23 Resonant Quasi-periodic and Periodic Orbits
Resonances Downloaded from pubs.acs.org by UNIV OF TEXAS AT EL PASO on 11/02/18. For personal use only.
For the Three-Dimensional Reaction of Fluorine Atoms with Hydrogen Molecules C. 1
C.
MARSTON
1
and
ROBERT
E.
WYATT
2
Departments of Physics and Chemistry and Institute for Theoretical Chemistry, University of Texas, Austin, TX 78712
2
Department
of Chemistry and Institute for Theoretical Chemistry, University of Texas, Austin,
TX 78712
Numerical methods are described for locating resonant quasiperiodic and periodic orbits in the 3D F+H reac tion with J=0. A number of plots of both types of resonant orbit are presented. This is the first time that resonant orbits have been found for a non-collinear reaction. These orbits are then used in the arbi trary trajectory semiclassical quantization scheme of DeLeon and Heller. The lowest resonance energy pre dicted using this procedure is in good agreement with all available quantal and adiabatic semiclassical re sults. 2
Over t h e p a s t few y e a r s , resonances i n c h e m i c a l r e a c t i o n s have been the f o c u s o f numerous t h e o r e t i c a l (_1) and e x p e r i m e n t a l s t u d i e s (_2). On t h e t h e o r e t i c a l s i d e , b o t h q u a n t a l and s e m i c l a s s i c a l methods have been used t o c a l c u l a t e resonance e n e r g i e s and w i d t h s , p r i n c i p a l l y f o r c o l l i n e a r r e a c t i o n s , a l t h o u g h t h e r e a r e a few s t u d i e s o f 3D r e a c t i o n s . I n q u a n t a l s t u d i e s o f 3D r e a c t i o n s , some c l o s e - c o u p l i n g c a l c u l a t i o n s on H+H^ have been r e p o r t e d ( I f ) , ( l i ) , b u t t h e l a r g e number o f chann e l s has n e c e s s i t a t e d approaches based upon t h e J - c o n s e r v i n g ( 3 ) , IOS ( 4 ) , BCRLM ( 5 ) , o r DWBA (6) a p p r o x i m a t i o n s . I n a d d i t i o n t o these q u a n t a l s t u d i e s , s e v e r a l s e m i c l a s s i c a l approaches have been a p p l i e d t o c o l l i n e a r (7) and 3D r e a c t i o n s ( 7 b ) , ( 8 ) . I n t h e P o l l a k - C h i l d theory, the energies of p e r i o d i c o r b i t s i n the c o l l i n e a r c o l l i s i o n complex a r e a d j u s t e d t o s a t i s f y i n t e g e r a c t i o n q u a n t i z a t i o n c o n d i t i o n s ( 7 c ) . These r e s o n a n t p e r i o d i c o r b i t s (RPO's) were f i r s t ment i o n e d i n an e a r l i e r study o f t r a j e c t o r i e s trapped i n e n t r a n c e o r e x i t r e g i o n s ( p e r i o d i c o r b i t d i v i d i n g s u r f a c e s - PODS) o r i n t h e c o l l i s i o n complex (9) o f c o l l i n e a r r e a c t i o n s . I n an e x t e n s i o n t o p r e d i c t resonance e n e r g i e s i n t h e 3D H+H and F+H r e a c t i o n s , P o l l a k and 2
2
Wyatt (8) developed an a d i a b a t i c r e d u c t i o n scheme i n w h i c h s e m i c l a s s i c a l l y q u a n t i z e d RPOs a t f i x e d v a l u e s o f t h e b e n d i n g a n g l e were com0097-6156/ 84/ 0263-0441 $06.00/ 0 © 1984 American Chemical Society
442
RESONANCES
puted i n t h e f i r s t s t e p . I n t h e second step of t h e r e d u c t i o n scheme, t h e s e e n e r g i e s then s e r v e d as an e f f e c t i v e p o t e n t i a l f o r t h e s l o w e r bending m o t i o n . S e m i c l a s s i c a l q u a n t i z a t i o n o f t h e p e r i o d i c bending o r b i t s l e d t o time-averaged e f f e c t i v e moments of i n e r t i a f o r t h e slow o v e r a l l t u m b l i n g motion. These and o t h e r s t u d i e s based upon PODS (10) o r RPOs have been e x t e n s i v e l y r e v i e w e d by P o l l a k (11). I n r e l a t e d s t u d i e s , D u c h o v i c , Swamy, and Hase found q u a s i p e r i o d i c o r b i t s above t h e d i s s o c i a t i o n t h r e s h o l d f o r t h e H-C-C-*H+C=C f r a g m e n t a t i o n (12). They used an i t e r a t i v e method t o s e m i c l a s s i c a l l y q u a n t i z