Reactive Resonances and Angular Distributions in the Rotating Linear

Jul 23, 2009 - We use the Bending-Corrected Rotating Linear Model (BCRLM) to investigate in detail the way in which resonances may affect the angular ...
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26 Reactive Resonances and Angular Distributions in the Rotating Linear Model E D W A R D F. HAYES

and

1

National

1

Resonances Downloaded from pubs.acs.org by YORK UNIV on 12/14/18. For personal use only.

2

ROBERT

B.

WALKER

2

Science Foundation, Washington, DC 20550

L o s Alamos National Laboratory, L o s Alamos, NM 87545

We use the Bending-Corrected Rotating Linear Model (BCRLM) to investigate in detail the way in which resonances may affect the angular distribution of reaction products. Using a lifetime matrix method, we separate the resonant and direct parts of the S matrix, and from the direct part we obtain angular distributions in the absence of the resonance. When applied to the F+H(v=0)->HF(v'=2)+H reaction on the M5 surface, at an energy for which the angular distri­ bution is strongly sideways peaked, we find the reso­ nance enhances the intensity of the sideways scat­ tering, and shifts the sideways peak toward the forward direction. We also describe conditions for which angular distributions have a significant forward scattering component at the reaction threshold. For the F+H(v=0)->HF(v'=2)+H reaction on an improved potential surface, there is forward scattering at threshold be­ cause a resonance builds in at large partial waves. For the He+H (v=3)->HeH(v'=0)+H reaction, there are many overlapping resonances in all partial waves, and the reaction proceeds without a barrier. Reactivity from many partial waves at the reaction threshold results in significant forward scattering. 2

2

+

+

2

I n t h i s paper, we w i l l p r e s e n t a d e t a i l e d a n a l y s i s of t h e way i n which resonances may a f f e c t t h e a n g u l a r d i s t r i b u t i o n of t h e p r o d u c t s of r e a c t i v e c o l l i s i o n s . To do t h i s , we have used an approximate t h r e e - d i m e n s i o n a l (3D) quantum t h e o r y of r e a c t i v e s c a t t e r i n g ( t h e B e n d i n g - C o r r e c t e d R o t a t i n g L i n e a r Model, o r BCRLM) t o generate t h e d e t a i l e d s c a t t e r i n g i n f o r m a t i o n (S m a t r i c e s ) needed t o compute t h e a n g u l a r d i s t r i b u t i o n of r e a c t i o n p r o d u c t s . We a l s o employ a v a r i e t y of t o o l s , n o t a b l y l i f e t i m e m a t r i x a n a l y s i s , t o c h a r a c t e r i z e t h e importance of a resonance mechanism t o t h e dynamics of r e a c t i o n s . As a r e s u l t , we hope t o g a i n i n s i g h t I n t o how the resonant component of t h e s c a t t e r i n g mechanism i s m a n i f e s t e d i n t h e a n g u l a r d i s t r i b u t i o n of r e a c t i o n p r o d u c t s . A p p l i c a t i o n s of these t e c h n i q u e s t o two r e a c t i v e systems, F+H and He+H^, w i l l be reviewed. 2

0097-6156/84/0263-0493506.50/0 © 1984 American Chemical Society

494

RESONANCES

The Bending-Corrected Rotating Linear Model The study of quantum effects such as resonances i n atom-molecule reactions has been largely confined to coupled-channel calculations for c o l l i s i o n s constrained to c o l l i n e a r geometries. Progress i n quantum reactive scattering techniques i s reviewed p e r i o d i c a l l y (1-4). A few 3D quantum calculations of simple reactions, some more approximate (5-17) than others (18-19), have been concerned with resonance features i n the reaction dynamics, and with the increasing s o p h i s t i c a t i o n and s e n s i t i v i t y of molecular beam experiments (20-23), i t has become evident that the angular d i s t r i b u t i o n of reaction products i s l i k e l y to be the most sensitive observable manifestation of resonant contributions to reaction mechanisms. Coupled channel methods for c o l l i n e a r quantum reactive calculations are s u f f i c i e n t l y well developed that calculations can be performed routinely. Unfortunately, c o l l i n e a r calculations cannot provide any insight into the angular d i s t r i b u t i o n of reaction products, because the impact parameter dependence of reaction p r o b a b i l i t i e s i s undefined. On the other hand, the best approximate 3D methods for atom-molecule reactions are computationally very intensive, and for this reason, i t i s impractical to use most 3D approximate methods to make a systematic study of the effects of potential surfaces on resonances, and therefore the effects of surfaces on reactive angular d i s t r i b u t i o n s . For this reason, we have become interested i n an approximate model of reaction dynamics which was proposed many years ago by Child (24), Connor and Child (25), and Wyatt (26). They proposed the Rotating Linear Model (RLM), which i s i n some sense a 3D theory of reactions, because the l i n e upon which reaction occurs i s allowed to tumble freely i n space. A f u l l three-dimensional theory would treat motion of the s i x coordinates ( i n the center of mass) associated with the two vectors r and R which specify the internuclear diatomic axis and the atom-molecule separation respectively. In a space-fixed frame, these vectors have polar coordinates ( r , ^ , ^ ) and (R,9,). In the RLM, two degrees of freedom are eliminated by requiring the 1

polar coordinates of the r and R vectors to coincide. The Hamiltonian for this system treats the four degrees of freedom associated with the coordinates (R,r,8,4>). As a model for angular d i s t r i b u t i o n s f o r reactive c o l l i s i o n dynamics, the RLM possesses two a t t r a c t i v e features. F i r s t , the angular degrees of freedom are handled i n a p a r t i a l wave expansion i n the t o t a l angular momentum index which introduces an impact parameter into the theory. Second, the RLM coupled-channel equations for each p a r t i a l wave, when expressed i n either Cartesian, polar, or hyperspherical coordinates, are i d e n t i c a l to those of a purely c o l l i n e a r formulat i o n , with the addition of an e f f e c t i v e centrifugal potential

R L M

2

2

2y(R +r )

to the c o l l i n e a r potential energy surface f o r reaction. Consequently, the solution of the reaction dynamics i n the RIM Is only as

26.

