Resonance Phenomena in Quantal Reactive Infinite-Order Sudden

24 Resonance Phenomena in Quantal Reactive Infinite-Order Sudden Calculations Z.

H.

ZHANG , 1

Department

1

N.

ABUSALBI , 1

M.

BAER ,

D.

2

J.

KOURI , 1

and

J.

JELLINEK

3

of Chemistry and Department of Physics, University of Houston-University Park,

Houston, TX 77004 Applied

Downloaded by UNIV OF SYDNEY on September 21, 2015 | http://pubs.acs.org Publication Date: September 28, 1984 | doi: 10.1021/bk-1984-0263.ch024

2

Mathematics, Soreq Nuclear Research Center, Yavne, Israel 70600

Department

3

of Chemical Physics, Weizmann Institute of Science, Rehovot, Israel 76100

In this paper we discuss the resonance tuning hypothesis as an important mechanism whereby resonances are spread in theF+H andF+D reactionsystems and examine wheth er the shirt from backward to sideways scattering of the HF and DF products is a resonance signature. All results are obtained using the Muckerman 5 potential surface. 2

2

The development o f n o n p e r t u r b a t i v e q u a n t a l a p p r o x i m a t i o n s f o r 3d i m e n s i o n a l r e a c t i v e s c a t t e r i n g i s v e r y i m p o r t a n t and i s c u r r e n t l y b e i n g pursued by s e v e r a l groups.(1-12) Of t h e a v a i l a b l e methods, one o f t h e most w i d e l y a p p l i e d has been t h e r e a c t i v e i n f i n i t e o r d e r sudden (RIOS) approximation.(4-7,10,12) T h i s method has now been a p p l i e d t o t h e H+H , F+H , F+D , D+H and D+HC1 r e a c t i o n s . I n the case o f H+H , q u a l i t a t i v e l y c o r r e c t r e s u l t s were o b t a i n e d f o r a n g u l a r d i s t r i b u t i o n s and good q u a n t i t a t i v e agreement was o b t a i n e d f o r t o t a l r e a c t i v e c r o s s s e c t i o n s compared t o e x a c t c l o s e c o u p l i n g (CC) r e s u l t s . ( 4 - 5 , 1 3 ) I n t h e case o f t h e F+H and F+D systems,(6-7) the &-av RIOS r e s u l t s f o r t h e t o t a l i n t e g r a l r e a c t i v e c r o s s s e c t i o n s agreed w e l l w i t h Muckerman s c l a s s i c a l t r a j e c t o r y ( C T ) r e s u l t s , ( 1 4 ) as w e l l as w i t h CT r e s u l t s o f Ron, P o l l a k and Baer.(15) E v i d e n c e suggested t h a t t h e t o t a l i n t e g r a l r e a c t i v e c r o s s s e c t i o n i s n o t s u b j e c t t o l a r g e quantum e f f e c t s ( 6 ^ 7 ) and thus t h e RIOS appeared t o g i v e a c c e p t a b l e a c c u r a c y f o r such s t a t e summed q u a n t i t i e s . The b r a n c h i n g r a t i o s f o r d i f f e r e n t p r o d u c t v i b r a t i o n a l s t a t e s appeared however t o show l a r g e quantum e f f e c t s , ( 6 - 7 ) i n q u a l i t a t i v e agreement w i t h t h e j - c o n s e r v i n g r e s u l t s o f Redmon and Wyatt(3) ( o b t a i n e d w i t h a somewhat d i f f e r e n t p o t e n t i a l s u r f a c e ) . Of p a r t i c u l a r i n t e r e s t were t h e a n g u l a r d i s t r i b u t i o n s of t h e HF and DF p r o d u c t m o l e c u l e s as a f u n c t i o n o f f i n a l v i b r a t i o n a l s t a t e v . T h i s i s due t o t h e f a c t 2

2

2

2

2

2

2

f

z

f

0097-6156/84/0263-0457$06.50/0 © 1984 American Chemical Society

In Resonances; Truhlar, D.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

458

RESONANCES

t h a t o n l y f o r two systems (F+H^ and F+D«) i s such s t a t e - r e s o l v e d d a t a c u r r e n t l y a v a i l a b l e from e x p e r i m e n t . ( 1 6 ) The e x p e r i m e n t s show a f a s c i n a t i n g b e h a v i o r i n w h i c h a t t h e l o w e s t energy, a l l HF and DF p r o d u c t s were backward s c a t t e r e d w h i l e a t a s l i g h t l y h i g h e r energy, a l l HF and DF p r o d u c t m o l e c u l e s c o n t i n u e d t o be backward s c a t t e r e d except t h e HF v =2 and DF v =3 p r o d u c t s , w h i c h were sideways s c a t t e r e d . More r e c e n t s t u d i e s s u g g e s t , however, t h a t t h e r e i s a s i g n i f i c a n t f o r w a r d peak i n t h e v =3 HF and v =4 DF p r o d u c t s . ( 1 7 ) The b e h a v i o r f o r F+H was t e n t a t i v e l y i n t e r p r e t e d i n terms o f a mechanism i n w h i c h t h e w e l l known v =2 c o l l i n e a r resonance(18) f o r t h e F+H^ system was s h i f t e d t o h i g h e r o r b i t a l a n g u l a r momentum a s t h e energy was i n c r e a s e d u n t i l u l t i m a t e l y , a s h i f t t o sideways s c a t t e r i n g o c c u r r e d . ( 1 6 , 3 ) I n t h e case o f F+H t h e RIOS c a l c u l a t i o n s u s i n g t h e Muckerman 5 p o t e n t i a l ( 1 4 ) were i n q u a l i t a t i v e agreement w i t h t h i s s h i f t from backwards t o sideways s c a t t e r i n g o f t h e HF(v =2) p r o d u c t b u t d i d n o t unambiguously i n d i c a t e t h a t a resonance mechanism was r e s p o n s i b l e . ( 6 ) Thus, w h i l e t h e r e was e v i dence t h a t resonance e f f e c t s c o u l d be p r e s e n t , i t was n o t n e c e s s a r i l y c l e a r t h a t t h e y were s o l e l y r e s p o n s i b l e f o r t h e s h i f t i n t h e a n g u l a r p a t t e r n f o r t h e HF(v =2) p r o d u c t s . Other model c a l c u l a t i o n s have r e c e n t l y been r e p o r t e d t h a t a l s o show sideways p e a k i n g o f t h e v =2 HF p r o d u c t . ( l i b , 1 9 ) I n a d d i t i o n , t h e y show backward s c a t t e r i n g f o r t h e v =3 HF p r o d u c t a t b o t h t h e l o w and h i g h e r energy. Of t h e s e , o n l y t h e Bowman, e t . a l . ( 1 7 ) r e s u l t s i n c l u d e d a sum o v e r a l l f i n a l HF r o t a t i o n a l s t a t e s . C l a s s i c a l t r a j e c t o r y (CT) c a l c u l a t i o n s have been performed by Ron, Baer and P o l l a k (RBP)(20a) f o r a s i n g l e energy E ^=0.5 eV (which i s somewhat h i g h e r t h a n t h e e x p e r i m e n t a l e n e r g i e s ) where E i s t h e t o t a l energy o f t h e system r e l a t i v e t o t h e H ( D ) d i a t o m i c w e l l . The RBP s t u d y c o n s i s t e d o f b o t h f o r w a r d and r e v e r s e CT c a l c u l a t i o n s . The f o r w a r d s t u d y showed o n l y backwards s c a t t e r i n g f o r a l l f o u r f i n a l v i b r a t i o n a l s t a t e s , and i n t h i s sense c o n f i r m e d t h e e a r l i e r f o r w a r d CT r e s u l t s o b t a i n e d by B l a i s and T r u h l a r . ( 2 0 b ) I n t h e case o f t h e r e v e r s e CT c a l c u l a t i o n s , a d i f f e r e n t p i c t u r e was o b t a i n e d . There a r e many a c c e s s i b l e r o t a t i o n a l s t a t e s o f t h e HF m o l e c u l e w h i c h can be p o p u l a t e d a t t h e n o m i n a l e x p e r i m e n t a l e n e r g i e s . T h e r e f o r e , t h e r e v e r s e CT s t u d i e s had t o be done f o r a v a r i e t y o f r o t a t i o n a l s t a t e s . I t was found t h a t a l l r o t a t i o n a l s t a t e s f o r HF i n t h e v=3 s t a t e y i e l d e d backwards s c a t t e r i n g , b u t s e v e r a l o f t h e most p r o b a b l e r o t a t i o n a l s t a t e s f o r HF w i t h v=2 y i e l d e d sideways s c a t t e r i n g . I t was f e l t by RPB t h a t t h e s e r e s u l t s p o i n t up a t e c h n i c a l problem w i t h t h e f o r w a r d CT c a l c u l a t i o n s h a v i n g t o do w i t h b o x i n g o f t h e HF r o t a t i o n a l - v i b r a t i o n a l s t a t e s . I n t h e r e v e r s e CT c a l c u l a t i o n s , one boxes on t h e f i n a l H« s t a t e s and b e g i n s t h e c a l c u l a t i o n i n a w e l l d e f i n e d HF r o t a t i o n a l - v i b r a t i o n a l s t a t e . Thus, RPB contend t h a t t h e CT r e s u l t s t a k e n as a whole cannot be c o n s i d e r e d t o s u p p o r t t h e e x i s t e n c e o f quantum e f f e c t s i n t h e v i b r a t i o n a l selected angular d i s t r i b u t i o n s . f

