Complex-Coordinate Coupled-Channel Methods for Predissociating

i s a Clebsch-Gordan coefficient. It i s expedient to define a ...... of Energy (Division of Chemical Sciences) and by the Alfred P. Sloan. Foundation...
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Complex-Coordinate Coupled-Channel Methods for Predissociating Resonances i n v a n der Waals Molecules SHIH-I CHU Department of Chemistry, University of Kansas, Lawrence, KS 66045 Complex-Coordinate Coupled-Channel (CCCC) methods are presented for the accurate and efficient treat­ ment of the resonance energies and widths (life­ -times) of multichannel rotationally predissociating van der Waals (vdW) molecule resonances. Algorithms for dealing with the complex scaling of numerical and piecewise analytical potentials are also presented. The CCCC methods for vdW complexes are formulated in both the space-fixed (SF) and the body-fixed (BF) coordinates. The SFCCCC method is more appropriate for the treatment of weak-coupling complexes (such as Ar-H ), whereas the BFCCCC method is better for strong-coupling complexes (such as Ar-HCl). These methods have been applied successfully to a number of vdW molecules, using reliable potential surfaces determined by experiments. In particular, the pre­ dicted widths for Ar-HD are in good agreement with the recent experimental data of McKellar. 2

It is well known that the resonance states characterized by complex energies correspond to poles of the resolvent operator (E-H)"l in the complex-energy plane on a non-physical higher Riemann sheet (1). Numerous techniques have been developed to compute these poles. These include close coupling, Feshbach, R-matrix, stabilization and other methods. In some methods, a complete scattering calculation is carried out, followed by a fitting of the results to a BreitWigner formula to extract the resonance parameters, while in others the pole of the resolvent is computed directly. In all of these methods the asymptotic form of the wave function plays an essential role in performing the calculation and/or extracting the scattering information. Recently, Aguilar and Combes (2), Balslev and Combes (3), and Simon (4), have suggested an elegant direct way to perform analytical continuation of the Hamiltonian into the complex plane. The method is based on the analytic properties of the Hamiltonian H(l£) under the complex coordinate transformation (also known as the complex scaling, coordinate-rotation, and dilatation analytical 0097-6156/ 84/0263-O263S07.50/0 © 1984 American Chemical Society

264

RESONANCES

ia

t r a n s f o r m a t i o n ) , ]£-*-l£e , where a can be r e a l o r complex. As a r e s u l t o f t h e complex c o o r d i n a t e t r a n s f o r m a t i o n ( 2 - 4 ) , t h e e i g e n ­ v a l u e s c o r r e s p o n d i n g t o t h e bound s t a t e s o f H s t a y i n v a r i a n t , w h i l e t h e b r a n c h c u t s a s s o c i a t e d w i t h t h e c o n t i n u o u s spectrum o f H a r e r o t a t e d about t h e i r r e s p e c t i v e t h r e s h o l d s by an a n g l e -2a (assuming 0 < a < TT/2), e x p o s i n g t h e complex r e s o n a n c e s t a t e s i n a p p r o p r i a t e s t r i p s o f t h e complex energy p l a n e . A c r u c i a l p o i n t from t h e computational p o i n t of view i s t h a t , although the e i g e n f u n c t i o n s a s s o c i a t e d w i t h the resonance s t a t e s normally d i v e r g e at i n f i n i t y , t h e y now become l o c a l i z e d , i . e . s q u a r e i n t e g r a b l e . The square i n t e g r a b i l i t y o f c o m p l e x - c o o r d i n a t e r e s o n a n c e wave f u n c t i o n s n a t u r a l l y l e d to the extension of w e l l - e s t a b l i s h e d bound-state t e c h n i q u e s t o t h e d e t e r m i n a t i o n o f r e s o n a n c e e n e r g i e s ( E ) and widths (T) of metastable s t a t e s . Recent a p p l i c a t i o n s o f t h e complex c o o r d i n a t e method t o v a r i o u s a r e a s o f atomic and m o l e c u l a r p h y s i c s d e m o n s t r a t e t h e power and u s e f u l n e s s o f such a d i r e c t t e c h n i q u e ( f o r r e c e n t r e v i e w s , see r e f e r e n c e s 5 and 6 ) . R