HAYES AND

Reactive Resonances & Angular Distributions

WALKER

495

d i f f i c u l t i n p r a c t i c e as s o l v i n g a f a m i l y of c o l l i n e a r problems, one f o r each p a r t i a l wave. I n e n f o r c i n g a s y m p t o t i c boundary c o n d i t i o n s , we use s p h e r i c a l B e s s e l f u n c t i o n s and i g n o r e the c e n t r i f u g a l z e r o p o i n t motion l / ( R + r ) i n E q u a t i o n 1. An obvious d e f e c t of the RLM i s t h a t i t n e g l e c t s motion of the r v e c t o r r e l a t i v e to the R v e c t o r . A s y m p t o t i c a l l y , the n e g l e c t e d degrees of freedom d e s c r i b e the r o t a t i o n a l motion of the d i a t o m i c , and i n the i n t e r a c t i o n r e g i o n , they d e s c r i b e the i n t e r n a l bending of the c o l l i s i o n complex. S i n c e the model o n l y samples the c o l l i n e a r p r o j e c t i o n of the e n t i r e p o t e n t i a l h y p e r s u r f a c e , one would expect the RU1 to be most r e l i a b l e f o r those r e a c t i o n s w h i c h proceed through l i n e a r c o l l i s i o n i n t e r m e d i a t e s w i t h s t r o n g l y h i n d e r e d bending motion. I n o r d e r to improve the q u a n t i t a t i v e a s p e c t s of the dynamics, e s p e c i a l l y i n the r e a c t i o n t h r e s h o l d r e g i o n , we have added a bending c o r r e c t i o n to the RLM, i n the same s p i r i t as d e s c r i b e d e a r l i e r (27-30) f o r o t h e r models. The model thus generated i s c a l l e d the B e n d i n g - C o r r e c t e d R o t a t i n g L i n e a r Model (BCRLM), and the r e s u l t s we d e s c r i b e here w i l l a l l be d e r i v e d from t h i s model. The d e t a i l s of the bending c o r r e c t i o n are d e s c r i b e d i n an e a r l i e r p u b l i c a t i o n (14). I n a s e r i e s of r e c e n t papers (14-17), we have used the BCRLM t o examine the m a n i f e s t a t i o n of resonances f o r s e v e r a l r e a c t i o n s , including 2

2

F+H (v) > HF(v') + H 2

H+H (v) 2

H (v') + H 2

and s e v e r a l of t h e i r i s o t o p i c v a r i a n t s . I n these papers, we have c o n s i d e r e d the importance of quantum e f f e c t s , e s p e c i a l l y resonances, upon i n t e g r a l and d i f f e r e n t i a l c r o s s s e c t i o n s and upon r a t e cons t a n t s f o r these r e a c t i o n s . The hope i s t h a t these approximate r e s u l t s c o n t a i n many of the i m p o r t a n t f e a t u r e s of more a c c u r a t e t h e o r e t i c a l methods. A t any t o t a l s c a t t e r i n g energy E, elements of the m u l t i c h a n n e l S m a t r i x i n the RLM are l a b e l l e d by the t o t a l a n g u l a r momentum i n d e x i and by the i n i t i a l and f i n a l v i b r a t i o n a l quantum numbers v and v*. E q u a t i o n s f o r p h y s i c a l o b s e r v a b l e s i n the BCRLM have been g i v e n p r e v i o u s l y (24-26), and we o n l y summarize the f i n a l r e s u l t s here, i n o r d e r to e s t a b l i s h a common n o t a t i o n . The o p a c i t y f u n c t i o n g i v e s the impact parameter dependence of the r e a c t i o n p r o b a b i l i t i e s , 9

P , ( E ) , a t f i x e d E, so t h a t V

V

P*, (E) = |s;, (E)| V

2

,

v

(2)

where the impact parameter i s r e l a t e d to the a n g u l a r momentum i n d e x by the s e m i c l a s s l c a l e x p r e s s i o n bk * (£+1/2), and k i s the t r a n s l a t i o n a l wavenumber. The a n g u l a r d i s t r i b u t i o n I ^ i ( 9 ) of products y

v

Is g i v e n by the d i f f e r e n t i a l c r o s s s e c t i o n

da

. ( e ; E ) / d f l = ( k . / k ) I f . ( 6;E) 1

2

v > v

v

v

v

v

= I

v > v

. ( 6)

(3)

496

RESONANCES

where the s c a t t e r i n g a m p l i t u d e i s

210JU/ I 12

f

v'v

( 9 ; E )

•

(

2

M

)

(

S

v ' v

(

E

)

- V v

)

P

(

) l

c

0

8

e

)

( 4 )

The i n t e g r a t e d c r o s s s e c t i o n i s





v

A l l a n g u l a r d i s t r i b u t i o n s r e p o r t e d i n t h i s paper a r e c a l c u l a t e d u s i n g E q u a t i o n s 3 and 4 above. However, i n i n t e r p r e t i n g f e a t u r e s of the a n g u l a r d i s t r i b u t i o n , i t i s o f t e n i n s t r u c t i v e t o a p p e a l t o the elements of the c l a s s i c a l f o r m u l a f o r a n g u l a r d i s t r i b u t i o n s (35) b P , (b)

I n E q u a t i o n 6, 6(b) i s the c l a s s i c a l d e f l e c t i o n f u n c t i o n , and s p e c i f i e s the a n g l e 8 a t which the p a r t i c l e s s e p a r a t e a f t e r a c o l l i s i o n a t an i n i t i a l impact parameter b. The d e f l e c t i o n f u n c t i o n has the quantum a n a l o g ( 3 5 ) dArg

