THEORETICAL STUDIES OF CHEMICAL REACTIONS IN A LASER FIELD

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THEORETICAL STUDIES OF CHEMICAL REACTIONS IN A LASER FIELD

Sr. Duffy, M. Hutchinson, Jian-Min Yuan, T. George

To cite this version:

Sr. Duffy, M. Hutchinson, Jian-Min Yuan, T. George. THEORETICAL STUDIES OF CHEMICAL REACTIONS IN A LASER FIELD. Journal de Physique Colloques, 1985, 46 (C1), pp.C1-1-C1-9.

�10.1051/jphyscol:1985101�. �jpa-00224470�

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JOURNAL DE PHYSIQUE

Colloque CI, supplement au n°l, Tome 16, Janvier 1985 page Cl-1

THEORETICAL STUDIES OF CHEMICAL REACTIONS IN A LASER FIELD Sr. K. Duffy*, M. Hutchinson+, Jian-Min Yuan* and T.F. George4"

^Department of Chemistry, University of Rochester, Rochester, New York 14627, U.S.A.

^Department of Physios and Atmospheric Science, Drexel University, Philadelphia, Pennsylvania 19104, U.S.A.

Résumé - Des approches semi-classique et quantique pour la description de la réaction F + H2 • HF + F dans un champ laser sont présentées brièvement.

Pour une réaction assistée par laser dans laquelle le système moléculaire forme un complexe de Van der Waals, comme He + I2, une théorie associant la formation d'une résonance induite par le rayonnement et la prédissocia- tion est proposée.

Abstract - Semiclassical and quantum mechanical approaches for describing the F + H -*• HF + H reaction in a laser field are summarized. For a laser-assisted reaction where the molecular system forms a van der Waals complex, such as He + T0, a theory which combines radiative resonance formation with predissociation is suggested.

During the past decade, both theory and experiment have indicated that laser radiation can significantly modify the dynamics of a molecular collision process, where the radiation need not be resonant with the asymptotic energy levels of the

collision system. The collision process which is perhaps of most interest to chemists is the chexn-Lcat fina.cJxon, where bonds are broken and new bonds are formed.

In this paper we discuss several theoretical treatments of chemical reactions in a laser field. We shall begin in Section II with a study of the F + H„ •* HF + H reaction and its isotopic variants. Our main focus will be on semiclassical techniques, although quantum mechanical techniques will also be mentioned. We shall look at effects due to the variation of the laser intensity and frequency, isotopic substitution and the variation of the collision energy. We shall then turn in Section III to a laser-assisted reaction where the molecular system forms a van der Waals complex, such as the He + I„ collision system. Our interest here is in^resonance formation, whereby the laser intensity can be low (less than a kW/cm ) . The mechanism which we shall consider is one where the van der Waals complex is con- verted to products by laser-induced predissociation. Section III is the Summary.

I. Semiclassical and quantum mechanical approaches

In this section, we review some of the theoretical results which we have obtained for the collinear chemical reaction

(1) Semiclassical results have been obtained by the application of semiclassical

theories of electronic transitions [1,2], where photon absorption and emission processes are described in terms of the electronic-field representation [3,4]. We shall discuss unpublished results of Duffy and Yuan and recently published results of Last, Baer, Zimmerman and George [5], where both studies use the surface-hopping model of Tully and Preston [1]. Some of the semiclassical results [5] are compared with quantum mechanical ones [6], generated by Gordon's propagation scheme in con- junction with the "most adiabatic" representation [7].

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985101

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Cl-2 JOURNAL DE PHYSIQUE

The e l e c t r o n i c - f i e l d s u r f a c e s a r e d e s c r i b e d e l s e w h e r e [ 4 ] . To u s e t h e s u r f a c e - hopping model, we f i n d i t c o n v e n i e n t t o f i t t h e d i a b a t i c seam, formed by t h e i n t e r s e c t i o n of t h e a d i a b a t i c g r o u n d - s t a t e e l e c t r o n i c s u r f a c e s h i f t e d by t h e e n e r g y of a photon and t h e f i r s t - e x c i t e d e l e c t r o n i c s u r f a c e , t o t h e s t r a i g h t l i n e L = ah.

3 1

+

b, where h i s t h e d i s t a n c e between F and t h e n e a r H atom and h i s t h a t between t h e two H atoms. 3 For photon e n e r g i e s of 1 . 1 7 , 0 . 5 and 0.469 eV, & h e f i t t i n g parame- t e r s a and b a r e found t o be ( i n a t o m i c u n i t s ) : a = 0.6398, 0.2843, 0.5102 and b = 1.448, 2.606, 2.720, r e s p e c t i v e l y .

