AIP CONFERENCE
PROCEEDINGS 275
ATOMIC PHYSICS 10
THIRTEENTH INTERNATIONAL CONFERENCE ON ATOMIC PHYSICS MUNICH, GERMANY 1992
EDITORS:
H. WALTHER T. W. HANSCH
D. NEIZERT
AAAX PLANCK INSTITUTE FOR QUANTUM OPTICS GARCHING, GERMANY
AND LUDWIG MAXIMILIAN UNIVERSITY MUNICH, GERAAANY
A I P
American Institute of Physics New York
LONG-LIVED RESONANT STATES IN PLANETARY ATOMS
K . R i c h t e r a n d D . W i n t g e n
F a k u l t a t fur P h y s i k , H e r m a n n - H e r d e r - S t r . 3, 7800 F r e i b u r g , F R G
A B S T R A C T
W e r e p o r t o n a class o f l o n g - l i v e d r e s o n a n t states o f d o u b l y e x c i t e d t w o - e l e c t r o n a t o m s w h i c h e x h i b i t d i s t i n c t a n g u l a r a n d r a d i a l e l e c t r o n c o r r e l a t i o n . T h e s t a t e s are c h a r a c t e r i z e d b y a h i g h l y p o l a r i z e d i n n e r e l e c t r o n l o c a t e d n e a r the a x i s b e t w e e n the n u c l e u s a n d a dynamically l o c a l i z e d o u t e r e l e c t r o n . C l a s s i c a l m e c h a n i c s s t u d i e s p r o v e t h e s t a b i l i t y o f the c o r r e s p o n d i n g c l a s s i c a l m o t i o n a n d a l l o w for E B K - q u a n t i z a t i o n to o b t a i n s e m i c l a s s i c a l energies. T h e resonance states are t r e a t e d f u r t h e r w i t h i n t h e f r a m e w o r k o f a s i n g l e c h a n n e l a d i a b a t i c a p p r o x i - m a t i o n . T h e a d i a b a t i c energies as w e l l as the s e m i c l a s s i c a l results are i n g o o d a g r e e m e n t w i t h r e s o n a n c e energies o b t a i n e d b y h i g h l y a c c u r a t e s o l u t i o n s o f the f u l l t h r e e - b o d y S c h r o d i n g e r e q u a t i o n . A p p r o x i m a t e q u a n t u m n u m b e r s d e r i v e d f r o m t h e s e m i c l a s s i c a l a n d f r o m the a d i a b a t i c a p p r o a c h e x p l a i n the n o d a l struc- t u r e s of the full q u a n t u m w a v e f u n c t i o n s . T h e d e c a y w i d t h s o f the resonances t u r n o u t to be e x t r e m e l y s m a l l .
I. I N T R O D U C T I O N
T h e n o n - s e p a r a b i l i t y o f t h e t h r e e - b o d y C o u l o m b p r o b l e m becomes e v i d e n t i n the case of h i g h l y d o u b l y - e x c i t e d a t o m s o r i o n s , w h e r e t h e e l e c t r o n - e l e c t r o n i n t e r a c t i o n is of c o m p a r a b l e i m p o r t a n c e to the e l e c t r o n - i o n i n t e r a c t i o n . T h e ef- fect o f i n t e r - e l e c t r o n r e p u l s i o n , i.e. e l e c t r o n c o r r e l a t i o n , t y p i c a l l y leads to the b r e a k d o w n of i n d e p e n d e n t p a r t i c l e a p p r o a c h e s a n d has f o c u s e d interest o n the s e a r c h for a p p r o x i m a t e s y m m e t r i e s u s i n g c o l l e c t i v e c o o r d i n a t e s of t h e t h r e e par- t i c l e s . T h u s the s t r u c t u r e a n d f o r m a t i o n o f h i g h l y c o r r e l a t e d e l e c t r o n i c states is o f t o p i c a l i n t e r e s t i n s p e c t r o s c o p y [1, 2, 3, 4] a n d t h e o r e t i c a l a t o m i c p h y s i c s [5, 6, 7, 8].
D u e t o the i n t r i n s i c n o n - s e p a r a b i l i t y o f t h e p r o b l e m , t h e r e exists n o g l o b a l c l a s s i f i c a t i o n s c h e m e w h i c h a l l o w s a n o v e r a l l d e s c r i p t i o n o f t h e huge v a r i e t y of d o u b l y - e x c i t e d states o c c u r r i n g . In t h i s c o n t r i b u t i o n w e r e p o r t o n a n o v e l class [9, 10] o f s t r o n g l y c o r r e l a t e d e l e c t r o n states ( "Planetary Atom11 states [11]) w h i c h d o n o t fit a n y o f t h e k n o w n c l a s s i f i c a t i o n schemes p r o p o s e d i n t h e l i t e r a t u r e . T h e s t a t e s are c o m p o s e d o f a s t r o n g l y p o l a r i z e d ( i n n e r ) e l e c t r o n l o c a t e d a l o n g t h e axis c o n n e c t i n g t h e n u c l e u s a n d t h e o u t e r e l e c t r o n w h i c h is dynamically l o c a l i z e d near s o m e f i x e d r a d i a l d i s t a n c e . T h u s b o t h e l e c t r o n s are l o c a t e d o n t h e s a m e side of the n u c l e u s ( i n c o n t r a s t to " s y m m e t r i c " c o l l i n e a r c o n f i g u r a t i o n s w i t h b o t h electrons o n different sides o f t h e n u c l e u s w h i c h are a s s o c i a t e d w i t h intra-shell resonances [7, 12]). T h e r e s o n a n c e s t a t e s h a v e the f o l l o w i n g p r o n o u n c e d p r o p e r t i e s :
388
© 1992 American Institute of Physics(i) d i s t i n c t a n g u l a r a n d r a d i a l c o r r e l a t i o n s ,
(ii) n o ( b o u n d ) i n d e p e n d e n t p a r t i c l e l i m i t ( n u c l e a r c h a r g e Z = o o ) ,
(Hi) q u a s i - s e p a r a b i l i t y o f the w a v e f u n c t i o n s i n c o l l e c t i v e s e m i c l a s s i c a l a n d m o l e - c u l a r c o o r d i n a t e s ,
(iv) e x t r e m e l y s m a l l ( p a r t i c l e - ) decay w i d t h s .
T h e s e s t a t e s e x h i b i t t h e i r t y p i c a l features for e x c i t i a t i o n s w h i c h w o u l d c o r r e s p o n d to i n d e p e n d e n t p a r t i c l e p r i n c i p a l q u a n t u m n u m b e r s TV > 5 o f the i n n e r e l e c t r o n . T h u s t h e states b e l o n g t o a n energy regime w h i c h is c h a r a c t e r i z e d b y a vast n u m b e r o f o v e r l a p p i n g resonances a n d i n t e r a c t i n g R y d b e r g series.