If we substitute Equation 9 Into Equation 11 and make the simplifying assumption that S i s independent of energy, we obtain a direct expression for Q i n the v i c i n i t y of the resonance Q

fcl^V

0

2 (E-y

2

__fcr

2

2

2

+ T /4

2 2

,

2

—

'

( 1 2 )

2

(E-E ) +r /A R

where the unit vector u> i s

2-r"

1 / 2

aS

/ 2

i

•

( 1 3 )

498

RESONANCES

E q u a t i o n 12 t e l l s us three t h i n g s about the l i f e t i m e m a t r i x near an INR when the energy dependence of S i s n e g l i g i b l e : ( i ) the t r a c e of Q has a L o r e n t z i a n form ~" Q

T r a c e (Q)

^

—

,

(14)

and ( i i ) a l l the e i g e n v a l u e s [q^] of Q a r e z e r o e x c e p t f o r one (Q i s a m a t r i x of rank 1 ) , and ( i l l ) , the e i g e n v e c t o r of Q c o r r e s p o n d i n g to the nonzero e i g e n v a l u e i s the u n i t v e c t o r a) d e f i n e d i n Equat i o n 13. Combining E q u a t i o n s 13 and 10 g i v e s a d i r e c t e x p r e s s i o n f o r S i n terms of the a p p r o p r i a t e e i g e n v e c t o r of Q, namely —R —

S

R

— E-E_ + R

im

(15)

ir/2

Combining E q u a t i o n 15 w i t h E q u a t i o n 8 a l l o w s a d e t e r m i n a t i o n of the background S m a t r i x S , and from S a t each p a r t i a l wave one can Q

Q

c a l c u l a t e the a n g u l a r d i s t r i b u t i o n i n the absence of a resonance, E q u a t i o n 3. I n p r a c t i c e , the energy dependence of compromises the p r a c t i c a l a p p l i c a t i o n of the p r e c e d i n g d i s c u s s i o n , but as we hope to show, the scheme o u t l i n e d n e v e r t h e l e s s seems q u i t e u s e f u l i n a n a l y z i n g resonances. Resonance E f f e c t s On Quantum D e f l e c t i o n F u n c t i o n s I n a r e c e n t paper (16) we e x p l o r e d the e f f e c t s of resonances on the phase b e h a v i o r of i n d i v i d u a l elements of the S m a t r i x , w i t h p a r t i c u l a r a t t e n t i o n p a i d to the quantum d e f l e c t i o n f u n c t i o n , d e f i n e d i n E q u a t i o n 7. F o l l o w i n g i n the s p i r i t of C h i l d ' s (35) a n a l y s i s , we assume the p a r t i a l wave dependence of the resonance energy to be E («

= E (0) + B*(*+l)

R

R

(16)

and f u r t h e r assume the resonance w i d t h depends q u a d r a t i c a l l y on I r

a "

r 0

(

1 + a i l + b

*

*

2 )

( 1 7 )

C o n s i d e r now the phase b e h a v i o r p r e d i c t e d by E q u a t i o n 9 f o r the elements of the S m a t r i x . F o r s i m p l i c i t y , we assume the background c o n t r i b u t i o n p r o v i d e d by i s weak compared to S, and we concent r a t e on o f f - d i a g o n a l t r a n s i t i o n s . I n t h i s case, the phase b e h a v i o r of i n d i v i d u a l S-matrlx elements i s i S*.

1

= IS .

|e

V

* v

,

(18)

26.

HAYES AND WALKER

\%

= v

Reactive Resonances & Angular Distributions

\ i ( b a c k g r o u n d ) + n (resonant)

,

v

tan n (resonant) =

E ^ O - E

*

499

(19)

^

2 0

^

E q u a t i o n s 18-20 resemble t h e s i n g l e - c h a n n e l c a s e , b u t d i f f e r by a f a c t o r o f two, s i n c e we a r e concerned here o n l y w i t h o f f - d i a g o n a l elements. Note t h a t i t a l s o f o l l o w s from E q u a t i o n 9 t h a t t h e r e s o n a n t p a r t of the phase s h i f t i s independent of the channel l a b e l s v and v*. E q u a t i o n s 16-20 then l e a d t o the f o l l o w i n g e x p r e s s i o n f o r the b e h a v i o r of the quantum d e f l e c t i o n f u n c t i o n near a resonance, .

n

v

v

v

2

r (a+2b£)(E -E) - BT (2A+1) (E («-E) R

2

+

I n an e a r l i e r paper ( 1 6 ) , we showed t h a t E q u a t i o n 21 ( w i t h a=b=0) was remarkably good i n modeling the r e s o n a n t b e h a v i o r of the deflection function. A p p l i c a t i o n to Angular

Distributions

I n t h i s s e c t i o n , we p r e s e n t an a p p l i c a t i o n o f the t e c h n i q u e s d e s c r i b e d i n the p r e v i o u s s e c t i o n s t o the a n g u l a r d i s t r i b u t i o n f o r the r e a c t i o n F+H (v=0) • HF(v'=2)+H 2