e t h e e n e r g i e s o f some of t h e s e o r b i t s . The p r e s e n t study i s concerned w i t h t h e a p p l i c a t i o n of n u m e r i c a l methods t o l o c a t e o r b i t s f o r c h e m i c a l r e a c t i o n s t h a t a r e n o t r e s t r i c t e d t o c o l l i n e a r geometry. I n c o n t r a s t t o t h e P o l l a k - W y a t t a d i a b a t i c r e d u c t i o n scheme ( 8 ) , t h e p r e s e n t treatment does n o t r e q u i r e an a d i a b a t i c s e p a r a t i o n o f m o t i o n s . A t a g i v e n energy E, t h e t r a j e c t o r y must be s t a r t e d a t a p o i n t such t h a t , a f t e r a time i n t e r v a l , i t n e a r l y ( q u a s i p e r i o d i c c a s e ) o r e x a c t l y ( p e r i o d i c case) r e t u r n s t o i t s s t a r t i n g p o i n t . The problem then i s t o s y s t e m a t i c a l l y l o c a t e t h e s e s t a r t i n g p o i n t s . I n p r a c t i c e , i t i s found t h a t t r a j e c t o r i e s i n i t i a t e d c l o s e t o t h e r e s o n a n t o r b i t q u i c k l y e v o l v e i n t o the a s y m p t o t i c r e a c t a n t o r p r o d u c t r e g i o n s ; t h e i n i t i a l c o n d i t i o n s must be a d j u s t e d t o m i n i m i z e t h e e s c a p i n g tendency o f t h e t r a j e c t o r y from the c o l l i s i o n complex i n t o b o t h t h e r e a c t a n t and p r o d u c t c h a n n e l s . Here, t h e e s c a p i n g tendency i s measured by t h e atom-molecule r e l a t i v e momentum a t a t u r n i n g p o i n t i n t h e m o t i o n ; t h i s momentum i s denoted z ° z ' * p r o d u c t s , r e s p e c t i v e l y . To l o c a t e RPOs f o r c o l l i n e a r r e a c t i o n s , P o l l a k and C h i l d (7c) d i s c u s s e d t h e t u r n i n g p o i n t (TP) and r e a c t a n t - p r o d u c t (RP) boundary methods. I n t h e TP method, t h e t r a j e c t o r y i s f o l l o w e d t o t h e f i r s t t u r n i n g p o i n t , where the s i g n o f t h e momentum component p e r p e n d i c u l a r t o i s examined. The s t a r t i n g p o i n t o f t h e t r a j e c t o r y i n t h e r e a c t a n t c h a n n e l i s then a d j u s t e d i n an attempt t o f o r c e t h i s momentum component t o be z e r o . On t h e o t h e r hand, t h e RP method was r e c e n t l y used by P o l l a k t o l o c a t e a q u a s i p e r i o d i c o r b i t near t h e e n t r a n c e c h a n n e l v = l v i b r a t i o n a l a d i a b a t i c b a r r i e r i n t h e 3D H+H r e a c t i o n ( 1 3 ) . U s i n g t h i s method, r
P
f
n
r
e
a
c
t
a
n
t
s
o
r
2
the t r a j e c t o r y i s i n t e g r a t e d from t h e r e g i o n o f t h e c o l l i s i o n complex l o n g enough t o see whether i t moves toward r e a c t a n t s o r p r o d u c t s . The i n i t i a l c o n d i t i o n i s then a d j u s t e d t o l o c a t e t h e boundary between o r b i t s d e c a y i n g i n t o r e a c t a n t s o r i n t o p r o d u c t s . A t t h e RP boundary, the o r b i t c o u l d be e i t h e r p e r i o d i c o r q u a s i p e r i o d i c . The method t h a t we d e s c r i b e i n t h e n e x t s e c t i o n i s a g e n e r a l i z a t i o n t o n o n c o l l i n e a r g e o m e t r i e s o f t h e TP method. I n t h e second p a r t o f t h e same s e c t i o n , we w i l l d e s c r i b e a method f o r l o c a t i n g RPOs f o r 3D r e a c t i o n s . The method i s based upon m i n i m i z a t i o n of an " a p e r i o d i c i t y i n d e x , " A, by again a d j u s t i n g the i n i t i a l c o n d i t i o n s . The H a m i l t o n i a n i s t h a t o f an atom-diatomic m o l e c u l e c o l l i s i o n a t t o t a l a n g u l a r momentum J=0, H =
i R [ P
+
P
+
P
r 7 '
]
+
V
(
R
r
' ^>»
where R and r a r e t h e s c a l e d (14) r e a c t a n t atom-molecule and molecul a r v i b r a t i o n a l c o o r d i n a t e s , r e s p e c t i v e l y , and where y i s t h e bending a n g l e between R and r ( I n t h e n e x t s e c t i o n , we w i l l a l s o use t h e no-
23.
