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MOLECULARPHYSICS,1983, VOL. 50, No. 5, 1025-1043

Qu an tu m d y n am ics o f n on-rig id sy s t em s com p r i s in g tw op o lya tom ic f ragm en ts

by G. BROCKS and A. VAN DER AVOIRD

Institute of Theoretical Chemistry, University of Nijmegen,

Toernooiveld, Nijmegen, The Netherlands

B. T. SUTCLIFFE

Chemistry Department, University of York, Heslington,

York Y01 5DD, England

and J. TENNYSON

SERC Daresbury Laboratory, Daresbury,

Warrington WA4 4AD, Cheshire, England

(Received 26 M a y 1983 ; accepted 4 f fu ly 1983)

We combine earlier treatments for the embedding of body-fixedcoordinatesin linear molecules with the close-coupling formalism developed for atom-diatom scattering and derive a hamiltonian which is most convenient fordescribing the nuclear motions in van der Waals complexes and other non-rigidsystems comprising two polyatomic fragments, .4 and B. This hamiltoniancan still be partitioned in the form I~A + I2IB+ I ~ I N T , just as the space-fixedhamiltonian. The body-fixed form, however, has several advantages. Wediscuss solution strategies for the rovibrational problem in non-rigid dimers,based on this partitioning of the hamiltonian. Finally, in view of thesize of the general polyatomic-polyatomic case, we suggest problems whichshould be currently practicable.

1. INTRODUCTION

The search for suitable coordinate systems and hamiltonians for the poly-

atomic vibration-rotation problem has excited much scientific interest over the

years [1-12]. For near-rigid systems, with localized or small amplit ude vibra-

tions, the use of Eckart coordinates has proved frui tfu l [1-5, 13, 14]. However,

for one or more large amplitude internal motions the simple Eckart hamiltonian

is unsatisfactory. Thi s feature was recognized very early by Sayvetz [2] who

modified the original approach of Eckart [1] to try and deal with the difficulties

inherent in treating large amplitude internal motions.

The problems arise because the transformation that leads from the hamil-

tonian in the laboratory-fixed coordinate system to the internal hamiltonian,

which is expressed in some suitably chosen body-fixed coordinate system, has a

jacobian which is singular for some values of the coordinates. Thus the trans-

formed internal hamiltonian is not everywhere well defined and, consequently,

has a domain that is more restricted than all square-intcgrablc functions of the

body-fixed coordinates. The difficulties associated with this restriction of the

domain usually manifest themselves in divergent expectation values of the internal

hamilton ian between seemingly reasonable functions. Thus it is well known



1 0 2 6 G . B r o c k s e t a l .

[ 1 5 - 1 8 ] t h a t i f a n a t t e m p t i s m a d e t o t r e a t a t r i a t o m i c t h a t h a s a l a rg e a m p l i t u d e

b e n d i n g m o d e , i n t h e E c k a r t a p p r o a c h , t h e n d i v e r g e n c e c a n o c c u r i n e x p e c t a t i o n

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

T h i s s o r t o f p r o b l e m c a n b e s o l v e d b y u s i n g t h e S a y v e t z m o d i f i c a t i o n o f t h eE c k a r t a p p r o a c h a n d s u c h a w a y o u t h a s b e e n a t t e m p t e d [ 1 9 - 2 2 ] . I t ca n a l so

b e s o l v e d b y a b a n d o n i n g t h e E c k a r t a p p r o a c h a l t o g et h e r , b a s e d a s i t i s o n t h e i d e a

o f a n e q u i l i b r i u m g e o m e t r y f o r t h e s y s t e m , a n d v a r i o u s d i f f e re n t f o r m u l a t i o n s

h a v e b e e n g i v e n [ 6 - 1 0 , 2 3 - 2 6 ] . H o w e v e r , b e c a u s e o f t h e c o m p l e x i t y o f t h e

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

a f e w s y m m e t r i c t e t r a - a t o m i c ( A B ) ~ v a n d e r W a a l s c o m p l e x e s [6 , 2 5 - 2 7 ] . W i t h

t h e e x c e p t i o n o f a r e c e n t c a l c u l a t i o n o n H e H F [ 28 ], a ll th e s e c a l c u l a t io n s h a v e

i n c l u d e d s o m e s im p l i f y i n g a p p r o x i m a t i o n i n v o l v i n g t h e d e c o u p l i n g o r f r e e z in g

o f c e rt a i n v i b r a t io n a l m o d e s .

A p o p u la r a p p r o a c h , a p p l i e d t o s e v e r a l v a n d e r W a a l s c o mp le x e s [ 6, 7 , 1 0,

2 3 - 2 8 ] a s w e l l a s H 2 0 [ 2 9] , K C N [ 10 , 1 6] , L i C N [ 3 0] a n d C H 2 + [ 3 1] , h a s b e e nt o w o r k i n s o - c a l le d ( d i ) a t o m - d i a t o m c o l li s io n c o o r d i n a t e s . T h e s e a r e d e f i n e d

a s t h e d i s t a n c e b e t w e e n t h e m o n o m e r c e n t r e s o f m a s s , t h e a n g le s d e s c r i b i n g th e

o r i e n t a t io n s o f t h e m o n o m e r s , t h e i n t er n a l m o n o m e r c o o r d i n a t e s a n d t h e o ve r a ll

r o t a t i o n a n g le s . S e v e r a l h a m i l t o n i a n s h a v e b e e n u s e d t o o b t a i n th e b o u n d

r o v i b r a t i o n a l s t a t e s o f s y s t e m s t r e a t e d a s c o l l i s i o n c o m p l e x e s u s i n g a p p r o a c h e s

w h i c h s h o w s t r o n g a n a l o g ie s t o a t o m - [ 3 2] , d i a t o m - [ 33 ] a n d e l e c t r o n - [ 3 4]

d i a t o m s c a t t e r in g p r o b l e m s . T h e m a i n d i f f e r e n c e b e t w e e n th e v a ri o u s

h a m i l to n i a n s i s t h e e m b e d d i n g o f t h e co o r d i n a te s y s te m . F o r a t o m - d i a t o m

s y s t e m s a l o ne , s p a c e f ix e d c o o r d i n a t e s [ 7] a n d a t l e a s t t h r e e d i f f e r e n t e m b e d d i n g s

o f b o d y - f i x e d c o o r d i n a t e s [ 8, 1 0, 3 0 ] h a v e b e e n u s e d .

I n r e c e n t w o r k w e h a v e f a v o u r e d t h e u s e o f c o o r d i n a t e s w h i c h h a v e R , t h ei n t e r a c ti o n o r c o l li s io n c o o r d i n a t e , e m b e d d e d a l o n g t h e z - a x is . T h i s s y s t e m h a s

c o m p u t a t i o n a l a d v a n t a g e s o v e r s p a c e - f i x e d c o o r d i n a t e s a n d a l l o w s t h e s i m p l i f y i n g

a p p r o x i m a t i o n o f n e g l e c t o f o f f - d i a g o n a l C o r io l i s in t e r a c ti o n s , w h i c h h a s p r o v e d

u s e f u l i n m a n y c a l c u l a ti o n s [1 0 , 2 6 , 3 0 , 3 5 ]. T h i s e m b e d d i n g i s f a v o u r e d o v e r

o n e w h i c h f i x e s t h e a x i s (e s ) in o n e f r a g m e n t o f t h e c o m p l e x , e l . I s t o m i n e t a l . [8] ,

as i t i s eas ie r to genera l ize .

