0(1), =_ .i.[v(l) 0 ] _ .!...(!..)2[V(1) ['0(1) ° ]] 3 2n- 2 ,11 12 n- l' 1, 11 ,

(17)

'0(2) I = .i. ['0 (2) 0 ] 2 2n- 1 , 22 ,

'0(3)1 _ i ['0(2) 0] 1 (i )2['0(2) ['0(2) ° ]] 3 - 2n- 2 , 22 + 12 Ii 1, 1, 22 ,

. .,. . In this manner, one obtains expressions for operators

!Pl and (J>2, which satisfy the differential relations, Eqs. (12).

From the last set of relations, and the Hermitian property of the operators, 0u and 0aa, Eq. (11), it fol­lows that '0 (1) and '0(2) are Hermitian:

'V(l)t = 'l)<1) , (18)

V(2)t = '0(2) •

One then finds from Eq. (14) that (J>1 and (J>a are unitary operators:

(J>1=!Pi1 ,

!P~ = !Pi1 •

(19)

Let us next define two additional functions T 1 (r) and T 2 (r) as solutions of the coupled equations

J. Chern. Phys., Vol. 67, No. 12, 15 December 1977

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5772 C. F. Curtiss: Rotational transitions in collisions

(20)

with the boundary conditions

T1(oo) ==e2i~1 , (21)

T2 (0) == 1P2'1 (0) Cl\ (0)T1 (0) •

With these definitions, it may be shown that the required solutions of Eqs. (5) are

eiQ (-) -IP T - 1 1,

.Q(+) e' ==1P2T2 •

(22)

The generalized phase shift H, Eq. (4), is then given (mod 7T) by the relation

e2iH == lim e iQ (+)

r~ 00

(23) ==T2(OO) •

A differential form of the operators 5'<*) is discussed at length in a previous paper.6 It is shown there [Eq. (5)] that

5'(*) == (n2/21I1)l/4 exp(i/If) S (*) ,

where [Eqs. (66) and (67)]

S(*) == L (ilf)nS~~) nk

(24)

(25)

and explicit expressions are given [Eqs. (68)-(71)] for the S~~). The s!:) are independent of If and of order k in the angular momentum operators.

In the previous development, the exponential operator is written as a series expansion

exp(i/If)S(*) ==e(iI~)S~*) e-4*) {4=(~ilf)n<R!*)} e-sg) (26)

and explicit expressions for the operators <R!*) are given [Eqs. (75)-(79)]. From this perturbation expansion and the definitions, Eqs. (10), it follows that

021 == 012 , (27)

022

== (M/2)1/2 e(//~)(st)-sJ+)*) e-st)-st)* L(- iIf)n0!22) , n

where

0(11) == '" (-l)"'<R (-) V(-)<R (-)t n.L....J n-m m, m

(-) (-)t V(-) ==e- 11 V(I) e-g11 ,

0(12) == '" (-1)~ (-) e-g~;) V(I) e-sf;)t<R (+)t n ~ n-m m ,

m

0(22) == '" (-1)~(+) V(+)<R(+)t n L...J n-m m'

m

(28)

For values of r greater than the classical turning point, ro, S~±), and 51*) are real. Therefore, in the classical limit,

tl12 - 0; r>ro (29)

and thus from Eqs. (20), T1 and T2 are constants for r >ro. It may be shown that in this limit, the boundary condition at the origin, Eq. (21) is to be replaced by a similar condition at the turning point. Thus the con­stant values are

r>ro (30) T2 ==Cl'21(ro)lPl(rO)e2i~l; r>ro

and from Eq. (23), the generalized phase shift, classical limit, is given by

in the

e2iH == 1P2'1 (ro) Cl'1 (ro) e2i~1 •

It follows from Eqs. (27) that for r>ro

022

== (M/2)1/2 e-2 st) L (- iIf)n0~22) • n

From Eqs. (28) and the explicit expressions for the <R~), one then finds that

(-) S (-) 0~11) ==V(-) ==e-sll V(I) ell,

(1 11 ) == 11 dt1 ([S~i)(tl)' V(-)]. +[S~2)(t1)' V(-)]} , o

0122 ) == fdt1 ([s~r)(t1)' V(·)]++[S~~)(tl)' V(+)]} ,

where [ ]. indicates the anticommutator, and

(31)

(32)

(33)

(34)

Finally let us return to the operators defined by Eqs. (16), and given implicitly by Eqs. (17). From the ex­panSions, Eqs. (32), it follows that

'0(1) == '" (_ ilf)m'O(l) n LJ nm ,

m (35)