MARSTON AND
WYATT
Resonant Quasi-periodic & Periodic Orbits
443
t a t i o n z and p f o r R and r , r e s p e c t i v e l y . ) I n a d d i t i o n , u i s the e f f e c t i v e reduced mass of the t h r e e atom system. The Muckerman V pot e n t i a l (15) was used f o r V ( R , r , y ) . Having computed q u a s i p e r i o d i c or p e r i o d i c r e s o n a n t o r b i t s , we t h e n use them i n a s e m i c l a s s i c a l q u a n t i z a t i o n scheme i n o r d e r t o p r e d i c t resonance e n e r g i e s . S i n c e t h e s e t r a p p e d t r a j e c t o r i e s a r e bound s t a t e s embedded i n the continuum of the c o l l i s i o n complex, one of the s e m i c l a s s i c a l q u a n t i z a t i o n schemes d e v i s e d f o r ( t r u l y ) bound systems may be used ( 1 6 ) . A l t h o u g h the s e m i c l a s s i c a l p r e d i c t i o n of resonance e n e r g i e s has been c o n s i d e r e d p r e v i o u s l y f o r the c o l l i n e a r H+H r e a c t i o n (7a) and f o r a model atom-diatom i n e l a s t i c c o l l i s i o n ( 1 7 ) , where ( u s i n g q u a s i p e r i o d i c t r a j e c t o r i e s ) the i t e r a t i v e s u r f a c e - o f - s e c t i o n method (18) was s u c c e s s f u l l y employed, we found the n o n i t e r a t i v e a r b i t r a r y - t r a j e c t o r y method of DeLeon and H e l l e r (19) t o a d m i r a b l y s u i t our needs. An i m p o r t a n t advantage of t h i s method i s t h a t " a r b i t r a r y t r a j e c t o r i e s ( r e l a t i v e l y c l o s e t o the quantum energy b e i n g sought) may be u s e d , thus e l i m i n a t i n g the need f o r d i f f i c u l t r o o t s e a r c h e s t o f i n d the " r i g h t " t r a j e c t o r i e s which s a t i s f y quantum c o n d i t i o n s on the action integrals. L a t e r , s e v e r a l F+H2 r e s o n a n t q u a s i p e r i o d i c o r 2
1 1
b i t s w i l l be i l l u s t r a t e d . Then, the s e m i c l a s s i c a l l y q u a n t i z e d r e s o nance energy, computed from two RPOs, i s compared t o r e s u l t s from a l l a v a i l a b l e q u a n t a l and s e m i c l a s s i c a l s t u d i e s . N u m e r i c a l Methods f o r L o c a t i n g Q u a s i p e r i o d i c
and P e r i o d i c O r b i t s
B e f o r e d e s c r i b i n g the n u m e r i c a l methods used t o l o c a t e q u a s i p e r i o d i c or p e r i o d i c r e s o n a n t o r b i t s , we w i l l d e f i n e s e v e r a l s e t s of c o o r d i n a t e s t h a t are u s e f u l i n s p e c i f y i n g the s i z e and shape of the t h r e e atom t r i a n g l e . L e t S be the ^ s c a l e d ) v e c t o r from the c e n t e r - o f - m a s s of H t o the F atom, and l e t r be the ( s c a l e d ) H s e p a r a t i o n v e c t o r 2
2