F i x i n g R a lo n g t h e z - a x i s i s n o t s u f f i c i e n t f u l ly t o b o d y - f i x t h e c o o r d i n a t e s a s

i t o n l y d e f i n es t w o o f t h r e e p o s s ib l e e m b e d d i n g E u l e r a n g le s . A fu l l y e m b e d d e d

h a m i l t o n ia n f o r t h e a t o m - d i a t o m p r o b l e m w a s r e c e n tl y g i v e n b y T e n n y s o n a n d

S u t c l i f f e [ 1 0 ] , b u t a s w e s h o w ( i n A p p e n d i x A ) t h i s i s n o t c o n v e n i e n t i n l a r g e r

s y s t e m s . H o w e v e r , as s h o w n b y T e n n y s o n a n d S u t c l i ff e , t h e r e is a s t r o n g

e q u i v a le n c e b e t w e e n t h e m a t ri x p r o b l e m s g e n e r a t e d b y t h e t w o e m b e d d i n g s .

I n t h i s p a p e r , w e p r e s e n t a g e n e r a l h a m i l t o n i a n f o r t h e r o v i b r a t i o n a l ( a n d

s c a t t e r in g ) p r o b l e m o f a s y s t e m A B , w h e r e A a n d B a r e p o l y a t o m i c m o l e c u l e s .

T h i s h a m i l t o n i a n c a n b e e x p r e s s e d i n t e r m s o f f r a g m e n t h a m i l t o n ia n s a n d a n

i n t e r a c t i o n h a m i l t o n i a n

/ t = / } a + ~ B + / l i n T (1 )

a n d i t i s d e r i v e d i n s u c h a m a n n e r t h a t / t A a n d / QB a r e t h e r o v i b r a t i o n a l h a m i l -

t o n i a n s o f t h e i s o l a te d f r a g m e n t s f o r w h i c h c o n v e n t i o n a l E c k a r t h a m i l t o n i a n s

[ 1 - 5 ] a r e a p p r o p r i a t e . O u r h a m i l t o n ia n i s s u i t a b l e f o r a n y s y s t e m w h i c h

c o n t a i n s t w o ( n e a r ) r i gi d f r a g m e n t s u n d e r g o i n g la r g e a m p l i t u d e m o t i o n s . T h u s

i t i s p a r t i c u l a r l y c o n v e n i e n t f o r v a n d e r W a a l s d i m e r s .



Q u a n t u m d y n a m i c s o [ v a n d e r W a a l s d im e r s 1027

B y e m b e d d i n g t h e h a m i l t o n i a n i n t h i s m a n n e r , w e fo l l o w s e v e ra l ea r li e r w o r k s

o n a t o m - d i a t o m a n d d i a t o m - d i a t o m s y s t em s . O u r h a m i l to n i a n e x te m d s t o

l a rg e r ( p o l y a t o m i c ) m o l e c u l e s , h o w e v e r , a n d , m o r e o v e r , w e h o p e t o c l a r if y t h e

s i t u a t i o n w i t h r e g a r d t o t h e f o r m a n d c o m m u t a t i o n r e l a t io n s h i p s o f t h e r e s u l t in gp s e u d o - a n g u l a r m o m e n t u m o p e r a t o rs ( § 2 ) . I n § § 3 a n d 4 w e d i s cu s s

c o n v e n i e n t a n d p r a c ti c a l s o l u t i o n s tr a t e g ie s f o r t h e p r o b l e m .

A n a p p r o a c h t o t h e s c a t t er i n g o f tw o p o l y a t o m i c m o l e c u l e s w h i c h i s si m i l a r to

t h e p r e s e n t o n e h a s b e e n p r o p o s e d i n 1 9 53 b y C u r t i s s [ 3 6 ], w h o b a s e d h i m s e l f o n

t h e p i o n e e r in g w o r k b y H i r s c h f e l d e r e t a l . [ 3 7, 3 8 ] . C u r t i s s s t a r t s o u t i n b o d y -

f i x ed c o o r d i n a t e s ( d e f i n e d b y t h r e e e m b e d d i n g a n g le s ). H o w e v e r , b y r o t a ti n g

a n d r e c o u p l i n g h is b a s i s f u n c t i o n s h e a c t u a ll y d e r i v e s t h e c l o s e - c o u p l e d e q u a t i o n s

i n s p a c e - f i x e d c o o r d i n a t e s.

2 . H A M IL T O N IA NW 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 x im a t i o n , t h e h am i l t o ni a n f o r n u c l e ar

m o t i o n i s

h 2

1 Vi2(xi ) + V, (2)=i= 1

wh e r e V i2 ( x i) i s t h e l a p l a c i an f o r t h e i t h n u c l e u s a n d V th e p o t e n t i a l . I f t h e

N - n u c l e u s s y s t e m i s d i v i d e d i n t o f ra g m e n t A w i t h N a n u c l e i a n d f r a g m e n t B

wi th N B n u c l e i , t h e n t h e s p a c e ( o r l a b o r a t o r y ) f i x e d c o o r d i n a t e s c a n b e o r d e r e d

. . . . , i . }i , i = l , 2 N ~ , N A + I , . .

(3 )

1 , . . . , N B .

T h e o v e r a ll t ra n s l a ti o n a l m o t i o n i s r e m o v e d a n d t h e r e la t iv e t r a n s la t i o n o f t h e

t w o m o n o m e r s A a n d B i s d e f i n e d b y t h e f o l l o w i n g t r a n s f o r m a t i o n [ 3 6]

t i a = x i - X A , i = 1 . .. , N a - 1 ]

l

t i B = X N A + i - XB, i = 1 . . . , N B - - 1

( 4 )

t o = X A - XB,

X = M - I ( M a X a + M B X B ) ,

w i t h

N ,i N .4 1XA=MA -1 E mixi, M a = ~ " m i ,

i=1 i=1N N

XB -~- MB -1 E mixi, MB = ~ mi'i=N,~+ i=NA+I

M = M a + M B ,

( 5 )

wh e r e x i d e n o te s a v e c to r o f c o o r d in a t e s i n t h e o r ig in al f r a m e a n d t i in t h e n e w

f r a m e .



1 0 2 8 G . B ro c k s e t a l .

U s e o f t h e c h a i n r u l e g i v e s

N 1E - - V ~ (x i) = M - t V ~ (X ) + /~ -1 V ~ (to )

i = l m s

+ EF = A , B

w h e r e t h e t e r m s i n t i F v a n i s h i f N F < 2 a n d

G i j F = ~ i i m g - 1 - - M F - 1 ,

t~ -1 = M A - 1 + M B - 1 .

NF-I

Ei , j = l

G i i F V ( t iF ) . V ( t iF ) , ( 6 )

( F = A , B ) , }

(7 )

T h e f i r s t t e r m i n ( 6) , M - 1 V 2 ( X ) , c o r r e s p o n d s t o t h e f r e e tr a n s l a ti o n o f t h e c e n t r e

o f m a s s a n d c a n b e s e p a r a t e d o ff . T h e r e m a i n i n g t e r m s f o r m th e k in e t i c e n e r g y

o p e r a t o r o f t h e r o v i b r a t io n a l h a m i l t o n i a n o f t h e A B s y s t e m , e x p r e s s e d i n ac o o r d i n a t e f r a m e w h i c h i s b o d y - f i x e d w i t h r e s p e c t t o t ra n s l a ti o n s , b u t s t il l p ar a ll e l

t o t h e l a b o r a t o r y f r a m e . T h i s f r a m e is u s u a l l y c a ll e d s p a c e - f ix e d .

I f t h e s e p a r a t i o n o f t h e m o n o m e r c e n t r e s o f m a s s is n o t d o n e f o r a l l p a r t ic l e s

in o n e s t e p , a s i n (4 ) , b u t p a i rw i s e i n ( N F - - 1 ) s t e p s f o r e a c h m o n o m e r F , w h i c h

d e f i n e s t h e s o - c a l l e d J a c o b i c o o r d i n a t e s t i , t h e n t h e m a t r i x G b e c o m e s d i a g o n a l

G ij F = ~,~jl~i 1 , (F = .4 , B ) , (7 ' )

w h e r e / ~ i i s t h e m a s s o f t h e i t h r e d u c e d p a r ti c le .