In t h e f o l l o w i n g we a p p l y c l a s s i c a l , s e m i c l a s s i c a l a n d q u a n t u m m e c h a n i c a l ( a p p r o x i m a t e a d i a b a t i c a n d l a r g e - s c a l e ab initio) m e t h o d s to i n v e s t i g a t e these states.
II. ( S E M I - ) C L A S S I C A L D Y N A M I C S
W e first g i v e a c l a s s i c a l a n a l y s i s of the relevant e l e c t r o n p a i r m o t i o n to get i n s i g h t i n t o t h e u n d e r l y i n g d y n a m i c a l p r o p e r t i e s .
T h e n o n - r e l a t i v i s t i c H a m i l t o n i a n of a t w o - e l e c t r o n a t o m (or ion) w i t h c h a r g e Z a n d n u c l e a r m a s s M = oo is g i v e n b y ( a t o m i c u n i t s used)
h =
p! + p | _ £ _ z
+_
L2 7"! 7 '2 7'! 2
rx a n d r2 are t h e e l e c t r o n distances f r o m the n u c l e u s , a n d r1 2 is the i n t e r - e l e c t r o n d i s t a n c e .
W e w i l l focus o n states w i t h t o t a l a n g u l a r m o m e n t u m L = 0 . T h e n the m o t i o n is c o n f i n e d to a fixed p l a n e i n c o n f i g u r a t i o n space a n d the H a m i l t o n i a n reduces e s s e n t i a l l y to t h r e e ( c o u p l e d ) degrees of freedom.
C o n s i d e r a c o l l i n e a r a r r a n g e m e n t of a n u c l e u s Z a n d o f t w o e l e c t r o n s , b o t h b e i n g o n the s a m e side o f the nucleus. T h e f u n d a m e n t a l p e r i o d i c m o t i o n o f s u c h a c o n f i g u r a t i o n is a c o h e r e n t o s c i l l a t i o n of b o t h electrons w i t h the s a m e frequency b u t , as i t t u r n s o u t , w i t h large differences i n t h e i r i n d i v i d u a l r a d i a l a m p l i t u d e s a n d v e l o c i t i e s as d e p i c t e d i n figure 1(a) for h e l i u m (Z—2)\ T h e o u t e r e l e c t r o n a p p e a r s t o s t a y n e a r l y frozen at some fixed r a d i a l d i s t a n c e . T h e l o c a l i z a t i o n of the o u t e r e l e c t r o n is a p u r e d y n a m i c a l effect due to e l e c t r o n c o r r e l a t i o n .
T h e s i g n i f i c a n c e o f a p e r i o d i c o r b i t for the c o r r e s p o n d i n g q u a n t i z e d s y s t e m d e p e n d s e s s e n t i a l l y o n the s t r u c t u r e of the c l a s s i c a l phase space i n the v i c i n i t y o f the o r b i t [13]. T h e p e r i o d i c t r a j e c t o r y of figure 1(a) is l i n e a r l y stable w i t h respect to v a r i a t i o n s i n t h e i n i t i a l c o n d i t i o n s . T h i s is d e m o n s t r a t e d i n figure 1 ( b ) , w h i c h shows t h e r e s u l t i n g r e g u l a r m o t i o n of the electrons w h e n t h e y are i n i t i a l l y i n a s l i g h t l y o f f - c o l l i n e a r a r r a n g e m e n t . T h e inner e l e c t r o n t h e n moves o n p e r t u r b e d K e p l e r ellipses a r o u n d the n u c l e u s , w h i l e the o u t e r e l e c t r o n r e m a i n s t r a p p e d at large r a d i a l d i s t a n c e s f o l l o w i n g the slow a n g u l a r precession of the i n n e r e l e c t r o n .
n z—r e2
: i ' 1 ! 1 1 ' i 1 J
- 2 - 1 0 1 2 3 4 5 6 7 X
F i g u r e 1: T h e r a d i a l e x t e n t s o f t h e electrons for (a) t h e p e r i o d i c t r a j e c t o r y , a n d (b) a n o n p e r i o d i c b u t r e g u l a r t r a j e c t o r y i n its n e i g h b o r h o o d .
T h e l o c a l i z a t i o n o f t h e o u t e r e l e c t r o n rests u p o n t h e i n t e r - e l e c t r o n r e p u l s i o n for s m a l l d i s t a n c e s o f t h e e l e c t r o n s a n d u p o n t h e a s y m p t o t i c a l d o m i n a n c e o f t h e n u c l e a r a t t r a c t i o n . A careful a n a l y s i s o f the c l a s s i c a l m e c h a n i c s reveals t h a t t h e f u n d a m e n t a l p e r i o d i c m o d e is e m b e d d e d i n a f u l l y s i x - d i m e n s i o n a l i s l a n d of s t a b i l i t y i n c l a s s i c a l p h a s e space. T h i s i m p l i e s t h e n e a r - i n t e g r a b i l i t y o f t h e t h r e e - b o d y C o u l o m b p r o b l e m for a s y m m e t r i c c o n f i g u r a t i o n s as s h o w n i n f i g . 1.
A s e m i c l a s s i c a l t r e a t m e n t o f t h e c l a s s i c a l m o t i o n suggests t h e e x i s t e n c e o f a R y d b e r g series o f r e s o n a n c e s c o n v e r g i n g to t h e t h r e e - p a r t i c l e b r e a k u p t h r e s h o l d [14].
Ssc2
Enki = " ( n + i + 2 ( t + l )7l + ( / + l )7 2)2 • ( 2)
S3C= 1.49150 is t h e (scaled) a c t i o n o f the p e r i o d i c o r b i t o f figure 1 for h e l i u m . 7 i= 0 . 4 6 1 6 4 a n d 72=0.06765 are t h e c l a s s i c a l w i n d i n g n u m b e r s . T h e R y d b e r g se- ries is c h a r a c t e r i z e d b y t h r e e q u a n t u m n u m b e r s n , fc, / w h i c h a r e to be i n t e r p r e t e d as n o d a l e x c i t a t i o n s a l o n g t h e o r b i t (n) a n d a l o n g t h e t w o d i r e c t i o n s p e r p e n d i c u - l a r t o t h e o r b i t , t h e b e n d i n g degree o f f r e e d o m (k) a n d t h e m o t i o n p e r p e n d i c u l a r t o t h e o r b i t p r e s e r v i n g c o l l i n e a r i t y (/). T h e s e m i c l a s s i c a l q u a n t u m n u m b e r s n , k a n d / reflect t h e a p p r o x i m a t e s e p a r a b i l i t y o f t h e a s s o c i a t e d s e m i c l a s s i c a l w a v e - f u n c t i o n s i n t h e l o c a l c o o r d i n a t e s p a r a l l e l a n d p e r p e n d i c u l a r t o t h e p e r i o d i c o r b i t .