In t h i s paper we f o c u s our i n t e r e s t on the e x t e n s i o n o f t h e complex c o o r d i n a t e method (CCM) t o t h e s t u d y o f m u l t i - c h a n n e l r o t a t i o n a l p r e d i s s o c i a t i o n r e s o n a n c e s i n van d e r Waals (vdW) molecules. Van d e r Waals m o l e c u l e s a r e weakly bound complexes o f atoms and m o l e c u l e s wjiich c h a r a c t e r i s t i c a l l y have s m a l l d i s s o c i a ­ t i o n e n e r g i e s (10 cm" - 500 cm" ) and l a r g e bond l e n g t h s and r e t a i n t h e i n d i v i d u a l p r o p e r t i e s of t h e m o l e c u l a r c o n s t i t u e n t s w i t h i n t h e vdW a g g r e g a t e . Van der Waals m o l e c u l e s i n m e t a s t a b l e s t a t e s can p r e d i s s o c i a t e by two d i f f e r e n t mechanisms. These a r e t u n n e l i n g t h r o u g h a p o t e n t i a l energy b a r r i e r and c o n v e r s i o n o f i n t e r n a l e x c i t a t i o n energy i n t o r e l a t i v e t r a n s l a t i o n o f f r a g m e n t s . V i b r a t i o n a l o r r o t a t i o n a l p r e d i s s o c i a t i o n o c c u r s when one o f t h e monomers i n a vdW m o l e c u l e i s i n t e r n a l l y e x c i t e d , and t h e i n t e r n a l v i b r a t i o n a l o r r o t a t i o n a l energy exceeds t h e i n t e r m o l e c u l a r (vdW) b i n d i n g energy. I n t h e c a s e o f r o t a t i o n a l p r e d i s s o c i a t i o n (RP), the a n i s o t r o p y of the i n t e r a c t i o n p o t e n t i a l permits t h i s excess i n t e r n a l r o t a t i o n a l energy t o f l o w i n t o t h e vdW bond, and t h e complex e v e n t u a l l y b r e a k s up. The m e t a s t a b l e s t a t e has a f i n i t e l i f e t i m e x ( r e l a t e d t o T by T=#/r), which can be measured i n p r i n c i p l e through the o b s e r v a b l e broadening of s p e c t r o s c o p i c l i n e s . The p r e d i s s o c i a t i o n l i n e w i d t h d a t a p r o v i d e v a l u a b l e i n f o r m a t i o n on t h e p o t e n t i a l w e l l d e p t h s and a n i s o t r o p i c s o f t h e i n t e r m o l e c u l a r i n t e r a c t i o n s , and t h e f r a g m e n t a t i o n o f t h e vdW bond v i a t h e v i b r a ­ t i o n a l / r o t a t i o n a l p r e d i s s o c i a t i o n mechanism p r o v i d e s an i l l u m i n a t i n g example o f i n t r a m o l e c u l a r dynamics on a s i n g l e e l e c t r o n i c p o t e n t i a l surface. There i s t h e r e f o r e c u r r e n t l y c o n s i d e r a b l e e x p e r i m e n t a l and t h e o r e t i c a l i n t e r e s t s i n the s t u d y o f t h e s t r u c t u r e and dynamics o f vdW m o l e c u l e s ( f o r r e c e n t r e v i e w s , see r e f e r e n c e s 7-9). In t h i s paper we p r e s e n t a s h o r t r e v i e w o f t h e c o m p l e x - c o o r d i n a t e c o u p l e d c h a n n e l (CCCC) methods r e c e n t l y d e v e l o p e d i n our l a b o r a t o r y f o r t h e a c c u r a t e d e t e r m i n a t i o n o f t h e r e s o n a n c e e n e r g i e s and w i d t h s ( l i f e t i m e s ) o f r o t a t i o n a l l y p r e d i s s o c i a t i n g vdW m o l e c u l e s . We complex studied so t h a t applied

n o t e t h a t i n s p i t e o f t h e i n c r e a s i n g p o p u l a r i t y o f the c o o r d i n a t e method (CCM) , most a t o m i c and m o l e c u l a r problems so f a r have i n v o l v e d t h e use o f s i m p l e a n a l y t i c p o t e n t i a l s , t h e complex s c a l i n g t r a n s f o r m a t i o n &*l£e can be d i r e c t l y (5, 6 ) . However, i n t h e c a s e o f more complex m o l e c u l a r