f

2

f


2

f

f

t

2

2

I n a d d i t i o n t o t h e F+H system, S p a r k s , e t . a l . (16,17) a l s o performed measurements on t h e F+D system. C o l l i n e a r c a l c u l a t i o n s f o r t h i s system a l s o show i n t e r e s t i n g resonance b e h a v i o r , a l t h o u g h the resonance appears t o be s i g n i f i c a n t l y b r o a d e r ( 2 1 ) t h a n i n t h e case o f c o l l i n e a r F+H The e x p e r i m e n t a l r e s u l t s f o r t h i s system showed p u r e backward s c a t t e r i n g f o r DF(v.) f o r a l l v ^ a t a r e l a t i v e k i n e t i c energy o f 2.34 k c a l / m o l e w h i l e a t 4.51 k c a l / m o l e , t h e p r o ?

2#


24.

Z H A N G E T AL.

Resonance Phenomena in Sudden Calculations

459

d u c t s a r e s c a t t e r e d p r o g r e s s i v e l y more f o r w a r d . The r e s u l t s show the v =4 DF p r o d u c t i s more s t r o n g l y f o r w a r d peaked than backward a t 4.51 k c a l / m o l e , w i t h t h e sequence v^=3,2,l as one goes t o l a r g e r s c a t t e r i n g a n g l e . Thus, t h e DF e x p e r i m e n t a l r e s u l t s agree q u a l i t a t i v e l y w i t h t h e newer HF e x p e r i m e n t a l r e s u l t s w h i c h appear t o show t h e v 3 HF p r o d u c t w i t h s i g n i f i c a n t f o r w a r d s c a t t e r i n g , f o l l o w ed by t h e v =2 and then t h e v^.=l HF p r o d u c t s . R e c e n t l y , Ron, P o l l a k , and B a e r ( 1 5 ; have performed a f o r w a r d CT c a l c u l a t i o n a l s o f o r t h e F+D system a t t h e two n o m i n a l e x p e r i m e n t a l e n e r g i e s . A g a i n , i n c o n t r a s t t o t h e e x p e r i m e n t a l r e s u l t s and t h e quantum RIOS r e s u l t s , ( 7 ) a l l t h e s t a t e - t o - s t a t e 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 were found t o be s c a t t e r e d backward. A b u S a l b i e t a l . ( 7 b ) have c a r r i e d out quantum RIOS c a l c u l a t i o n s f o r F+D and found t h a t t h i s system ( u s i n g t h e Muckerman 5 p o t e n t i a l ) behaves i n a manner c o m p l e t e l y analogous t o F+H so f a r as t h e f i x e d - Y r e a c t i o n p r o b a b i l i t i e s and s t a t e - r e s o l v e d a n g u l a r d i s t r i b u t i o n s a r e concerned. Thus, t h e v =3 s t a t e o f DF p l a y s t h e r o l e o f t h e v =2 s t a t e o f HF, w h i l e v f =5 HF c o r r e s p o n d s t o v = 4 DF. These h i g h e s t s t a t e s behave d i f f e r e n t from experiment i n b o t h FH^ and FD^ s i n c e t h e y a r e backward s c a t t e r e d a t b o t h low and h i g h e r e n e r g i e s . The q u e s t i o n o f resonances i n t h e FD system was answered u n e q u i v o c a b l y u s i n g t h e l i f e t i m e m a t r i x method o f Smith,(22) and i t was shown t h a t t h e energy o f t h e resonance d i d tune as Y o r I was changed where Y i s t h e i n t e r n a l a n g l e between t h e v e c t o r from t h e d i a t o m c e n t e r o f mass t o t h e atom (F) and t h e d i a t o m (H^ o r D^) a x i s and £ i s t h e CS o r b i t a l parameter (more d e t a i l s about £ w i l l be g i v e n b e l o w ) . However, t h e g r i d i n Y and £ was n o t f i n e enough t o t h o r o u g h l y s t u d y how t h e resonance s h i f t e d and changed i n i n t e n s i t y as Y and I were v a r i e d . I n a d d i t i o n , resonances i n FH^ a r e e x p e c t e d t o be s t r o n g e r t h a n i n FD^ and i t i s o f i n t e r e s t t o examine i n more d e t a i l how s e n s i t i v e t h e resonance i s t o t u n i n g i n Y and I. In p a r t i c u l a r , i t i s i m p o r t a n t t o examine j u s t how r a p i d l y t h e l i f e t i m e o f t h e c o l l i n e a r resonance d e c r e a s e s as Y and/or £ a r e increased. =

f


2

2

f

f

I n t h i s p a p e r , we w i s h t o examine t h e r e s u l t s o f e a r l i e r RIOS c a l c u l a t i o n s f o r t h e FH^ and FD^ systems. Our purpose i n t h i s i s t o show how t h e two systems behave i n a p a r a l l e l f a s h i o n . We a l s o summarize t h e r e s u l t s o f t h e l i f e t i m e s t u d i e s o f t h e FD^ system. Next we r e p o r t new, more d e t a i l e d l i f e t i m e c a l c u l a t i o n s c a r r i e d out f o r t h e F H system. These r e s u l t s show i n much more d e t a i l how t h e energy o f t h e resonance s h i f t s w i t h changes i n Y and £. A l s o , we show how t h e magnitude o f t h e d e l a y t i m e changes w i t h Y and I. F i n a l l y , we compare t h e F H d e l a y t i m e s f o r v a r i o u s Y , * w i t h t h e c h a r a c t e r i s t i c t i m e s o f m o l e c u l a r r o t a t i o n and v i b r a t i o n i n o r d e r t o get a b e t t e r f e e l i n g f o r how s t r o n g l y d e l a y e d t h e system i s . A l l t h e s e r e s u l t s show i n d e t a i l t h e t u n i n g mechanism. Finally, t h e q u e s t i o n o f whether t h e resonance phenomenon i s r e s p o n s i b l e f o r t h e sideways s h i f t o f t h e F H ( v =2) a n g u l a r d i s t r i b u t i o n i s a d d r e s s ed. 9

?

