17 Classical, Semiclassical, a n d Q u a n t u m D y n a m i c s o f L o n g - L i v e d H i g h l y E x c i t e d V i b r a t i o n a l States o f Triatoms
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R. M. HEDGES, JR., R. T. SKODJE, F. BORONDO, and W. P. REINHARDT Department of Chemistry, University of Colorado and Joint Institute for Laboratory Astrophysics, National Bureau of Standards and University of Colorado, Boulder, CO 80309 Triatoms with doubly vibrationally excited predisso ciating states of exceptionally long lifetime are theo retically investigated using several techniques. For a two-degrees-of-freedom model of H O fully converged quantum estimates of resonance lifetimes are made, con firming the correspondence principle expectation that exceptionally long lived states exist and display non -RRKM behavior in the sense that lifetimes do not always decrease with increasing energy above the dissociation limit. Using the quantum results as a benchmark, the validity of a Golden-Rule-type formula is demonstrated, and the formula is then applied for physically realis tic values of frequency, anharmonicity, and well depth: States with lifetimes of up to 0.1 sec are found. The paper ends with presentation of preliminary adiabatic semiclassical estimates of resonance energies for HOD in two and three degrees-of-freedom models. 2
The possibility that highly excited molecules can display strongly non-statistical behavior has been an important theme of experimen tal and theoretical research during the past decade. To name only two examples motivating such interest, the predissociation of very weakly coupled van der Waals molecules provides a possibility of long lived high energy (relative to dissociation) systems; and, the possibility of infrared laser directed chemistry depends of at least short term localization of energy, defying statistical ran domization. The questions to be addressed here may be simply stated: How do dissociation times for covalently bonded systems depend on excitation mechanism? Namely, are there specific modes of highly excited polyatomic molecules that live longer than iso energetic ones? Are there sequences of modes where lifetimes in crease with increasing energy? Classical mechanical studies of such questions date from the work of Bunker (1) and co-workers. Our attention was drawn to such problems by the work of Wolf and Hase (2,3) who showed that model systems of two and three degrees of freedom displayed large appar ently trapped volumes of classical phase space well above the clas0097-6156/ 84/0263-0323S06.00/0 © 1984 American Chemical Society
In Resonances; Truhlar, D.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
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sical dissociation limit. Such volumes o f n o n - d i s s o c i a t i n g phase space gave r i s e t o s t r o n g l y n o n s t a t i s t i c a l (non-RRKM) c l a s s i c a l d i s s o c i a t i o n k i n e t i c s ( 3 ) • Hase (4) s u b s e q u e n t l y i d e n t i f i e d such n o n - d i s s o c i a t i n g c l a s s i c a l t r a j e c t o r i e s as b e i n g q u a s i - p e r i o d i c , and thus q u a n t i z a b l e u s i n g the u n d e r l y i n g i n v a r i a n t t o r i ( 5 ) • Noid and Koszykowski (6) and Hase (4) c a r r i e d out such p r i m i t i v e q u a n t i z a t i o n s , o b t a i n i n g s e m i c l a s s i c a l estimates of the energies of the h i g h l y e x c i t e d s t a t e s . The q u e s t i o n remained as t o the l i f e times o f the s t a t e s . An analogous s e r i e s o f c l a s s i c a l c a l c u l a t i o n s has been c a r r i e d out by Uzer, Hynes and R e i n h a r d t ( 7 ) , i n a study o f d i s s o c i a t i o n f o l l o w i n g m o d e - s p e c i f i c c l a s s i c a l e x c i t a t i o n , modeling the e x p e r i ments of R l z z o e t a l . (8) on t h e l a s e r d i s s o c i a t i o n o f HOOH v i a pumping o f the h