( 1 4 ) . I n a d d i t i o n , l e t y be the a n g l e between R and r , such t h a t y=0 or 7T denote c o l l i n e a r c o n f i g u r a t i o n s , w h i l e y=ir/2 or 3ir/2 denote p e r p e n d i c u l a r c o n f i g u r a t i o n s . I n o r d e r to o r i e n t the m o l e c u l a r v e c t o r r r e l a t i v e t o S, we may a l s o use C a r t e s i a n c o o r d i n a t e s , x=rcosy and y = r s i n y , so t h a t y measures the d e v i a t i o n from c o l l i n e a r i t y (y=0 thus d e f i n e s c o l l i n e a r g e o m e t r i e s , y=0 or IT). Thus, the t h r e e C a r t e s i a n c o o r d i n a t e s ( R , x , y ) , where R i s the l e n g t h of v e c t o r or the c y l i n d r i c a l c o o r d i n a t e s (R,p,y), where p(=r) i s the l e n g t h of v e c t o r r , may be used t o s p e c i f y the s i z e and shape of the F H three-atom t r i angle. C o n t i n u i t y of c l a s s i c a l dynamics w i t h r e s p e c t t o i n i t i a l c o n d i t i o n s s u g g e s t s t h a t the s e a r c h f o r q u a s i p e r i o d i c t r a j e c t o r i e s i n 3D should b e g i n by s e l e c t i n g i n i t i a l c o n d i t i o n s c l o s e t o the known c o l l i n e a r RPOs, but r o t a t e d s l i g h t l y out of the c o l l i n e a r p l a n e . To a f i r s t a p p r o x i m a t i o n , the c y l i n d r i c a l c o o r d i n a t e s R and p a r e s e t 2
q
equal to R and
q
Q
and X , r e s p e c t i v e l y , of the known c o l l i n e a r RPO q
(
v
= o
0)
the o r i e n t a t i o n of the r v e c t o r i s d e t e r m i n e d by the v a l u e of y J
o ( i n i t i a l l y 1.0°); t h a t i s t o say, y ^ p ^ c o s y ^ T h i s a p p r o x i m a t i o n t o the i n i t i a l p o s i t i o n v e c t o r must t h e n be a d j u s t e d w i t h i n the y ^ p l a n e a l o n g the component of VV l y i n g i n t h a t p l a n e t o w i t h i n 10 of the d e s i r e d energy. N u m e r i c a l i n t e g r a t i o n of the e q u a t i o n s of
eV
444
RESONANCES
m o t i o n i s a l l o w e d t o proceed through the f i r s t two extremes i n the p motions, or t u r n i n g p o i n t s (the t r a j e c t o r y i s again^near i t s s t a r t i n g p o i n t ) and t h e v a l u e of the d i s s o c i a t i v e momentum |P | i s compared z
w i t h t h a t of the p r e v i o u s t r a j e c t o r y so t h a t subsequent d i s p l a c e m e n t s ( A R , w i t h y f i x e d ) w i l l be i n the d i r e c t i o n of d e c r e a s i n g f i n a l q
Q
d i s s o c i a t i v e momentum. T h i s p r o c e d u r e o f advancing the i n i t i a l p o s i t i o n v e c t o r i n the d i r e c t i o n of d e c r e a s i n g f i n a l d i s s o c i a t i v e momentum i s c o n t i n u e d u n t i l the v a l u e o b t a i n e d i s no l o n g e r l e s s t h a n t h a t of the p r e v i o u s d i s p l a c e m e n t , a t w h i c h p o i n t the d i s p l a c e m e n t d i r e c t i o n ( A R ) i s r e v e r s e d and t h e s e a r c h i s c o n t i n u e d i n a convergent sequence. The i n t e g r a t o r i s then a l l o w e d t o proceed t o the n e x t t u r n i n g p o i n t i n the p m o t i o n and the m i n i m i z a t i o n p r o c e d u r e ( t h e adj u s t m e n t of A R ) i s r e p e a t e d as b e f o r e , but w i t h an a p p r o p r i a t e d e f i n i t i o n of the d i s s o c i a t i v e momentum d i r e c t i o n , depending upon whether the t r a j e c t o r y i s t e m p o r a r i l y t e r m i n a t e d i n the e n t r a n c e o r e x i t c h a n n e l of the p o t e n t i a l energy s u r f a c e . The procedure of m i n i m i z i n g the d i s s o c i a t i v e momentum a f t e r an i n c r e a s i n g number o f t u r n i n g p o i n t s