Numerical Evaluation of Barrier Penetration and Resonance Effects on Phase Shifts

Abstract

Quantum mechanical c a l c u l a t i o n s , based on t h e Lennard-Jones (12,6) p o t e n t i a l , a r e presented showing t h e dependence of t h e reduced phase s h i f t , 7* , on t h e quantum parameter, h * , f o r a f i x e d reduced e f f e c t i v e p o t e n t i a l and v a r i o u s reduced e n e r g i e s , E". The observed o s c i l l a t o r y behavior of ?*[.') i s due p r i m a r i l y t o t h e i n c l u s i o n of t h e p h y s i c a l l y unimportant c o n t r i b u t i o n of wv t o t h e phase s h i f t , where i s t h e number of quasi-bound ( v i r t u a l ) s t a t e sof energy l e s s than E*. A modified reduced phase s h i f t , ?* 9 d e f i n e d by e x c l u d i n g t h i s c o n t r i b u t i o n , d i s p l a y s only t h e s h a r p i n f l e c t i o n s a s s o c i a t e d w i t h b a r r i e r p e n e t r a t i o n under resonance c o n d i t i o n s . Except f o r t h e resonance c o n t r i b u t i o n , the phase s h i f t s may be a c c u r a t e l y reproduced by a second-order JWKB procedure. T h i s method a l s o a c c u r a t e l y p r e d i c t s t h e r e s o n a n t e n e r g i e s ( i . e . t h e e n e r g i e s of t h e quasi-bound s t a t e s ) . The f i r s t -o r d e r JWKB treatmentof t h e b a r r i e r p e n e t r a t i o n problem by Ford, H i l l , Wakano, and Wheeler s u f f i c e s f o r t h e purpose of e s t i m a t i n g t h e l e v e l widths and l i f e t i m e s of t h e v i r t u a l s t a t e s a s w e l l a s t h e main f e a t u r e s of t h e r e s o n a n t phase s h i f t s , b u t does n o t a c c u r a t e l y reproduce t h e q u a n t a l c a l c u l a t i o n s .W T h i s r e s e a r c h r e c e i v e d f i n a n c i a l support from t h e N a t i o n a l Aeronautics fi t t i h f r-----e and Space A d m i n i s t r a t i o n Grant . W P r e s e n t a d d r e s s : N a t i o n a l Bureau of Standard https://ntrs.nasa.gov/search.jsp?R=19670008526 2018-05-09T17:30:18+00:00Z I n t r o d u c t i o n I n t h e p r e s e n t paper, we p r e s e n t t h e r e s u l t s of a numerical s t u d y of phase s h i f t s which a r i s e i n c o l l i s i o n s between molecules which i n t e r a c t according t o a Lennard-Jones (12,6) p o t e n t i a l . W e c o n s i d e r , i n p a r t i c u l a r , t h e behavior of t h e phase s h i f t s under c o n d i t i o n s such t h a t t h r e e c l a s s i c a l t u r n i n g p o i n t s e x i s t . Although t h e q u a n t i t a t i v e r e s u l t s apply t o t h e p r e s e n t model p o t e n t i a l , most of t h e q u a l i t a t i v e f e a t u r e s a r e c h a r a c t e r i s t i c of any r e a l i s t i c i n t e r m o l e c u l a r p o t e n t i a 1.The L. -J. (12,6) p o t e n t i a l f u n c t i o n v m a y be w r i t t e n where c p Y ( v * ) = 4['" -1 2 -p -6 3 i s t h e depth of t h e p o t e n t i a l minimum, Ui s t h e s e p a r a t i o n a t which t h e p o t e n t i a l i s zero, and s e p a r a t i o n . The phase s h i f t s , *lp , are defined by t h e asymptotic form o f t...

Effect of orbiting resonance on the elastic differential cross sections

“…To calculate two of the resonance parameters, namely the resonance energy and width, we used the quantum mechanical method [ 5 ] . The resonance energy was obtained from the position of the inflection point of the phase shifts vs. energy curve and the resonance width was obtained from the slope of the curve at the resonance energy.…”

On the phase shift for orbiting collisions

“…However, other first-order semi-classical phase shifts derived for this problem [3][4][5][6][7][8] have shown no such pairing effect (nor have calculations involving numerical integration of the radial wave equation [9]). It is shown below that whilst Livingston's expression for the phase shift is essentially correct, the pairing effect arises from an incorrect specification of the resonance conditions in [2].…”

Diatomic predissociation linewidths

“…Advances have been made by both computational [14][15][16][17][18][19][20][21] and semi-classical methods [13,. Some computational workers have used the resonant jump in the scattering phase shift [14][15][16][17], or the closely associated peak in the collision delay time (see below) [20,[33][34][35] to determine the positions and energy widths of the quasi-bound states.…”

“…Advances have been made by both computational [14][15][16][17][18][19][20][21] and semi-classical methods [13,. Some computational workers have used the resonant jump in the scattering phase shift [14][15][16][17], or the closely associated peak in the collision delay time (see below) [20,[33][34][35] to determine the positions and energy widths of the quasi-bound states. Others have pointed to the spectroscopically important peak in the relative amplitude of the wavefunction inside and outside the barrier region [14,[18][19][20]; however the intimate connection [12] between this characteristic and the behaviour of the phase shift has received relatively little emphasis in the molecular literature.…”

“…where [14,15,17,19,20,22,23] iHD and D 2 [20], OH [19] HeH + , 3HeH , 4HeD , 3HeH [5], and ArK and RbK [31] systems and for a standard model based on the Lennard-Jones potential [16,21,29].…”