I I I . A D I A B A T I C A P P R O X I M A T I O N
A s t r i k i n g p r o p e r t y o f t h e c l a s s i c a l p e r i o d i c o r b i t o f f i g u r e 1 is t h e l a r g e dif- ference i n t h e e l e c t r o n i c v e l o c i t i e s . T h i s i n d i c a t e s t h a t a n a d i a b a t i c q u a n t u m m e c h a n i c a l t r e a t m e n t s i m i l a r t o t h e B o r n - O p p e n h e i m e r ( B O ) a p p r o x i m a t i o n i n m o l e c u l a r p h y s i c s s h o u l d be a p p l i c a b l e . W e use t h e a x i s r i b e t w e e n t h e n u c l e u s a n d t h e o u t e r e l e c t r o n as a d i a b a t i c c o o r d i n a t e . A d e t a i l e d d e s c r i p t i o n o f o u r
a d i a b a t i c m e t h o d c a n b e f o u n d i n R e f . [10].
A s a first s t e p t h e S c h r o d i n g e r e q u a t i o n has to be s o l v e d for t h e i n n e r e l e c t r o n i n t h e field o f t h e t w o fixed C o u l o m b centers w i t h charges Z a n d —1 a n d d i s t a n c e R = 7*1. A s is w e l l k n o w n f r o m m o l e c u l a r p h y s i c s [15] t h i s S c h r o d i n g e r e q u a t i o n is s e p a r a b l e i n p r o l a t e s p h e r o i d a l c o o r d i n a t e s A , / i , w h i c h for o u r c o o r d i n a t e s r e a d
x r2 -I- r i 2 r2 - r1 2
T h e r e s u l t i n g m o l e c u l a r o r b i t a l ( M O ) eigenfunctions <f) for t h e i n n e r e l e c t r o n sepa- r a t e i n p r o l a t e s p h e r o i d a l c o o r d i n a t e s , (f)(\,fi;R) = £n A ( A ) 7 /n M( / z ) . ( T h e a z i m u t h a l q u a n t u m n u m b e r m o f t h e M O f u n c t i o n s is zero for L = 0.) T h e f u n c t i o n £n A( A ) has e l l i p t i c a l n o d a l surfaces w i t h t h e nucleus a n d t h e o u t e r e l e c t r o n as f o c i . T h e f u n c t i o n rjntl(fi) possesses a c o r r e s p o n d i n g h y p e r b o l i c n o d a l s t r u c t u r e . T h e n o d a l q u a n t u m n u m b e r s n\ a n d nM are c o n s e r v e d for a r b i t r a r y p a r a m e t e r R. I n t h e l i m i t o f l a r g e R ( e q u i v a l e n t t o t h e s e p a r a t e d a t o m l i m i t i n m o l e c u l a r p h y s i c s ) t h e q u a n t u m n u m b e r s n\ a n d c o i n c i d e w i t h p a r a b o l i c c o o r d i n a t e q u a n t u m n u m b e r s r i i a n d n2 [7]. T h e effect o f the o u t e r e l e c t r o n is t h e n t o p r o d u c e a n e l e c t r i c field w h i c h is n e a r l y c o n s t a n t over t h e s p a t i a l r a n g e e x p e r i e n c e d b y t h e i n n e r e l e c t r o n . T h u s t h e i n n e r w a v e f u n c t i o n s m e r e l y b e c o m e S t a r k - l i k e s t a t e s o f t h e r e m a i n i n g H e + i o n .
T h e q u a n t u m a n a l o g u e o f t h e a s y m m e t r i c ( c o l l i n e a r ) c l a s s i c a l c o n f i g u r a t i o n of figure 1 c o n s i s t s o f a n i n n e r e l e c t r o n i n a m o l e c u l a r t y p e s t a t e o f m a x i m a l p o l a r i z a t i o n a l o n g t h e a x i s R . F o r a p r i n c i p a l h y d r o g e n i c q u a n t u m n u m b e r N =
nA + 7 iM- f 1 o f t h e i n n e r e l e c t r o n t h i s i m p l i e s rt\ = 0 ( m i n i m a l o f f - r a d i a l e x c i t a t i o n ) a n d 71^ = N — I ( m a x i m a l n u m b e r o f nodes a l o n g R ) .
0 . 0 3
t—r F i g u r e 2:
B o r n - O p p e n h e i m e r
0) c LU
0 . 0 5 0 . 0 4
0 . 0 6 i - 0
p o t e n t i a l surfaces for t h e M O s t a t e s ( o f he- l i u m ) b e l o n g i n g t o t h e N = 7 , 8 - m a n i f o l d .
5 0 1 0 0 1 5 0 2 0 0 2 5 0
R (a.u.)
for a l l - h e l i u m M O states b e l o n g i n g to the Ar= 7 , 8 - m a n i f o l d s . £nxn»(R) denotes the e n e r g y e i g e n v a l u e o f t h e t w o - c e n t e r H a m i l t o n i a n . T h e p o t e n t i a l c u r v e s for fixed N are l a b e l e d b y n\ a n d nM = N — n\ — 1 w i t h n\ r u n n i n g f r o m zero ( u p p e r c u r v e ) to N — 1 ( l o w e r c u r v e ) . A s c a n be seen the u p p e r p o t e n t i a l c u r v e of e a c h . V - m a n i f o l d ( i n w h i c h the i n n e r e l e c t r o n is m a x i m a l l y p o l a r i z e d a l o n g the a x i s r i ) d e v e l o p s a b r o a d p o t e n t i a l well at R % 100 a.u. i n figure 2. It is t h i s m i n i m u m i n the r a d i a l p o t e n t i a l w h i c h leads to b o u n d e d v i b r a t i o n a l m o t i o n of the o u t e r e l e c t r o n . T h e s o l u t i o n o f the r a d i a l S c h r o d i n g e r e q u a t i o n for the o u t e r e l e c t r o n i n t h e B O p o t e n t i a l s (4) y i e l d s the t o t a l e n e r g y o f the t h r e e - b o d y c o m p l e x a n d ( r a d i a l ) w a v e f u n c t i o n s for the m o t i o n o f the o u t e r e l e c t r o n . T h e w a v e f u n c t i o n s o f t h e v i b r a t i o n a l m o t i o n of the s l o w e l e c t r o n i n the o u t e r p o t e n t i a l w e l l of the u p p e r p o t e n t i a l c u r v e s e x h i b i t o s c i l l a t o r - l i k e c h a r a c t e r . T h e y are l o c a l i z e d i n t h e r e g i o n o f the o u t e r p o t e n t i a l m i n i m u m i n close c o r r e s p o n d e n c e w i t h t h e l o c a l i z e d v i b r a t i o n a l m o t i o n of the t r a p p e d ' c l a s s i c a l1 e l e c t r o n i n figure 1 ( b ) . T h e ( a d i a b a t i c ) p o t e n t i a l b a r r i e r c o r r e s p o n d s to t h e d y n a m i c a l b a r r i e r a p p e a r i n g i n the c l a s s i c a l s o l u t i o n a n d it prevents the o u t e r e l e c t r o n f r o m r e a c h i n g the r e g i o n a r o u n d the n u c l e u s .