The main purpose of o u r s t u d i e s i s t o e s t i m a t e t h e o r d e r of magnitude of t h e l a s e r f i e l d i n t e n s i t y a t which f i e l d - i n d u c e d dynamical e f f e c t s f i r s t a p p e a r . S e v e r a l p a r a m e t e r s f o r R e a c t i o n (1) c a n b e c o n t r o l l e d by t h e e x p e r i m e n t a l s e t u p , s u c h a s l a s e r i n t e n s i t y and f r e q u e n c y , i s o t o p i c v a r i a t i o n of t h e masses, c o l l i s i o n

( r e l a t i v e t r a n s l a t i o n a l ) e n e r g y and t h e i n i t i a l s t a t e s of t h e r e a c t a n t s . I n t h e f o l l o w i n g s u b s e c t i o n s , e f f e c t s of v a r y i n g t h e s e p a r a m e t e r s on t h e r e a c t i o n dynamics a r e d i s c u s s e d .

A. F i e l d i n t e n s i t y e f f e c t s . I n T a b l e I we l i s t t h e t o t a l r e a c t i v e p r o b a b i l i t i e s (Pa) f o r t h e F

+

HH, HD, DH and DD r e a c t i o n s a s f u n c t i o n s of t h e f i e l d i n t e n s i t y ( I ) f o r a photon energy .fiw = 1.17 eV. The c o l l i s i o n energy of 0.049 eV f o r t h e F

+

^HH

Table I . R e a c t i v e p r o b a b i l i t i e s f o r 5 w = 1.17 eV.

F i e l d I n t e n s i t y F

+

^HH ^F

+

^HD ^F

+

^DH ^F

+

^DD

0 W* 0.72 0 . 2 4 1.00 0 . 7 3

**

T h i s number i s f o r 10 GW.

system i s chosen s u c h t h a t t h e i n e l a s t i c c h a n n e l F*( P 2

,

⁾

+

^HHi s n o t e n e r g e t i c a l l y 1 2 .

a c c e s s i b l e . C o l l i s i o n e n e r g i e s f o r t h e o t h e r systems age f l x e d s u c h t h a t a l l s y s t e m s have t h e same t o t a l energy w i t h r e s p e c t t o t h e a s y m p t o t i c p o t e n t i a l energy.

As I i n c r e a s e s , we s e e t h a t P d e c r e a s e s f o r t h e F

+

HH. DH and DD systems and

increases f o r F

+

HD, and a l l h P seem t o approach 0 . 5 . The i n t e n s i t y dependences of Ph f o r Xw = 0.5 eV have s i m i f a r t r e n d s b u t w i t h g r e a t e r f l u c t u a t i o n s .

The r e s u l t s f o r t h e most i n t e n s e f i e l d show t h e most pronounced dynamic e f f e c t s . The r e a s o n i s t h a t a t t h i s i n t e n s i t y t h e r e p r e s e n t a t i v e p a r t i c l e on t h e e l e c t r o n i c - f i e l d s u r f a c e s h a s i t s l o w e s t v e l o c i t y around t h e photon r e s o n a n c e r e g i o n . A s t h e i n t e n s i t y i n c r e a s e s , a p o r t i o n of t h e d i a b a t i c seam f a l l s i n t o t h e n o n c l a s s i c a l r e g i o n which a t r a j e c t o r y can r e a c h o n l y by t u n n e l i n g . I n t h e c a l c u l a t i o n s we have n o t allowed f o r t u n n e l i n g , and i f 10% o r more of t h e t r a j e c t o r i e s a r e r e f l e c t e d back by t h e p o t e n t i a l b a r r i e r , t h e n t h e r e s u l t s a r e c o n s i d e r e d u n r e l i a b l e and marked s o by 2 i n T a b l e s 1-111. Another f a c t o r which may c o n t r i b u t e t o t h e uncer- t a i n t y a s t h e i n t e n s i t y i n c r e a s e s i s t h a t some t r a j e c t o r i e s t e n d t o wander i n t h e c o l l i s i o n r e g i o n , presumably d u e t o t h e upper e l e c t r o n i c - f i e l d s u r f a c e becoming f l a t t e r i n t h a t r e g i o n . Such t r a j e c t o r i e s can c r o s s t h e seam many t i m e s , and i n o r d e r t o o b t a i n a c c u r a t e t o t a l r e a c t i o n p r o b a b i l i t i e s we have had t o sum o v e r a l l p o s s i b l e c o m b i n a t i o n s of p a t h s . T h i s can make t h e c a l c u l a t i o n q u i t e t i m e consuming, and we have n o t c o n s i d e r e d t r a j e c t o r i e s which c r o s s t h e seam more t h a n 41 t i m e s . B. Frequency e f f e c t s . R e a c t i o n p r o b a b i l i t i e s a t two photon e n e r g i e s , 5 w = 1.17 and 0.5 e V , a r e l i s t e d i n T a b l e s I and 11, r e s p e c t i v e l y . I n t h e l a t t e r c a s e t h e d i r e c t