N e x t , w e w i s h t o s e p a r a t e o f f t h e o v e r a l l r o ta t i o n s o f t h e s y s t e m b y d e f i n i n g

a b o d y - f i x e d c o o r d i n a t e fr a m e . T h e q u e s t i o n i s h o w t o f i x t h i s f r a m e o n a n o n -

r i g i d s y s t e m , w h e r e t h e e q u i l i b r i u m s t r u c t u r e , w h i c h c o u l d b e u s e d t o d e f i n e t h eE c k a r t e m b e d d i n g c o n d i t i o n s [1 ], m a y n o t b e m e a n i n g f u l . I t s e e m s n a t u r a l t o

s i ng l e o u t t h e v e c t o r t 0 = R w h i c h c o n n e c t s t h e c e n t r e s o f m a s s o f t h e t w o

f r a g m e n t s .4 a n d B a n d t o e m b e d t h e b o d y - f i x e d f r a m e w i t h t h e z - a x is a lo n g R

( f o ll o w i n g t h e P a c k a n d H i r s c h f e l d e r [ 3 4] tr e a t m e n t o f d i a t o m i c s y s t e m s ) .

I f R h a s t h e p o la r a n g le s ( f i , a ) w i th r e s p e c t t o t h e s p a c e - f ix e d s y s t e m , t h i s

e m b e d d i n g i s a c h i e v e d b y t h e o r th o g o n a l t r a n s f o r m a t i o n

( i )0 = ¢ ~ t o = , ( 8 a )

z i F = C r t i E , ( F = A , B ) , (8 b )

w i t h t h e m a t r i x E , t h a t c o r r e s p o n d s t o a r o t a t i o n o v e r t w o E u l e r a n g l e s , a a n d f l,

g i v e n b y

/ c o s f i c o s a - s i n a s i n f i c o s a \

C = / [ c o s f i s i n ~ c o s ~ s in f i s i n a ) . ( 9 )

\ - s i n f l 0 c o s] 3

I t i s p o s s i b l e to d e f i n e a t h i rd e m b e d d i n g a n g l e Y b u t , a s w e s h o w i n A p p e n d i x A ,

th i s l e a d s t o a l e s s c o n v e n ie n t f o rm o f t h e h a m i l to n i a n .



Q u a n t u m d y n a m i c s o [ v a n d e r W a a l s d i m e r s 1 0 2 9

W e h a v e t o c o n s i d e r t h e e f f e c t o f t h e r o t a t i o n ¢ o n t h e h a m i l t o n i a n , i n

p a r t i c u l a r o n t h e k i n e t i c e n e r g y o p e r a t o r . I n o r d e r to e x p r e s s t h e d i f f e r e n t ia l

o p e r a t o r s o f th e o l d c o o r d i n a t e s ( 4 ) i n t h e n e w o n e s , w e h a v e t o u s e t h e c h a i n r u l e.

I n a p p l y i n g th i s r u le , h o w e v e r , i t m u s t b e r e m e m b e r e d t h a t th e e m b e d d i n g c o n d i -t io n ( 8 ) m a k e s e a c h o f t h e n e w c o o r d i n a t e s z f a f u n c t i o n o f t 0 , b e c a u s e t h e m a t r ix

/3 w h i c h o c c u r i n t o = R = ( R s i n / 3 c o s ~ ,d e p e n d s o n t h e an g l es a a n d

R s i n 13 s i n ~ , R c o s / 3 ) . T h u s

O O R O 0 ~ O

a t o , ~ - O t o , ~ O R l - a -~ o ,~ ~ - I - - - - -

a n d w e c a n s u b s t i t u t e

a n d

w i t h

0 /3 b N , - , O z i ,~ O ( 1 0 )

Oto,,O/3 + ~ i~=, ~ O to. , O zi, , F

O ROto,~ C~z ,

0~Oto,----~= ( R s i n / 3 ) - ' C~u

o / 3- R - 1 C ~ x

ato,~

( 1 1 )

Ozi, yF OC~y • _ OC ~Oto.~ = ~ - ~ o , t i , , " = g~" - ~ o j C g , z i .p F , ( 1 2 )

- - - F - ( 1 3 )0 to , ~ aa 0 to ,g 0/3 ato, g"

T h e r e s u l ts o f t h e s e s u b s t i t u t io n s c a n b e w r i t t e n m o r e s i m p l y i n t e r m s o f t h e

f o l lo w i n g a n g u l a r m o m e n t u m o p e r a t o r s a n d t h e ir t r a n s f o r m a t i o n p r o p e r t i e s

(~a~,v i s t h e L e v i - C i v i t a a n t i s y m m e t r i c t e n s o r ; A ,/~ , v = x , y o r z )

h O; a ( t i F ) = ~ - ~ ' , ~ "a.vti.#F Oti , F , ( 1 4 a )

N~-t

i F = Z l ( t iF ) , ( 1 4 b )i = l

i = E i r = i ~ + i ~ , ( 1 4 c )F

h 0 h ~ R 0

la = ~" ~ Ea ""t ° '" ~ t o ~ , ~ R ~( 1 4 d )

a n d t h e to ta l an g u la r m o m e n t u m

J = l + j = l - l - - j a + j B . ( 1 4 e )

T h e t r a n s f o r m a t i o n o f t h e o p e r a t o r s j ( t i F ) , j R a n d j i s e a s y b e c a u s e C i s n o t a

f u n c t i o n o f t h e c o o r d i n a t e s t i F a n d w e c a n w r i t e , u s i n g ( 8 )

0 O Z i , y F 0

O t i j F = ~ O t i,g ~ Z i , ~ F

a= Y ~ C ~ . ( 1 5 )

OZi ,~IF "7



1030 G. Brock s e t a l .

U s i n g t h i s r e s u l t a n d t h e t r a n s f o r m a t i o n p r o p e r t i e s o f t h e L e v i - C i v i t a t e n s o r ,

th i s y ie lds

- - - - E 1 4 "

h ~,~ ~ (16 )= - 'z ~ C a ~ , E ~ Z i '~ F ~ Z i,~ '

w h i c h s h o w s t h a t t h e c o m p o n e n t s o f j t r a n s f o r m a s t h e c o m p o n e n t s o f a v e c t o r

j ( t iF ) = Ci (z iF) , (17)

w i t h t h e s a m e e x p r e s s i o n s ( 1 4 a - c ) f o r | h o l d i n g i n t h e b o d y - f i x e d c o o r d i n a t e s

Zi F .

T h e t r a n s f o r m a t i o n o f | , e q u a t i o n ( 14 d ) , i s m o r e c o m p l i c a t e d b e c a u s e o n em u s t u s e t h e c h a i n r u l e e x p r e s s i o n s ( 1 0 ) t o ( 1 3 ) .

F i r s t , w e s u b s t i t u t e (1 1 ) i n t o ( 1 3 ) a n d r e w r i t e ( 1 0 ) as

e =R_IC ~ x ~ _ ~ j y + R _ ~ C ~ " c o s e c f l~ _ ~ a _ c o t f l ~ ]~ + ~ L~ t o , ~ -~ f l

+ C~, ~-~, (18 )

w i t h t h e a n gu l a r m o m e n t u m o p e r a to r j = j A + j B e x p r e ss e d i n b o d y - f i x e d

c o o r d i n a t e s , i F . T h e n , u s i n g ( 8 a ) , i t i s s t r a i g h t f o r w a r d t o d e r i v e t h e a n g u l a r

m o m e n t u m o p e r a t o r | , ( 1 4 d ) , a ls o i n b o d y - f i x e d c o o r d i n a te s . I n s t e a d , w e w r i t et h e r e s u l t f o r t h e t o ta l a n g u l a r m o m e n t u m ) = | + j , w h i c h i s s l ig h t l y s i m p l e r .

J u s t a s j , s e e ( 1 7 ), 3 t r a n s f o r m s a s

3( t i ) = C3 (z i ) , (19)

w i t h t h e c o m p o n e n t s o f 3 i n t h e b o d y - f i x e d c o o r d i n a t e s Zi F, O~and f l , g iven by

J~ = - _ cosec + co t f l j ~ ,• z ~

j r = h ~ ji @ '

J = L .