F o r l a r g e R each M O m a n i f o l d merges i n t o a S t a r k m u l t i p l e t as was d i s c u s s e d a b o v e a n d w h i c h c a n be seen e x p l i c i t l y i n figure 2. T h e degree of p o l a r i z a t i o n of the i n n e r e l e c t r o n d e p e n d s o n N a n d n\. O n l y t h o s e M O c u r v e s i n figure 2 w h i c h are r e l a t e d a s y m p t o t i c a l l y to the S t a r k states o f m a x i m a l o r i e n t a t i o n a l o n g the " f i e l d axis"' R s h o w a p o t e n t i a l w e l l . F o r n,\ = 0 a n d N > 6 the B o r n - O p p e n h e i m e r p o t e n t i a l s s h o w a m i n i m u m s u f f i c i e n t l y p r o n o u n c e d to a l l o w for q u a n t i z e d v i b r a t i o n a l s t a t e s . B e l o w t h i s v a l u e t h e o f f - r a d i a l e x t e n t of the i n n e r e l e c t r o n M O w a v e f u n c t i o n is t o o large. F o r the s a m e reason w a v e f u n c t i o n s w i t h one o r m o r e n o d a l e x c i t a t i o n s p e r p e n d i c u l a r to R (n,\ ^ 0) do not s u p p o r t a p o t e n t i a l m i n i m u m i n figure 2. In g e n e r a l the o c c u r r e n c e o f m i n i m a i n the B o r n - O p p e n h e i m e r p o t e n t i a l s is n o t r e s t r i c t e d to M O states w i t h n\=Q. If N is large e n o u g h (i.e. N > 16 i n t h e case o f h e l i u m ) , the p o l a r i z a t i o n of a n i n n e r - e l e c t r o n s t a t e w i t h one o f f - r a d i a l n o d e (n,\ = l ) is s t r o n g e n o u g h to p r o d u c e a p o t e n t i a l w e l l i n t h e a d i a b a t i c p o t e n t i a l .
F o r n o n - v a n i s h i n g t o t a l a n g u l a r m o m e n t u m L the e n t i r e r o t a t i o n a l energy o f t h e t h r e e - b o d y s y s t e m leads to a n a d d i t i o n a l r a i s i n g o f the a d i a b a t i c p o t e n t i a l b a r r i e r a n d to a shift o f t h e m i n i m a t o w a r d s l a r g e r R. H o w e v e r , the o v e r a l l s t r u c t u r e is not affected.
C a l c u l a t e d energies for d o u b l y e x c i t e d states o b t a i n e d w i t h i n t h i s s i n g l e c h a n - n e l a d i a b a t i c a p p r o x i m a t i o n are c o m p a r e d w i t h e x a c t r e s u l t s i n s e c t i o n V .
I V . A B I N I T I O C A L C U L A T I O N S
In t h i s s e c t i o n we d e s c r i b e o u r n u m e r i c a l m e t h o d to s o l v e the S c h r o d i n g e r e q u a t i o n for h i g h l y d o u b l y - e x c i t e d e l e c t r o n states. A f u l l s o l u t i o n o f t h i s S c h r o d i n - ger e q u a t i o n is a n o n - t r i v i a l p r o b l e m . H e r e we use a t r a n s f o r m a t i o n of the S c h r o d i n g e r e q u a t i o n i n t o p e r i m e t r i c c o o r d i n a t e s [16, 17]. W e o b t a i n r e s o n a n c e
p o s i t i o n s a n d r e s o n a n c e w i d t h s w i t h i n n e a r - m a c h i n e p r e c i s i o n even for h i g h l y e x c i t e d s t a t e s [9]. T h i s a l l o w s us to check v e r y a c c u r a t e l y t h e p r e d i c t i o n s o f the c l a s s i c a l , s e m i c l a s s i c a l a n d a d i a b a t i c a p p r o x i m a t i o n s d e s c r i b e d i n the p r e v i o u s s e c t i o n s .
U s i n g p e r i m e t r i c c o o r d i n a t e s defined a s l l 6f1 7 ]
x = n - f r2 - r1 2 ; y = rx - r2 + r1 2 ; z = - rx + r2 + r1 2 (5) t h e H a m i l t o n i a n (1) for L= 0 reads ( w i t h ( x i , x2, : r 3 ) d e f i n e d as (x,y,z))
1 v
—
p( 3 )r d z z 1~ (z + y)(x + z)(y + z) ^ d x i " [X^Z)d x j x + y~x + z + y + z ' (6) T h e P-p are p o l y n o m i a l s of degree 3 a n d c a n be f o u n d , e.g., i n Ref. [18]. W e e x p a n d e a c h degree o f freedom i n a c o m p l e t e S t u r m i a n basis set a n d ( a n t i - ) s y m m e t r i z e the p r o d u c t f u n c t i o n s . In this r e p r e s e n t a t i o n a l l the m a t r i x e l e m e n t s are o f s i m p l e a n a l y t i c a l f o r m . T h e i r c a l c u l a t i o n requires m o s t l y integer a r i t h m e t i c a n d is fast a n d a c c u r a t e . In a d d i t i o n , s e l e c t i o n rules g u a r a n t e e t h a t m o s t o f t h e m v a n i s h . T h e r e s u l t i n g m a t r i x e q u a t i o n is of b a n d e d , s p a r s e s t r u c t u r e a n d a l l o w s for efficient d i a g o n a l i z a t i o n .
W e use the m e t h o d o f c o m p l e x r o t a t i o n [19, 20] to c a l c u l a t e a c c u r a t e p o s i t i o n s a n d d e c a y w i d t h s o f the a u t o i o n i z i n g t w o - e l e c t r o n resonances. T o g i v e a n e s t i m a t e of t h e e n e r g y r e g i o n c o v e r e d b y o u r c a l c u l a t i o n s we n o t e t h a t d o u b l y e x c i t e d intra-shell resonances w i t h .'V r a n g i n g f r o m 6 to 18 c o v e r t h i s energy r e g i o n . W e d i a g o n a l i z e m a t r i c e s up to d i m e n s i o n s o f a p p r o x i m a t e l y 7000 to o b t a i n a n a c c u r a c y of t h e c o m p l e x energies of at least 10 s i g n i f i c a n t d i g i t s .