A s e r i e s of c a l c u l a t i o n s , u s i n g the BCRLM, was performed on t h e M5 s u r f a c e ( 4 0 ) , as r e p o r t e d e a r l i e r ( 1 6 ) . Here we w i l l c o n c e n t r a t e on the a n g u l a r d i s t r i b u t i o n a t a f i x e d t o t a l energy E-1.807 eV [measured from the minimum o f the HF v i b r a t i o n a l p o t e n t i a l f o r asymp t o t i c H+HF g e o m e t r i e s ] . The i n i t i a l r e l a t i v e k i n e t i c energy o f ground s t a t e r e a c t a n t s a t t h i s t o t a l energy i s 0.162 eV [ o r 3.73 k c a l / m o l ] . We show the impact parameter dependence o f the magnitudes and phases o f the a p p r o p r i a t e elements o f the S m a t r i x i n F i g u r e s 1 and 2. F i g u r e 1 shows the o p a c i t y f u n c t i o n , P20 v s . I [see E q u a t i o n 1 ] , and F i g u r e 2 shows the quantum d e f l e c t i o n f u n c t i o n ©20 » * [see E q u a t i o n 7 ] . The presence of a r e s o n a n t c o n t r i b u t i o n v s

to the r e a c t i o n mechanism i s e v i d e n t as a peak i n t h e o p a c i t y f u n c t i o n near £=14 and as a d i p i n the d e f l e c t i o n f u n c t i o n near I = 16. The d i p i n the d e f l e c t i o n f u n c t i o n i s a consequence o f the enhanced l i f e t i m e of the r e s o n a n t complex, w h i c h r o t a t e s toward more f o r w a r d a n g l e s b e f o r e i t decays. The a n g u l a r d i s t r i b u t i o n w h i c h r e s u l t s from these c a l c u l a t i o n s i s shown i n F i g u r e 3. L e t us now a n a l y z e the resonance c o n t r i b u t i o n t o t h i s a n g u l a r d i s t r i b u t i o n more c l o s e l y . S i n c e F i g u r e 2 shows t h a t a t E-1.807 eV the r e s o n a n t c o n t r i b u t i o n t o the d e f l e c t i o n of p r o d u c t s i s g r e a t e s t near £=16, we show i n F i g u r e 4 a p l o t of the e i g e n v a l u e s of the £=16 l i f e t i m e m a t r i x [ E q u a t i o n 11] as a f u n c t i o n of t o t a l energy E. Over

500

RESONANCES

F i g u r e 2. Quantum d e f l e c t i o n f u n c t i o n , 0(Jt), as d e f i n e d by E q u a t i o n 7, f o r the r e a c t i o n F+H (v=0) HF(v =2)+H a t t o t a l E = 1.807 eV. ,:

2

26.

HAYES A N D WALKER

Reactive Resonances & Angular Distributions

501

0.03

0.00

-0.03

-0.06

-

1.70

1.85

1.80

1.75

E (eV) F i g u r e 4. E i g e n v a l u e s of the c o l l i s i o n l i f e t i m e m a t r i x Q v s . t o t a l energy f o r the £=16 p a r t i a l wave of the F+H r e a c t i o n . ""The e i g e n v a l u e f o r the resonant e i g e n l i f e t i m e c h a n n e l i s drawn as a s o l i d c u r v e , and a l l the o t h e r e i g e n v a l u e curves a r e d o t t e d . 2

502

RESONANCES

t h i s range of t o t a l e n e r g i e s , t h e r e a r e a r e f i v e open channels i n the S m a t r i x [v=0 of r e a c t a n t s and v'=0,1,2,3 of p r o d u c t s ] . I t i s c l e a r i n F i g u r e 4 t h a t a L o r e n t z l a n shaped f e a t u r e i s indeed c e n t e r e d near E=1.807 eV i n one of the e i g e n l i f e t l m e s . I f t h i s f e a t u r e i s f i t t e d to the form of E q u a t i o n 16, w i t h a q u a d r a t i c background b e h a v i o r , the resonance energy ^ ( 1 6 ) and w i d t h r can be determined. V a l u e s of E (£) and f o r other I values are determined s i m i l a r l y . I n F i g u r e 5, we show a p l o t of the energy dependence of the magnitudes of the elements of the e i g e n v e c t o r of g 1 6

R

c o r r e s p o n d i n g to the e i g e n v a l u e w i t h the L o r e n t z l a n p r o f i l e . These a r e r e l a t e d t o the p a r t i a l w i d t h s by E q u a t i o n 13. F i g u r e s 4 and 5 g i v e an i n d i c a t i o n of the degree of v a l i d i t y of the assumpt i o n s d i s c u s s e d from E q u a t i o n s 8 to 15. A l t h o u g h one of the e i g e n l i f etimes shows the resonance f e a t u r e much more s t r o n g l y than the o t h e r s , the o t h e r e i g e n l i f e t i m e s a r e n e i t h e r z e r o over the w i d t h of the resonance, nor a r e they even c o n s t a n t . S i m i l a r l y , i n F i g u r e 5, we see t h a t the r e s o n a n t p a r t i a l w i d t h s a r e n o t n e c e s s a r i l y s l o w l y v a r y i n g w i t h energy over the w i d t h of the resonance. U s i n g the data of F i g u r e 5 t o generate the p a r t i a l w i d t h s , we c a l c u l a t e the r e s o n a n t p a r t of the S m a t r i x u s i n g E q u a t i o n 16, and the background S m a t r i x u s i n g E q u a t i o n 8. Because the elements of the a) v e c t o r I n E q u a t i o n 16 come from the n u m e r i c a l d i a g o n a l l z a t i o n of Q c a l c u l a t e d u s i n g E q u a t i o n 11, we o b t a i n u> w i t h an undetermined phase e~*+. P o s s i b l e c h o i c e s of a r e determined i t e r a t i v e l y by r e q u i r i n g the background S m a t r i x S be u n i t a r y [ i n a d d i t i o n to Q

b e i n g s y m m e t r i c ] . A t e n e r g i e s E = E ^ ( i l ) , o n l y one v a l u e of makes S u n i t a r y , and a t o t h e r v a l u e s of E, a u n i t a r y S Is o b t a i n e d Q

Q

from e i t h e r of two c h o i c e s of . The p h y s i c a l l y s a t i s f a c t o r y c h o i c e i s the one f o r w h i c h the r e s o n a n t b e h a v i o r of S^ i s e l i m i n a t e d ( t h e o t h e r c h o i c e tends to a c c e n t u a t e the resonance). An example of procedure i s demonstrated i n F i g u r e 6, where we have p l o t t e d