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problems, such as the p r e d i s s o c i a t i o n o f vdW m o l e c u l e s , v e r y o f t e n a c c u r a t e p o t e n t i a l energy s u r f a c e s a r e a v a i l a b l e o n l y as t a b u l a t e d numbers o r as s o p h i s t i c a t e d p i e c e w i s e a n a l y t i c forms. T h e r e i s no p r e v i o u s d e m o n s t r a t i o n t h a t the CCM can be a p p l i e d s u c c e s s f u l l y t o r e a l i s t i c c o m p l i c a t e d systems. In t h e n e x t s e c t i o n , we p r e s e n t a s t a b l e c o m p u t a t i o n a l a l g o r i t h m t o a l l e v i a t e the problems a s s o c i a t e d w i t h n u m e r i c a l and p i e c e w i s e p o t e n t i a l s (10)• Then we p r e s e n t the c o m p l e x - c o o r d i n a t e c o u p l e d - c h a n n e l (CCCC) f o r m a l i s m i n b o t h the s p a c e - f i x e d (SF) and b o d y - f i x e d (BF) c o o r d i n a t e s . The SFCCCC method i s then a p p l i e d to the d e t e r m i n a t i o n o f t h e resonance e n e r g i e s and w i d t h s ( l i f e t i m e s ) o f Ar-H^ and Ar-HD vdW complexes, u s i n g r e l i a b l e a n i s o t r o p i c p o t e n t i a l s u r f a c e s r e c e n t l y determined from e x p e r i m e n t s . Comparison o f the SFCCCC r e s u l t s w i t h o t h e r t h e o r e t i c a l and e x p e r i ­ m e n t a l d a t a a r e made. T h i s i s f o l l o w e d by a c o n c l u d i n g remarks i n the f i n a l s e c t i o n . Complex S c a l i n g Method f o r N u m e r i c a l

and P i e c e w i s e

Potentials

2 L e t {(J)^(R)} be an a p p r o p r i a t e L (square i n t e g r a b l e ) b a s i s s e t f o r a s p e c i f i c problem under c o n s i d e r a t i o n . One o f t h e main e f f o r t s i n complex s c a l i n g c a l c u l a t i o n s ^ t ° compute e f f i c i e n t l y and r e l i a b l y the m a t r i x elements . (R) |V(Re )|.(R)>, o f t h e c o m p l e x - r o t a t e d p o t e n t i a l V ( R e ) o b t a i n e d by a p p l y i n g R-*Re t o the p o t e n t i a l V(R) . However, i n t h e c a s e t h a t V(R) i s o n l y a v a i l a b l e i n t a b u l a t e d numerc a l form o r i n some p i e c e w i s e a n a l y t i c a l form, i t i s n o t o b v i o u s how one can compute t h e s e complex m a t r i x elements d i r e c t l y and accurately. F o r example, a s t r a i g h t f o r w a r d a p p l i c a t i o n of t h e complex c o o r d i n a t e t r a n s f o r m a t i o n t o c u b i c - s p l i n e i n t e r p o l a t e d complex n u m e r i c a l p o t e n t i a l s i n d i c a t e s t h a t d i l a t a t i o n a n a l y t i c i t y i s b a d l y d e s t r o y e d (10). To bypass t h e s e n u m e r i c a l i n s t a b i l i t i e s , we have d e v e l o p e d a method (10) which t a k e s advantage o f s e v e r a l w e l l - e s t a b l i s h e d t r a n s f o r m a t i o n theorems and s t a b l e q u a d r a t u r e a l g o r i t h m s . The p r o c e d u r e c o n s i s t s o f the f o l l o w i n g t h r e e s t e p s : Step ( i ) . The i d e n t i t y i s

10t

a

«j>.(R) | v ( R e

i0t

) | t

= e" i0t

:La

l a

i

(1)

i s used t o t r a n s f o r m the complex r o t a t e d p o t e n t i a l back t o the r e a l potential. Step ( i i ) . The i n n e r p r o j e c t i o n t e c h n i q u e (11) i s adopted such t h a t i0

«t, (Re" )|v(R)|

.«

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J

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J J

a

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v v

j J ' W

a

j r

W

.