i g h OH v i b r a t i o n a l o v e r t o n e s . These workers ( 7 ) found t h a t t r a j e c t o r i e s w i t h c l a s s i c a l i n i t i a l c o n d i t i o n s c o r r e s ponding t o i n i t i a l h i g h e x c i t a t i o n i n the OH s t r e t c h seem not t o d i s s o c i a t e , energy b e i n g " t r a p p e d " i n a Fermi resonance c o u p l i n g the OH s t r e t c h t o the H00 bend i n t h i s s i x - d e g r e e s - o f - f r e e d o m problem. T h i s s u g g e s t s non-RRKM l i f e t i m e s o f d i s s o c i a t i o n f o r t h e m o d e - s p e c i f i c e x c i t a t i o n used i n the e x p e r i m e n t s . U n f o r t u n a t e l y , t h e s e experiments (8) g i v e no i n d i c a t i o n o f a c t u a l d i s s o c i a t i o n time s c a l e s , but o n l y t h a t the OH i s produced i n a d i s t r i b u t i o n o f excited states consistent with s t a t i s t i c a l . As r e p e a t e d l y p o i n t e d out by Hase ( 9 ) , t h i s type o f time independent o b s e r v a t i o n does not p r e c l u d e a non-RRKM d i s t r i b u t i o n o f d i s s o c i a t i o n l i f e t i m e s . The c l a s s i c a l work on HOOH a g a i n r e q u i r e s t h a t a t e c h n i q u e be d e v e l o p e d f o r d e t e r m i n i n g t h e l i f e t i m e s o f t h e quantum r e s o n a n c e s suggested by t h e c o r r e s p o n d e n c e p r i n c i p l e . As t h e t r a p p e d volumes of phase space o f t e n appear s t a b l e , t e c h n i q u e s based on c l a s s i c a l s t a b i l i t y a n a l y s i s o f t r a p p e d but u n s t a b l e p e r i o d i c o r b i t s ( 1 0 ) y i e l d no e s t i m a t e s o f l i f e t i m e s , and a quantum or s e m i c l a s s i c a l method must be employed. R e a l i s t i c quantum s t u d i e s o f h i g h l y e x c i t e d systems w i t h many v i b r a t i o n a l degrees o f freedom a r e c u r r e n t l y p r o h i b i t i v e i f a l l c o u p l i n g s and degrees of freedom a r e t o be accounted f o r . S e v e r a l two-degrees-of-freedom systems have been i n v e s t i g a t e d a t " l a r g e " v a l u e s o f P l a n c k s c o n s t a n t : Waite and M i l l e r (11,12) have l o o k e d at t h e t u n n e l i n g p r e d i s s o c i a t i o n i n t h e c u b i c Henon-Heiles problem; Numrich and Kay (13) and C h r i s t o f f e l and Bowman (14) have c a r r i e d out c a l c u l a t i o n s o f t h e r e s o n a n c e widths o f an harmonic o s c i l l a t o r c o u p l e d t o a Morse o s c i l l a t o r ; Hedges and R e i n h a r d t (15,16) have i n v e s t i g a t e d Feshbach type r e s o n a n c e s i n ABA t r i a t o m i c s modeled as two mass-coupled Morse o s c i l l a t o r s . L a r g e v a l u e s of P l a n c k ' s c o n s t a n t a r e used as fewer b a s i s s t a t e s , or c h a n n e l s , a r e needed t o d e s c r i b e the system, a l l o w i n g f u l l y converged quantum e s t i m a t e s o f w i d t h s t o be made. However, such h i g h l y quantum models need not c o r r e l a t e w i t h c l a s s i c a l dynamics (5,17-19) and may, i n d e e d , miss many f e a t u r e s o f t h e c l a s s i c a l - q u a n t u m c o r r e s p o n d e n c e . T h i s i s w e l l i l l u s t r a t e d by t h e r e a n a l y s i s o f the Henon-Heiles problem a t s m a l l e r R, by B a i e t a l . ( 2 0 ) , who come t o q u i t e d i f f e r e n t con c l u s i o n s t o those o f Waite and M i l l e r ( 1 1 , 1 2 ) , who found no mode specificity. In the f i r s t two p a r t s o f the p r e s e n t paper the c l a s s i c a l and quantum work of R e f s . (15,16) i s r e v i e w e d , and extended ( 2 1 , 2 2 ) , v i a a G o l d e n - R u l e - t y p e a n s a t z ( 2 3 ) t o g i v e resonance widths f o r the d o u b l y e x c i t e d s t a t e s of ABA t r i a t o m s f o r n - 1. I n agreement w i t h
In Resonances; Truhlar, D.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
17.