( 2 0 )

I d a , d o ] = [ d y , d , ] = 0 , ]

[ J x ,J u j = h c o t l f i x + ] , l (21)

T h e f o r m o f t h i s o p e r a t o r i s u n u s u a l , i t i s d i f fe r e n t f r o m t h e m o r e f a m i l ia r

e x p r e s s i o n f o r J i n a b o d y - f i x e d f r a m e [ 34 ] w h i c h is e m b e d d e d b y t h r e e E u l e r

a n g le s , ~ , fl a n d 7 , r a t h e r t h a n j u s t t w o . W e o b s e r v e t h a t i ts c o m p o n e n t s d o n o t

e v e n s a t i s f y a n g u l a r m o m e n t u m c o m m u t a t i o n r e l a t i o n s



Q u a n t u m d y n a m i c s o [ v a n d e r W a a l s d im e r s 1031

s o t h a t w e m u s t ca ll d a p s e u d o - a n g u l a r m o m e n t u m o p e r a t o r . A l s o , t h e c o m -

p o n e n t s o f 3 d o n o t c o m m u t e w i t h j

[ ' ]x , ]a] = - - c ot f l % a j ~ ,

[ J u ' £ ] = 0 , ( 22 )

h j7 e~aaJa w ith A, ~ = x , y , z .

W a t s o n [ 5 ] , i n h i s i somorph ic h a m i l t o n i a n f o r l i n e a r mo le c u l e s , r e s to r e s t h e u s u a l

b o d y - f i x e d e x p r e s s i o n s f o r d b y i n t r o d u c i n g a n a r ti fi c ia l t h i r d r o t a t i o n a n g l e y .

I n t h e n e x t s e c t i o n w e s h o w t h a t t h e a c t io n o f o u r d o n a s u i ta b l y c h o s e n b a s i s is

a c t u a ll y q u i t e s i m p l e , h o w e v e r , s o th a t t h e r e i s n o n e e d t o i n v o k e t h i s e x t r a n e o u s

a n gl e. I n A p p e n d i x A w e d e m o n s t r a t e t h at e m b e d d i n g w i t h t h r e e ( p h y s ic a l ly

d e f i n e d , r a t h e r t h a n e x t r a n e o u s ) E u l e r a n g l e s o n l y l e a d s t o a f o r m a l i s m w h i c h i s

l e ss tr a n s p a r e n t a n d m o r e d i f f i c u lt to a p p l y .

Af t e r a l l t h i s p r e l im in a r y wo r k i t i s n o t t o o t e d io u s t o wr i t e t h e k in e t i c e n e r g y

o p e r a to r ( 6 ) i n b o d y - f ix e d c o o r d in a t e s , = , f l , R , l i r "~ w e o n l y h a v e t o s u b s t i t u t e

( 1 5 ) f o r t h e d e r iv a t i v e s V(ti F) a n d (1 8 ) f o r V( t0 ) . T h e d e r iv a t i v e s V(t iF )c o m m u t e w i t h t h e m a t r i x C a n d s o w e e a s il y f i n d ( u s i n g C T C = 1) t h a t t h e t e r m s

h 2 Nv- ~

- -- -2 G ij F V ( zi F ) V ( z j F) , ( F = A , B ) ( 2 3)/~ F = i,i~=l

a r e f o r m - i n v a r i a n t . I f N F < 2 t h e s e t e r m s v a n i s h .

S u b s t i t u t i n g ( 1 8 ) i n t o V 2 ( t 0 ) w e m u s t r e m e m b e r t h a t C d e p e n d s o n = a n d f l( b u t i t c o m m u t e s w i t h j ) a n d t h e r e s u l t b e c o m e s

h 2/~L~T = - ~ V2(t0)

h ~ ~ ~ 1 r- 2 ~ R ~ ~ -R R 2 - ~ + 2 - - - - ~~ l ( Y = - J = ) 2 + ( J u - J Y ) 2

h - ] ~ ) ] ( 24 )+ - c o t f l ( J~l

T h i s c a n s t il l b e s i m p l i f i e d s o m e w h a t f u r th e r . W e n o t e , a f t e r s u b s t i t u t i n g ( 1 9 )

a n d ( 2 0 ) , th a t a l s o t h e e x p r e s s i o n f o r j 2 h a s a n u n u s u a l f o r m i n th e b o d y - f i x e df r a m e

j 2 ( t , ) = c 3 ( z , ) , c , ] ( , , ) = J 2 + J 2 + c ( 3 • , ]

h= J : : + J ~ ': + J : : + 7 c o t f l J u

= j = 2 + c os ec f l J u s in f lJ u + j 2 . ( 2 5 )

I f w e u s e t h i s f o r m , i n c o m b i n a t i o n w i t h t h e r e s u l t t h a t ,]~ = j~ a n d t he c o m m u t a -

t i o n r e l a t i o n s ( 2 2 ) , we o b t a in

h ~ ~ R 2 ~ 1g IN W -- 2 # R 2 ~ n ~ ' R + 2 ~ [ J 2 + / 2 - 2 j " J ] (2 6)

w i t h j 2 g i v e n b y ( 2 5 ) .



1 0 3 2 G . B r o c k s e t a l .

S o f a r w e h a v e o n l y a n a l y s e d t h e k i n e t ic e n e r g y p a r t o f t h e h a m i l t o n i a n ( 2 ).

T h e p o t e n t i a l i s, o f c o u r s e , n o t i n g e n e r a l s e p a r a b l e , b u t i t i s a lwa y s p o s s ib l e t o

d e f i n e a n i n t e r a c t i o n p o t e n t i a l

V I N T ( R , z i ~ , z i B ) = V ( R , z : 4 , z i B) - Va (z i a ) - VB(z iB) , (27)

w h e r e V F ( z i F ) i s t h e p o t e n t ia l f u n c t i o n o f t h e is o l a te d f r a g m e n t F . E q u a t i o n

( 2 7 ) i s i n t h e s p i ri t o f t h e S o r b i e - M u r r e l l a p p r o a c h t o p o t e n t i a l e n e r g y s u r f a c e

f i t ti n g [ 3 9] . T h i s a l l o w s t h e h a m i l t o n i a n t o b e w r i t t e n i n t h e f o r m o f ( 1 ) w i t h

/ ~ F = / ~ F + V F , ( F = A , B , I N T ) . ( 2 8 )

T h i s f o r m o f t h e h a m i l t o n i a n h a s b e e n g i v e n f o r t h e a t o m - d i a t o m c a s e

( N A = 2 , N B = 1 ). H o w e v e r , li tt le m e n t i o n h a s b e e n m a d e o f t h e u n u s u a l f o r m

o f t h e p s e u d o - a n g u l a r m o m e n t u m o p e r a t o rs J a n d ]2 , a s g i v e n b y ( 2 0 ) - ( 2 2 ) a n d

( 2 5 ). I n d e e d , t h e c o m m u t a t i o n r e l a t io n s h a v e o f t e n b e e n i g n o r e d le a d i n g t o

t h e l o o s e e x p r e s s io n ( j _ ] ) 2 [ 2 0 , 3 5 ] o r e v e n t h e i n c o r r e c t o n e ( d 2 + f l - 2 J . j ) ,( 2 9 ) a n d ( 3 0 ) o f [ 24 ] . T h e l a t t er e x p r e s s i o n b e c o m e s f o r m a l l y c o r re c t , h o w e v e r ,

i f o n e i n t r o d u c e s a n e x t r a n e o u s t h i r d r o t a t i o n a n g l e , as W a t s o n [ 5] d o e s i n h i s

i s o m o r p h i c h a m i l t o n i a n f o r l in e a r m o l e c u l e s:

E x p r e s s i o n s s i m i l a r t o o u r s w e r e g i v en b y I s t o m i n e t a l . [ 8] fo r th e a t o m - d i a t o m

c a se w i t h t h e d i a to m b o n d v e c t o r e m b e d d e d a lo n g t h e z - ax i s. T e n n y s o n a n d

v a n d e r A v o i r d u s e d a h a m i l t o n i a n w i t h t h e f o r m o f / ~I nT f o r th e i r d i a t o m -

d i a t o m c a l c u l a t io n s o n t h e v a n d e r W a a l s d i m e r o f n i t r o g e n [ 2 6 ]. H o u g e n [ 40 ]

a n d H o w a r d a n d M o s s [ 4 1 ] d e r i v e d e q u i v a l e n t e x p r e s s i o n s f o r l i n e a r m o l e c u l e s

b y a p p l y i n g t h e P o d o l s k y t r a n s f o r m a t i o n t o t h e c l a ss ic a l e x p r e s s io n .