i n the p m o t i o n and w i t h e v e r - i n c r e a s i n g p r e c i s i o n i s c o n t i n ued u n t i l the a c c u r a c y o f t h e n u m e r i c a l i n t e g r a t o r i s exhausted ( t y p i c a l l y a f t e r 14 t u r n i n g p o i n t s ) . I n t h i s way, t h e e s c a p i n g tendency of the t r a j e c t o r y (toward F+H r e a c t a n t s o r FH+H p r o d u c t s ) i s m i n i m i z e d , a t a g r a d u a l l y i n c r e a s i n g number of t u r n i n g p o i n t s . Having thus l o c a t e d the s t a r t i n g c o n d i t i o n R f o r q u a s i p e r i o d i c Q
Q
2
q
dynamics i n i t i a t e d a t a p a r t i c u l a r v a l u e of Y > q
the i n i t i a l
position
v e c t o r may be r o t a t e d t o a h i g h e r i n i t i a l a n g l e and the s u c c e s s i v e m i n i m i z a t i o n p r o c e d u r e r e p e a t e d a t t h i s new v a l u e of y . P r o c e e d i n g o through 2° r o t a t i o n a l i n c r e m e n t s , i t was p o s s i b l e t o f i n d q u a s i p e r i o d i c o r b i t s a t a n g l e s below Y , w i t h o u t e n c o u n t e r i n g the b a r r i e r t o 'max h i g h a n g l e bending expected on t h e FR^ p o t e n t i a l energy s u r f a c e . In o r d e r t o l o c a t e 3D p e r i o d i c o r b i t s , a s l i g h t l y d i f f e r e n t p r o cedure was used. For q u a s i p e r i o d i c o r b i t s a t E=0.4 eV, a comparison of y v s . p p l o t s (as i n F i g u r e 3B) f o r d i f f e r e n t s t a r t i n g c o n d i t i o n s ( R , Y ) r e v e a l e d t h a t the t u r n i n g p o i n t s of the t r a j e c t o r y i n i t i a t e d a t 24.8° appeared t o s e p a r a t e i n t o f o u r d i s t i n c t s e t s i n the t r a j e c t o r y i n i t i a t e d a t Y =24.0°. The q u a i s p e r i o d i c t r a j e c t o r y i n i t i a t e d a t 21.1° was l o c a t e d and i t s Y v s . p p l o t was found t o be c o n s i s t e n t w i t h t h i s t r e n d i n t h a t the g r o u p i n g e f f e c t was even more d i s t i n c t than t h a t observed a t h i g h e r a n g l e s . These o r b i t s a r e thus becoming i n c r e a s i n g l y p e r i o d i c . T h i s p r o g r e s s i o n toward an e x a c t s u p e r p o s i t i o n of t u r n i n g p o i n t s i n t o f o u r c l u s t e r s was q u a n t i f i e d by d i v i d i n g the d i f f e r e n c e of the y v a l u e s a t t u r n i n g p o i n t s 8 and 10 (see F i g u r e 3B) by the i n i t i a l v a l u e of Y t o o b t a i n an " a p e r i o d i c i t y i n d e x , " A, f o r w h i c h a v a l u e of z e r o would i m p l y e x a c t p e r i o d i c i t y . By p l o t t i n g A as a f u n c t i o n of Y » i t was p o s s i b l e t o e x t r a p o l a t e t o A=0 to obt a i n s u c c e s s i v e l y b e t t e r e s t i m a t e s o f the v a l u e of Y l e a d i n g t o exa c t p e r i o d i c i t y . At Y 1 7 . 7 ° , the v a l u e was c o n s i d e r e d a c c e p t a b l y &
Q
O
o
Q
q
=
o
s m a l l to assume e s s e n t i a l l y e x a c t p e r i o d i c i t y of the dynamics. The same method was used t o f i n d r e s o n a n t p e r i o d i c o r b i t s a t o t h e r s t a r t i n g a n g l e s and e n e r g i e s . An example w i l l be p r o v i d e d l a t e r .
23.