V . R E S U L T S
In t a b l e 1 we s u m m a r i z e o u r results for the energies o f t h e r e s o n a n t states (l Se a n d 35e) w h i c h are d e s c r i b e d b y the set o f M O q u a n t u m n u m b e r s {n\^n^tj = ( n , 0 , 0 ) . T h e t a b l e gives the ( n u m e r i c a l l y ) e x a c t r e s u l t s o f t h e q u a n t u m c a l c u l a - t i o n s as w e l l as t h e a p p r o x i m a t e values Esci p r e d i c t e d b y t h e s i m p l e s e m i c l a s s i c a l f o r m u l a (2) a n d t h e a p p r o x i m a t e values EBO o b t a i n e d b y s o l v i n g the a d i a b a t i c s i n g l e - c h a n n e l e q u a t i o n s w i t h the a d i a b a t i c p o t e n t i a l s s u p p o r t i n g a m i n i m u m at l a r g e R (n > 5 ) . S t a t e s w i t h n <C 6 m a y be c a l l e d p r e c u r s o r s , since t h e y possess a c h a r a c t e r t h a t t r a n s f o r m s s m o o t h l y i n t o t h a t o f t h e h i g h e x c i t e d states.
A s c a n be seen f r o m t a b l e 1 the s i m p l e s e m i c l a s s i c a l f o r m u l a is s u p e r i o r to t h e m o r e e l a b o r a t e a d i a b a t i c c a l c u l a t i o n s i n p r e d i c t i n g a c c u r a t e l y the q u a n t u m energies. T h e errors o f the s e m i c l a s s i c a l energies are b e l o w 1% for a l l resonances ( e x c e p t t h e l o w l y i n g n = 2 s t a t e of the 15e s u b s p a c e ) a n d b e l o w 0 . 1 % for states w i t h n > 7.
C o m p a r e d to e n e r g y eigenvalues, a d i r e c t e x a m i n a t i o n o f the n o d a l s t r u c t u r e of t h e a s s o c i a t e d w a v e f u n c t i o n s is a m o r e s t r i n g e n t test o f the different a p p r o x i - m a t i o n s . F i g u r e 3(a) d e p i c t s the p r o b a b i l i t y d i s t r i b u t i o n of the w a v e f u n c t i o n for the ( 6 , 0 , 0 ) s t a t e o f the p r i n c i p a l series in the ( 0 = 0 ) - p l a n e ( 0 represents the
T a b l e 1: E n e r g i e s o f t h e ( n , 0 , 0 ) c o n f i g u r a t i o n s o b t a i n e d b y f u l l q u a n t u m s o l u t i o n s ( s i n g l e t a n d t r i p l e t s t a t e s r e s p e c t i v e l y ) , t h e s e m i c l a s s i c a l t r i p l e R y d b e r g f o r m u l a (2) (Esci), a n d t h e s i n g l e c h a n n e l a d i a b a t i c a p p r o x i m a t i o n (EBO)- I n a d d i t i o n t h e t o t a l d e c a y w i d t h s o b t a i n e d f r o m c o m p l e x r o t a t i o n a r e g i v e n . ( T h e n u m b e r s are t r u n c a t e d , n o t r o u n d e d ) .
-E ('Se) r/2 (*5g) - £ (3Sg) T / 2 (35e) E3cl EB0
2 0.257 371 609 0.000 010 564 0.249 964 615 0.000 006 7S9 0.247 923 3 0.141 064 156 0.000 011 739 0.140 088 483 0.000 004 409 0.139 351 4 0.089 570 804 0.000 002 024 0.089 467 826 0.000 000 179 0.089 144 5 0.062 053 558 0.000 000 560 0.062 041 278 0.000 000 033 0.061 887
6 0.045 538 667 0.000 000 202 0.045 539 242 0.000 000 376 0.045 458 0.045 956 7 0.034 842 642 0.000 000 368 0.034 843 857 0.000 000 143 0.034 798 0.035 109 8 0.027 517 599 0.000 001 184 0.027 519 289 0.000 000 022 0.027 491 0.027 612 9 0.022 284 5S7 0.000 000 525 0.022 283 665 0.000 000 035 0.022 265 0.022 413 10 0.018 411 985 0.000 000 058 0.018 411 896 0.000 000 030 0.018 400 0.018 507 11 0.015 468 259 0.000 000 023 0.015 468 265 0.000 000 019 0.015 460 0.015 541 12 0.013 178 121 ' 0.000 000 022 0.013 178 140 0.000 000 010 0.013 172 0.013 235 13 0.011 361 442 0.000 000 014 0.011 361 444 0.000 000 005 0.011 357 0.011 406 14 0.009 896 121 0.000 000 004 0.009 896 120 0.000 000 002 0.009 893 0.009 932
a n g l e b e t w e e n t h e e l e c t r o n i c r a d i a l v e c t o r s ) . T h i s c o r r e s p o n d s t o t h e c o l l i n e a r a r r a n g e m e n t o f t h e e l e c t r o n s . T h e o f f - c o l l i n e a r p a r t o f t h e p r o b a b i l i t y d e n s i t y , n o t s h o w n here, decreases e x p o n e n t i a l l y i n d i c a t i n g a z e r o p o i n t m o t i o n i n t h e 0 - b e n d i n g degree o f f r e e d o m . T h e zero p o i n t m o t i o n is e x p r e s s e d b y t h e as- s i g n m e n t n\ = 0 w i t h i n t h e M O d e s c r i p t i o n r e s p e c t i v e l y . T h e c o o r d i n a t e rx ( r2) d e n o t e s t h e r a d i a l d i s t a n c e o f t h e o u t e r ( i n n e r ) e l e c t r o n . T h e o u t e r e l e c t r o n p r o b - a b i l i t y is s t r o n g l y l o c a l i z e d i n t h e r e g i o n rx « 1 2 0 a . u . , r e f l e c t i n g t h e d y n a m i c a l l o c a l i z a t i o n o f t h e " f r o z e n " e l e c t r o n . N o t e also t h e l a r g e differences i n t h e r a d i a l e x t e n t s rt. T h e n o d a l e x c i t a t i o n s a r e a l l d i r e c t e d a l o n g t h e p e r i o d i c o r b i t o f fig. 1, w h i c h is a n e a r l y s t r a i g h t l i n e a l o n g t h e r a d i u s o f t h e f r o z e n o u t e r e l e c t r o n i n d i - c a t e d b y a n a r r o w i n t h e figure. R e c a l l i n g t h e t y p i c a l q u a d r a t i c s p a c i n g o f n o d a l lines i n C o u l o m b i c s y s t e m s , w e a c h i e v e n e a r l y c o n s t a n t n o d a l d i s t a n c e s b y u s i n g q u a d r a t i c a l l y s c a l e d axes as d o n e i n figure 3 ( d ) . T h e n u m b e r o f nodes a l o n g t h e o r b i t is n = 6 i n a g r e e m e n t w i t h t h e s e m i c l a s s i c a l p r e d i c t i o n s . T h e w a v e f u n c t i o n does n o t s h o w a n y o f f - o r b i t e x c i t a t i o n s , w h i c h agrees w i t h t h e s e m i c l a s s i c a l l o c a l c o o r d i n a t e c l a s s i f i c a t i o n ( n , fc,/) = ( 6 , 0 , 0 ) .