0

2

1

Summary o f Theory The t h e o r y o f t h e q u a n t u a l RIOS was i n t r o d u c e d i n an e a r l i e r s e r i e s o f p a p e r s . (4^, 6) Here we s i m p l y summarize some o f t h e s a l i e n t p o i n t s of the formalism.


460

RESONANCES

D i f f e r e n t i a l and i n t e g r a l c r o s s s e c t i o n s . The degeneracy averaged d i f f e r e n t i a l c r o s s s e c t i o n from a g i v e n i n i t i a l s t a t e ( ^ k ^ ) f i n a l v i b r a t i o n a l s t a t e v and a l l r o t a t i o n a l s t a t e s i s g i v e n by v v

da(vJv j m |e,i|i)

m

t

o

a

2

^7777

/J

f (

e

Vv\lVA^ -*>l

( 2 j . + 1 ) J.m X X X where X and v denote t h e i n i t i a l and f i n a l arrangement s,u) i s a s o l i d a n g l e c o m p r i s e d o f t h e s c a t t e r i n g a n g e l 0 , and t h e a z i m u t h a l s c a t t e r i n g a n g l e ip; v , j and m , a=X,v, a r e t h e v i b r a t i o n a l , r o t a t i o n a l and p - h e l i c i t y quantum numbers, r e s p e c t i v e l y , and f ( v > J I >J x XI^ * iJO i s t h e s c a t t e r i n g a m p l i t u d e . I n what f o l l o w s , we drop t h e v ^ and v indices t o s i m p l i f y the notation. The s c a t t e r i n g a m p l i t u d e i s r e l a t e d t o t h e b o d y - f i x e d s c a t t e r i n g m a t r i x element a c c o r d i n g t o oo)

(

a

a

)

a

v


1

m

v

V

m

A

v

,h~iv m |j m |6,#) = i

j

i m (9)S ^

J [J]d

+1

f(j

I

T

,

X

(2)

J where [X] stands f o r (2X + 1) and m m (8) i s the Wigner rotation matrix (the notion of Rose i s employed (23)). The body-fixed S matrix elements are related to the Arthurs-Dalgarno (AD)(24) matrix elements by d

x

1 m S _ v

3

v

h

m

l.-l / T O P ] i ^ v X v 1

i

\

v

i i v

J 1

J

[j]

x

s x

x

v

x

v

(3)

v

A A

and t h e AD S m a t r i x elements a r e approximated w i t h i n t h e RIOS as (4., 6)

J j

X*X

fl

n

^ v O J ^ J j a ^ 2ir L

d

[ M Y

a

[J] dY.sinY^*^

, B ) ] S , ( Y

X

s

X

) Y ( Y

X

[Y (B.Y ).0]

, 0 )

V

x

,

(4)

X X

defined as

1

a

o

J

X v where Y , a=X,v i Y

J

= cos" (r ,R ); a

a

a=X,v

(5)

and A i s an a n g l e d e f i n e d a s 1

A = cos" (R ,R ) x

v

.

(6)

B i s a parameter w h i c h o r i g i n a t e s from t h e m a t c h i n g o f t h e X


24.

Z H A N G E T AL.


461

channel wave function with the v channel wave function (more d e t a i l s about B w i l l be given i n the next section) and £ i s the CS o r b i t a l parameter(25) which i s i d e n t i f i e d either as the i n i t i a l value of the angular momentum quantum number i n the X arrangement channel, namely or as the average value; i . e . , £ = ( £ + £ A

)/2 .

(7)

V


In the f i r s t case the I labeling i s known as the I-initial labeling and i n the second as the il-average labeling. In the i n t e g r a l we have also a collinear-type y-dependent (and also B-and ^-dependent) S matrix element. Lifetime Matrix. In the analysis of our RIOS r e s u l t s for both FH^ and FD-, we s h a l l employ the l i f e t i m e matrix analysis due to Smith.T22) In t h i s approach, the l i f e t i m e s are given by the diagonal elements of the time delay matrix

where E i s either the t o t a l or r e l a t i v e t r a n s l a t i o n a l energy. By going to a diagonal representation of the £(E) matrix, one obtains the most compact description of the system. The eigendelays can be negative (corresponding to a process i n which the c o l l i s i o n i s a c c e l lerated by the p o t e n t i a l , r e l a t i v e to no potential at a l l ) or positive (in which the potential causes a delay r e l a t i v e to the time required to traverse the c o l l i s i o n region i n the absence of a p o t e n t i a l ) . When one or more eigendelay time i s positive and substantial, one may speak of a resonant process occurring. In our calculations reported herein, a 5-point numerical d i f f e r e n t a t i o n formula has been used. The £(E) matrix was found to be diagonally dominant so that following the eigenvalues did not prove at a l l d i f f i c u l t for FH and FT^. 2

Comparison of FH^ and FD

2

We s h a l l begin by f i r s t comparing a variety of RIOS results for the FH and FD systems. One of the advantages of the RIOS method i s the fact that i t permits us to examine how the reaction responds to changes i n the r e l a t i v e configuration of the system, as well as i n the r e l a t i v e o r b i t a l angular momentum. Although not of direct physical significance, the so called "primitive reaction p r o b a b i l i t i e s " obtained from the y-dependent S-matrix elements 2

x v

x

v afford insight into the "tuning" of the reaction as a function of £ and Y. In Figures 1-2 we give the reactive state-to-state p r o b a b i l i t i e s P

Y

* v v V E

rt

n

e

as a function of Y for d i f f e r e n t I values and > f° system FH (6b). In Figures 1(a) and 2(a) are shown the°results for f l > t Q t