MARSTON AND WYATT
Resonant Quasi-periodic
Semiclassical Quantization
& Periodic Orbits
445
Using A r b i t r a r y T r a j e c t o r i e s
The s e m i c l a s s i c a l q u a n t i z a t i o n p r o c e d u r e o f DeLeon and H e l l e r (19) was used t o o b t a i n q u a n t i z e d resonance e n e r g i e s because o f i t s capab i l i t y o f y i e l d i n g a c c u r a t e r e s u l t s from " a r b i t r a r y ' t r a j e c t o r i e s ( i . e . , root searches f o r the " r i g h t " q u a n t i z i n g t r a j e c t o r i e s are not r e q u i r e d a t a l l ) . The method r e c o g n i z e s t h a t i n t e g r a b i l i t y o f t h e dynamics p e r m i t s t h e energy t o be e x p r e s s e d a s a f u n c t i o n o f o n l y the N a c t i o n v a r i a b l e s o b t a i n a b l e from a system o f N degrees o f freedom. A f i r s t o r d e r e x p a n s i o n o f t h e energy from one s e t o f a c t i o n v a r i a b l e s t o a n o t h e r i s then p o s s i b l e u s i n g t h e e x p r e s s i o n : 1
E = E
o il. J +
6
( 1 )
aj I n o r d e r t o o b t a i n an approximate energy e i g e n v a l u e , 6 J must be s e l e c t e d t o be a p p r o p r i a t e f o r a n e x p a n s i o n t o a s e t o f a c t i o n v a r i a b l e s c o n s i s t e n t w i t h t h e q u a n t i z a t i o n c o n d i t i o n s . I f we now l e t where J =vti denotes t h e s e t o f q u a n t i z e d
action integrals
(V i s a s e t o f quantum numbers), and where J^°^ denotes t h e a c t i o n s o f the s t a r t i n g t r a j e c t o r y , and u s i n g 8E/93=uJ from H a m i l t o n - J a c o b i theor y , t h e energy q u a n t i z a t i o n e x p r e s s i o n becomes (o)
E(V)-E(3°)-hS.tf-3 )ti Thus, i f J^°^ and u> can be o b t a i n e d
(2)
from t r a j e c t o r i e s a t energy E°=
E ( J ^ ) , then t h e approximate q u a n t i z e d resonance energy, E ( V ) , l a b e l e d by t h e s e t o f quantum numbers ^, can be p r e d i c t e d . The a c t i o n s may be o b t a i n e d from a c o n s i d e r a t i o n o f t h e average phase o f t h e t r a j e c t o r i e s . For one t r a j e c t o r y , 1 +
-> 1
+
+ 1
T
a
)
i + ->
where i i n d e x e s t h e t o p o l o g i c a l l y d i s t i n c t p a t h s on t h e t o r u s manif o l d , and n_^ i s t h e number o f w i n d i n g s o f t h e i - t h t o p o l o g i c a l l y d i s t i n c t p a t h b e f o r e (exact o r n e a r l y e x a c t ) c l o s u r e . The a c t u a l t r a j e c t o r y i s assumed t o wind back on i t s e l f ( e x a c t l y o r a p p r o x i m a t e l y ) i n a time T. A l s o , i n conforming t o t h e n o t a t i o n o f DeLeon and H e l l e r , we i n t r o d u c e d a denominator o f 2ir i n t o the d e f i n i t i o n o f t h e a c t i o n i n t e g r a l . I n t r o d u c i n g t h e i n d e x (k) t o s p e c i f y a p a r t i c u l a r t r a j e c t o r y w i t h i n a s e t , a l l a t energy E(J^°^), t h i s becomes,
^ ) (k).-j(k) k
4
Assuming t h a t the v a l u e s o f
( 4 )
f o r the d i f f e r e n t t r a j e c t o r i e s are
446
RESONANCES
e s s e n t i a l l y equal ( f o r a f i x e d value of energy), the s e t of vectors may be r e p l a c e d by t h e s i n g l e v e c t o r 3^°^ t o o b t a i n t h e a p p r o x i mation ^(k) (k). (o) 4
5
( 5 )
The s e t o f e q u a t i o n s i m p l i e d by t h e above n o t a t i o n may be e x p r e s s e d i n the s i n g l e matrix equation
$
and
=
2
J
(
O
)
(6)
t h e a c t i o n v a r i a b l e s a r e then o b t a i n e d
matrix:J^°^=5
by i n v e r t i n g t h e frequency
I n E q u a t i o n 6, Q..