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c o r r e s p o n d e n c e p r i n c i p l e e x p e c t a t i o n s , s t a t e s of e x t r a o r d i n a r i l y l o n g l i f e t i m e are r o u t i n e l y found, the l o n g e s t of the p r e s e n t s t u d y b e i n g 0.1 seconds. I n the t h i r d p a r t , p r e l i m i n a r y n u m e r i c a l s t u d i e s of a "new" a d i a b a t i c s e m i c l a s s i c a l q u a n t i z a t i o n t e c h n i q u e (24,25) a r e p r e s e n t e d (26) and a p p l i e d to d e t e r m i n a t i o n s of resonance p o s i t i o n s f o r two- and t h r e e - d e g r e e s - o f - f r e e d o m models of HOD ( 2 7 ) . C l a s s i c a l Dynamics Above D i s s o c i a t i o n : ABA Two Degrees of Freedom
L o c a l Mode Systems w i t h
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A u s e f u l model system, r e p r e s e n t i n g the i n t e r a c t i o n of two " l o c a l mode" o s c i l l a t o r s as might occur i n a l i g h t - h e a v y - l i g h t system, such as water, i s d e s c r i b e d by the H a m i l t o n i a n
+ D(l - e x p ( - ( r r ° ) ) ) r
2
+ D(l - exp(-(r -r°)))
2
(1)
2
where the c o u p l i n g between the Morse type l o c a l mode o s c i l l a t o r s i s s i m p l y the momentum c o u p l i n g through the c e n t r a l mass. The H a m i l t o n i a n of E q u a t i o n (1) can r e p r e s e n t a c o l l i n e a r t r i a t o m or a bent system, such as H2O, f o r a p p r o p r i a t e c h o i c e s of the e f f e c t i v e masses 12> 2» F i g u r e 1 shows a P o i n c a r e s u r f a c e of s e c t i o n ( d e f i n e d by z e r o e s of the v a r i a b l e Q2 - *\~*2) f o r c l a s s i c a l dynamics d e t e r mined by the H a m i l t o n i a n above — u s i n g parameters a p p r o p r i a t e f o r H2O — f o r an energy e q u a l to the c l a s s i c a l d i s s o c i a t i o n energy. I t i s e v i d e n t t h a t the dynamics are governed by the presence of i n v a r i a n t t o r i even at the d i s s o c i a t i o n energy. As the energy i s i n c r e a s e d , the a r e a o c c u p i e d by t o r i on the composite s u r f a c e of section slowly decreases: F i g u r e 2 i n d i c a t e s the s i t u a t i o n a t 5/4 of the d i s s o c i a t i o n energy. T o r i are s t i l l e a s i l y v i s i b l e at 7/4 D, a l t h o u g h the a r e a on the s u r f a c e of s e c t i o n o c c u p i e d by t o r i i s decreased (15,16). The presence of i n v a r i a n t t o r i a t , and above, the c l a s s i c a l d i s s o c i a t i o n l i m i t i n d i c a t e s t h a t whole volumes of the c l a s s i c a l phase space are n o n - d i s s o c i a t i n g f o r the model H a m i l t o n i a n . T h i s s u g g e s t s the presence of l o n g - l i v e d quan tum r e s o n a n c e s , which i f the correspondence p r i n c i p l e h o l d s , s h o u l d be r e c o g n i z a b l e w e l l above d i s s o c i a t i o n . u
m
Quantum Dynamics of ABA
Systems
Complex C o o r d i n a t e S t u d i e s . The H a m i l t o n i a n of E q u a t i o n a n a l y t i c f u n c t i o n of p o s i t i o n s and momenta. We may thus t h a t the a n a l y t i c a l l y c o n t i n u e d H a m i l t o n i a n
+ D[l - e x p - ( ( r , e 1
i 9
- r?))] 1
2
+ D[l - e x p ( - ( r e 9
1
1 9
-
(1) i s an expect
2
r°))] (2)
1
In Resonances; Truhlar, D.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
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-20.04
-10.02
0
10.02
20 04
F i g u r e 1. Composite o f s e v e r a l Poincare" s u r f a c e s o f s e c t i o n f o r the mass r a t i o y 1 2 ^ 2 = 1:64 ( a p p r o p r i a t e t o H 0 ) , a t t h e c l a s s i c a l d i s s o c i a t i o n energy, D. I n v a r i a n t t o r i dominate the phase space s t r u c t u r e . Reproduced from r e f . ( 1 6 ) , w i t h p e r m i s s i o n . C o p y r i g h t 1983, American I n s t i t u t e o f P h y s i c s . 2
-ae.m
-11.20
.00
11.20
aa.m
Pi F i g u r e 2. As i n F i g . 1, but E = 5/4 D. The f r a c t i o n o f phase d i r e c t l y c o u p l e d t o d i s s o c i a t i v e c h a n n e l s grows as a f u n c t i o n o f energy above D, but trapped i n v a r i a n t t o r i r e m a i n , even a t q u i t e h i g h energy. Reproduced from r e f . ( 1 6 ) , w i t h p e r m i s s i o n . C o p y r i g h t 1983, American I n s t i t u t e o f P h y s i c s .