v

2


=

462

RESONANCES

i n F i g u r e s 1(b) and 2(b) f o r v =2 and i n F i g u r e s 1 ( c ) and 2 ( c ) f o r v =3. I n F i g u r e 1 a r e shown t h e r e s u l t s f o r E =0.34 eV and i n Figure 2 f o r E =0.423 eV. The main f e a t u r e s t o be n o t i c e d a r e : (a) t h e p r o b a b i l i t i e s f o r v = l , 2 and f o r v =3 e x h i b i t a d i f f e r e n t Y dependence f o r most o f t h e it v a l u e s . P r o b a b i l i t i e s f o r v = l and v =2 tend t o peak a t a n g l e s which a r e f a r removed from t h e c o l l i n e a r arrangement w h i l e the v^=3 p r o b a b i l i t i e s a l l have a maximum a t 7=0, and then s l o w l y decay t o z e r o as Y i n c r e a s e s , (b) The d i f f e r e n c e s between t h e r e s u l t s i n F i g u r e s 1 and 2 a r e minor and b o t h f i g u r e s e x h i b i t v e r y s i m i l a r p a t t e r n s . The o n l y n o t i c e a b l e changes a r e t h a t a l l t h e p r o b a b i l i t i e s , as a f u n c t i o n o f Y, extend t o l a r g e r Y v a l u e s when t h e energy i s i n c r e a s e d from E =0.34 eV t o E =0.423 eV. A l s o t h e v = l , 2 r e s u l t s tend t o s h i f t t h e i r peak r e g i o n t o h i g h e r a n g l e s . As f o r v =3, most of t h e h i g h e s t p r o b a b i l i t i e s a r e s t i l l c o n c e n t r a t e d around t h e c o l l i n e a r arrangement. The FD^ r e a c t i v e s t a t e - t o - s t a t e p r o b a b i l i t i e s as f u n c t i o n s o f Y f o r d i f f e r e n t I values are analogous. As i n the F H system, t h e main f e a t u r e s a r e : (a) t h e p r o b a b i l i t i e s f o r v =1,2,3 a r e q u a l i t a t i v e l y d i f f e r e n t as f u n c t i o n s o f y and H from t h o s e f o r v =4. The p r o b a b i l i t i e s f o r v^=l,2,3 t e n d t o peak a t a n g l e s away from t h e c o l l i n e a r w h i l e t h o s e f o r v =4 tend t o have t h e i r maximum a t =0°. F u r t h e r , t h e q u a l i t a t i v e f e a t u r e s a r e e s s e n t i a l l y t h e same a t b o t h t h e low and h i g h e r energy. (b) Any d i f f e r e n c e s between the r e s u l t s at r ' 91 * P P e a r t o be o f an e s s e n t i a l n a t u r e . Once a g a i n , t h e main changes a r e t h a t a l l p r o b a b i l i t i e s extend t o l a r g e r y v a l u e s a t t h e h i g h e r energy. (The v =4 r e s u l t s w i t h 1-0 and £=5 dp_ show a s l i g h t maximum away from the 1=0 a n g l e but h i g h e r £'s remain peaked a t Y=0.) Thus, comparing the F+D r e s u l t s w i t h t h e c o r r e s p o n d i n g F+H r e s u l t s ( 6 b ) , i t i s n o t e d t h a t q u a l i t a t i v e l y , t h e two s e t s o f r e s u l t s a r e the same, w i t h t h e v = 3 HF and £ 4 DF i n correspondence w i t h each o t h e r and t h e v =2 HF and v = 3 DF i n correspondence w i t h each o t h e r . Very s i m i l a r r e s u l t s a r e o b t a i n e d f o r t h e r e a c t i v e p r o b a b i l i t y f o r f i x e d Y as a f u n c t i o n o f £, f

f


t

f

t Q

f

2

f

f

E

Q

2

e V

a n d

E

tot

= 0

3 8 5

e V

d o

n o t

a

t o t

f

2

f

v

=

f

f

'

f o r both FH for

9

V

and FD .

I n g e n e r a l , t h e HF

9

Z

z

(v,-=2) and DF(v -=3) r e s u l t s 4

1

X v

P

X A

YVV

1

(£) V

peak a t £ v a l u e s s i g n i f i c a n t l y l a r g e r than &=0, w h i l e those f o r HF(v^=3) and DF(v =4) peak a t £=0. These q u a l i t a t i v e f e a t u r e s a g a i n a r e t h e same a t e n e r g i e s b o t h below and above t h e c o l l i s i o n energy at which t h e DF(v =3) and HF(v =2) p r o d u c t s s h i f t from backward t o sideways p e a k i n g . However, i t i s found t h a t the v a l u e s o f £ f o r which f

f

f

p

W

A

YVv

v

y

v has i t s maximum ( f o r HF(v =2) and D F ( v = 3 ) ) a r e l a r g e r f o r t h e h i g h e r energy t h a n f o r t h e low energy. These r e s u l t s suggest t h a t t h e f

f



24.

Z H A N G ET AL.


10

r

e

a

20 30 X(degrees) c

t

i

v

40

463

50

e

F i g u r e 1. P r i m i t i v e F + l ^ state-to-state probabilities P(v=o-*v') as a f u n c t i o n o f Y> f o r d i f f e r e n t ^ - v a l u e s . The energy i s 0.34 eV. (a) v =0->v =l (b) v =0->v =2 (c) v =0->v =3. Reproduced by p e r m i s s i o n from Ref. 6b, C o p y r i g h t ±983, American I n s t i t u t e o f Physics. ±

f



464

RESONANCES

7 (degrees)

F i g u r e 2. Same as F i g u r e 1 except E p e r m i s s i o n from Ref. 6b, C o p y r i g h t

=0.423 eV. Reproduced by American I n s t i t u t e o f P h y s i c s .

l5§5,


24.


Z H A N G ET AL.

465

resonance t u n i n g s p r e a d s any resonance e f f e c t s o v e r a range o f e n e r g i e s , b u t t h e r e i s s t i l l a c r i t i c a l energy r e g i o n where t h e s c a t t e r i n g w i l l s h i f t from backwards t o sideways. In F i g u r e 3 a r e shown F H r e a c t i v e t r a n s i t i o n p r o b a b i l i t i e s as a f u n c t i o n o f energy (E =0.28-0.7 eV) f o r a few v a l u e s o f Y and The s t a t e - t o - s t a t e P ( v =8 •> v =2) p r o b a b i l i t i e s a r e shown i n F i g u r e s 3 ( a ) , 3 ( c ) and 3 ( e ) , and t h e s t a t e - t o - s t a t e P ( v =0 v =3) p r o b a b i l i t i e s a r e shown i n F i g u r e s 3 ( b ) , 3 ( d ) , and 3 ( f J . I n each f i g u r e , three curves which correspond t o three d i f f e r e n t £ v a l u e s are given. We f i r s t c a l l a t t e n t i o n t o t h e c l e a r p r e s e n c e o f t h e (y=0,£=0) resonance a t E 0.28 eV t h a t was d e t e c t e d l o n g ago.(26) This ( l M ) , M ) ) resonance i s c h a r a c t e r i z e d by a s t e e p r i s e i n p r o b a b i l i t y i n t h e t h r e s h o l d r e g i o n . F o r y=0, v =2 [ F i g u r e 3 ( a ) ] , i n c r e a s i n g causes a s i g n i f i c a n t b r o a d e n i n g o f t h e resonance peak. By c o n t r a s t , t h e y=0, v = 3 [ F i g u r e 3 ( b ) ] r e s u l t s show t h e n o r m a l r i s e a t t h r e s h o l d f o l l o w e d by a s t e a d y d e c l i n e i n t h e p r o b a b i l i t y . In F i g u r e 3 ( c ) we have t h e r e s u l t s f o r v =2, y=30° and we o b s e r v e t h a t the 2=0 case i s q u a l i t a t i v e l y l i k e t h e y=0, v =2, £=10 o r 20 cases i n F i g u r e 3 ( a ) . One has t h e resonance peak w h i c h i s a g a i n broadened and s h i f t e d t o h i g h e r energy compared t o t h e y=0, £=0, v =2 c a s e . In f a c t , f o r £=10 and 20, Y 30°, and v = 2 , one sees t h e Broadened resonance peak f o l l o w e d by t h e s h o u l d e r produced by t h e background nonresonant r e a c t i o n p r o b a b i l i t y . The v =3, Y=30° r e s u l t s a g a i n show no r e s o n a n t peak. F i n a l l y , when y=50°, t h e v =2 r e s u l t s do n o t seem t o show a resonance f e a t u r e . R e g a r d i n g t h e (v^=0 v =3) r e s u l t s , an i m p o r t a n t t h i n g t o n o t e i s t h e appearance o f a resonance f o r t h e c o l l i n e a r arrangement and £=0 a t t h e v i c i n t y o f E =0.7 eV (see a l s o R e f e r e n c e ( 2 6 ) ) . A n o t h e r f e a t u r e i s t h e r e l a t i v e l y h i g h t h r e s h o l d e n e r g i e s as compared w i t h t h o s e o f ( v ^ O v =2) t r a n s i t i o n s once t h e a n g l e y becomes l a r g e . In F i g u r e 3 t h e p r i m i t i v e p r o b a b i l i t y f u n c t i o n s c l e a r l y show i n d i c a t i o n s o f t h e resonance t u n i n g e f f e c t a l l u d e d t o above. It is a l s o seen t h a t t h e r e s o n a n t peak broadens and s h i f t s t o h i g h e r energy as t h e v a l u e i s i n c r e a s e d ( f o r f i x e d y ) . However, f o r y=50°, t h e r e does not appear t o be a resonance. F i n a l l y , we n o t e t h a t t h e h e i g h t of t h e y=0, £=10 r e s u l t s u g g e s t s an even l o n g e r l i v e d resonance t h a n i n £=0,y=0 (however, we s h a l l see t h a t i n f a c t , t h e t i m e d e l a y i s l e s s f o r y=0, £=10). For F+D-, a p p r o x i m a t e l y i d e n t i c a l r e s u l t s a r e o b t a i n e d f o r t h e energy dependence o f t h e f i x e d y FD p r o b a b i l i t i e s f o r t h e v ^ O + v = 3 and v ^ O v =4 r e a c t i v e t r a n s i t i o n s f o r t h e o r b i t a l a n g u l a r momentum quantum numbers £=0, 10 and 20. J u s t as f o r F H t h e FD^ r e s u l t s show t h e p r e s e n c e o f a resonance w h i c h has p r e v i o u s l y been found i n y=0, £=0 c a l c u l a t i o n s . ( 2 1 ) Furthermore as y i n c r e a s e s , the t h r e s h o l d f o r r e a c t i o n goes up i n energy and a l s o as £ i n c r e a s e s , the t h r e s h o l d energy i n c r e a s e s . A g a i n i n a n a l o g y t o FH_, we f i n d t h a t i n i t i a l l y as y i n c r e a s e s , t h e maximum v a l u e or t h e P;: r e a c t i o n p r o b a b i l i t y i n c r e a s e s . The same e f f e c t i s seen when £'is i n c r e a s e d ( a t l e a s t a t l o w e r y a n g l e s ) . I n t h e case o f t h e v =0 •+ v =4 r e a c t i o n , however, i n c r e a s i n g £ causes a d e c r e a s e i n t h e maximum r e a c t i o n p r o b a b i l i t y , as does a l s o i n c r e a s i n g y f o r f i x e d £. In F i g u r e 4, we g i v e Argand p l o t s o f t h e f i x e d - y F D reactive S m a t r i x elements f o r t h e v =0 -v v = 3 r e a c t i o n p r o c e s s (we i n c l u d e r e s u l t s f o r £=0, 10, and 20;. R e s u l t s f o r y=0° and y=15° a r e shown. 2