W i t h i n t h e M O d e s c r i p t i o n t h e w a v e f u n c t i o n is c h a r a c t e r i z e d as f o l l o w s : T h e i n n e r e l e c t r o n is m a x i m a l l y p o l a r i z e d a l o n g t h e n u c l e u s - f r o z e n e l e c t r o n - a x i s R (n\ = 0, nM = 6) w h i l e t h e o u t e r e l e c t r o n is i n i t s v i b r a t i o n a l g r o u n d s t a t e (£ = 0) o f t h e effective p o t e n t i a l w e l l . N o t e t h e absence o f n o d a l l i n e s i n rx for t h e e x a c t w a v e f u n c t i o n i n figure 3 ( a ) . T h u s t h e w a v e f u n c t i o n r e v e a l s t h e e q u i v a l e n c e o f t h e s e m i c l a s s i c a l q u a n t u m n u m b e r n a n d t h e M O q u a n t u m n u m b e r nM.
F i g u r e 3: C o n d i t i o n a l p r o b a b i l i t y densities o f ( n , 0 , / ) s t a t e s i n h e l i u m for n = 6 . T h e a n g l e 0 b e t w e e n i*i a n d r2 is fixed to 0 = 0 . T h e axes h a v e a l i n e a r (left p a r t ) a n d a q u a d r a t i c scale ( r i g h t p a r t ) , r e s p e c t i v e l y . S h o w n are / = 0 ( a , d ) , /=1 (b,e), a n d 1=2 ( c , f ) . N o t e t h e a s y m m e t r y i n the scales o f t h e axes. O n l y t h e p a r t s
ri > r2 are s h o w n . T h e full w a v e f u n c t i o n is s y m m e t r i c i n r\ a n d r2.
T h e s e m i c l a s s i c a l t r i p l e R y d b e r g f o r m u l a (2) suggests t h e e x i s t e n c e o f q u a n - t u m s t a t e s w i t h n o d a l e x c i t a t i o n s t r a n s v e r s e to the p e r i o d i c o r b i t ( l a b e l e d b y / ) , w h i c h preserves t h e c o l l i n e a r c h a r a c t e r o f the m o t i o n . S u c h c o n f i g u r a t i o n s are s h o w n i n figure 3 for / = 1 (b,e) a n d / = 2 ( c , f ) . In t h e a d i a b a t i c M O d e s c r i p t i o n s u c h e x c i t a t i o n s i n rx = R r e p r e s e n t v i b r a t i o n a l levels o f t h e o u t e r e l e c t r o n . It follows t h a t the q u a n t u m n u m b e r s / i n the s e m i c l a s s i c a l a n d £ i n the m o l e c u l a r c l a s s i f i c a t i o n s c h e m e are i d e n t i c a l .
U p to n o w we i n v e s t i g a t e d t h e ( r a d i a l ) v i b r a t i o n a l m o d e o f t h e o u t e r elec- t r o n d e s c r i b e d b y t h e q u a n t u m n u m b e r /, i.e. e x c i t a t i o n s w i t h i n the c o l l i n e a r ar- r a n g e m e n t o f the e l e c t r o n s . W e n o w focus o n the s t r u c t u r e o f the i n n e r e l e c t r o n w a v e f u n c t i o n . In t h e a d i a b a t i c t r e a t m e n t the d y n a m i c s d e c o u p l e s i n t o m o t i o n o f the o u t e r e l e c t r o n ( v i b r a t i o n a l 7 ? - m o t i o n ) a n d o f the i n n e r e l e c t r o n , w h i c h for fixed R s e p a r a t e s i n p r o l a t e s p h e r o i d a l c o o r d i n a t e s A , / i . T o test these p r e d i c t i o n s we c o m p a r e i n figure 4 the p r o b a b i l i t y densities of a t w o - c e n t e r M O w a v e f u n c - t i o n o f h e l i u m o b t a i n e d w i t h i n the a d i a b a t i c a p p r o a c h w i t h the c o r r e s p o n d i n g ab initio t h r e e - d i m e n s i o n a l q u a n t u m w a v e f u n c t i o n for fixed d i s t a n c e s R o f t h e o u t e r e l e c t r o n . T h e figure t h e n d e p i c t s the c o n d i t i o n a l p r o b a b i l i t y for finding t h e i n n e r e l e c t r o n i n t h e c o o r d i n a t e s p a c e r e l a t i v e to the a x i s R . W e choose R as t h e c l a s s i c a l e x p e c t a t i o n v a l u e for t h e r a d i a l d i s t a n c e of the o u t e r e l e c t r o n a l o n g t h e c l a s s i c a l p e r i o d i c o r b i t o f figure 1(a).
P a r t (a) o f t h e figure 4 i n d e e d shows t h a t the i n n e r e l e c t r o n w a v e f u n c t i o n is o f S t a r k - t y p e c h a r a c t e r w i t h m a x i m a l p o l a r i z a t i o n a l o n g the a x i s b e t w e e n n u c l e u s a n d o u t e r e l e c t r o n . T h e s t a t e e x h i b i t s no o f f - r a d i a l e x c i t a t i o n s a n d is d e s c r i b e d b y nM = 6 , n A = 0 i n t h e M O c l a s s i f i c a t i o n s c h e m e . A c o m p a r i s o n o f p a r t (a) a n d (b) e x h i b i t s the close s i m i l a r i t y of the a p p r o x i m a t e a d i a b a t i c a n d the f u l l q u a n t u m w a v e f u n c t i o n . T h i s p r o v e s the v a l i d i t y of t h e a d i a b a t i c a p p r o x i m a t i o n a n d reveals the q u a s i - s e p a r a b i l i t y o f the full q u a n t u m w a v e f u n c t i o n s i n m o l e c u l a r o r b i t a l c o o r d i n a t e s . A s s h o w n i n R e f . [10] full q u a n t u m w a v e f u n c t i o n s w i t h n\ ^ 0 a p p r o x i m a t e l y s e p a r a t e i n M O c o o r d i n a t e s , t o o .