'0 (2) == '" (_ iti)m '0 (2) n L..J nm ,

m

where

'0 1!)I == - (M/2j1/2 e-2 s1-) 0~l1) ,

1 (M/2)1/2 e-2 s1-) '" ['0(1) ['0(1) 0(11)]] - ii L..." 1,n-m'+2, 1, m'-m, m , mm'

(36)

J. Chem. Phys., Vol. 67, No. 12, 15 December 1977

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C. F. Curtiss: Rotational transitions in collisions 5773

V (Z)' = .!.(M/2)1/2 e-2sf+) "['0(2) 0 (22)] 2n 2 ~ 1,n-m+b m ,

no

The standard "5 matrix" determined by the asymptotic form of the functions determined by Eqs. (1) is related to an analogous 5 matrix determined by the asymptotic form of the functions determined by the transformed equations, Eqs. (2) by the same unitary transformation. Thus, one finds that

5J(j.jbjl;j~j~J' n = (- l)J+l'+J [(2j. + 1}(2jb + 1}(21 + l)(2j + 1)

X (2J'+1)Jl!Z L (-1)L'[(2L!+1){2L~+1)(2L'+l)]1/2 L'L'L' ab ~~

{IJ.' 1 L,}~j. j~ L~~

j' J ljb j~ L~(. 5(XA') ,

~ j j' L"

where the braces indicate 6-j and 9-j symbols. It fol­lows from the second transformation, Eq. (3), that the elements of the transformed 5 matrix are given terms of the generalized phase shift by

(38)

Thus the generalized phase shift determines all of the scattering information. The expressions developed above may be used to develop expressions for the classi­cal limit of those quantities associated with the scatter­ing which have such limits.

As an example of the quantities which are determined by the generalized phase shift, we consider the degen­eracy averaged cross section which may be written in the form

I(jajb;j~j~) = 1~2 L (21 + l)P(jajbl;j~j~I') , (39) fC" 1/'

where the final factor may be interpreted, loosely, as the probability that, in a collision, a transition occurs in which the three angular momenta change from the un­primed to the primed values. This transition probability is given by

P(X; x') = L 1 Il(X; X/)Il(L~L~L'; 000) - 5(AA') 12 . L~LbL'

(40)

Next let us define a commuting set of Hermitian opera­tors 'Yl by

'Yl =x(a) ,

'Yz = X(b) ,

'Y3 =61 , 'Y _£(a)! + o(a)2 +£(a)2

4- 1 ""'2 3,

'Ys =£}b)2 +~b)2 +£~b)2,

'Y6 =3R~ +~ +3R~ ,

(41)

where the £~a', £~b), and 3Ri are angular momentum op­erators defined in an earlier paper7 and x(a), X(b), and 6J are sums of products of these operators also defined earlier. Then it may be shown that

'Y i X(XA'S)* = Y i (XA') X(XA'5)* , (42)

where

Yl(AA')=[j:(j~+1)-ja(ja+1)]n2 ,

Y!(XA') = [j~(j~ + 1) - jb(jb + 1)]n2 ,

Y3(AA') = [l'(l' + 1) -1(1 + l)]n Z ,

Y4{AA') = L!{L! + 1)1i2 ,

Y5(AA') =L~(L~ +l)n 2 ,

Ys(AA')=L'(L' +l)nZ •

(43)

Let us now define an operator 'Y which is a sum of products of the 'Y i • Then since the 'Yi form a commuting set of Hermitian operators

(44)

where Y(A-A') is a similar sum of products of the eigen­values, Yi(XA'). One then finds, using Eq. (38), once again that

~5(AA')*Y(AA')5(AA')= (8:!)zJdSe-2i1t*'Ye2iH •

From Eq. (31), it follows that

where

(5' = (5'a1 (ro) (5'1 (ro)

is a unitary operator [see Eq. (19)],

(5't =(5'-1 •

(45)

(46)

(47)

(48)

The last result, Eq. (40), may thus be written in the form

(49)

Because of the form of this result it is convenient to de­fine a set of transformed operators

Yi = (5'-l'Y j (5'. (50)

(51)

is a sum of products of the 'Y; similar to that used in the definition of 'Y, and from Eq. (49)

:EY(AA')IS(AA')12=-1-fdSe-2i~l~e2i~/. (52) A' (8'/1"2)2

Since the 'Y j are sums of products of the angular momen­tum operators, the '9 j may be written as similar sums of products of the transformed operators

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5774 C. F. Curtiss: Rotational transitions in call isions

EJa) ::(I'-l,c}a)(J'

E~b) :: (I'-1,c\b)(I' J ]

3Uj ::cP-1;m(P.

(53)

The essential problem in the evaluation of the integrals on the right of Eq. (52), is that of evaluating these trans­formed operators.