f

f

s=

f

f

f

fc

f

f

f

2 >

Q

f

2

f


RESONANCES


466

|

0.28

0.40

0.52

0.64

040

0.28

0.52

0.64

eV

Etot < >

F i g u r e 3. P r i m i t i v e r e a c t i v e s t a t e - t o - s t a t e p r o b a b i l i t i e s P ( v =0->v ) as a f u n c t i o n o f energy f o r d i f f e r e n t Y.and & v a l u e s . (a) Y = 0 ° , v =2 (b) Y - 0 ° , v =3 ( c ) Y =30°, v =2(d? Y =30°,v -3 (e) Y,=50°, v =2 (f) =50°, v = 3 . Reproduced by p e r m i s s i o n from R e f . 6b, C o p y r i g h t 1983, American I n s t i t u t e o f P h y s i c s . f

x

x

x

x

f



467


Z H A N G ET AL.

R

EAL

S,

ft

30

F i g u r e 4. Argand p l o t o f t h e f i x e d - y S - m a t r i x v e r s u s t o t a l energy f o r t h e p r o c e s s F+D« (v =0)-*DF(v =3)+D f o r v a r i o u s Jl-values. (a) y=0° (b) Y=15°. C i r c l e s r e p r e s e n t p o i n t s spaced by 1 meV; t r i a n g l e s r e p r e s e n t p o i n t s spaced by 2 meV; and s q u a r e s r e p r e s e n t p o i n t s spaced by 2.5 meV. Reproduced by p e r m i s s i o n from R e f . 7b, C o p y r i g h t 1984, American I n s t i t u t e o f P h y s i c s . f


RESONANCES

468

The c h a r a c t e r i s t i c feature of these r e s u l t s i s the occurrence of a flattened portion of the Argand p l o t . E a r l i e r c o l l i n e a r (Y=0°) studies of F+D with *=0 showed t h i s same behavior(21) and i t was found i n that case to signal a resonance i n the presence of strong background scattering. In Figure 5, we give Argand plots of the f i x e d - Y FH reactive S matrix elements f o r the v^=0 •+ v^=2 reaction process. Here, the resonance f o r £=0 causes a greater f l a t t e n i n g of the Argand plot while f o r £=10, the plot a c t u a l l y shows a complete loop (counterclockwise). One might be tempted to conclude from t h i s , plus the fact that the maximum i n the Y=0, £=10 reactive p r o b a b i l i t y curve i n Figure 3a i s much higher than f o r Y=0, £=0, that the Y=0, £=10 resonance i s stronger than f o r Y=0, £=0. (Similar behavior i s seen for FD .) As we s h a l l see, t h i s i s not true and the resonances f o r both the FH^ and FD^ systems are the longest l i v e d f o r Y=0, £=0. The reason the resonance i n Y=0, £=0 shows up only as a f l a t t e n i n g of the Argand plot i s due to the stronger background f o r t h i s case compared to that f o r Y=0, &=10. The most precise characterization of the resonances i s obtained using the l i f e t i m e matrix analysis of Smith.(22) In Figure 6 we give the eigendelay times f o r FH« with Y=0, &=0 and Y=0, £=10. I t i s immediately recognized that the time delay f o r Y=0, £=0 i s over twice as large as that f o r Y=0, £=10. This i s so i n spite of the fact that the Argand plots i n Figure 5 show a complete counterclockwise loop f o r Y=10, ^=0 and not f o r Y=0, £=0. The time delay f o r FH Y=0, ^=0 i s about 3.4x10 s. This corresponds to about 45 vibrations of the (free) H molecule. The Y=0, £=10 resonance l a s t s about 20 vibrations of the (free) H molecule. These are very subs t a n t i a l l i f e t i m e s . S i m i l a r l y , FD calculations f o r given y and £ values as a function of energy also show that there i s a s i g n i f i c a n t delay i n d i c a t i n g the occurrence of a resonance complex being formed i n the F+D^ system. However, the resonance delay i s not as great as that occurring i n FH . In the case-of FD , the Y=0, £=0 resonance l a s t s 1.4x10" s (compared to 3.4x10 s f o r FH with Y=*=0). This corresponds to about 13 vibrations of a free D molecule. The Y=0, £ =10 FD resonance l a s t s about 11 vibrations so i t appears that the l i f e t i m e of the resonance i n FD does not decrease as rapidly with £ as does that i n FH « When Y=0 , i t i s found that the time delay experienced decreases ( r e l a t i v e l y slowly) as a function of o r b i t a l angular momentum. Similar behavior i s also shown f o r y±0 . Furthermore, the magnitude of the time delay also decreases as increases. These r e s u l t s for both FH and FD show unequivocably that resonances are occurring over a range of y and £ values. We also point out that the nonresonant time delays are a l l negative, which indicates that the duration of the reactive c o l l i s i o n i s shorter than the t r a n s i t time that would occur i f no i n t e r a c t i o n were present. This i s l i k e l y a r e f l e c t i o n of the large exothermicity of the HF exit channel which strongly accelerates the product molecules. 2

2


2

2

2

?

?

?

2

2

2

?