3 . MATRIXELEMENTS

I n t h i s s e c t i o n w e d i s c u s s a s u i t a b l e s o l u t i o n s t r a t e g y f o r t h e ( t w o - a n g l e )

b o d y - f i x e d h a m i l t o n i a n j u s t d e r i v e d . W e s u g g e s t b a s i s f u n c t i o n s w h i c h r e fl e c t t h e

p a r t i ti o n i n g o f t h i s f o r m o f t h e h a m i l t o n i a n . W e s t a rt b y l o o k i n g a t t h e

r o v i b ra t i on a l w a v e f u n c t io n s o f t h e m o n o m e r s A a n d B . I f t h e s e m o n o m e r s a re

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

h a m i l t o n i a n s 1-17 = I ~ F + ~ 'F( l i , F ) i n t h e W a t s o n f o r m [ 4 ] a n d t o a s s u m e t h a t t h e

v i b r a t io n a l a n d r o ta t i o n a l p r o b l e m s c a n b e so l v e d s e p a r a t el y . A p ra c t ic a l w a y

t o d o th i s fo r t r ia t o m i c m o l e c u le s ha s b e e n p r o p o s e d b y W h i t e h e a d a n d H a n d y

[ 1 3] a n d d e s c r i b e d a ls o b y T e n n y s o n a n d S u t c l i ff e [ 15 ]. I n p r i n c i p l e t h i s m e t h o d

c a n b e g e n e r a l iz e d t o l a r g e r s y s t e m s ; i n p r a c t ic e o n e w i ll h a v e t o m a k e f u r t h e r

a p p r o x i m a t i o n s to k e e p t h e c a l c u la t i o n s t r a c t a b le . T h e m e t h o d a m o u n t s to

a s s u m i n g t h at t h e m o n o m e r r o v i b ra t io n a l w a v e f u n c t i o n s c a n b e w r i t t e n a s

p r o d u c t s q )v ~ ( Q F ) ~ d k F n n F . k v ( J V ) * ( t O F ) ( 2 9 )kp

w h e r e t h e v i b ra t i o n a l w a v e f u n c t i o n s O vF d e p e n d o n t h e in t e r n a l m o n o m e r

c o o r d i n a t e s Q F a n d t h e r o t a t i o n a l f u n c t i o n s a r e li n e a r c o m b i n a t i o n s o f n o r m a l i z e d

r o t a t i o n m a t r i x e le m e n t s in t h e c o n v e n t i o n o f B r i n k a n d S a t c h l e r [4 2 ]. T h e

r o t a t i o n a n g l e s t o F = ( ~ F , O F , ~ b F ) d e s c r i b e t h e o r i e n t a t i o n s o f c o o r d i n a t e f r a m e s

f i x e d o n t h e m o n o m e r s , f o r i n s ta n c e v i a t h e E c k a r t c o n d i t i o n s [1 ] , w i t h r e s p e c t t o

t h e b o d y - f i x e d d i m e r f r a m e i n t r o d u c e d i n § 2 . T h e a s s u m p t i o n o f ( 2 9 ) i s

e q u i v a l e n t t o s t a t i n g t h a t t h e m o n o m e r h a m i l t o n i a n s / 4 F a r e d i a g o n a l i n t h ev i b r a t i o n a l f u n c t i o n s O v a.



Q u a n t u m d y n a m i c s o ~ v a n d e r W a a l s d i m e r s 1033

T a k i n g t h e e x p e c t a t i o n v a l u e s o f /- iF o v e r O rE l e a d s t o e f f e c t i v e ro t a t i o n a l

h a m i l t o n i a n s [4 3 ] :

/ ~ F R O T = [ X ~ v ~ . ( ia F ) 2 .9 v B v F ( J b F ) 2 .-~ C v , ( i v F ) 2 ]

= ½ [(Av, + B , , , . ) j F 2 + ( 2 C ~ , - A , ,, ~ - B , F ) j ~ F 2

+ ½ ( A , , - B , , ,~ ) ( j + F 2 + 1 f2 ) ] . ( 3 0 )

H e r e i t h as b e e n a s s u m e d t h a t t h e v i b r a t i o n a ll y a v e r a g e d i n v e r s e in e r ti a t e n s o r o f

m o n o m e r F ( i n v i b r a t i o n a l s t a t e r E ) h a s b e e n t r a n s f o r m e d t o p r i n c i p a l a x e s a , b

and c ; t he ro t a t iona l con s t an t s A~. , , B~.v and Cv~ are it s p r inc ip a l va lue s . T he

q u a n t u m n u m b e r k F i s t h e c o m p o n e n t o f t h e a ng u la r m o m e n t u m j R o n t h e

m o n o m e r c - ax is , ~ F i t s c o m p o n e n t o n t h e d i m e r z - a x is . T h e c o e f f ic i e n ts d k , :

i n ( 2 9 ) c o u l d b e d e t e r m i n e d f o r t h e f r e e m o n o m e r s b y d i a g o n a l i s i n g t h e

h a m i l t o n i a n s / t F R O T.

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

v i b r a t io n s b y t h e h a r m o n i c o s c i ll a to r m o d e l a n d t h e i r r ig i d b o d y r o t a t io n s b y t h e

h a m i l t o n i a n (3 0 ) . T h e r i g id ro t o r m o d e l i m p l i e s t h a t o n e r e p l a c e s t h e v i b ra -

t i o n a l l y a v e ra g e d ro t a t i o n a l c o n s t a n t s A ~ .~ , B ~ .F a n d C , . ~ b y t h e i r v a l u e s A e ,

B e a n d C e a t th e e q u i l i b r i u m s t r u c t u r e . T h i s a p p r o x i m a t i o n h a s b e e n m a d e

a l m o s t u n i v e rs a l ly , e v e n i n c a l c u la t io n s o n a t o m - d i a t o m s y s t e m s . I t s a c c u r a c y

h a s b e e n d e m o n s t r a t e d f or H e H F b y T e n n y s o n a n d S u t c li ff e [ 28 ]. F o r la r ge r

m o l e c u l e s o r f r a g m e n t s i t w i ll u n d o u b t e d l y b e r e q u i r e d . T h e a p p r o a c h w i th t h e

v i b r a t io n a l l y a v e r a g e d m o n o m e r g e o m e t r ie s h a s b e e n u s e d b y L e R o y a n d V a n

K r a n e n d o n k [ 7] f o r H 2 - r a r e g a s v a n d e r W a a l s d i m e r s i n s e v e ra l H 2 v i b r a ti o n a l

s t a t es .