It follows from Eq. (14) and a standard operator iden­tity that

(P2(rO),cJa)(pzl(ro) ::e-(i/,,)'U (Z) (ro) .c}a) e (i/1>),,,(2) (TO)

(54)

::.c(a) + £[,cla) UIZ){r)J _ _ l_[[,c~a) U(Z)(r)] j If } , 0 2lf2 J' 0 ,

and thus from Eqs. (14), (47), and (51)

- ~[[.cJa),u(1)(ro)];VI1)(ro)]

+ ~2 [[.cJa),U(Z) (ro)), 'O(l){ro)]

_1_[[(1(0) 'O(2)(r ») 'O(Z)(r ») + - 2lf2 ""j , 0 , o· .• (55)

An examination of this series shows that it contains no terms in inverse powers of If and that all of the terms of zero order in If are purely multiplicative terms, which do not involve differentiation. Thus

E(a) =L~a) + o (If) J J ,

(56)

where LJa) is the function

L (a) :: _ £ '"' ([,c~a) '0 (1) (r )] _ [,c~a) U(2) (r )]} + i- L ([[.c~a) 1.: (1) (r )] U (I) (r )] + [[.c~a) u(l)(r ) J U(l) (r )] 1 If Ln- J' nO 0 J' nO 0 2fi nn' J' nO 0, n'1 0 J ,.1 0, n'O 0

- 2[[.c }a) ,U~~) (rO)), u~~l (rO)] - 2[[.cJa) ,U~~) (rO) l;o~}~(ro)] + [[.c Ja>, U~~) (ro)];\)!~t (ro) J + [(.cJa) ,U!~) (rO)], U~~~(ro)]} + ••.

(57)

From Eq. (41) and the arguments given above, it fol­lows that

(58)

where

Yl (>.5) = L~a)2 + L~a)2 _ L~a)2 _ 2Lia)[jaUa + 1)lf2 _ Lia )2p!2 ,

Yz(>.5) = L{b)2 + Lib)2 _ Lib)2 _ 2Lib)[jb(jb + 1)fi2 _ L~b)211/2 ,

Y3 (>.5) =M i +M ~ -M; + 2M2 [1(l + 1)lf2 _M~]lI2, (59)

Y6 {>.5) = L:M~ , j

where Ljb) and M J are functions given by expressions similar to Eq. (57). Thus, one may calculate the clas­sicallimit of the integral on the right of Eq. (52) and in particular any moment of IS{AA/) 12 with respect to the quantities on the right of Eqs. (43). Using the first three of these quantities one may evaluate the moments of the transition probability given by Eq. (40). As an example, the nth moment of the transition probability with respect to the change in rotational energy of the molecule a is for n 2: 1,

M ~.) "" L: {[ja(j. + 1) - j;(j: + 1) ]fi2 }np(A; A') A' 21a

"" (21.)-" 2: Y l (XA/)" \S(XA/)1 2 (60) A'

where Yl (>.5) is the function given in the previous equa­tion. The experimental rotational relaxation time de­pends on an integral of Ml or M2 over the impact param­eter and a thermal average over the initial collision en­ergies. 8 In a previous development,9 we obtained an ex­pression for this quantity to second order in the aniso­tropic portion of the interaction potential. The present expression is valid to all orders in the anisotropy.

ACKNOWLEDGMENT

The author wishes to thank R. D. Olmsted, K. R. Squire, and R. R. Wood for reading and commenting on the manuscript.

tG. Giournousis, "Molecular Scattering and the Kinetic Theory of Gases," dissertation, University of WisconSin, 1955; G. Gioumousis and C. F. Curtiss, J. Math. Phys. Z. 96 (1961). Eq. (3.10); A. M. Arthurs and A. Dalgarno, Proc. R. Soc. London Ser. A 256, 540 (1960).

2C. F. Curtiss, J. Chern. Phys. 49, 1952 (1968), Eq. (4). 3Reference 2, Eq. (26). See also, C. F. Curtiss, Mol. Phys.

34, 441 (1977). 4C. F. Curtiss, J. Chero. Phys. 52, 4832 (1970). Eqs. (8) and

(17). 5Reference 4, Eq. (22). 6C. F. Curtiss, J. Chero. Phys. 63, 2738 (1975), Eq. (8). 7Reference 6, Eqs. (17), (27), (28), and (29). 8L. Monchick, A. N. G. Pereira, and E. A. Mason, J. Chern.

Phys. 42. 3241 (1965). 9C. F. Curtiss and R. D. Olmsted, J. Chern. Phys. 56, 5706

(1972); R. D. Olmsted and C. F. Curtiss, J. Chern. Phys. 57, 3298 (1972); J. D. Russell, R. B. Bernstein, and C. F. Curtiss, J. Chern. Phys. 57, 3304 (1972).

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