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T a b l e 11. R e a c t i v e p r o b a b i l i t i e s f o r . h w = 0.5 eV.

F i e l d I n t e n s i t y F

+

^HH F

+

^HD ^F

+

^DH

1 GW 0.70 0.28 1 .OO

pumping of t h e HF v i b r a t i o n a l s t a t e s i n t h e a s y m p t o t i c r e g i o n h a s been n e g l e c t e d , s o t h a t t h e comparison between t h e two t a b l e s can b e made on t h e same l a s e r - i n d u c e d dynamical e f f e c t s . L e t us d e f i n e a t h r e s h o l d i n t e n s i t y ( I ) a s t h a t f o r which P

t h

changes by 10% from t h e f i e l d - f r e e v a l u e . The r e s u l t s show t h a t I o c c u r r i n g h a t 0 . 1 TW f o r f i w = 0 . 5 eV, i s one t o two o r d e r s of magnitude l e s s ti%% t h a t ( - 5 TW) f o r fiw = 1.17 eV. T h i s i s a t t r i b u t a b l e t o t h e s h i f t i n g o f t h e r e s o n a n c e r e g i o n on t h e e l e c t r o n i c - f i e l d s u r f a c e s a s t h e photon e n e r g y changes. A s s e e n i n F i g . 1 , t h e f i e l d - i n d u c e d avoided c r o s s i n g i s l o c a t e d on t h e r e a c t a n t s i d e o f t h e t r a n s i t i o n s t a t e f o r ?iw = 0.5 eV and on t h e p r o d u c t s i d e f o r .ho = 1.17 eV. S i n c e t h e r e s o n a n c e r e g i o n o c c u r s a t t h e f l a t p a r t of t h e f i e l d - f r e e g r o u n d - s t a t e p o t e n t i a l s u r f a c e f o r fiw = 0.5 eV, t h e e f f e c t i v e l e n g t h of t h i s r e g i o n i s l o n g e r and t h e l o c a l v e l o c i t y s l o w e r t h a n f o r fiw = 1.17 eV. Also due t o t h e f l a t n e s s of t h e s u r f a c e , t h e d i a b a t i c seam f a l l s i n t o t h e n o n c l a s s i c a l r e g i o n f a s t e r a s I i n c r e a s e s . A l l t h e s e f a c t s c o n t r i b u t e t o t h e l o w e r i n g of Ith f o r fiw = 0.5 eV.

C. I s o t o p e e f f e c t s . A s shown i n T a b l e s I and 11, P v a r i e s d r a m a t i c a l l y i f one of t h e H atoms i s r e p l a c e d by a D atom. T h i s i s b a s i c a f l y a k i n e m a t i c e f f e c t . We s e e t h a t Ith i s lower f o r F

+

HD t h a n f o r F

+

^HH.

D. C o l l i s i o n e n e r g y e f f e c t s . The c o l l i s i o n e n e r g y ( E 11) dependence of Ph f o r t h e F

+

^HDsystem f o r 5 w = 1.17 e V i s p r e s e n t e d i n T a b l e If?. A s t h e c o l l i s i o n energy d e c r e a s e s , Ith a e c r e a s e s by two t o t h r e e o r d e r s of magnitude. T h i s i s c o n s i s t e n t w i t h t h e e x p e c t a t i o n t h a t s l o w e r p a r t i c l e motion i n c r e a s e s t h e e f f e c t i v e i n t e r a c t i o n

t i m e w i t h t h e f i e l d and hence l e a d s t o more pronounced l a s e r - i n d u c e d dynamical e f f e c t s .