the j - t h frequency f o r
t r a j e c t o r y i , so t h a t t h e f r e q u e n c i e s f o r a g i v e n t r a j e c t o r y r u n a c r o s s a row. For t h i s r e a c t i o n , t h e two a c t i o n s w i l l be denoted J f o r t h e &
h i g h - f r e q u e n c y t r a n s l a t i o n - v i b r a t i o n (asymmetric) m o t i o n and the l o w - f r e q u e n c y bending m o t i o n . a r e used f o r b o t h a c t i o n s , (v
V
Integer q u a n t i z a t i o n
n
+1
for
conditions
1)
(7)
V a' b>=( a 'V
For J , a n i n t e g e r q u a n t i z a t i o n c o n d i t i o n i s used by analogy t o t h e a
c o l l i n e a r RPO s t u d i e s o f P o l l a k and C h i l d ( 7 c ) ; they found t h a t v = &
4, 6, 8, ... l e d t o s e m i c l a s s i c a l resonance e n e r g i e s w h i c h were c l o s e to the exact quantal values. F o r J ^ , i n t e g e r q u a n t i z a t i o n i s used because we a r e t r y i n g t o o b t a i n t h e ground s t a t e energy o f a doubly degenerate bending degree o f freedom. To p r e d i c t t h e l o w e s t r e s o nance energy o f 3D F+H^, we w i l l thus use v =4 and v ^ - l . Higher e n ergy resonances c o u l d be p r e d i c t e d w i t h v = 4 , v^=2, v = 6 , v ^ = l , e t c . For t h e c u r r e n t problem, E q u a t i o n 2 becomes &
a
( o )
E , J^°\ and a), , J^°\ I n order t o a a b b s i m p l i f y t h e F o u r i e r a n a l y s i s , t r a j e c t o r y number one w i l l be t h e c o l l i n e a r RPO, and t r a j e c t o r y number two w i l l be t h e n o n c o l l i n e a r RPO. I n E q u a t i o n 8, t h e two f r e q u e n c i e s and t h e two a c t i o n s w i l l b o t h r e f e r t o t r a j e c t o r y number two. R e c e n t l y , M i l l e r has shown how j u s t one t r a j e c t o r y may be used t o p r e d i c t an e i g e n v a l u e i n t h e a r b i t r a r y t r a j e c t o r y method ( 2 0 ) . the f r e q u e n c i e s
and a c t i o n s :
23.
MARSTON AND WYATT
Resonant Quasi-periodic
& Periodic Orbits
447
Quasiperiodic Resonant Orbits In t h i s Section, we w i l l i l l u s t r a t e several quasiperiodic resonant o r b i t s for F+H « Using the numerical methods discussed e a r l i e r , the 2
quasiperiodic o r b i t at E=0.9 eV and Y =17° was computed. o
The t o t a l
energy, E, i s measured from the f l o o r of the entrance v a l l e y on the FH^ p o t e n t i a l surface. In Figure 1, t h i s o r b i t i s i l l u s t r a t e d i n (R,x,y) Cartesian i n t e r n a l coordinate space. In addition, projections are shown of the o r b i t upon the three coordinate planes (R»x), (R,y), and (x,y). R e c a l l that (R,x) i s the c o l l i n e a r plane. In part A of the f i g u r e , the orbit has been integrated through 9 turning points, while i n part B the integration time was extended to 18 turning points. The projection of the o r b i t a l motion i n the c o l l i n e a r (R,x) plane i s a "blurred" or thickened version of the c o l l i n e a r resonant periodic o r b i t s i l l u s t r a t e d by Pollak and C h i l d (6c). This thickening a r i s e s because the o r b i t evolves on a two-dimensional curved surface embedded i n the three-dimensional space. Figure 2 shows another quasiperiodic resonant o r b i t , t h i s time for E=0.4 eV and Y = 2 0 ° . Part A again shows the o r b i t (integrated through 14 o
turning points i n (R,x,y) space, while part B shows the o r b i t within the FI^ p o t e n t i a l space. The p o t e n t i a l surface i n Figure 2B i s the locus of points with V=0.4 eV; the reactant region on the l e f t i s connected to the product region on the right by the FHH i n t e r a c t i o n region i n the middle of the f i g u r e . In reactants or products, the c l a s s i c a l l y allowed region (V