1

S

V S

E

T h e

s

o

l

i

d

l

i

n

e

s

h

o

w

s

t h e

this

20 ^ | * " magnitudes of the f u l l - m a t r i x elements, the s o l i d c i r c l e s show the background S - m a t r i x elements c o r r e s p o n d i n g to the p h y s i c a l r o o t , and the open c i r c l e s show background S - m a t r i x elements c o r r e s p o n d i n g to the n o n p h y s i c a l root. A s s e m b l i n g the magnitudes of the background S - m a t r i x elements f o r the p h y s i c a l r o o t s a t a l l p a r t i a l waves from £=0 t o 18, we c o n s t r u c t the background o p a c i t y f u n c t i o n shown as the s o l i d curve i n F i g u r e 7, a t E - 1.807 eV. The l a r g e d i p i n t h i s o p a c i t y f u n c t i o n near £=12 i s the r e s u l t of the c r o s s i n g of two e i g e n v a l u e s of the l i f e t i m e m a t r i x I n t h i s range of I and E. The c r o s s i n g e i g e n v a l u e s induce i n t e r a c t i o n s between the e i g e n v e c t o r s and p e r t u r b the c a l c u l a t i o n of the background S m a t r i x . C o n s e q u e n t l y , we have a l s o used the o p a c i t y f u n c t i o n g i v e n by the dashed l i n e i n F i g u r e 7 to produce an e s t i m a t e of the background a n g u l a r d i s t r i b u t i o n . Furthermore, because of these p r a c t i c a l d i f f i c u l t i e s w i t h the d e t e r m i n a t i o n of the magnitudes of the background S - m a t r i x elements, we e l e c t e d n o t to use the phases determined i n t h i s manner, but to use phases from the d e f l e c t i o n f u n c t i o n a n a l y s i s d e s c r i b e d e a r l i e r . The s e t of resonance e n e r g i e s i s r e a s o n a b l y d e s c r i b e d by the form of

26.

HAYES A N D WALKER

1.00

Reactive Resonances & Angular Distributions

-

0.75

3

>

0.50

-

0.25

-

0.00 1.70'

1.86

F i g u r e 5. Energy dependence of the magnitudes of the elements of the e i g e n v e c t o r of Q c o r r e s p o n d i n g to the r e s o n a n t e i g e n v a l u e . Near the r e s o n a n t energy E = 1.806 eV, the c h a n n e l s w h i c h p a r t i c i p a t e most i n the resonance dynamics a r e c l e a r l y V J ^ O of the H r e a c t a n t s and v =2 of the HF p r o d u c t s . 2

p

1.00

0.75

-

i 0.50

-

0.25

0.00 1.70

F i g u r e 6. Magnitudes of background S - m a t r i x elements f o r the r e a c t i o n F+H (v=0) HF(v'=2)+H, p a r t i a l wave £=16. The s o l i d curve shows the f u l l S - m a t r l x elements, and the open and f i l l e d c i r c l e s show two c h o i c e s of background elements c o r r e s p o n d i n g to the two phase c h o i c e s f o r the p a r t i a l w i d t h s d i s c u s s e d i n the t e x t . The open c i r c l e s a r e elements c o r r e s p o n d i n g to a n o n p h y s i c a l r o o t , the c l o s e d c i r c l e s c o r r e s p o n d to the p h y s i c a l r o o t . 2

504

RESONANCES

1

E q u a t i o n 18 w i t h E ( 0 ) - 1.7234 eV and B - 3.05X10" * eV, and the I dependence of the w i d t h s i s a d e q u a t e l y d e s c r i b e d by E q u a t i o n 17 w i t h T = 0.031 eV, and a - b = 3.11X10" . We then use E q u a t i o n 21 to produce the background d e f l e c t i o n f u n c t i o n , and we show r e s u l t s i n F i g u r e 8 as the s o l i d c u r v e . The d o t t e d curve i s the f u l l d e f l e c t i o n f u n c t i o n of F i g u r e 2, shown f o r comparison. Phases of the background S - m a t r i x elements a r e determined, up to an unimp o r t a n t c o n s t a n t , by i n t e g r a t i n g the background d e f l e c t i o n f u n c t i o n . H a v i n g determined a s e t of background S - m a t r i x elements, we c a l c u l a t e an a n g u l a r d i s t r i b u t i o n due to the background s c a t t e r i n g , shown i n F i g u r e 9 as the s o l i d c u r v e . The dashed curve shows the a n g u l a r d i s t r i b u t i o n of u s i n g the smoothed o p a c i t y f u n c t i o n of F i g u r e 7, and the d o t t e d curve reproduces the d i s t r i b u t i o n of F i g u r e 3 f o r comparison. The e f f e c t of the resonance c o n t r i b u t i o n to the a n g u l a r d i s t r i b u t i o n i n F i g u r e 9 i s t w o f o l d : (1) the resonance enhances the i n t e n s i t y of the sideways peak, and ( i i ) the p o s i t i o n of the s i d e way speak i s s h i f t e d toward the f o r w a r d d i r e c t i o n by a p p r o x i m a t e l y 20°, an a d d i t i o n a l d e f l e c t i o n which q u a n t i t a t i v e l y agrees w i t h the s i z e of the d i p i n the d e f l e c t i o n f u n c t i o n shown i n F i g u r e 8. There i s a l s o a s m a l l r e d u c t i o n i n the i n t e r f e r e n c e o s c i l l a t i o n s i n the f o r w a r d d i r e c t i o n . I t i s s i g n i f i c a n t to n o t e , however, t h a t the background a n g u l a r d i s t r i b u t i o n i s sideways peaked, as a consequence of the g e n e r a l l y I n c r e a s i n g b e h a v i o r of the o p a c i t y f u n c t i o n s of F i g u r e 7 a t low £. R