2 2

We have a l s o c o n s i d e r e d R e a c t i o n ( 1 ) where F( P ) i s r e p l a c e d by F*( P ) f o r I = 0.1 TW and .hw = 0.469 eV, which i s t h e dominant3& l a s e r e m i s s i o n [ 5 , 6 ] .114he main f i n d i n g h e r e i s t h e p r e s e n c e of a f i e l d - i n d u c e d p o t e n t i a l b a r r i e r of o v e r 0 . 4 eV on t h e lower e l e c t r o n i c - f i e l d s u r f a c e . [A s i m i l a r k i n d of b a r r i e r i s shown i n F i g . 1.1 T h i s c a u s e s t h e r e a c t i o n t o be d e l a y e d u n t i l t h e c o l l i s i o n e n e r g y i s i n c r e a s e d t o 0 . 4 3 eV. Both q u a n t a l and s e m i c l a s s i c a l r e s u l t s e x h i b i t s u c h a d_e&.

b u t t h e q u a n t a l and s e m i c l a s s i c a l r e s u l t s t h e n show a s h a r p i n c r e a s e d u e t o t h e opening of t h e HF(v=3)

+

^Ec h a n n e l . T h i s k i n d o f t h r e s h o l d e f f e c t i s m i s s i n g f r o 2 t h e s e m i c l a s s i c a l method, whYhfch must b e m o d i f i e d t o i n c l u d e t r a n s l a t i o n a l - v i b r a t i o n a l r e s o n a n c e b e h a v i o r . A s t h e c o l l i s i o n e n e r g y i n c r e a s e s w e l l beyond 0.42 eV, t h e s e m i c l a s s i c a l and q u a n t a l r e s u l t s come i n t o agreement.

T a b l e 111. C o l l i s i o n e n e r g y dependence of t o t a l r e a c t i o n p r o b a b i l i t e s of F

+

^HDf o r -6o = 1.17 eV.

F i e l d I n t e n s i t y 0.049 eV 0.055 eV 0.065 eV 0.086 eV

O W 0.37 0.37 0.31 0.24

1 GW 0.53 0 . 4 5 0.36 0 . 2 6

0.1 TW 0.44 0.41 0.41 0.31

1 TW 0.49 ¹0.51 1 0 . 4 1 0.28

11. Resonance f o r m a t i o n and a p r e d i s s o c i a t i v e mechanism

We o u t l i n e h e r e some work which i s a l r e a d y under way t o d e s c r i b e a three-body

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C1-4 J O U R N A L DE PHYSIQUE

chemical r e a c t i o n d r i v e n by r a d i a t i o n . The k i n d of r e a c t i o n we have i n mind is one which o c c u r s a t v e r y low t e m p e r a t u r e s , s a y i n a s u p e r s o n i c a l l y expanded m o l e c u l a r beam o r i n t h e i n t e r s t e l l a r medium. Moreover, we s h a l l c o n s i d e r a r e a c t i o n which proceeds by way of a l o n g - l i v e d i n t e r m e d i a t e s t a t e . The i m p o r t a n c e of t h e s e two c o n d i t i o n s i s t h a t b o t h s e r v e t o l e n g t h e n t h e d u r a t i o n of t h e c o l l i s i o n , g i v i n g t h e system more time t o a b s o r b a photon and t h e r e b y r e d u c i n g t h e i n t e n s i t y r e q u i r e m e n t s of t h e r a d i - a t i o n f i e l d . P r e v i o u s work on atom-atom c o l l i s i o n s i n a l a s e r f i e l d [ 8 1 i n d i c a t e s t h e huge e f f e c t resoilance f o r m a t i o n and low t e m p e r a t u r e s can h a v e on l a s e r - induced c o l l i s i o n p r o c e s s e s .

-

^S

^(bohr)

F i g . 1

-

P o t e n t i a l e n e r g y ( E ) , i n a t o m i c u n i t s r e l a t i v e t o t h e three-body d i s s o c i a - t i o n t h r e s h o l d , p l o t t e d a l o n g t h e f i e l d - f r e e g r o u n d - s t a t e r e a c t i o n c o o r d i n a t e ( S ) . The dashed c u r v e (----) i s t h e f i e l d - f r e e ground s t a t e . The d o t t e d c u r v e s (....) a r e t h e e l e c t r o n i c - f i e l d p o t e n t i a l s f o r ?iw = 0 . 5 eV and I = 1 TW; t h e r e p u l s i v e upper c u r v e h a s n o t been p l o t t e d o u t c o m p l e t e l y f o r c l a r i t y of p r e s e n t a t i o n . The s o l i d c u r v e s (-) and dot-dashed c u r v e s (.-.-.) a r e t h e e l e c t r o n i c - f i e l d p o t e n t i a l s f o r .fiw = 1.17 eV and I = 1 TW and 10 TW, r e s p e c t i v e l y . The c o l l i s i o n e n e r g y i s i n d i c a t e d by Ecoll, and max s p e c i f i e s t h e t r a n s i t i o n s t a t e on t h e f i e l d - f r e e p o t e n t i a l .