2

2

2

D i f f e r e n t i a l and Integral Cross Sections. In Figure 7 are shown the state-to-state £-average reactive d i f f e r e n t i a l cross sections f o r F+H (degeneracy-averaged over m,, summed over m^ and but resolved with respect to the f i n a l v i b r a t i o n a l state v ). In each f i g u r e , 2


24. ZHANG ET AL.


E = 0 . 2 6 8 0 4 eV


Q

1.0 ImS 0.8 V=0

i=10

0.6 E = 0.26804 eV 0.4

38f 0.2

-0.8

-0.6

-0.4

-0.2

28* 14 meV 0 0.2

ReS 0.4

0.6

0.8

-0.2

-0.4

-0.6

-0.8

-1.0 F i g u r e 5. Argand p l o t o f t h e f i x e d y S-matrix v e r s u s t o t a l energy f o r t h e p r o c e s s F « HF(v -2)« f o r Y=0°, (a) £=0 (b) £-10. 2

t

f


469

RESONANCES


470

1 4

F i g u r e 6. E i g e n d e l a y t i m e s i n u n i t s o f 1 0 " s e c f o r Y=0°, £=0 and £=10 f o r F + H ( - 0 ) + H F ( v = 2 ) + H . 2

Vj

f



24.

ZHANG ET AL.


TRANSITION 1.40 ^

471

-0J5eV

.. 0-1 0-2 0-3

1.20 G

•o \

D

1.00

i

I ' '

20.0

40.0

60.0

' I '

' ' I '

1

' I

80.0 100.0 120.0140.0 160.0

180.0

e F i g u r e 7. V i b r a t i o n a l S t a t e r e s o l v e d F-ffl ">HF+H d i f f e r e n t i a l c r o s s section. A l l c r o s s s e c t i o n s a r e n o r m a l i z e d t o one a t 0=180°. (a) E =0.36 eV (b) E =0.5 eV. Reproduced w i t h p e r m i s s i o n from Ref. 6b° C o p y r i g h t 1983,°American I n s t i t u t e o f P h y s i c s . 2


472

RESONANCES

v

=

t h r e e c u r v e s a r e g i v e n f o r the r e a c t i v e c r o s s s e c t i o n s i n t o l > 2, and 3 a t a g i v e n t o t a l energy. A l l t h e r e s u l t s a r e s c a l e d t o be e q u a l t o one a t 0=*rr, and d i s p l a y t h e f o l l o w i n g f e a t u r e s : (a) 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 f o r v =3 i s backwards peaked a t 0=TT f o r a l l energies studied ( E =0.31, 0.34, 0.36, 0.423, 0.50 eV) (b) 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 f o r v = l i s backwards peaked (0£TT) f o r a l l e n e r g i e s s t u d i e d but has a s i g n i f i c a n t sideways component ( 0 ^ / 2 ) a t t h e h i g h e s t energy. (c) 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 f o r v =2 i s backwards peaked (0=TT) o n l y f o r t h e t h r e e l o w e s t e n e r g i e s ; at t h e f o u r t h i t shows a secondary peak a t 0^100° and a t the f i f t h i t peaks a t 9 W 3 . (d) I n a d d i t i o n , 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 f o r v^=2 has o s c i l l a t i o n s , w i t h t h e most pronounced o c c u r r i n g a l i t t l e f o r w a r d o f TT. Sparks e t . a l . ( 1 6 ) performed m o l e c u l a r beam experiments i n which they measured t h e s t a t e - t o - s t a t e 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 found t h e v^=2 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 t o behave d i f f e r e n t l y from t h e o t h e r two ( i . e . , v^=l,3) 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 . Whereas f o r a low energy measurement ( E 0 . 1 ^ where ^tot~^0* EQ i s t h e z e r o - p o i n t v i b r a t i o n a l i n i r g y o f t h e d i a t o m ^ ^ o r D^;) they found t h a t a l l t h r e e a n g u l a r d i s t r i b u t i o n s peaked backwards, a t a h i g h e r r e l a t i v e t r a n s l a t i o n a l energy (^.15 eV) they n o t i c e d a d r a m a t i c change i n t h e v =2 a n g u l a r d i s t r i b u t i o n . The o t h e r two d i s t r i b u t i o n s remained backward peaked but t h e v =2 peaked sideways (^120°). (More r e c e n t l y S p a r k s , e t . a l . r e p o r t t h a t the v = 3 HF p r o duct i s found t o s c a t t e r f o r w a r d . ( 1 7 ) ) T h i s b e h a v i o r was c o n s i s t e n t w i t h t h e d i f f e r e n c e i n t h e p a r t i a l wave r e a c t i v e p r o b a b i l i t i e s o f RW(3) f o r the two e n e r g i e s E ^ %2 and 3 k c a l / m o l . That i s , a t low — trans energy, RW found a monotonic d e c l i n e w i t h J?, o f t h e r e a c t i v e v^=2 p r o b a b i l i t y while at E ^3 k c a l / m o l , they observed a maximum f o r v =2 a t £=10. Our QM-l6§ ingular d i s t r i b u t i o n s f o r v = l , 2 , 3 q u a l i t a t i v e l y f i t the experimental f i n d i n g s . In p a r t i c u l a r f o r the three e n e r g i e s below o r e q u a l t o 2.14 k c a l / m o l a l l t h r e e a n g u l a r d i s t r i b u t i o n s peak backwards as can be seen i n F i g u r e 7a f o r E =0.36 eV but for E =0.156 eV, which corresponds t o E =3.54 k c a l / m o l a change ? n t h e v =2 d i s t r i b u t i o n i s n o t i c e d . t r ? R c o n t r a s t t o t h e two o t h e r s , i t shows a s t r o n g sideways component a t about 80°. At t h e energy E ~0»5 eV, t h e sideways p e a k i n g i s v e r y e v i d e n t as can be seen i n F?gure 7b. S i m i l a r r e s u l t s a r e o b t a i n e d f o r t h e RIOS v i b r a t i o n a l - s t a t e r e s o l v e d F+D a n g u l a r d i s t r i b u t i o n s . We may summarize t h e r e s u l t s at 2.91 k c a l / m o l as f o l l o w s : (a) t h e DF m o l e c u l e s a r e back s c a t t e r e d f o r v = l , 2 , 3 , 4 ; (b) t h e l e a s t backward peaked i s v = 3 and t h e most i s v^=4; and (c) i n t e r e s t i n g u n d u l a t i o n s a r e observed. The g e n e r a l f e a t u r e s o f t h e 4.51 k c a l / m o l r e s u l t s a r e : (a) t h e v = l , 2 , 3 DF m o l e c u l e s have a l l developed pronounced sideways peaks; (b) t h e o r d e r of t h e peaks (as t h e s c a t t e r i n g a n g l e i s i n c r e a s e d from z e r o toward TT) i s f i r s t v = 3 , then v =2 and f i n a l l y v =1; and (c) t h e v =4 DF p r o d u c t c o n t i n u e s t o show v e r y s t r o n g backward p e a k i n g and shows a pronounced minimum i n t h e f o r w a r d d i r e c t i o n . Thus, the v^=3 HF r e s u l t s a r e analogous t o t h e v =4 DF r e s u l t s and s i m i l a r l y f o r t h e v = 2 , l HF and v = 3 , 2 , l DF r e s u l t s . Thus, we a g a i n f i n d s t r o n g resemblances between t h e a n g u l a r d i s t r i b u t i o n s f o r the DF and HF p r o d u c t s , w i t h the v =4 DF and v = 3 HF behaving s i m i l a r l y and t h e v =3,2,1 DF and v =2,1 HF p r o d u c t s f