,

(18)

where

VJ'.J.O)

= ("I)" [ ( 2

j +

1 / 2

l>(2r l)] +

(J' J J)(_J'

I J)

,

(19)

and

Xj(a,o*) =

+ [J(J+D

[J(J+D

- n(fl+i)]

- «(n-i)]

1 / 2

1 / 2

• rjcj+i)

•

[j(j+D - n ( f i + i ) ]

-oG»-i>

1 / 2

V,o,i

1 / 2

6 . n

f

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and ( ) i s a 3-j symbol. The r e s u l t i n g m a t r i x o f H i s a g a i n a symmetric complex one. A l t h o u g h t h e BFCCCC method d e s c r i b e d i n t h i s s e c t i o n i s form­ a l l y e q u i v a l e n t t o t h e SFCCCC f o r m u l a t i o n , t h e SF t h e o r y appears more a p p r o p r i a t e f o r w e a k - c o u p l i n g complexes, whereas t h e BF t h e o r y i s more n a t u r a l f o r s t r o n g - c o u p l i n g complexes. I n a d d i t i o n , f o r s t r o n g - c o u p l i n g vdW complexes which a r e o n l y s l i g h t l y asymmetric t o p s , such a s A r - H C l ( 2 4 ) , t h e " t u m b l i n g " a n g u l a r momentum quantum number ft i n t h e BF frame i s n e a r l y c o n s e r v e d , and t h e c e n t r i f u g a l d e c o u p l i n g (CD) a p p r o x i m a t i o n (namely, i g n o r i n g t h e J j - / 2 p R c o u p l i n g terms i n e q u a t i o n 13 o r A terms i n e q u a t i o n " ^ 18) w i l l be e f f e c t i v e . T h i s d r a s t i c a l l y r e d u c e s t h e number o f c o u p l e d c h a n n e l s and p r o v i d e s a tremendous s i m p l i f i c a t i o n and c o m p u t a t i o n a l advantage i n t h e BF frame. E x t e n s i o n o f t h e BFCCCC-CD method t o t h e s t u d y o f r o t a t i o n a l p r e d i s s o c i a t i o n o f t h e A r - H C l vdW complex w i l l be d i s c u s s e d e l s e w h e r e (16) . a

+

Rotational Predissociation o f Ar-H

2

and Ar-HD Van Der Waals Complexes

The c o m p l e x - c o o r d i n a t e c o u p l e d - c h a n n e l (CCCC) f o r m u l a t i o n s d e s c r i b e d i n t h e l a s t s e c t i o n have been a p p l i e d t o t h e d e t e r m i n a t i o n o f r e s o ­ nance e n e r g i e s and w i d t h s ( l i f e t i m e s ) o f s e v e r a l r o t a t i o n a l l y p r e ­ d i s s o c i a t i n g vdW m o l e c u l e s , i n c l u d i n g A r - H ( 1 9 ) , Ar-HD ( 2 0 ) , A r - N ( 1 0 ) , and A r - H C l ( 1 6 ) . I n t h i s s e c t i o n , we p r e s e n t t h e r e ­ s u l t s f o r t h e A r - H and Ar-HD systems. The A r - H system i s one o f t h e most e x t e n s i v e l y s t u d i e d vdW m o l e c u l e s , and s e v e r a l r e l i a b l e a n i s o t r o p i c p o t e n t i a l energy s u r f a c e s d e r i v e d from e x p e r i m e n t a l data a r e a v a i l a b l e . A l t h o u g h r o t a t i o n a l p r e d i s s o c i a t i o n (RP) l i n e w i d t h s f o r Ar-H2 systems have n o t been measured, M c K e l l a r (25) has 2