I f w e a s s u m e t h a t t h e r e i s n o s t r o n g c o u p l i n g b e t w e e n t h e i n t e r n a l m o n o m e rv i b r a t i o n s a n d t h e d i m e r m o d e s , w h i c h i s j u s t i f ie d i f t h e g a p b e t w e e n t h e

f r e q u e n c i e s i s su f f i c i e n t l y la rg e , t h e n w e c a n a l so a v e ra g e t h e i n t e r a c t i o n p o t e n t i a l

o v e r t h e m o n o m e r v i b r a ti o n a l w a v e f u n c t i o n s . T h i s y i e ld s a n e f f e c ti v e p o t e n ti a l

f o r t h e d i m e r r o v i b r a t i o n a l p r o b l e m w i th t h e m o n o m e r s A a n d B i n s ta t e s v A

a n d V B , r e sp e c t i v e l y

VINT(R, ', '~, ~B)

=(%A(QD~B(Q~)IVINT(R,Z / , Z~)I¢~.~(QA)¢~B(QB)). ( 3 1 )

F o r b r e v i t y t h e v i b r a t i o n a l s t a t e l a b e ls v A a n d v B h a v e b e e n o m i t t e d f r o m V I N T -

I f w e c o n s i d e r r i g id b o d y i n t e ra c t io n s , t h e n w e c a n r e p la c e O ~ a n d ( I) ~ b y d e l ta

f u n c t i o n s a t t h e e q u i l i b r i u m s t r u c t u r e s .I n o r d e r t o e x p r e s s e x p l i c it ly t h e d e p e n d e n c e o f t h e p o t e n t i a l ( 3 1 ) o n t h e

m o n o m e r o r i e n t a ti o n s t o ~ a n d r OB , w e c a n e x p a n d i t i n t e r m s o f a c o m p l e t e

a n g u l a r b a s is . A g e n e ra l f o r m u l a f o r s u c h a n e x p a n s i o n h a s b e e n g i v e n b y

v a n d e r A v o i r d e t a l . [44]

VINT( R , ~A , rOB)= ~ V L . , , 1 . , L . . a . , L ( R ) A L . , , t ~ , , , L . , K . , L ( t O A , r o B ) . (3 2 )La, K,4

LB, Km L

I n o u r b o d y - f i x e d f r a m e t h e a n g u l a r b a s i s f u n c t i o n s o f [4 4 ] s i m p l i f y to

= 8rr2(2L + 1)1/2 ~ DM.K(La)*(toA)D_MaKa(LB)*(tOB),3 3 )

M , , \ M 4 - M , ~ 0



1034 G. Brock s e t a l .

w h e r e t h e b r a c k e t s d e n o t e a 3 - j s y m b o l . T h e e ff e c ti v e h a m i l t o n i a n f o r t h e

d i m e r r o v i b r a t i o n a l p r o b l e m c a n n o w b e w r i t t e n a s

~ i~ / ~ A R O T - } -/ ~ B O T q - / ~ I N T -}- ) 'I NT ( 3 4 )

w i t h t h e t e r m s / ~ F R ° T ( F = A , B ) , ~ I N T a n d I ~I N g i v e n b y ( 3 0 ) , ( 2 6 ) a n d ( 3 2 )

r e s p e c t i v e l y .

N e x t w e w i s h t o i n t r o d u c e a c o n v e n i e n t b a s is f o r t h i s h a m i h o n i a n . L e t u s

s t a r t in t h e s p a c e - f i x e d f r a m e a g a i n , j u s t a s i n § 2 . W i t h r e s p e c t t o t h i s f r a m e t h e

m o n o m e r A a n d B o r i e n ta t i o n s a re d e f i n e d b y th r e e E u l e r a n g l es e ac h , w h i c h w e

d e n o t e b y t o ' ~ a n d t o ' ~ , a n d t h e o r i e n t a t i o n o f t h e r e l a t i v e p o s i t i o n v e c t o r R i s

g i v e n b y t h e p o l a r a n g l e s ( /3 , ~ ) . A c o m p l e t e a n g u l a r b a s is w o u l d b e

E E D n ~ . ~ ( ~ a ) * ( t ° ' A ) D n ~ k . U ~ ) * ( t ° ' B ) ( J a ~ I A J B f l B [ Jm i >m ~ , m ~ . 4 , ~

x Y l m ( f l, ~ ) < j m ~ I m l J M > . (35)

W e h a v e c o u p l ed t h e a n g u l a r f u n c t io n s b y m e a n s o f st a n d a r d C l e b s c h - G o r d a n

c o e f f i c i e n t s [ 4 2 ] , b e c a u s e J a n d M a r e g o o d q u a n t u m n u m b e r s f o r t h e o v e r a l l

d y n a m i c a l p r o b l e m . N o w w e t r a n s f o r m t h i s b a si s t o t h e b o d y - f i x e d f r a m e b y

t h e t w o E u l e r r o t a t i o n s o v e r ~ a n d / 3 , w h i c h a r e r e p r e s e n t e d i n c o o r d i n a t e s p a c e

b y t h e m a t r i x ¢ , ( 9 ). T h e o r i e n t a t i o n s o f l o ca l f r a m e s o n A a n d B w i t h r e sp e c t

t o t h i s d i m e r f r a m e h a v e b e e n d e n o t e d b e f o r e b y t o A a n d t o B ; t h e p o l a r a n g l e s

o f R i n t h i s f r a m e a r e ( 0 , 0 ) . T h e r e s u l t i s

E ~ ' . E D nAk~ (JA)* ( tO A )D nr , r , ( JB)* (eO B)< JAD ajB~ Bl jm j>

× Y lm ( O , O ) < j m / m l J ~ > D M n ( S ) * ( ~ , /3 , 0 ) . (36)

R e m e m b e r i n g t h a t Y l m (0 , 0 ) = ( 2 l + 1 /4 7r)1/2 8m0, we can s im pl i fy th i s resu l t and ,

i n s t e a d o f t h e b a s i s ( 3 6 ) , u s e t h e e q u i v a l e n t b o d y - f i x e d b a s i s

Z D n ~ , k . ( J a ) * ( t o a ) D n . k , , ( J " ) * ( t O B ) < J a f l - 4 J B ~ l B l J g l >

× D M n< J)* (= , /3 , 0) . (37)

T h i s b a s is i s v e r y c o n v e n i e n t i n d e e d f o r c a l c u l a ti n g t h e a n g u l a r m a t r i x e l e m e n t s

o v e r t h e h a m i l to n i a n (3 4) . T h e o p e r a t o r s / ~ F n ° T (3 0 ) w o r k o n ly o n t h e m o n o m e r

r o t a t i o n f u n c t i o n s D n ~ q r) *( C0 F) a n d w e g e t

< j ' a k '~ t j ' B k ' B J ' £ 2 ' I / ~ A R O T - ~ - - ISIBROT[jAkAjBkBj~>

= ~ r Z f i y . j f l j , ~ ' ~ n , n [ ~ ¥ . k . h j ~ k , ~ k ~ ~ k . ~ h j ~ k , . ~ . ] ( 3 8 a )

w i t h ( F = A , B )

h 2

h i ~ , ~ = -~ 8 ~ , ~ [ ( A + B ) j F ( j F + 1 ) + (2C - A - B ) k v 2]

h 2 h ~

+-¢ a~,~,~÷~(A-B)C% ~C+~k~+,+-£ a~,~,~_~(A B )C - ~ C -¢ ~ _ I (38 b)

a n dC ~ ± = [ j ( j + 1 ) - k ( k + 1)] 1/2. (3 9)

T h e o p e r a t o r / i ~ i NT i s e x p r e s s e d b y ( 2 6) i n t e r m s o f t h e a n g u l a r m o m e n t u m

o p e r at o r ] = j a + ] B a n d t he p s eu d o - a ng u l ar m o m e n t u m o p e r a to r ~ , g iv e n b y



Qua ntum dynam ics o[ van der W aals dimers 1035

( 2 0 ) . I t c a n b e w r i t t e n i n t h e f o r m

/~INT - - 2 /~R2 OR ~ 2 t~R 2 [ j g . + j 2 - 2 j j ~ - j + J + - j _ J _ ] ( 4 0 )

w i t h t h e s h i f t o p e r a t o r s d e f i n e d a s

j±=~+_i jv a n d J ± = J z - T - J u. ( 4 1 )

D e s p i t e t h e p e c u l i a r f o r m o f t h e c o m p o n e n t s o f d , s e e (2 0 ), a n d a ls o o f j z , s e e (2 5 ),

i t c a n b e s h o w n , s e e a p p e n d i x B , t h a t t h e o p e r a t o r s o c c u r r i n g i n (4 0 ) a c ti n g o n t h e

b a s i s ( 3 7 ) o b e y t h e s a m e r e l a t i o n s a s n o r m a l a n g u l a r m o m e n t u m o p e r a t o r s .