f


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e

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t

r

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n

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f

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behaving s i m i l a r l y . I n t h e case o f t h e e x p e r i m e n t a l r e s u l t s , i t a l s o would appear t h a t t h e F+D and F+H systems behave q u a l i t a t i v e l y t h e same. Thus i n F + H t h e most r e c e n t r e s u l t s i n d i c a t e t h a t t h e v = 3 i s f o r w a r d peaked w h i l e i n F+D , i t i s t h e v =4 w h i c h i s f o r w a r d peaked.(17) Then a s one moves away from t h e f o r w a r d d i r e c t i o n , t h e v =2 HF and v =3 DF a r e n e x t i n o r d e r o f i n c r e a s i n g s c a t t e r i n g a n g l e , f o l l o w e d by t h e v = l HF and v =2 DF.(17) Thus, t h e l a t e s t e x p e r i mental d a t a shows a s i m i l a r correspondence between t h e v ^ s t a t e o f HF w i t h t h e v + l s t a t e o f DF. The p r e s e n t t h e o r e t i c a l r e s u l t s agree q u a l i t a t i v e l y w i t h t h e e x p e r i m e n t a l r e s u l t s so f a r as t h e v = 3 , 2 , l DF p r o d u c t and v = 2 , l HF p r o d u c t a n g u l a r d i s t r i b u t i o n s b u t d i s a g r e e w i t h experiment f o r t h e V£=4 DF and v^=3 HF p r o d u c t s . However, even i n t h i s i n s t a n c e , t h e t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s agree t h a t t h e s e p r o d u c t s behave s i m i l a r l y . Regarding t h e backward a n g u l a r d i s t r i b u t i o n o f t h e v =4 DF p r o d u c t (and a n a l o g o u s l y t h e v =3 HF p r o d u c t ) , t h e r e seems t o be l i t t l e chance, i n o u r o p i n i o n , t h a t t h e Muckerman 5 s u r f a c e can r e s u l t i n b e h a v i o r o t h e r t h a n t h i s . I f the e x p e r i m e n t a l f i n d i n g s a r e i n d e e d c o r r e c t , then we b e l i e v e t h i s i s c l e a r e v i d e n c e t h a t t h e Muckerman 5 s u r f a c e i s n o t c a p a b l e o f f u l l y describing the v i b r a t i o n a l - s t a t e - r e s o l v e d angular d i s t r i b u t i o n s . 2>

f

2

f

f

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f

f

Resonance Tuning i n t h e F+H^ System I t would appear on t h e b a s i s o f t h e r e s u l t s d i s c u s s e d above t h a t t h e r e a r e d e f i n i t e l y resonances p r e d i c t e d w i t h i n t h e RIOS method f o r b o t h t h e FD and F H systems. Here, we w i s h t o examine t h e s e resonances t o r F+H- i n more d e t a i l w i t h t h e p a r t i c u l a r a i m o f e x p l o r i n g t h e resonance t u n i n g c o n c e p t . We have chosen t h e F+H system r a t h e r t h a n F+D because t h e former appears t o show l o n g e r l i v e d resonances and t h e t u n i n g e f f e c t seems t o be a c c e n t u a t e d . I n F i g u r e 8 we g i v e a t w o - d i m e n s i o n a l p l o t o f t h e maximum t i m e d e l a y a s a f u n c t i o n o f Y and £. I t i s v e r y c l e a r l y seen t h a t i n c r e a s i n g e i t h e r y o r £ causes a d e c r e a s e i n t h e t i m e d e l a y e x p e r i e n c e d i n t h e F+H c o l l i s i o n . However, t h e r a t e o f d e c r e a s e o f t h e maximum d e l a y t i m e i s n o t such as t o c o m p l e t e l y e l i m i n a t e t h e resonance b e f o r e f a i r l y s u b s t a n t i a l v a l u e s o f y and/or £ a r e a t t a i n e d . Of c o u r s e , i n an a c t u a l c o l l i s i o n , t h e a n g l e y i s n o t i n f a c t c o n s t a n t ( a s i s assumed i n t h e RIOS). I n F i g u r e 9 we g i v e a two d i m e n s i o n a l p l o t showing how t h e t o t a l energy E o f t h e maximum time d e l a y v a r i e s w i t h y and £. T h i s shows v e r y c l e a r l y t h e t u n i n g mechanism r e f e r r e d t o e a r l i e r . ( 6 - 7 ) I t i s e a s i l y seen t h a t i n c r e a s i n g e i t h e r Y o r £ causes a s h i f t t o h i g h e r energy o f t h e maximum d e l a y t i m e , t h e r e b y i m p l y i n g t h a t t h e resonance o c c u r s a t h i g h e r c o l l i s i o n energy. This i s the essence o f t h e resonance t u n i n g h y p o t h e s i s and i t i m p l i e s t h a t a s t h e c o l l i s i o n energy i s r a i s e d , t h e r e a c t i o n w i l l be i n f l u e n c e d by t h e resonance o c c u r r i n g i n h i g h e r o r b i t a l a n g u l a r momenta and f o r l a r g e r Y v a l u e s . Very s i m i l a r b e h a v i o r i s found f o r t h e F+D system. 2

2

?

2

R o l e o f t h e Resonance i n t h e F+H

2

and F+D

2

Angular

Distributions

The c e n t r a l r e m a i n i n g q u e s t i o n t o be addressed i s what i s t h e r o l e o f resonance t u n i n g i n t h e a n g u l a r d i s t r i b u t i o n o f t h e F+H r e a c t i o n . In p a r t , t h i s question a l s o i n v o l v e s the question of the relevance of t h e Muckerman 5 p o t e n t i a l s u r f a c e f o r t h e F + H 2 r e a c t i o n . 2


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Figure 9. The resonance energy E (defined as the t o t a l energy at which the time delay i s maximal) a f a function of y and £. S


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To an extent, the Muckerman 5 surface does give r e s u l t s that are i n substantial agreement with experiment. As was discussed e a r l i e r i n this and previous papers, (6,2.) both theory and experiment indicate that the FH n d FD« systems are completely analogous, with the state v i n HF corresponding i n behavior to state v + 1 of DF. However, we strongly believe that the fact that experiment shows the 3 HF and v =4 DF products strongly forward scattered(17) must be taken as pointing up a central defect i n the Muckerman 5 surface. We believe the RIOS r e s u l t s for the state-resolved angular d i s t r i b u t i o n s do represent at least q u a l i t a t i v e l y (and perhaps even quantitatively) what results from the Muckerman 5 surface. In the case of the RIOS state-resolved angular d i s t r i b u t i o n s , we also believe that the resonance tuning does have an important influence. However, i t s influence i s primarily to accelerate (as a function of energy) the s h i f t from backward to forward scattering which normally occurs whether or not there i s any resonance. That i s , any reaction w i l l show a s h i f t from backward to forward scattering as the c o l l i s i o n energy i s increased from low to higher values. However, t h i s s h i f t normally occurs over a broader energy range than i s experienced i n FH and F D We believe the resonance tuning i s responsible for sharpening the energy range over which t h i s s h i f t occurs. We also believe that the most s i g n i f i c a n t aspect of the comparison between experiment and the RIOS r e s u l t s i s the fact that experiment shows the v^=3 HF and v =4 DF products more forward scattered than any other product. This suggests that i t i s these product states which are most strongly influenced by resonant phenomena and t h i s should prove extremely useful i n obtaining further refinements of the FH and FD potentials(28) (beyond the Muckerman 5 surface). a

2

f

f

v

=


f

2

2#

f

2

2

Acknowledgments P a r t i a l support of t h i s research under National Science Foundation Grant CHE82-151317 i s g r a t e f u l l y acknowledged. Acknowledgment i s made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for p a r t i a l support of t h i s research. Literature Cited