2

2

2

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r e c e n t l y s u c c e s s f u l l y o b t a i n e d t h e RP w i d t h s f o r t h e A r - H D ( v , j ) vdW m o l e c u l e s from t h e (improved) I R - h i g h r e s o l u t i o n s p e c t r a . The d i r e c t comparison between 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 a l l o w a c r i t i c a l assessment o f t h e q u a l i t y o f the t h e o r e t i c a l methods as w e l l as t h e r e l i a b i l i t y o f t h e p o t e n t i a l s u r f a c e s u s e d i n the c a l ­ culations. SFCCCC C a l c u l a t i o n s o f A r - H A t o m - R i g i d R o t o r vdW Complex ( 1 9 ) . C a l c u l a t i o n s f o r t h e A r - H 2 system were performed by i n v o k i n g the s p a c e - f i x e d c o m p l e x - c o o r d i n a t e c o u p l e d - c h a n n e l (SFCCCC) f o r m u l a t i o n and t h e use o f s e v e r a l r e a l i s t i c a t o m - r i g i d r o t o r a n i s o t r o p i c p o t e n ­ t i a l s ( d e r i v e d from e x p e r i m e n t a l data) o f t h e form, 9

V(R,

r = r , 6) = V ( R ) + V ( R ) P ( c o s e

Q

2

2

9).

(21)

Six d i f f e r e n t p o t e n t i a l functions f o r V and V 2 were employed ( 1 9 ) , i n c l u d i n g t h r e e Lennard-Jones L J (12,6) t y p e s , L J ( I ) , L J ( I I ) and L J ( I I I ) , two Buckingham-Corner BC(6,8) t y p e s , BC(I) and B C ( I I ) , and t h e s e m i e m p i r i c a l p o t e n t i a l o f Tang and T o e n n i e s ( T T ) . The L J ( I ) p o t e n t i a l (both V and V ) i s t h a t o b t a i n e d by LeRoy and van Kranendonk (26) . I n U ( I I ) , V i s t h a t o f LeRoy and v a n Kranendonk ( 2 6 ) , whereas V i s d e t e r m i n e d by Zandee and Reuss (27) . I n L J ( I I I ) , V i s t h a t o b t a i n e d by H e l b i n g e t . a l . ( 2 8 ) , and V2 i s t h a t o f Zandee and Reuss (27) . I n t h e f i r s t BC p o t e n t i a l , V i s that obtain­ ed by LeRoy e t . a l . ( 2 9 ) , whereas V2 i s d e t e r m i n e d by Zandee and Reuss ( 2 7 ) . F o r B C ( I I ) p o t e n t i a l , b o t h V and V 2 a r e d e t e r m i n e d by LeRoy and C a r l e y (30) from the s p e c t r o s c o p i c d a t a . I t i s now gen­ e r a l l y b e l i e v e d t h a t t h e Buckingham-Corner-type p o t e n t i a l s , BC(I) and B C ( I I ) , and the TT p o t e n t i a l (31) p r o v i d e more r e a l i s t i c de­ s c r i p t i o n o f t h e A r - H system t h a n t h e L J p o t e n t i a l s . D e t a i l e d de­ s c r i p t i o n o f t h e p o t e n t i a l parameters has been g i v e n i n r e f e r e n c e 19. F i g u r e 3 d e p i c t s two o f t h e p o t e n t i a l f u n c t i o n s , BC (I) and TT, used. F o r a l l the s i x p o t e n t i a l s , we c o n s i d e r t h e m e t a s t a b l e s t a t e c o r r e ­ l a t i n g w i t h t h e i s o t r o p i c c h a n n e l |j=2, £=2, J=M=0>. Because o f t h e symmetry o f H 2 , o n l y t h e f o l l o w i n g a n g u l a r b a s i s need t o be c o n s i d ­ ered: (j=0, £ = 0 ) , (j=2, £ = 2 ) , (j=4, £ = 4 ) , . . . . e t c . F o r t h e A r - H system, we found t h a t t h e r e s o n a n c e energy and w i d t h ( l i f e t i m e ) o f t h e m e t a s t a b l e s t a t e can be d e t e r m i n e d s a t i s f a c t o r i l y by i n c l u d i n g t h e o n l y open c h a n n e l (j=0, £=0, J=0, M=0) and one c l o s e d c h a n n e l (j=2, £=2, J=0, M=0). V e r y h i g h p r e c i s i o n can be a c h i e v e d by i n ­ c l u d i n g one a d d i t i o n a l c l o s e d c h a n n e l (j=4, £=4, J=0, M=0) i n t h e basis set. Q