3

0

Forward Peaked A n g u l a r D i s t r i b u t i o n s I n the remainder of t h i s paper, we w i s h to c o n s i d e r the c o n d i t i o n s w h i c h may l e a d to broad, or even f o r w a r d peaked, a n g u l a r d i s t r i b u t i o n s of p r o d u c t s f o r e n e r g i e s near the r e a c t i o n t h r e s h o l d . Such a n g u l a r d i s t r i b u t i o n s have been observed i n r e c e n t F+H beam e x p e r i ments ( 2 3 ) , and w i t h i n the c o n t e x t of the BCRLM, we have observed t h i s phenomenon i n s e v e r a l systems. 2

F+H . We see t h i s s o r t of b e h a v i o r i n the r e a c t i o n F+H (v=0) HF(v*=2)+H on a m o d i f i c a t i o n (41) of the M5 p o t e n t i a l s u r f a c e . T h i s m o d i f i e d s u r f a c e i s c a l l e d s u r f a c e 3 i n r e f e r e n c e 41; i t d i f f e r s from the M5 s u r f a c e i n the f o l l o w i n g t h r e e q u a l i t a t i v e ways: ( i ) the new s u r f a c e has a l o w e r c o l l i n e a r b a r r i e r to r e a c t i o n , ( i i ) i t has a s o f t e r bending p o t e n t i a l i n the e n t r a n c e v a l l e y , and ( i l l ) has a much l o w e r a d i a b a t i c b a r r i e r i n the e x i t v a l l e y f o r r e a c t i o n i n t o the HF(v'=3) f i n a l s t a t e . We show i n F i g u r e s 10 and 11 a p l o t of the r e a c t i o n p r o b a b i l i t i e s f o r t h i s system, f o r r e a c t i o n s i n t o HF(v'=2) and HF(v'=3) r e s p e c t i v e l y . The q u a l i t a t i v e shape of the p r o b a b i l i t y curves f o r r e a c t i o n to H F ( v 3 ) resembles t h a t f o r the M5 p o t e n t i a l s u r f a c e , e x c e p t f o r a reduced t h r e s h o l d d e l a y a t low p a r t i a l waves [a consequence of the lower a d i a b a t i c b a r r i e r ] . There i s a much l a r g e r d i f f e r e n c e i n the dynamics l e a d i n g to H F ( v 2 ) p r o d u c t s . A t low p a r t i a l waves, the r e a c t i o n i n t o HF(v'=2) i s suppressed r e l a t i v e t o the M5 c a s e , and a t l a r g e r p a r t i a l waves, t h e r e i s an obvious onset of a resonance f e a t u r e . Presumably, the e x i t - v a l l e y a d i a b a t i c 9

2

,js

, =

26.

HAYES A N D WALKER

0

Reactive Resonances & Angular Distributions

10

20

505

30

TOTAL ANGULAR MOMENTUM I F i g u r e 7. Background o p a c i t y f u n c t i o n f o r the r e a c t i o n F+H (v=0) + HF(v'=2)+H a t E - 1.807 eV. The s o l i d c u r v e i s the d i r e c t r e s u l t of the e x t r a c t i o n procedure d i s c u s s e d i n the t e x t , and the dashed curve i s a smoothed v e r s i o n . The d o t t e d c u r v e reproduces the o p a c i t y f u n c t i o n of F i g u r e 1 f o r comparison. 2

0

10

20

30

TOTAL ANGULAR MOMENTUM / F i g u r e 8. Background d e f l e c t i o n f u n c t i o n f o r the r e a c t i o n F+H > HF(v'=2)+H a t E = 1.807 eV. The s o l i d c u r v e removes the resonance c o n t r i b u t i o n u s i n g E q u a t i o n 21. The d o t t e d curve reproduces the d e f l e c t i o n f u n c t i o n of F i g u r e 2 f o r comparison. 2

506

RESONANCES

ANGLE (deg) F i g u r e 9. Background a n g u l a r d i s t r i b u t i o n s f o r the r e a c t i o n F+H (v=0) HF(v'=2)+H a t E - 1.807 eV. The s o l i d curve i s based on the o p a c i t y f u n c t i o n shown i n F i g u r e 7 as a s o l i d c u r v e , and the smoother, dashed curve shows the a n g u l a r d i s t r i b u t i o n w h i c h r e s u l t s from the smoothed o p a c i t y f u n c t i o n of F i g u r e 7. The d o t t e d curve reproduces the f u l l a n g u l a r d i s t r i b u t i o n o f F i g u r e 3 f o r comparison. 2

1

1

!

y

Z=30 1=25

1=20

Z=15 Z=10 1=5 1=0

0 E

1

1

1

2

4

6

TRANS

8

(kcal/mol)

F i g u r e 10. R e a c t i o n p r o b a b i l i t i e s v s . r e l a t i v e t r a n s l a t i o n a l energy f o r the r e a c t i o n F+H (v=0) HF(v'=2)+H on s u r f a c e 3 o f r e f e r e n c e 41. 2

26.

HAYES A N D W A L K E R

Reactive Resonances & Angular Distributions

507

b a r r i e r which Is present In the M5 surface, but has been reduced i n this newer surface, i s replaced by a s i m i l a r [but centrifugal] barr i e r at intermediate p a r t i a l waves, thereby restoring the resonance mechanism. The e f f e c t on angular d i s t r i b u t i o n s i s obvious. Figure 12 shows opacity function plots at several energies, and Figure 13 shows the corresponding angular d i s t r i b u t i o n s . The resonant mechanism In this case Is considerably stronger than for the corresponding M5 c a l c u l a t i o n , and the e f f e c t on the angular distributions is to induce much stronger interference o s c i l l a t i o n s , since there are many fewer p a r t i a l waves contributing s i g n i f i c a n t l y to the reaction. In the case of the reaction to produce HF(v 3) on this surface, It is evident that the v'=3 state of products p a r t i c i pates more strongly i n the resonance mechanism, as can be seen i n the angular d i s t r i b u t i o n plots of Figure 14. The angular d i s t r i b u t i o n s h i f t s from backward to sideways peaked, and has noticeable interference o s c i l l a t i o n s , i n d i c a t i v e of resonance dynamics, at a l l energies. The F+H reaction on this newer potential surface demonstrates one way i n which the BCRLM w i l l produce an angular d i s t r i b u t i o n at the reaction threshold which is not smooth and backward-peaked. In this case, the absence of a s i g n i f i c a n t reaction p r o b a b i l i t y for low p a r t i a l waves, and the appearance of a resonance feature at larger p a r t i a l waves, combine to produce an opacity function which peaks at large p a r t i a l waves, and hence an angular d i s t r i b u t i o n which has a predominant forward d i s t r i b u t i o n of reaction products. We have seen results s i m i l a r to these (41) i n the angular d i s t r i b u t i o n f o r the ,sa