1. Baer, M. Adv. Chem. Phys. 1982, 49, p. 191; Wyatt,R.E. In AtomMolecule Collision Theory: A Guide for the Experimentalist, Bernstein, R.B., Ed. Plenum: N.Y., 1979 p. 447. 2. Elkowitz, A.B., Wyatt, R.E. Mol. Phys., 1976, 31, 189. 3. Redmon, M.J.; Wyatt, R.E. Int. J. Quant. Chem. 1977, 11, 343; Chem. Phys. Lett 1979, 63, 209. 4. Khare, V.; Kouri, D.J.; Baer, M. J. Chem. Phys. 1979, 71, 1188; Baer, M.; Khare, V.; Kouri, D.J. Chem. Phys. Lett. 1979, 68, 378; Baer, M.; Mayne, H.; Khare, V.; Kouri, D.J.; ibid. 1980, 72, 269; Jellinek, J.; Baer, M.; Khare, V.; Kouri, D.J. ibid. 1980, 75, 460; Khare, V.; Kouri, D.J.; Jellinek, J.; Baer, M. In Potential Energy Surfaces and Dynamics Calculations, Truhlar, D.G. Ed. Plenum: N.Y., 1981 p. 475; Baer, M.; Kouri, D.J.; Jellinek, J. J. Chem. Phys. (in press). 5. Bowman, J.M.; Lee, K.T. J. Chem. Phys. 1978, 68, 3940; Chem. Phys. Lett. 1979, 64, 29; J. Chem. Phys. 1980, 72, 5071.



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6. (a) Jellinek, J.; Baer, M.; Kouri, D.J. Phys. Rev. Lett. 1981, 47, 1588; (b) Baer, M. Jellinek, J.; Kouri, D.J. J. Chem. Phys. 1983, 78, 2962; Jellinek, J.; Baer, M.; Kouri, D.J. J. Phys. Chem. 1983, 87, 3370. 7. (a) Shoemaker, C.L.; Kouri, D.J., Jellinek, J; Baer, M. Chem. Phys. Lett. 1983, 94, 359; (b) AbuSalbi, N.; Shoemaker, C. L.; Kouri, D.J.; Jellinek, J.; Baer, M. J. Chem. Phys. 1984, 80, 3210. 8. Barg, G.D.; Drolshagen, G. Chem. Phys. 1980, 47, 209. 9. (a) Clary, D.C.; Mol. Phys. 1981, 44, 1067 and 1083; (b) Clary, D. C.; Garrett, B.C.; Truhlar, D.G. J. Chem. Phys., 1983, 78, 777. 10. Clary, D.C.; Drolshagen, G. J. Chem. Phys. 1982, 76, 5027. 11. (a) Walker, R.B.; Hayes, E.F. J. Phys. Chem. 1982, 86, 85; (b) Walker, R.B.; Hayes, E.F.; J. Phys. Chem. 1983, 87, 1255. 12. Jellinek, J.; Kouri, D.J. In Theory of Chemical Reaction Dynamics, Baer, M., Ed. CRC Press; Boca Raton, Fla., 1983; Jellinek, J., Ph.D. Thesis, Weizmann Institute of Science, Rehovot, Israel, 1983, 13. Schatz, G.C.; Kuppermann, A. J. Chem. Phys. 1976, 65, 4624, 4642, 4668. 14. Muckerman, J.T. In Theoretical Chemistry: Advances and Perspectives, Vol. 6A, Henderson, D.H., Eds.; Academic, N.Y., 1981, p.1. 15. Ron, S.; Pollak, E.; Baer, M. J. Chem. Phys. (in press). 16. Sparks, R.K.; Hayden, C.C.; Shobatake, D.; Neumark, D.M.; Lee, Y.T. In Horizons in Chemistry, Fukui, K.; Pullman, B.; Eds.; D. Reidel, N.Y., 1980, p. 91. 17. Sparks, R.F.; Shobatake, K.; Lee, T.Y.; personal communication. 18. Schatz, G.C.; Bowman, J.M.; Kuppermann, A.; J. Chem. Phys. 1973, 56, 1024; ibid. 1975, 63, 674. 19. Bowman, J.M.; Lee, K.-T.; Ju, G.-Z. Chem. Phys. Lett. 1982, 86, 384. 20. (a) Ron, S.; Baer, M.; Pollak, E. J. Chem. Phys. 1983, 78, 4414; (b) Blais, I.C., Truhlar, D.G. J. Chem. Phys. 1982, 76, 4490. 21. Kuppermann, A. In Potential Energy Surfaces and Dynamics Calculations, Truhlar, D.G. Ed. Plenum, N.Y., 1981, p. 375. 22. Smith, F.T. Phys. Rev. 1960, 118, 349. 23. Rose, M.E. Elementary Theory of Angular Momentum, Wiley, J. and Sons, N.Y., 1963, p.52. 24. Arthurs, A.M.; Dalgarno, A. Proc. Roy. Soc. 1960, A256, 540. 25. McGuire, P.; Kouri, D.J. J. Chem. Phys. 1974, 60, 2488; Pack, R.T. ibid. 1974, 60, 633; Secrest, D. ibid. 1975, 62, 710; Shimoni, Y.; Kouri, D.J. ibid. 1976, 65, 3372 and 3958 and 1977, 66, 675 and 2841; Truhlar, D.G.; Poling, R.E.; Brandt, M.A. ibid. 1976, 64, 26; Parker, G.A.; Pack, R.T. ibid. 66, 2850; Kouri, D.J.; Shimoni, Y. ibid. 1977, 67, 86; Goldflam, R.; Kouri, D.J. ibid. 1977, 66, 542; Khare, V. ibid. 1977, 67, 3897; Monchick, L. ibid. 1977, 67, 4534; Fitz, D.E. Chem. Phys. 1977, 24,133; Khare, V.; Kouri, D.J.; Pack, R.T. J. Chem. Phys. 1978, 69, 449; Fitz, D.E. Chem. Phys. Lett. 1978, 55, 202; Schinke, R.; McGuire, P. Chem. Phys. 1978, 28, 129; Monchick, L.; Kouri, D.J. J. Chem. Phys. 1978, 69, 3262; Coombe, D.A.; Snider, R.F. ibid. 1979, 72, 4284; Stolte, S.; Reuss, J. In Atom-Molecule Collision Theory: A Guide for the Experimentalist, Bernstein, R.B. Plenum, N.Y., 1979; Kouri, D.J. ibid.; Khare, V.; Kouri,


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D.J.; Hoffman, D.K. J. Chem. Phys. 1981, 74, 2275; Khare, V.; Fitz, D.E.; Kouri, D.J. ibid. 1980, 73, 2802and 4148; Khare, V.; Kouri, D.J. Chem. Phys. Lett. 1981, 80, 262; Khare, V.; Kouri, D.J.; Hoffman, D.K. J. Chem. Phys. 1981, 74, 2656. 26. Wu, S.F.; Johnson, B.R.; Levine, R.D. Mol. Phys. 1973, 25, 839; Schatz, G.C.; Bowman, J.M.; Kuppemann, A. J. Chem. Phys. 1973, 56, 1024 and 1975, 63, 674. 27. Siegbahn, P.; Liu, B. J. Chem. Phys. 1978, 68, 2457; Truhlar, D.G.; Horowitz, C.J. ibid. 1978, 68, 2466 and 1979, 71, 1514. 28. Truhlar, D.G.; Garrett, B.C.; Blais, N.C. J. Chem. Phys. 1984, 80, 232. June 11, 1984


RECEIVED


Resonance Phenomena in Quantal Reactive Infinite-Order Sudden

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