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In the case of a chemical reaction it is perhaps not surprising that resonance formation leads to an enhancement of the cross section, but it may seem rather paradoxical that a decrease in collision energy should have a similar effect. After all, the Arrhenius Law implies that chemical reactions proceed at a slower rate as the temperature is lowered. However, it must be remembered that in a chemical reaction there is always an activation energy barrier, and the reason that reactions become faster at higher energies is simply that more reactants have sufficient energy to overcome this barrier. But if the activation energy is supplied by a photon, there is then no need for a high collision energy-indeed, low energies are favored since they entail longer collision times. At exceptionally low collision temperatures (less than a few degrees Kelvin), the collision must be treated quantum mechanically since the scattering wavelengths become comparable to the molecular dimensions, except for very heavy atomic and molecular reactants.

We first briefly review the theory of atom-atom collisions in a laser field and shall later see how this may be extended to treat a radiatively driven chemical reaction. To treat resonance scattering, we consider a Breit-Wigner separation of the T-matrix. The radiative interaction-gives rise to resonances in the scattering spectrum, while the field-free interaction may be considered to be a slowly-varying function of the scattering energy. It is therefore convenient to separate the T-matrix into slowly and rapidly varying parts:

P .

where T 1s the slowly-varying, potential scattering T-matrix and T~ represents the resonance scattering. Following the formulation of Feshbach [9],we define the following projection operators:

Here, a is a complete set of angular momentum quantum numbers, and E' is the energy of the continuum state specified by a. The n index is a set of vibrational and angular momentum quantum numbers which specify a bound state of the system. The space of Q can be further subdivided into R and S, such that R + S = Q :

It is now possible to write the T-matrix for resonance scattering as

HXY = XHY.

+ .

G 1s the Green's function for potential scattering, i.e., with the r diative interaction switched off and the scattering determined by the term Hef in the P Hamiltonian. This term may have off-diagonal elements in a diabatic basis which are indirectly due to spin-orbit coupling. Such terms can induce Curve switching even in the absence of radiation. The scattering eigenfunctions $. and $f are, respectively, the out-wave for this potential scattering problem, :tarting In the initial state i, and the in-wave starting in the final state f. These are

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CI-6 J O U R N A L DE PHYSIQUE

t o t a l p o t e n t i a l s c a t t e r i n g wavefunctions c o n s i s t i n g of a p l a n e wave i n t h e - s p e c i f i e d channel and s p h e r i c a l waves i n a l l channels. These may be w r i t t e n i n terms of t h e standing-wave s o l u t i o n s of t h e p o t e n t i a l s c a t t e r i n g , /oaE>,by u s i n g t h e o r t h o g o n a l m a t r i x W which d i a g o n a l i e s t h e p o t e n t i a l s c a t t e r i n g S m a t r i x [ 8 ] . The r e s u l t i s

T (DWBA) R = I:

f i n

mm' Hfm mm' H m l i '

where m and m' a r e bound i n t e r m e d i a t e s t a t e s of t h e c o l l i s i o n complex, and

where i s a standing-wave e i g e n f u n c t i o n of H. I n t h e distogted-wave Born a p p r o x i @ t i o n (DWBA), W i s t h e u n i t m a t r i x , and furthermore, T f i = Hfi.

Therefore, we have f o r t h e t o t a l T-matrix

Tfi (DWBA) = H f i + ^&,^t H f m ^{Q m m t} H m t i .

The r e s u l t s of c a l c u l a t i o n s performed on rare-gas halogen systems i n d i c a t e t h a t e l e c t r o n i c t r a n s i t i o n s may be e f f e c t e d v e r y e f f i c i e n t l y by r a d i a t i v e t r a n s i t i o n d u r i n g an atomic c o l l i s i o n e v e n t . M

2

r e o v e r , t h i s i s accomplished by l a s e r s o p e r a t i n g i n t h e range of a few 1cW/cm

.