Q

2

Q

2

Q

Q

Q

2

2

The m a t r i x s t r u c t u r e i n t h e \yn> r e p r e s e n t a t i o n i s o f t h r e e b y - t h r e e b l o c k form ( f o r t h e t h r e e c h a n n e l case) as d e p i c t e d i n F i g u r e 4. W i t h i n each d i a g o n a l b l o c k s p e c i f i e d by t h e c h a n n e l quantum number ^ ( J & J M ) , we u s e d t h e o r t h o n o r m a l harmonic o s c i l l a t o r L^-basis *n

( R )

=

1

1/2 TT

n ] 2 n!

1 / 2

H (g*)e~f ^

, n-l,2...,N

n

r

(22)

t o expand t h e r a d i a l wave f u n c t i o n ^ n - ( ) d e f i n e d i n e q u a t i o n 8. I n e q u a t i o n 22, H i s a H e r m i t e p o l y n o m i a l , x=R-R , and 3 i s an a d ­ j u s t a b l e n o n - l i n e a r parameter. The p o t e n t i a l m a t r i x elements f o r T

Q

274

RESONANCES

14.

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Predissociating Resonances in van der Waals Molecules

ax

c

O

60

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cd P CO CO d •H * x cd d T3 £ =£"+3, J »J"+l. O n l y t h e N- and T- b r a n c h s p e c t r a l l i n e s a r e w e l l r e s o l v e d and t h e i r w i d t h s can be d e t e r m i n e d e x p e r i m e n t a l l y . The SFCCCC c a l c u l a t i o n s were c a r r i e d out f o r a l l N- and Tbranch p r e d i s s o c i a t i n g s t a t e s observed e x p e r i m e n t a l l y . The b a s i s s e t o f HD i n c l u d e s a l l c h a n n e l s c o r r e s p o n d i n g t o v = l , j=0, 1, 2, and 3. Channels c o r r e s p o n d i n g t o HD(v=0,j) s t a t e s can be s a f e l y i g n o r e d , as t h e 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 (VP) r a t e s from v = l t o v=0 s t a t e s were found t o be l e a s t t h r e e o r d e r s o f magnitude s m a l l e r than t h e r o t a t i o n a l p r e d i s s o c i a t i o n (RP) r a t e s i n t h e v = l s t a t e s . S i n c e t h e r a d i a l p o t e n t i a l s V^(R) were g e n e r a t e d i n n u m e r i c a l r a t h e r than i n a n a l y t i c form, t h e r a d i a l c o u p l i n g m a t r i x elements < X » (R) | v ^ ( R e ) |x (R)>, i n harmonic o s c i l l a t o r b a s i s , were computed u s i n g t h e t h r e e - s t e p p r o c e d u r e a f o r e m e n t i o n e d . Shown i n F i g u r e 10 i s a t y p i c a l SFCCCC a - t r a j e c t o r y whose s t a t i o n a r y p o i n t i s t a k e n as t h e r e s o n a n c e p o s i t i o n sought. The converged r e s o n a n c e e n e r g i e s E R and w i d t h s T c o r r e s p o n d i n g t o N- and T- b r a n c h p r e d i s s o c i a t i n g s t a t e s o f Ar-HD(v=l,j=2) were t a b u l a t e d i n r e f e r e n c e 20. There a g a i n we found t h e SFCCCC w i d t h s a r e i n e x c e l l e n t agreement w i t h t h e r e c e n t independent c a l c u l a t i o n s o f LeRoy e t a l . (32, 34) u s i n g t h e CCS method. Shown i n F i g u r e 11 i s t h e comparison o f t h e SFCCCC c a l c u l a t e d w i d t h s w i t h t h e e x p e r i m e n t a l d a t a r e c e n t l y o b t a i n e d by M c K e l l a r ( 2 5 ) . The agreement appears g r a t i f y i n g , c o n s i d e r i n g t h e s e n s i t i v i t y of the width c a l c u l a t i o n s with respect to p o t e n t i a l s u r f a c e u s e d . The d i s c r e p a n c y between 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 w i d t h s as w e l l as t h e r e c e n t h y p e r f i n e s p e c t r o s c o p y o f Ar-H2 (35) suggest t h a t f u r t h e r r e f i n e m e n t o f t h e BC3 (6,8) a n i s o t r o p i c p o t e n t i a l may be p o s s i b l e . , = =