F i n a l l y , we focus o n t h e c o r r e s p o n d e n c e b e t w e e n t h e c l a s s i c a l a n d t h e q u a n t u m d y n a m i c s b y s h o w i n g i n figure 5 a c o m b i n e d p l o t o f t h e i n n e r a n d o u t e r e l e c t r o n i c d e n s i t i e s , w h i c h i m a g e s t h e c h a r g e d i s t r i b u t i o n o f t h e e n t i r e t w o e l e c t r o n a t o m . A g l o b a l s p a c e filling c h a r g e d i s t r i b u t i o n is o b t a i n e d b y a n o v e r a l l r o t a t i o n a r o u n d t h e c e n t e r o f m a s s ( n u c l e u s ) . T h e p r o b a b i l i t y d e n s i t y o f the o u t e r e l e c t r o n o f t h e s t a t e ( 6 , 0 , 0 ) is o b t a i n e d i n a n a l o g y to figure 4 b y d r a w i n g a c u t t h r o u g h t h e f u l l w a v e f u n c t i o n at a fixed r a d i a l d i s t a n c e r2 of the i n n e r e l e c t r o n . T h e w a v e f u n c t i o n o f the o u t e r e l e c t r o n (left h a n d side o f figure 5) a p p e a r s as t h e b u m p far a w a y f r o m the n u c l e u s Z. It j u s t resembles t h e g r o u n d s t a t e o s c i l l a t o r - l i k e w a v e f u n c t i o n (/ = 0) i n t h e o u t e r w e l l o f t h e B O - p o t e n t i a l a n d t u r n s o u t to be e x t r e m e l y n o n - h y d r o g e n i c . T h e i n n e r w a v e f u n c t i o n is t a k e n f r o m figure 4 ( b ) . T h e s e s t a t e s e x h i b i t s t r o n g e l e c t r o n c o r r e l a t i o n : Radial c o r r e l a t i o n leads to t h e o s c i l l a t o r - l i k e w a v e f u n c t i o n o f the o u t e r e l e c t r o n , angular c o r r e l a t i o n is v i s i b l e f r o m t h e p o l a r i z e d i n n e r M O t y p e w a v e f u n c t i o n w h i c h is a c o h e r e n t s u p e r p o s i t i o n o f a l l
F i g u r e 4: C o n d i t i o n a l p r o b a b i l i t y densities for t h e i n n e r e l e c t r o n w i t h respect to t h e f i x e d a x i s R b e t w e e n the n u c l e u s a n d t h e o u t e r e l e c t r o n . T h e (n\,n^^() = ( 6 , 0 , 0 ) s t a t e o b t a i n e d f r o m t h e s o l u t i o n of the t w o - c e n t e r S c h r o d i n g e r e q u a t i o n w i t h i n t h e B o r n - O p p e n h e i m e r a p p r o a c h (a) is c o m p a r e d w i t h t h e c o r r e s p o n d i n g ab initio q u a n t u m w a v e f u n c t i o n ( b ) . T h e p o s i t i o n o f t h e n u c l e u s (Z(= 2)) is i n d i c a t e d i n p a r t s (a) a n d (b), ( P a r t (a) f r o m R e f . [20]).
s i n g l e - p a r t i c l e a n g u l a r m o m e n t a /,-. A c o m p a r i s o n w i t h t h e q u a s i p e r i o d i c c l a s s i c a l m o t i o n s h o w n i n figure 5 ( b ) i l l u s t r a t e s the c l a s s i c a l - q u a n t a l c o r r e s p o n d e n c e of t h e e l e c t r o n - p a i r m o t i o n .
T h e r e are i n p r i n c i p a l t w o m e c h a n i s m s w h i c h l e a d to d e c a y o f the p l a n e t a r y c o n f i g u r a t i o n s d e s c r i b e d i n the p r e c e d i n g sections: r a d i a t i v e a n d n o n - r a d i a t i v e
F i g u r e 5: I m a g e of t h e c h a r g e d i s t r i b u t i o n o f the h e l i u m a t o m i n t h e ( 6 , 0 , 0 ) s t a t e , p a r t (a). T h e r i g h t p a r t is the p r o b a b i l i t y d i s t r i b u t i o n o f the i n n e r e l e c t r o n at fixed r a d i a l d i s t a n c e R o f the o u t e r e l e c t r o n ; t h e left p a r t is t h e p r o b a b i l i t y for t h e o u t e r e l e c t r o n at fixed i n n e r e l e c t r o n r a d i u s r2. T h e fixed values w e r e c h o s e n as t h e c l a s s i c a l o u t e r t u r n i n g p o i n t s of the e l e c t r o n s . P a r t (b) shows a t y p i c a l c l a s s i c a l t r a j e c t o r y o f the t w o e l e c t r o n s . T h e t r a j e c t o r y is c o n f i n e d to a t o r u s i n p h a s e space. T h e p o s i t i o n o f t h e n u c l e u s (Z) is i n d i c a t e d i n p a r t (a).
decay. W e w i l l focus o n n o n - r a d i a t i v e p a r t i c l e decay, i.e. a u t o i o n i z a t i o n o f t h e res- o n a n c e s . S i n c e the c l a s s i c a l m o t i o n c o r r e s p o n d i n g to t h e q u a n t u m states is s t a b l e (see figure 1) t h e y are c l a s s i c a l l y b o u n d . H o w e v e r , i n a n a l o g y to the ( s e m i c l a s s i - c a l ) p e n e t r a t i o n t h r o u g h a " s t a t i c " p o t e n t i a l b a r r i e r these s t a t e s c a n a u t o i o n i z e s e m i c l a s s i c a l l y b y " d y n a m i c a l " t u n n e l i n g [22], b u t the d e c a y w i d t h s for s u c h p r o - cesses decrease e x p o n e n t i a l l y w i t h the n o d a l e x c i t a t i o n a l o n g t h e o r b i t [10].
In t a b l e 1 we s u m m a r i z e the w i d t h s o f the 1Se a n d 3Se resonances w i t h q u a n - t u m n u m b e r s ( n , 0 , 0 ) . T h e y were c a l c u l a t e d fully q u a n t u m m e c h a n i c a l l y u s i n g the c o m p l e x r o t a t i o n m e t h o d ( s e c t i o n I V ) . T h e i m a g i n a r y p a r t s are e x t r e m e l y s m a l l . T h e w i d t h s T / 2 i n d e e d decrease e x p o n e n t i a l l y w i t h i n c r e a s i n g n o d a l ex- c i t a t i o n n ~ 1 /y/—E, l n T / 2 ~ —0.71 n , a l t h o u g h t h e y fluctuate r a t h e r s t r o n g l y a r o u n d t h e a v e r a g e t r e n d . T h u s t h e resonances a p p e a r as b o u n d states i n t h e c o n t i n u u m i n t h e l i m i t o f l a r g e e x c i t a t i o n even t h o u g h the n u m b e r o f o p e n c h a n - nels increases t r e m e n d o u s l y w i t h i n c r e a s i n g energy ( m o r e t h a n 100 c h a n n e l s are o p e n for t h e n = 14 state!).