This encourages us t o

b e l i e v e t h a t a full-blown chemical r e a c t i o n may b e d r i v e n by l o w - i n t e n s i t y l a s e r s . There have r e c e n t l y been experiments which appear t o demonstrate t h a t chemical r e a c t i o n s can be d r i v e n by l a s e r s which a r e n o t r e s o n a n t w i t h any asymptotic s t a t e s

[ l l - 1 6 1 . This work h a s helped m o t i v a t e us t o c o n s i d e r a p a r t i c u l a r kind of chemical r e a c t i o n which h a s a high y i e l d , namely, a r e a c t i o n which forms a long-lived complex, and a l t h o u g h we have s o f a r t r e a t e d r a d i a t i v e resonances, i t might be u s e f u l t o begin with a system t h a t forms resonances i n t h e absence of r a d i a t i o n . It i s assumed t h a t t h e complex i s s u f f i c i e n t l y long-lived t h a t i t can absorb a photon w i t h high prob- a b i l i t y , undergoing an i n t e r n a l conversion i n t o a r e a c t i v e channel.

Consider a c o l l i n e a r r e a c t i o n , A

+

^BC^+.ha⁺^AB*

+

^C, where t h e bonds BC and AB*

a r e s t r o n g and t h e bonds AB and BC* a r e r e l a t i v e l y weak. This kind of c o n d i t i o n would be expected i n a t y p i c a l van d e r Waals molecule, such a s He1 shown i n Fig. 2.

In t h e absence of r a d i a t i o n , resonances occur i n each i n d i v i d u a l e l e c t r o n i c s t a t e . For example, t h e BC bond may g a i n a quantum of v i b r a t i o n a l energy through coupling t o t h e AB bond, which t h e n e n t e r s a bound s t a t e of motion. The atoms A, B and C then c o - e x i s t i n a quasibound s t a t e which i s , however, u n s t a b l e w i t h r e s p e c t t o t h e r e v e r s e p r o c e s s , p r e d i s s o c i a t i o n of atom A . This e f f e c t i s known a s " v i b r a t i o n a l p r e d i s s o c i a t i o n " [17-191.

Let us o u t l i n e t h e c o n s t r u c t i o n of a s c a t t e r i n g t h e o r y i n which t h e p r e d i s s o c i a - t i o n t a k e s t h e form of a Feshbach resonance. The r a d i a t i v e i n t e r a c t i o n between e l e c t r o n i c s t a t e s i s t r e a t e d i n much t h e same way a s f o r t h e atom-atom problem. The two e l e c t r o n i c s t a t e s have t h e c o o r d i n a t e r e p r e s e n t a t i o n

x

( [ p ] ; R,r) and x2([pI;

R , r ) , r e s p e c t i v e l y , where [ p l a r e t h e e l e c t r o n i c c o o r d i n a t e s and 1 r and R a r e , r e s p e c t i v e l y , t h e l e n g t h of t h e s t r o n g bond and t h e d i s t a n c e from t h e more weakly- bound atom t o t h e c e n t e r of mass of t h e two strongly-bound atoms ( s e e Fig. 2).

Averaging t h e t o t a l Hamiltonian H over t h e e l e c t r o n i c c o o r d i n a t e s , we o b t a i n t h e e f f e c t i v e (Born-Oppenheimer) Hamiltonian f o r t h e n u c l e a r d e g r e s s of freedom:

(8)

Here T i s t h e k i n e t i c e n e r g y o p e r a t o r of t h e s u b s c r i p t e d s p e c i e s , and m and p a r e Horse f u n c t i o n s r e p r e s e n t i n g t h e p o t e n t i a l e n e r g y a s s o c i a t e d w i t h t h e two "bonds".

V and V r e p r e s e n t t h e three-body p o r t i o n s of t h e ground and e x c i t e d p o t e n t i a l s u r f a c e s : which a r e coupled o n l y t h r o u g h t h e r a d i a t i v e term V 1

12'

We s t a r t by d e f i n i n g o p e r a t o r s P and Q which p r o j e c t , r e s p e c t i v e l y , o n t o t h e open and c l o s e d c h a n n e l s of t h e system:

where 1n.E.z

+

= ln.>lE:>, e t c . , j b e i n g t h e e l e c t r o n i c s t a t e i n d e x . IE'> i s t h e s t a t e v e s t a r f o r

inelastic

s c a t t e r i n g i n t h e ground e l e c t r o n i c s t a t e wAose co- o r d i n a t e r e p r e s e n t a t i o n c o n s i s t s of a p l a n e wave i n t h e c h a n n e l l a b e l e d by t h e a s y m p t o t i c e n e r g y E , p l u s o u t g o i n g s p h e r i c a l waves i n a l l o t h e r c h a n n e l s . S i m i l a r l y ,

[E;> i s r e p r e s e n t e d by a n incoming wave f o r t h e r e a r r a n g e m e n t c h a n n e l . The quantum numbers n . and v . l a b e l , r e s p e c t i v e l y , t h e quantum s t a t e s of t h e Morse o s c i l l a t o r s m a n d p . ? We c a n s i d e r i n a d d i t i o n t h e p r o j e c t o r s R and S s u c h t h a t R

+

^S⁼^Q.