T h u s w e f i n d

( j ' A k 'A J 'B k ' . j ' n ' I g m T [ j a k a j . k . j n )

=h 2 8J'Ma3k'xka3J'aJn3k'ak"~J'i 3n 'n 2 /L R2 0 R 0---R

J ( J + l ) + j ( j + l ) - 2 ~ 2} 1 {+ 2ILR~ + 2 - - ~ 3 n " n+ l C Jn + Can+

+ 3 n , n _ , C , n - C s n - } ] . ( 4 2 )

T h e a n g u l a r m a t r i x e l e m e n t s o v e r t h e p o t e n t i a l V I N T c a n b e c o m p l e t e l y e v a l u a t e d

b y v e c t o r c o u p l i n g t e c h n i q u e s i f w e e x p a n d t h i s p o t e n t i a l (3 2 ) i n t e r m s o f t h e

a n g u l a r f u n c t i o n s ( 3 3) . T h e re s u l t i s

(J 'a k 'a j 'B k 'B j ' ~ ' [ VIN T[ jaka jBkB j~ )

= 3n. n ~ VL.,K.,L.K.L(R)L a , K a

L B , K s , L

×gLAKaLsKaL(J'A j 'B j ' ; jaJI~J ; kA k'B k 'a kB ; f~ ), ( 43 )

w h e r e t h e g e n e r a l i z e d G a u n t c o e f f i c i e n t i s

g = ( - 1 j ' + j ' A + j ' B +LA+LB - k ' A - k ' B - f l

× [ (2 j ' a + 1 ) (2 j a + 1 ) (2L .4 + 1) ( 2 j ' B + 1 ) ( 2 j n + 1 ) ( 2 L n + 1 )

× ( 2 j ' + 1 ) ( 2 j + 1 ) ( 2 L + 1 )] x/2

\ k ' a K ~ k a J k k B K B kB - f ~ 0[ j ' j

i n t e r m s o f 3 - j a n d 9 - j s y m b o l s [4 2] .

( 44 )

W i t h a ll t h e a n g u l a r m a t r i x e l e m e n t s o f ~ c o m p l e t e l y e v a l u a t e d w e c a n i m -

m e d i a t e l y w r i te d o w n t h e c l o s e - c o u p l e d e q u a t i o n s [ 2 4 , 3 2 , 3 3] f o r t h e r a d ia l

w a v e f u n c t i o n s - J M jAkAiBkS~a(R)n a g e n e r a l iz e d c o l li s io n c o m p l e x i n t h e b o d y -

f ix e d a p p r o a c h . T h e c o r r e s p o n d i n g e q u a t i o n s i n s p a c e - f ix e d c o o r d i n a t e s h a v e

b e e n o b t a i n e d b y C u r t i s s [3 6 ]. T h e s p a c e - f i x e d e q u a t i o n s a r e e a s ie r t o d e r iv e ,

b u t o u r b o d y - f i x e d f o r m a l i s m i s f i n a l l y s i m p l e r a n d h a s s e v e r a l a d v a n t a g e s i n

p r a c t i c a l a p p l i c a t i o n s .



1036 G. Broc ks e t a l .

T h e r e a r e t h r e e g e n e r a l m e t h o d s o f o b t a i n i n g b o u n d s t a t e s o l u t i o n s o f t h e

c l o s e - c o u p le d e q u a t i o n s . F i r st , b y d i r e c t n u m e r i c a l s o l u ti o n o f t h e c o u p l e d

d i f f e re n t i a l e q u a t i o n s , a s d o n e b y D u n k e r a n d G o r d o n [ 45 ] a n d a l so b y S h a p i r o

a n d B a l i n t - K u r t i [ 46 ] i n th e i r a r t i f ic i a l -c h a n n e l s - s c h e m e , a n d r e c e n t l y b y D a n b ya n d F l o w e r [4 7] . T h i s , h o w e v e r , i s u n l i k e l y t o b e fe a s i b le f o r p r o b l e m s w h e r e a

l a rg e n u m b e r o f c h a n n e l s m u s t b e i n c l u d e d . S e c o n d l y , b y e x p a n s i o n in a r a d ia l

bas i s se tW a r n ~ o ~ X t R ~ C a M (45)

2 a k a j B k B j f l ~ l k ) = E n \ 1 J A k Aj B kB j ~' ~ ; ~1n

a n d s o l v i n g a m a t r i x e i g e n v a l u e p r o b l e m .

T h e r a d i a l b a s i s X n ( R ) c a n b e e i t h e r n u m e r i c a l [ 1 6 , 2 4 , 3 0 ] o r a n a l y t i c a l

[10 , 26, 31 , 48] ; the rad ia l m at r ix e lem ent s over /~7INT, (42) and ove r the po te n t ia l

e x p a n s i o n c o e f f i c i e n t s V L a K A L , K , L ( R ) i n ( 4 3 ) c a n b e c a l c u l a t e d a c c o r d i n g l y .

T h i s a p p r o a c h i s c a l l e d b y L e R o y e t a l . [ 2 4 ] t h e s e c u l a r e q u a t i o n m e t h o d ,

w h e r e a s T e n n y s o n a n d v a n d e r A v o i r d [ 2 6 ] h a v e p r o p o s e d t h e a c r o n y mL C - R A M P ( L i n e a r C o m b i n a t i o n o f R a d i a l a n d A n g u l a r M o m e n t u m P r o d u c t s ) .

T h i r d l y , o n e c a n in t r o d u c e f u r t h e r a p p r o x i m a t i o n s s u c h a s B O A R S ( B o r n

O p p e n h e i m e r A n g u l a r a n d R a d ia l S e p a r a t i o n ) [4 9] a n d s o l v e t h e e f f e c t iv e r a d ia l

e q u a t i o n n u m e r i c a l l y .

C l e a r l y , w h i c h e v e r m e t h o d i s c h o s e n , t h e p r o b l e m w i l l i n g e n e r a l b e v e r y

l ar ge . W i t h o u t s y m m e t r y t h e r e a r e ( N + 1 ) ( 1 6 N a + 6 4 N Z + 9 6 N 2 + 6 4 N + 15)/15

a n g u l a r b a s i s f u n c t i o n s f o r a J = 0 c a lc u l a t i o n w h i c h i n c l u d e s a l l a l l o w e d f u n c t i o n s

w i t h j a , J B < < -N . F o r N = 6 , f o r e x am p l e , th i s g iv e s 17 92 7 a n g u l a r c h a n n e l s.

F u l l c a l c u l a t i o n s w i t h J > 0 w i l l b e e v e n m o r e e x p e n s i v e .

A s i m p l i f i c a t i o n , w h i c h i s o n l y p o s s i b l e i n t h e b o d y - f i x e d f o r m a l i s m , i s

c a u s e d b y t h e n e g l e c t o f t h e o f f - d i a g o n a l C o r i o l i s t e r m , t h e l a s t t e r m i n ( 4 2 ) ,w h i c h i s t h e o n l y o n e t h a t c o u p l e s ba s is f u n c t i o n s w i th d i f f e r e n t f L T h i s

a p p r o x i m a t i o n m a k e s s o l u t i o n s d i f f e r i n g o n l y i n p a r i t y , d e g e n e r a t e a n d i t h a s

b e e n m a d e o n s e v e r a l o c c a s i o n s [ 1 0 , 2 6 , 3 0 , 3 5 ] s i n c e t h e s e C o r i o l i s t e r m s o f t e n

a p p e a r t o b e s m a l l .