2

reactions F+D (v=0) 2

,ss

DF(v 3,4)+D on this same surface.

1

He+H?"". We performed a set of BCRLM calculations for the reaction He+H HeH +H using the DIM representation (42) of the p o t e n t i a l surface. A great many resonance features have been seen (43) i n the c o l l i n e a r reactive dynamics of this system, and many have been i d e n t i f i e d (44) with the a t t r a c t i v e well which i s present on this surface i n the entrance v a l l e y . Because the bending potential predicted by the DIM surface Is very weak, the BCRLM dynamics (for the £=0 p a r t i a l wave) resembles that of the c o l l i n e a r c a l c u l a t i o n . Reaction p r o b a b i l i t i e s for the reaction He+H (v=3) • HeH (v =0)+H are shown i n Figure 15 f o r several p a r t i a l waves. The predominance of many resonances i n each p a r t i a l wave i s evident In the figure. As written, this reaction i s endothermic by 0.81 eV, but there i s no b a r r i e r over and above the endotherm!city of reaction. Consequentl y , reaction occurs from many p a r t i a l waves at the reaction threshold. There i s l i t t l e or no evidence of a c e n t r i f u g a l b a r r i e r to reaction appearing i n the dynamics u n t i l the £=15 p a r t i a l wave. For this reason the opacity function, at the reaction threshold, contains contributions from a large number of p a r t i a l waves. The r e s u l t i n g angular d i s t r i b u t i o n s of products are therefore quite broad, and show the effects of the multitude of resonances through the very o s c i l l a t o r y nature of the d i s t r i b u t i o n s , as i s shown i n Figure 16. Results s i m i l a r to these, except for reaction from other i n i t i a l v i b r a t i o n a l states of the H + reactant molecule, are q u a l i t a t i v e l y the same — reaction from many p a r t i a l waves at threshold makes possible a very broad angular d i s t r i b u t i o n of products. +

2

+

2

2

+

,

RESONANCES

2 E

4

6

8

(kcal/mol)

TRANS

F i g u r e 11. As i n F i g u r e 10, e x c e p t v

J

1

i

3.

l_

A

J

Q_

'

1

, s s

eV

\E=4.65

E=3.73 eV

o — I O

A

E=2.81 eV

J

< Q_

O

E=1.88 eV

E=0.96 eV

A

1

F i g u r e 12. O p a c i t y r e a c t i o n F+H (v=0) 2

,

1

,

1

:

0

10

20

TOTAL

ANGULAR

1

,

1

30

40

MOMENTUM

I

f u n c t i o n s a t s e v e r a l e n e r g i e s f o r the HF(v'=2)+H on s u r f a c e 3 o f r e f e r e n c e

Reactive Resonances & Angular Distributions

HAYES A N D WALKER

i ii i ii i

0

iiiiii i i

60

120

180

ANGLE ( d e g ) F i g u r e 13. A n g u l a r d i s t r i b u t i o n s a t the same e n e r g i e s as F i g u r e 12 f o r the r e a c t i o n F+H (v=0) HF(v'=2)+H on s u r f a c e 3 of r e f e r e n c e 4 1 . 2

II

| i

iii i|iiiii |

E=1.88 eV

E=0.96 eV I

0

I,•,•,I

60

120

i

180

ANGLE ( d e g ) F i g u r e 14.

As i n F i g u r e 13, except

v

, = !

3.

510

RESONANCES

0.9

1.0 1.1

1.2 1.3 1.4 1.5

E (eV) F i g u r e 15. R e a c t i o n p r o b a b i l i t i e s v s . energy f o r the r e a c t i o n He+H (v=3) + HeH (v'=0)+H on the DIM s u r f a c e of r e f e r e n c e 42. T o t a l e n e r g i e s a r e measured from the minimum of the H entrance valley well. +

+

2

+

2

J

0

i

i

i

i

i—I

i—i—i

60

i—i—I—i—i—i—i—i—I—

120

180

ANGLE (deg) +

F i g u r e 16. A n g u l a r d i s t r i b u t i o n s f o r the r e a c t i o n He+H (v=»3) • HeH (v 0)+H. T o t a l e n e r g i e s a r e measured from the minimum o f the H entrance v a l l e y w e l l . 2

+

,s

+

2

26.