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Conclusion The r e s u l t s d e s c r i b e d i n t h i s paper f o r vdW m o l e c u l e p r e d i s s o c i a t i o n demonstrate t h a t t h e CCCC f o r m a l i s m s p r o v i d e a c c u r a t e and e f f i c i e n t methods f o r t h e d i r e c t p r e d i c t i o n o f r e s o n a n c e e n e r g i e s and w i d t h s of metastable s t a t e s . I n work p u b l i s h e d e l s e w h e r e ( 3 6 ) , t h e CCCC method has a l s o been extended and a p p l i e d t o t h e f i r s t d e t e r m i n a t i o n o f t h e e n e r g i e s and w i d t h s o f t h e a u t o i o n i z i n g r e s o n a n c e s o f t h e hydrogen atom i n i n t e n s e magnetic f i e l d s . The u t i l i t y and advan­ t a g e s o f t h e CCCC methods may be summarized as f o l l o w s : (1) I t i s an ab i n i t i o method ( g i v e n a d e f i n e d " e x a c t " h a m i l t o n i a n ) . (2) Only

284

RESONANCES

12.0.

F i g u r e 9. The d i a g o n a l v i b r a t i o n a l l y a v e r a g e d a n i s o t r o p y s t r e n g t h r a d i a l f u n c t i o n s V ( v , j | R ) f o r Ar-HD(v=l, j=2) . Reproduced w i t h p e r m i s s i o n from Ref. 20. C o p y r i g h t 1983, North-Holland Physics Publishing. A

Predissociating Resonances in van der Waals Molecules

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285

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Re(E), CM-1 Figure 10 A t y p i c a l SFCCCC a - t r a j e c t o r y f o r t h e complex e i g e n ­ v a l u e a s s o c i a t e d w i t h m e t a s t a b l e (J=5, £=3) s t a t e o f t h e Ar-HD(v=l, j=2) system. Reproduced w i t h p e r m i s s i o n from R e f . 20. C o p y r i g h t 1983, N o r t h - H o l l a n d P h y s i c s P u b l i s h i n g . 0

286

RESONANCES

1

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F i g u r e 11. Comparison o f c a l c u l a t e d (SFCCCC) and e x p e r i m e n t a l w i d t h s f o r r o t a t i o n a l l y p r e d i s s o c i a t i n g s t a t e s o f Ar-HD(v=l,j=2). Reproduced w i t h p e r m i s s i o n from R e f . 20. C o p y r i g h t 1983, North-Holland Physics P u b l i s h i n g .