T A ~

s p a c 4 of R i s d e f i n e d by t h o s e b o u n d - s t a t e v e c t o r s which g i v e rise t o s t r o n g r e s o n a n c e s ( t h e s e can be c a l l e d "doorway s t a t e s " [ 2 0 ] ) , where S i s comprised of a l l bound-state v e c t o r s . It i s now p o s s i b l e t o w r i t e t h e g e n e r a l T-matrix e l e m e n t c o n n e c t i n g t h e i n i t i a l and f i n s 1 s t a t e s a s

Equation (19) c o n t a i n s a l l t h e i n f o r m a t i o n r e l e v a n t t o a c h e m i c a l r e a c t i o n , i n c l u d i n g t h e e f f e c t of t h e r a d i a t i o n f i e l d . It may a t f i r s t s i g h t appear analogous t o t h e DWBA p i c t u r e of a c h e m i c a l r e a c t i o n , b u t i t i s n o t . It d o e s i n f a c t r e p r e s e n t t h e e x a c t s o l u t i o n t o t h e Lippmann-Schwinger e q u a t i o n , w i t h t h e c o m p l i c a t i o n t h a t t h e e f f e c t i v e ( o p t i c a l ) p o t e n t i a l , H QH i s n o t o n l y complex b u t a l s o n o n l o c a l i n t h e c o o r d i n a t e s R and r . The p r o g y e m q i n g e s on t h e r e p r e - s e n t a t i o n of ?. T h i s i s g i v e n i n t e r m s of t h e w i d t h m a t r i x T, t h e l e v e l - s h i f t m a t r i x F and ehe m a t r i x V whose o f f - d i a g o n a l e l e m e n t s d e s c r i b e r a d i a t i v e bound- bound t r a n s i t i o n s . The o f f - d i a g o n a l e l e m e n t s of i" c o n s i s t of t h e ( n u c l e a r ) bound-continuum i n t e r a c t i o n ( b o t h r a d i a t i v e and n o n r a d i a t i v e ) . I n o t h e r words,

r

c o u p l e s t h e open and c l o s e d c h a n n e l s . Under c e r t a i n c o n d i t i o n s we e x p e c t t h e s e m a t r i c e s t o b e r e p r e s e n t a t i o n s of l o c a l o p e r a t o r s , w i t h t h e s e c o n d i t i o n s depending on t h e v a l i d i t y of t h e Markov a p p r o x i m a t i o n used i n time-dependent many-body t h e o r y . Here a n o n l o c a l k e r n e l K ( t , t l ) i n t h e r a t e e q u a t i o n f o r a subsystem i s r e p l a c e d by a lo'cal f u n c t i o n of time: K ( t , t ' ) 2 ~ ( t ) G ( t - t ' ) . T h i s means p h y s i c a l l y t h a t t h e motion of t h e subsystem a t one time i s c o m p l e t e l y u n c o r r e l a t e d w i t h i t s motion a t some o t h e r t i m e , i . e . , t h e r e i s no memory i n t h e motion (due t o r a n d o m i z a t i o n on a s h o r t t i m e s c a l e ) . S i m i l a r s i t u a t i o n s a r e e x p e c t e d t o p e r t a i n f o r l o n g - l i v e d r e s o n a n c e s i n three-body c o l l i s i o n s , which a r e q u i t e d i f f e r e n t from what would b e e x p e c t e d i n , s a y , a d i r e c t c o l l i s i o n o r s t r i p p i n g r e a c t i o n where t h e i n i t i a l and f i n a l s t a t e s a r e much more c o r r e l a t e d and t h e n o n l o c a l i t y of t h e p o t e n t i a l i s more i m p o r t a n t .