HAYES A N D WALKER

Reactive Resonances & Angular Distributions

511

Summary We have attempted t o d e s c r i b e here the way i n w h i c h resonances i n r e a c t i v e s c a t t e r i n g may a f f e c t the a n g u l a r d i s t r i b u t i o n of the p r o d u c t s o f r e a c t i o n . To do t h i s , we have employed a s i m p l e 3D model of r e a c t i o n s , the B e n d i n g - C o r r e c t e d R o t a t i n g L i n e a r Model, and have computed a n g u l a r d i s t r i b u t i o n s f o r s e v e r a l r e a c t i v e systems i n w h i c h resonances c o n t r i b u t e to the s c a t t e r i n g dynamics. The most s t r a i g h t f o r w a r d way to determine the e f f e c t o f r e s o nances on a n g u l a r d i s t r i b u t i o n s i s to d e f i n e a p a r t i t i o n of the a n g u l a r d i s t r i b u t i o n i n t o i t s d i r e c t and r e s o n a n t components. We d e s c r i b e d an a t t e m p t to do t h i s f o r the r e a c t i o n F+H (v=0) + H F ( v 2 ) + H a t an energy where the c a l c u l a t e d a n g u l a r d i s t r i b u t i o n i s d e c i d e d l y sideways peaked. I n t h i s a n a l y s i s , we found the e f f e c t of the r e s o n a n t p a r t of the s c a t t e r i n g was to enhance the i n t e n s i t y of the sideways peak and to s h i f t i t t o a more f o r w a r d a n g l e . Howe v e r , even w i t h o u t the r e s o n a n t c o n t r i b u t i o n , we f i n d the background a n g u l a r d i s t r i b u t i o n to be sideways peaked. T h i s o b s e r v a t i o n need n o t be s u r p r i s i n g , however, f o r i t has been observed f o r many y e a r s t h a t a d i r e c t s c a t t e r i n g mechanism can l e a d t o sideways peaked a n g u l a r d i s t r i b u t i o n s as the c o l l i s i o n energy i s i n c r e a s e d (45-52, 2

,s

15>ii>We have a l s o sought to d e s c r i b e the c o n d i t i o n s under w h i c h the a n g u l a r d i s t r i b u t i o n of r e a c t i o n p r o d u c t s c o u l d be forward-peaked a t the r e a c t i o n t h r e s h o l d , w i t h i n the c o n t e x t o f the BCRLM. Two s o u r c e s of t h i s e f f e c t have been seen i n our c a l c u l a t i o n s , and the e s s e n t i a l f e a t u r e w h i c h they have i n common i s t h a t the r e a c t i o n must, a t t h r e s h o l d , c o n t a i n c o n t r i b u t i o n s from s c a t t e r i n g a t l a r g e p a r t i a l waves. I n the example shown f o r t h e F+H r e a c t i o n on a m o d i f i c a t i o n (41) of the M5 s u r f a c e , a f o r w a r d peaked d i s t r i b u t i o n a t t h r e s h o l d was seen to r e s u l t from a r e s o n a n t mechanism w h i c h I s a b s e n t i n low p a r t i a l waves, and becomes s i g n i f i c a n t f o r l a r g e r p a r t i a l waves. I n the second example, f o r the H e + H r e a c t i o n , f o r w a r d peaked a n g u l a r d i s t r i b u t i o n s a t t h r e s h o l d r e s u l t from the f a c t t h a t the r e a c t i o n proceeds w i t h o u t a b a r r i e r ( o t h e r than t h a t due t o the o v e r a l l e n d o t h e r m i c i t y o f r e a c t i o n ) . I n the H e + H r e a c t i o n , there a r e i n d e e d many resonances w h i c h c o n t r i b u t e t o the dynamics, b u t I t i s p o s s i b l e t h a t the a n g u l a r d i s t r i b u t i o n would have been peaked i n the f o r w a r d d i r e c t i o n even w i t h o u t the resonances. 2

+

2

+

2

Literature Cited

1. Walker, R. B.; Light, J. C. Ann. Rev. Phys. Chem. 1980, 31, 401. 2. Schatz, G. C. In "Potential Energy Surfaces and Dynamics Calculation"; Truhlar, D. G., Ed.; Plenum: New York, 1981; Chap. 12. 3. Kuppermann, A. In "Theoretical Chemistry: Advances and Perspectives, Vol. 6A"; Academic: New York, 1981; p. 79-164. 4. Baer, M. Adv. Chem. Phys. 1982, 49, 191. 5. Wyatt, R. E. In "Horizons in Quantum Chemistry"; Fukui, K.; Pullman, B., eds.; Reidel: Dordrecht, 1980; p. 63. 6. Redmon, M. J.; Wyatt, R. E. Chem. Phys. Letters 1979, 63, 209. 7. Wyatt, R. E.; McNutt, J. F; Redmon, M. J. Ber. Bunsenges. Phys. Chem. 1982, 86, 437.

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26.

41.

42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52.

HAYES A N D W A L K E R

Reactive Resonances & Angular Distributions

513

Steckler, R.; Truhlar, D. G.; Garrett, B. C.; Walker, R. B.; Blais, N. C., to be published. This modified surface is called new surface number 3; it is obtained by a slight modification of new surface number 2 of Truhlar, D. G.; Garrett, B. C.; Blais, N. C. J. Chem. Phys. 1984, 80, 232; the value for the Sato parameter is increased from 0.106 to 0.168. The new value removes the exit-channel adiabatic barrier for FD(v'=4)+D. Kuntz, P. J. Chem. Phys. Letters 1972, 16, 581. Adams, J. T. Chem. Phys. Letters 1975, 33, 275. Chapman, F. M., Jr.; Hayes, E. F. J. Chem. Phys. 1975, 62, 4400. Tang, K. T. Ph.D. Thesis, Columbia University, New York, 1965. Karplus, M.; Tang, K. T. Disc. Faraday Soc. 1967, 44, 56. Tang, K. T.; Karplus, M. Phys. Rev. A 1971, 4, 1844. Truhlar, D. G.; Dixon, D. A. in "Atom-Molecule Collision Theory"; Bernstein, R. B., Ed.; Plenum: New York, 1979; Chap. 18. Agmon, N. Chem. Phys. 1981, 61, 189. Suck, S. H.; Emmons, R. W. Phys. Rev. A 1981, 24, 129. Emmons, R. W.; Suck, S. H. Phys. Rev. A 1982, 25, 178. Salk, S. H. Suck; Emmons, R. W.; Klein, C. R. Phys. Rev. A 1984, 29, 1135.

RECEIVED

June 19, 1984