In Resonances; Truhlar, D.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
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defined by the scale transformation r r exp(i0), w i l l have the usual properties of d i l a t a t i o n analytic (or "complex-scaled") Hamiltonians (see the reviews of References (28-30)). Namely: (1) the r e a l bound state eigenvalues are independent of the trans formation; (2) complex resonance eigenvalues are "exposed" as cuts of the corresponding resolvent are rotated off the r e a l axis; and (3) the eigenfunctions corresponding to the resonance eigenvalues are square integrable, and thus may be approximated by usual linear v a r i a t i o n a l methods. Hedges and Reinhardt (15,16) have carried out a series of such v a r i a t i o n a l calculations using a v a r i a t i o n a l spline basis contracted by prediagonalization of the one-degree-of-freedom Morse problem. Use of this non-analytic basis resulted from a com promise between accuracy and computational f l e x i b i l i t y . The basis allowed easy spanning of the required coordinate space near the tops of the anharmonic wells, but gave only "asymptotic" convergence (such as that discussed i n (31)) for narrow resonances: imaginary parts of eigenvalues less than 10" i n absolute magnitude were unreliable. As these calculations have been described i n d e t a i l elsewhere (15,16), we present results only for the mass r a t i o 1:64:1, which corresponds to H2O with a bond angle of 104.5°. Figure 3 shows the quantum eigenvalues, plotted as lifetime (decreasing upwards) ver sus energy i n units of the d i s s o c i a t i o n energy, D. The well depth i n these calculations corresponds to ten bound states per l o c a l bond mode, or, put another way, to an e f f e c t i v e value of n = 2.4 atomic units, where n = 1 would correspond to experimental r e a l i t y . Nevertheless, the figure indicates the existence of very long l i v e d states (up to 10 harmonic v i b r a t i o n a l periods) even for this excessively quantum-like model. Several comments are i n order: the c l a s s i c a l bound states i n the continuum of Figures 1 and 2 correspond to doubly excited vibrations with approximately equal energies i n each bond mode, with the asymmetric stretch normal mode type c l a s s i c a l motion (see (32) for a discussion of the difference between c l a s s i c a l l o c a l and normal mode behavior) being the most stable. The quantum results mirror this i n that the doubly excited states with bond mode quantum numbers (n,n) display high s t a b i l i t y above d i s s o c i a t i o n . For example, the longest lived state i n Figure 3 i s the (4,4), where (0,0) denotes the l o c a l mode ground state. Such doubly excited states are familiar i n nuclear physics as com pound state, or Feshbach, resonances, and i n atomic spectroscopy as autoionizing, or Auger, states of atoms and molecules (33). Just as i n the atomic case (33,34), sequences of states with quantum numbers of the form (n,n+m) do not necessarily have shorter l i f e times as a function of increasing m, i n contradistinction to s t a t i s t i c a l expectations. This follows from the increase i n period of the l o c a l bond modes as d i s s o c i a t i o n i s neared, and the detuning of any near frequency resonances as m increases i n a sequence of (n,n+m) states. 5
6
Approximate Quantum Studies: The Zeroth Order Golden Rule. De creasing the value of n below ~2.4 i s prohibitive i f a l l complex eigenvalues are to be simultaneously determined, as has been done in the above studies. It i s of course possible to determine i n dividual complex eigenvalues of much larger complex symmetric matrices using inverse i t e r a t i o n (35), or, larger matrices might be treated using an altogether different type of algorithm ( i . e . ,
In Resonances; Truhlar, D.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
RESONANCES
328
t h a t of Ref. ( 3 6 ) ) r a t h e r than the complex symmetric v e r s i o n of Givens used h e r e . However, i t seems p r e f e r a b l e to use the n = 2.4 c a l c u l a t i o n s as benchmarks to t e s t s i m p l e approximate methods, r a t h e r than to d e v e l o p d i r e c t methods f o r working w i t h the v e r y l a r g e complex m a t r i c e s which would a r i s e i n systems of more than two degrees of freedom. Rosen (23) i n a v e r y e a r l y study o f c o u p l e d Morse systems suggested use of a s i m p l i f i e d Golden R u l e , which we now e x p l o r e . G i v e n d o u b l y e x c i t e d s t a t e s , | i j > , of an u n c o u p l e d Morse system l y i n g above d i s s o c i a t i o n , decay i s induced by the p e r t u r b a t i o n p
p
l 2 m
(3)
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2
w i t h Che p a r t i a l w i d t h i n t o the bound ( £ ) , f r e e (k) channel |fck> b e i n g g i v e n by u n c o u p l e d Golden Rule e x p r e s s i o n
r
k>
( i j ) " T T I l< |pil*>
