28 A New Helicity Representation for Reactive
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Atom-Diatom Collisions C. W. McCURDY and W. H. MILLER Department of Chemistry, University of California, Berkeley, CA 94720
There have recently been several accurate quantum mechanical scattering calculations for the H + H reaction i n three dimen sions, (1,2) and this provides interesting data to test approxi mate theoretical models. In p a r t i c u l a r , we would l i k e to see the extent to which " c l a s s i c a l S-matrix" theory (3) provides a good description of the threshold region of this reaction; e a r l i e r calculations (4) for the c o l l i n e a r version of the reaction showed the semiclassical model to be quite accurate. Preliminary semiclassical calculations for the three dimensional H + H reaction have been reported, (5) but these were not extensive enough and of too preliminary a nature to allow a very useful comparison with the l a t e s t quantum mechanical reactive scattering calculations. To keep the semiclassical calculation as manageable as possible, we would like to take advantage of the helicity, or "j conserving" approximation (6,7) that has been seen to work well quantum mechanically (8,9). This effectively eliminates one degree of freedom from the problem, and semiclassically this means that the quantizing boundary conditions that must be applied to the c l a s s i c a l trajectories require only a two dimensional root search rather than a three dimensional one (5). The usual j -conserving approximation is s p e c i f i c a l l y designed for i n e l a s t i c collision processes, however, and it i s not immediately obvious how such an approximation should be applied to reactive c o l l i s i o n s . Thus, i f the helicity K, the projection of the total angular momentum J along the initial r e l a t i v e translational coordinate R , κ = J-R, i s assumed to be conserved along the incoming part of the trajectory, and i f the projection κ ' of J along the translational coordinate R ' of the f i n a l arrangement i s assumed to be conserved for the outward part of the trajectory, then at some point during the trajectory one must stop conserving κ and s t a r t conserving κ , since the two are not the same. The particular point at which one switches from conserving the heli city in one arrangement to conserving i t i n the other, however, i s somewhat a r b i t r a r y . 2
2
z
z
1
239
Brooks and Hayes; State-to-State Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1977.
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240
STATE-TO-STATE
CHEMISTRY
To avoid t h i s i l l - d e f i n e d s i t u a t i o n , we have considered a new h e l i c i t y r e p r e s e n t a t i o n : the b o d y - f i x e d q u a n t i z a t i o n a x i s we choose i s n e i t h e r £ nor , but r a t h e r the p r i n c i p a l a x i s of the ABC t r i a n g l e of atoms which corresponds to the s m a l l e s t p r i n c i p a l moment of i n e r t i a . This q u a n t i z a t i o n a x i s becomes the t r a n s l a t i o n a l coordinate R a s y m p t o t i c a l l y when the i n i t i a l atom A i s f a r from the diatom'BC, i t becomes the f i n a l t r a n s l a t i o n a l coordinate R" when the f i n a l atom C i s f a r from AB, and i t v a r i e s smoothly between them i n the intermediate r e g i o n . I f the atoms are c o l l i n e a r , the a x i s i s the c o l l i n e a r a x i s . The p r o j e c t i o n of j onto the b o d y - f i x e d a x i s of the s m a l l e s t p r i n c i p a l moment of i n e r t i a thus becomes the o r d i n a r y h e l i c i t y i n the i n i t i a l and f i n a l asymptotic r e g i o n s , and i t v a r i e s smoothly between the two during the c o l l i s i o n . In t h i s regard the new h e l i c i t y i s somewhat l i k e a "natural c o l l i s i o n coordinate" (10) f o r t h i s v a r i a b l e . By analogy with the well known behavior of r i g i d asymmetric r o t o r s ( V [ ) , i t i s a l s o c l e a r t h a t the p r o j e c t i o n of j onto the a x i s of the s m a l l e s t p r i n c i p a l moment of i n e r t i a i s the b o d y - f i x e d p r o j e c t i o n t h a t should be most nearly conserved. The c l a s s i c a l Hamiltonian can be w r i t t e n i n t h i s new r e p r e s e n t a t i o n by t a k i n g the c l a s s i c a l l i m i t of the quantum mechanical Hamiltonian worked out by Diehl e t a K ( 1_2) ; u n l i k e the usual d e f i n i t i o n {]3) the three E u l e r angles i n t h i s case o r i e n t the p r i n c i p a l axes of the t r i a n g l e of atoms with respect to a spacef i x e d a x i s . The c l a s s i c a l Hamiltonian so obtained can be w r i t t e n as 9
H(p,q) = £
(p -Ap ) R
R
2
^
+
(P -Ap ) r
j 2
j 2
V(R,r,Y) •
+
r
2
(-J-2 +
+
^)(Ρ -ΑΡ ) γ
γ
2
j 2 +
2
f -
.