14. CHU

287 Predissociating Resonances in van der Waals Molecules

bound-state structure calculations are required and no asymptotic conditions need to be enforced. (3) The resonance energies are obtained directly from eigenvalue analysis of appropriate nonhermitian matrices, the imaginary parts of the complex eigenvalues being related directly to the lifetimes of the metastable states. (4) The CCCC method is applicable to many-channel problems involving multiply coupled open continua as well as close channels. Combining with the non-hermitian Floquet theory (also called the complex quasi-energy formalism) (37, 38), the CCCC method can be extended to the study of photodissociation or multiphoton dis­ sociation of vdW molecules. Work in this direction is in progress. Acknowledgments The author acknowledges fruitful collaborations with several of his coworkers, particularly, Dr. K.K. Datta and Dr. S.K. Bhattacharya. This work has been supported in part by the United States Department of Energy (Division of Chemical Sciences) and by the Alfred P. Sloan Foundation. The author is grateful to the United Telecom Computing Group (Kansas City) for generous support of the CRAY computer time. Literature Cited 1. M.L. Goldberger and K. Watson, "Collision Theory", Wiley: New York, 1964. 2. J . Aguilar and J.M. Combes, Commun. Math. Phys. 22, 269 (1971). 3. E. Balslev and J.M. Combes, Commun, Math. Phys. 22, 280 (1971). 4. B. Simon, Ann. Math. 97, 247 (1973). 5. W.P. Reinhardt, Ann. Rev. Phys. Chem. 33, 223 (1982). 6. B.R. Junker, Adv. At. Mol. Phys. 18, 208 (1982). 7. D.H. Levy, Advan. Chem. Phys. 47, 323 (1981). 8. J.A. Beswick and J . Jortner, Advan. Chem. Phys. 47, 363 (1981). 9. R.J. LeRoy and J.C. Carley, Advan. Chem. Phys. 42, 353 (1980). 10. K.K. Datta and S.I. Chu, Chem. Phys. Lett. 87, 357 (1982). 11. B. Schneider, Chem. Phys. Lett. 31, 237 (1975). 12. D.O. Harris, G.G. Engerholm and W.D. Gwinn, J . Chem. Phys. 43, 1515 (1965). 13. B. Simon, Phys. Lett. 71A, 211 (1979). 14. J.D. Morgan and B. Simon, J. Phys. B14, L167 (1981). 15. C.W. McCurdy, Phys. Rev. A21, 464 (1980). 16. K.K. Datta and S.I. Chu (in preparation). 17. R. Lefebvre, "Siegert Quantization, Complex Rotation and Molecular Resonances", (preprint). 18. S.I. Chu, J. Chem. Phys. 72, 4772 (1980). 19. S.I. Chu and K.K. Datta, J . Chem. Phys. 76, 5307 (1982). 20. K.K. Datta and S.I. Chu, Chem. Phys. Lett. 95, 38 (1983). 21. Z. Bacic and J . Simons, Int. J . Quantum Chem. 14, 467 (1980). 22. I.C. Percival and M.J. Seaton, Proc. Cambridge Phil. Soc. 53, 654 (1957). 23. S.I. Chu, Chem. Phys. Lett. 88, 213 (1982). 24. S.L. Holmgren, M. Weldman and W. Klemperer, J . Chem. Phys. 69, 1661 (1978).

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RESONANCES

A.R.W. McKellar, Faraday Discuss. Chem. Soc. 73, 89 (1982). R.J. LeRoy and J . van Kranondonk, J . Chem. Phys. 61, 4750 (1974) L. Zandee and J. Reuss, Chem. Phys. 26, 345 (1977). R. Helbing, W. Gaide, and H. Pauly, Z. Phys. 208, 215 (1968). R.J. LeRoy, J.S. Carley, and J.E. Grabenstetter, Faraday Dis­ cuss. Chem. Soc. 62, 169 (1977). J.S. Carley, Ph D. Thesis, University of Waterloo, Waterloo, Ontario, 1978. K.T. Tang and J.P. Toennis, J . Chem. Phys. 74, 1148 (1981). R.J. LeRoy, G.C. Corey, and J.M. Hutson, Faraday Discuss. Chem. Soc. 73, 339 (1982). S.I. Chu, J. Chem. Phys. 62, 4089 (1975). J.M. Hutson and R.J. LeRoy, J . Chem. Phys. 78, 4040 (1983). M. Waaijer, Ph.D. thesis, Katholieke University, Nijmegen, The Netherlands, 1981. S.K. Bhattacharya and S.I. Chu, J . Phys. B16, L471 (1983). S.I. Chu, J. Chem. Phys. 75, 2215 (1981) and references therein. S.I. Chu, C. Laughlin and K.K. Datta, Chem. Phys. Lett. 98, 476 (1983).

RECEIVED June 11, 1984