V I . C O N C L U S I O N
In t h e present w o r k we h a v e s t u d i e d p r o p e r t i e s of a c e r t a i n class o f c o r r e l a t e d r e s o n a n t states o f t w o - e l e c t r o n a t o m s a n d ions o f h i g h l y d o u b l y - e x c i t e d elec- t r o n s . T h e p r o b l e m has been i n v e s t i g a t e d f r o m several different p o i n t s o f v i e w
— c l a s s i c a l l y , s e m i c l a s s i c a l l y a n d q u a n t u m m e c h a n i c a l l y ( b o t h e x a c t l y a n d i n a n a d i a b a t i c a p p r o a c h ) . S u m m a r i z i n g the results w e h a v e e s t a b l i s h e d a classifica- t i o n o f these r e s o n a n t states i n t e r m s of s e m i c l a s s i c a l q u a n t u m n u m b e r s ( n , fc, /) a s s o c i a t e d w i t h l o c a l c o o r d i n a t e s of the p e r i o d i c o r b i t a n d the M O set (nM,n\,£) a s s o c i a t e d w i t h m o l e c u l a r - t y p e c o o r d i n a t e s , w h i c h ( l o c a l l y ) are i d e n t i c a l . T h e e x a c t w a v e f u n c t i o n s o f t h e p r o b l e m show q u a s i - s e p a r a b l e b e h a v i o u r i n these co- o r d i n a t e s . T h e c o r r e s p o n d i n g u n d e r l y i n g d y n a m i c a l s y m m e t r y a p p e a r s due to the d i s t i n c t e l e c t r o n c o r r e l a t i o n . E n e r g i e s c a l c u l a t e d w i t h i n the t w o a p p r o x i m a t e a p p r o a c h e s r e p r o d u c e t h e e x a c t q u a n t u m results q u i t e a c c u r a t e l y .
T h e r e s o n a n t states possess e x t r e m e l y s m a l l w i d t h s w h i c h decrease e x p o n e n - t i a l l y w i t h i n c r e a s i n g e x c i t a t i o n .
A n a l t e r n a t i v e a p p r o a c h w o u l d be to l o o k for s i m i l a r c o n f i g u r a t i o n s i n o t h e r t h r e e - b o d y s y s t e m s . It is i m m e d i a t e l y o b v i o u s f r o m the c l a s s i c a l a n a l y s i s t h a t the d y n a m i c a l l y l o c a l i z e d o u t e r e l e c t r o n can be r e p l a c e d b y h e a v y n e g a t i v e l y c h a r g e d p a r t i c l e s ( s u c h as k a o n s K~ o r a n t i p r o t o n s p) w i t h o u t c h a n g i n g i n n e r e l e c t r o n d y n a m i c s e s s e n t i a l l y . I n d e e d , the present m e c h a n i s m has been p r o p o s e d as a t r a p for a n t i - p a r t i c l e s [23] a n d u n e x p e c t e d l y l o n g - l i v e d states h a v e been f o u n d e x p e r i m e n t a l l y i n s u c h s y s t e m s r e c e n t l y [24].
A C K N O W L E D G M E N T S
W e w o u l d l i k e to t h a n k J S B r i g g s , E A S o l o v ' e v , U E i c h m a n n , J M R o s t , W S a n d n e r a n d R T h i i r w a c h t e r for s t i m u l a t i n g d i s c u s s i o n s . T h i s w o r k was s u p p o r t e d b y the D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t u n d e r c o n t r a c t W i 8 7 7 / 2 a n d w i t h i n t h e S o n d e r f o r s c h u n g s b e r e i c h 276 at the U n i v e r s i t y of F r e i b u r g .
R E F E R E N C E S
[1] C a m u s P , G a l l a g h e r T F , L e c o m t e J M , P i l l e t P a n d B o u l m e r J , P h y s . R e v . L e t t . 62, 2365 (1989)
[2] E i c h m a n n U , L a n g e V a n d S a n d n e r W, P h y s . R e v . L e t t . 64, 274 ( 1 9 9 0 ) , P h y s . R e v . L e t t . 68, 21 (1992)
[3] S a n d n e r W , t h i s v o l u m e [4] H o g e r v o r s t , t h i s v o l u m e
[5] F a n o , U , P h y s . R e p . 46, 97 (1983)
[6] H e r r i c k D E , A d v . C h e m . P h y s . 52, 1 (1983)
[7] R o s t J M a n d B r i g g s J S, J . P h y s . B 24, 4 2 9 3 (1991) [8] G r e e n , C H , t h i s v o l u m e
[9] R i c h t e r K a n d W i n t g e n D , J . P h y s . B 24, L 5 6 5 (1991)
[10] R i c h t e r K , B r i g g s J S, W i n t g e n D a n d S o l o v ' e v E A , J . P h y s . B , i n press (1992)
[11] P e r c i v a l I C , P r o c . R o y . S o c . L o n d . A 353, 289 (1977)
[12] E z r a G S, R i c h t e r K , T a n n e r G a n d W i n t g e n D , J . P h y s . B 24, L 4 1 3 (1991) [13] G u t z w i l l e r M C , Chaos in Classical and Quantum Mechanics ( N e w Y o r k :
S p r i n g e r ) (1990)
[14] R i c h t e r K a n d W i n t g e n D , P h y s . R e v . L e t t . 65, 1965 (1990) [15] H e l f r i c h K , T h e o r . C h i m . A c t a 24, 271 (1972)
[16] J a m e s H M a n d C o o l i d g e A S, P h y s . R e v . 51, 857 (1937) [17] P e k e r i s C L , P h y s . R e v . 112, 1649 (1958)
[18] Z h e n Z , P h y s . R e v . A 41, 87 (1990)
[19] R e i n h a r d t W P , A n n . R e v . P h y s . C h e m . 33, 223 (1982) [20] H o Y K , P h y s . R e p . 99, 1 (1983)
[21] T h u r w a c h t e r R , p r i v a t e c o m m u n i c a t i o n (1992)
[22] D a v i e s M J a n d H e l l e r E J , J . C h e m . P h y s . 75, 246 (1981)
[23] R i c h t e r R , R o s t J M , T h u r w a c h t e r R , B r i g g s J S, W i n t g e n D a n d S o l o v ' e v E A , P h y s . R e v . L e t t . 66, 149 (1991)
[24] I w a s a k i M et a l . , P h y s R e v . L e t t . 67, 1246 (1991)