We could i n i t i a l l y assume a l c c a l o p t i c a l p o t e n t i a l . Then t h e m a t r i x r e p r e s e n t a - t i o n of R i n t h e s p a c e p r o j e c t e d by R w i l l c o n s i s t of r a d i a t i v e bound-bound t r a n s i - t i o n s between r e a c t a n t and p r o d u c t s t a t e s ; i t w i l l a l s o c o n s i s t o f n o n r a d i a t i v e p r e d i s s o c i a t i o n w i d t h s which c o u p l e t h e bound and s c a t t e r i n g s t a t e s f o r t h e p r o d u c t o r r e a c t a n t c h a n n e l s t a k e n s e p a r a t e l y . Roughly s p e a k i n g , t h e p r o b a b i l i t y of r e a c t i o n may be r e g a r d e d a s t h e p r o b a b i l i t y of a s s o c i a t i o n m u l t i p l i e d by t h e p r o b a b i l i t y of d i s s o c i a t i o n i n t o a p a r t i c u l a r p r o d u c t c h a n n e l . [ T h i s p i c t u r e s h o u l d n o t b e c a r r i e d

(9)

C1-8 JOURNAL DE PHYSIQUE

too far, however, because it is essentially perturbative and we are not using perturbative theory.] The rather complicated effect of nonlocality will eventually have to be addressed. Fortunately,there is evidence [ 2 1 ] that at sufficiently low collision energies the nonlocal terms are separable.

Fig. 2

-

Schematic representation of the He12 moleculeinthe X and B electronic states. The motion of He is in a potential defined with respect to the distance, r, from the He atom to the center of mass of 12. The other coordinate, R, is the separation of the I atoms. In the absence of radiation, there are certain colli- sional energies, E . , for which the system undergoes resonance scattering. Here the motion in the

tend

of 1-1 is strongly coupled to the motion in the bond of He-I. In the excited excimer state, the I-I- bond is isoelectronic with the XeI ground state (which is weakly bound), wherein the roles of the strong and weak bonds are switched. After formation of the vibrationally predissociative resonance in the X state, a photon may be adsorbed to excite the system into a predissociative eximer state. Subsequently, predissociation occurs : vibrational energy from the I--~e+

bond is transferred to the weak I-I- bond,resulting in bond dissociation.

111. Summary

Although the threshold laser intensity (Ith) for laser-induced effects in re ctive collisions of

1

F with H and its isotopic variants is high (100 gigawatts/ 2 cm ), we have been able to understand dynamical features which should be relevant to other chemical reactions. For example, Ith can change by one to two orders of magnitude as the laser frequency or the collision energy is varied. Furthermore, reactive probabilities can change dramatically by isotopic substitution. These effects can be explained in terms of the shapes of the potential energy surfaces, where a flat- ness in the region of resonant photon absorption leads to increased time of

(10)

i n t e r a c t i o n b e t w e e n t h e l a s e r f i e l d a n d t h e m o l e c u l a r s y t e m a n d

9

hence a l o w e r i n g o f Ith. I n o r d e r t o s i g n i f i c a n t l y r e d u c e I t o a kW/cm o r l e s s , i t i s necessary t o l o o k f o r r e s o n a n c e f o r m a t i o n , d u e t o e i t k k r r a d i a t i v e c o u p l i n g o r t h e f o r m a t i o n o f a f i e l d - f r e e c o m p l e x . A n e x a m p l e o f t h e l a t t e r i s t h e van d e r W a a l s c o m p l e x f o r m e d b y H e

+

I2 c o l l i s i o n s , w h e r e t h e p r o d u c t s H e 1

+

^Ican b e o b t a i n e d b y m e a n s o f l a s e r - i n d u c e d p r e d i s s o c i a t i o n o f t h e H e 1 2 c o m p l e x .

A c k n o w l e d g m e n t s

T h i s w o r k w a s s u p p o r t e d i n p a r t b y t h e A i r F o r c e O f f i c e o f S c i e n t i f i c R e s e a r c h ( A F S C ) , U n i t e d S t a t e s Air F ~ r c e ~ u n d e r G r a n t AFOSR-82-0046, a n d t h e N a t i o n a l S c i e n c e F o u n d a t i o n u n d e r G r a n t C H E - 8 0 2 2 8 7 4 .

Jm

a c k n o w l e d g e s t h e D o n o r s o f t h e P e t r o l e u m R e s e a r c h F u n d , a d m i n i s t e r e d b y t h e A m e r i c a n C h e m i c a l S o c i e t y , f o r p a r t i a l s u p p o r t o f t h i s research. TFG a c k n o w l e d g e s t h e C a m i l l e a n d H e n r y D r e y f u s F o u n d a t i o n f o r a T e a c h e r - S c h o l a r A w a r d ( 1 9 7 5 - 8 4 ) a n d t h e J o h n S i m o n G u g g e n h e i m M e m o r i a l F o u n d a t i o n f o r a F e l l o w s h i p ( 1 9 8 3 - 8 4 ) .

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