(1)
where the canonical v a r i a b l e s are q = ( R , r , y , q ) , and ρ = p ^ , p , p , K ) ; R, r, and γ are the usual t r a n s l a t i o n a l c o o r d i n a t e , v i b r a t i o n a l c o o r d i n a t e , and r e l a t i v e angle (cos γ Ξ r - R ) , respectively. J J , J 3 are the components of t o t a l angular momentum along the p r i n c i p a l axes o f r o t a t i o n of the t r i a n g l e of p a r t i c l e s , and { Ι · } , i = 1,2,3 are the p r i n c i p a l moments of i n e r t i a ordered so that I < I < I Ξ Ι + I ; s p e c i f i c a l l y K
r
L
5
2
η
x
I
2
- I-j = [ ( y R )
I
2
+ I-, = yR
2
2
2
2
+ (mr )
+ mr
The terms Δρ , Δρ , and Δρ
3
2
2
2
λ
2
+ 2yR mr 2
2
cos 2y]
h
. are
Brooks and Hayes; State-to-State Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1977.
(2a) (2b)
28.
MCCUBDY
=
A N D MILLER
.
2 Sin γ pR mr 2
j
R
3
d -ι ) 2
=
^
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r
A«
γ
Δ ρ
2sinγ
j
/τ
(i
CO|Y_
2
2
R
2
cosjr.
2
U
R mr
3
A
)
- I,)
τ
δ
τ
(
3
b
)
r
2 2 τ 2 s i n γ yR mr "" 3 , Λ (I - 1^ J
(
Ί
τ \ 1 ( J - K )
2
2
2
2
(5a)
2
2
.
(5b)
The r e s u l t i n g approximate, h e l i c i t y - c o n s e r v i n g Hamiltonian i s
P r e l i m i n a r y t e s t s o f some o f these ideas have been c a r r i e d out. Reactive c l a s s i c a l t r a j e c t o r i e s f o r the three dimensional Η + H r e a c t i o n have been examined, f o r example,^and we see t h a t the "new" h e l i c i t y , J and the "old" h e l i c i t y , J - R , a r e about 2
l
f
Brooks and Hayes; State-to-State Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1977.
STATE-TO-STATE CHEMISTRY
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242
e q u a l l y well conserved f o r the incoming part of the t r a j e c t o r y . The c l a s s i c a l r e a c t i v e cross s e c t i o n computed ( v i a a t r a j e c t o r y c a l c u l a t i o n ) from the Hamiltonian of Eq. ( 6 ) , however, i s not i n as good agreement with the exact c l a s s i c a l r e s u l t as i s t h a t obtained from the "old" h e l i c i t y - c o n s e r v i n g Hamiltonian. This seems to be due to the f a c t that the e f f e c t i v e c e n t r i f u g a l p o t e n t i a l i n Eq. (6) i s too small f o r l a r g e R, a l l o w i n g too much r e a c t i o n f o r l a r g e J. For small and intermediate values of J the r e a c t i o n p r o b a b i l i t y given by Eq. (6) i s i n reasonably good agreement with the exact c l a s s i c a l r e s u l t s . Other approximations to the c l a s s i c a l Hamiltonian of Eq. (1) are being explored to t r y to c o r r e c t t h i s d e f i c i e n c y of Eq. ( 6 ) .
Literature Cited 1. Elkowitz, A. B., and Wyatt, R. F . , J. Chem. Phys. (1975) 62, 2504. 2. Schatz, G. C., and Kuppermann, Α., J. Chem. Phys. (1976) 65, 4642, 4668. 3. For reviews, see Miller, W. H., Adv. Chem. Phys. (1974) 25, 69; (1975) 30, 77. 4. George, T. F., and Miller, W. H., J. Chem. Phys. (1972) 56, 5722; (1972) 57, 2458. 5. D o l l , J. D . , George, T. F . , and M i l l e r , W. H., J. Chem. Phys. (1973) 58, 1343. 6. McGuire, P., and Kouri, D. J., J. Chem. Phys. (1974) 60, 2488. 7. Pack, R. T . , J. Chem. Phys. (1974) 60, 633. 8. Kuppermann, Α., Schatz, G. C., and Dwyer, J. P., Chem. Phys. Lett. (1977) 45, 71. 9. Wyatt, R. E., private communication 10. Marcus, R. Α., J. Chem. Phys. (1966) 45, 4493, 4500; (1968) 49, 2610. 11. Townes, C. H., and Schawlow, A. L., Microwave Spectroscopy, McGraw-Hill, New York, 1955, pp. 83-109. 12. Diehl, H., Flugge, S., Schroder, U., Volkel, Α., and Weigury, Α., Z. Physik. (1961), 162, 1. 13. M i l l e r , W. H., J. Chem. Phys. (1969) 50, 407. Supported in part by the National Science Foundation under grant GP-41509X.
Brooks and Hayes; State-to-State Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1977.