4. CONCLUSIONS

I n t h e p r e c e d i n g s e c t i o n s w e h a v e d e r i v e d a g e n e r a l r o v ib r a t io n a l o r s c a t t e r i n g

h a m i l t o n i a n f o r a co l li si o n c o m p l e x f o r m e d b y t w o p o l y a t o m i c s . F r o m o u r

d i s c u s s i o n o f a p o s si b l e s o l u t i o n s t r a t e g y , i t is c l e a r t h a t , e v e n i n t h e b o d y - f i x e d

f r a m e w h e r e t h e c l o s e - c o u p l e d e q u a t i o n s a r e s i m p l e r t h a n t h e i r s p a c e - f ix e dc o u n t e r p a r t s , t h e s o l u t i o n o f t h e m o s t g e n e r a l p r o b l e m i s l i k e l y t o r e m a i n

p r o h i b i t iv e l y e x p e n si v e f o r t h e f o r e s e e ab l e f u t u r e . H o w e v e r , t h e r e a r e c e r t a in

p o s s i b i l i t i e s w h i c h m a k e t h e p r o b l e m t r a c t a b l e .

( 1 ) I f t h e f r a g m e n t s A a n d B a r e i d e n ti c a l, p e r m u t a t i o n s y m m e t r y ca n r e d u c e

t h e si ze o f t h e p r o b l e m . T h i s , t o g e t h e r w i t h ( 2) , h a s a l lo w e d t h e v a n d e r

W a a l s c o m p l e x e s ( H F ) z [ 2 5 ] a n d ( N 2 ) 2 [ 2 6] t o b e t a c k l e d . F o r t h e

e th y le ne d im er (C~H4) 2 the re i s a po ten t ia l su r fa ce ava i lab le a l so [50 , 51 ]

a n d t h e s o l u t i o n o f t h e d y n a m i c a l p r o b l e m m i g h t b e f e a si b le .

( 2 ) I f o n e o f t h e f r a g m e n t s o r b o t h a r e e i th e r s y m m e t r i c o r li n e ar , t h e r e i s a

c o r r e sp o n d i n g r e d u c t i o n i n t h e c o m p l e x it y o f t h e p r o b le m . T h e s y m -

m e t r y , w h i c h i n v o lv e s t h e o c c u r r e n c e o f i d e n ti c a l n u c le i , c a n b e i n t r o -

d u c e d v i a t h e g r o u p o f f e a s i b l e p e r m u t a t i o n s [4 3, 5 2 ] . F o r l i n e a r



Quantum dynamics o f van der Waals d imers 1037

m o l e c u l e s F = A , B we c a n s e t t h e l a b e l s k F i n t h e b a s i s ( 3 7 ) a n d K e in

th e p o t e n t i a l ( 3 2 ) e q u a l t o z e r o a n d u s e t h e p r o p e r ty t h a t DM 0 tL )* (= , f l, 7 )

= ( 4 w / 2 L + 1 ) 1/2 YL~J(fl, ~). A l s o t h e r o t a t i o n o p e r a t o r I 4 F m ) T = B j F2

( 3 0 ) , a n d i t s m a t r i x e l e m e n t s ( 3 8 ) b e c o m e c o n s i d e r a b l y s i m p l e r ,h i F = h 2 B j F ( j F + 1 ). T h i s s u g g e s t s t h a t C O - H 2, f o r w h i c h a fu l l

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

p u b l i s h e d [ 5 3 ] , c o u l d r e a s o n a b l y b e t a c k l e d .

( 3 ) I f o n e o f t h e f r a g m e n t s , s a y B , is a n a t o m a v e r y g re a t s i m p l i f i c a ti o n

r e su l ts ( L B = 0 , L = L A ; j B = O , J = JA )" A l t h o u g h , t o o u r k n o w l e d g e

o n l y a t o m - d i a t o m p r o b l e m s h a v e b e e n tr e a te d , a t o m - p o l y a t o m p r o b l e m s

a r e c e r t a in l y f e a s ib l e . E x a m p l e s a r e t h e m o t i o n o f T I a r o u n d R e O 4 ,

s u g g e s t e d b y I s t o m i n et al. [ 8 ] a n d S F n - A r , f o r w h i c h a s u r f a c e h a s

r e c e n t ly b e e n p u b l i s h e d [ 54 ]. T h e M + - L H 4 - p r o b l e m h a s r e c e n tl y

b e e n t r e a t e d i n a n a p p r o x i m a t i o n w h i c h i n v o l v e d f r e e z i n g t h e d i s t a n c e

R [ 5 5 ] .

( 4 ) I f t h e r o t a t io n a l c o n s t a n t s f o r o n e o f t h e f r a g m e n t s o r b o t h a r e la r g e

c o m p a r e d w i t h t h e s t re n g t h o f t h e a n i s o t r o p i c c o n t r i b u t i o n s i n th e

p o t e n t ia l , t h e n t h e n u m b e r o f b a s i s f u n c t i o n s r e q u i r e d f o r t h a t f r a g m e n t

i s g r e a t ly r e d u c e d . T h i s i s t h e c as e w i t h m o s t h y d r o g e n i c s y s t e m s su c h

a s H 2 - H 2 , H 2 - r a r e g a s, w h e r e t h e H 2 m o n o m e r s a r e p r a c ti c a l ly u n p e r -

tu r b e d i n t e r n a l r o to r s , a n d , t o a s ma l l e r e x t e n t , a ls o w i th H C 1 - A r , e t c .

( 5 ) O t h e r s i m p l i f i c a t i o n s c a n a r i s e f r o m c o n s t r a i n t s t h a t c a n b e p u t o n t h e

m o t i o n s o f t h e f r a g m e n t s . 4 a n d B , d u e t o t h e w a y i n w h i c h t h e i n t e r a c t i o n

p o t e n t i a l d e p e n d s u p o n t h e i r o r i e n t a t i o n s . . A s i m p l e e x a m p l e is t h e c a sew h e r e t h e r e l a t iv e m o t io n o f .4 a n d B i s c o n f in e d t o o n e t o r s io n a l a n g l e

a r o u n d t h e R - a xi s, s u c h a s t h e i n t e r n al r o t a t io n a r o u n d s i ng l e b o n d s i n

m o l e c u l e s , f o r e x a m p l e e t h a n e C 2 H e . T h e n , w e c a n p u t K A = K B = 0 ,

u s e t h e q u a n t u m n u m b e r s A ~ = IM A I, A,~=F M ~ I a n d m a i n t a i n o n e

in t e r n a l a n g l e o n ly , ~ = ~ B - - ~ .4 , i n t h e b a s i s ( 3 7 ) . T h i s c a s e h a s b e e n

t r e a t e d b y H o u g e n [ 5 6 ]. A si m i l a r c a s e i s t h e ( s e m i - ) ri g id b e n d e r

m o d e l o f B u n k e r et al. [20 , 21] .

S o , a l t h o u g h t h e s o l u t i o n o f t h e r o v i b r a t i o n a l p r o b l e m f o r t h e i n t e r a c t i o n o f t w o

g e n e r a l p o l y a t o m i c s is s ti ll s o m e w a y o f f, t h e r e a r e s e v e r a l in t e r e s t in g p r o b l e m s

t h a t c a n b e t a c k l e d w i t h i n t h e s c o p e o f t h e f o r m a l i s m p r e s e n t e d h e r e . I t s h o u l d

b e r e m e m b e r e d , m o r e o v e r , t h a t b e f o r e a r o v i b r a t io n a l c a l c u l a t io n c a n b e p e r -

f o r m e d i t i s n e c e s s a r y t o h av e a p o t e n t i a l su r f a c e f o r t h e s y s t e m . T h i s p o t e n t i al

c a n b e o b t a i n e d f r o m ab initio c a l c u l a t i o n s [ 4 4 ] o r f r o m e x p e r i m e n t a l d a t a b u t ,

i f i t is to b e i n a n y wa y r e al i st i c , i t s c o n s t r u c t i o n i s li k e ly t o b e c o n s id e r a b ly m o r e

e x p e n s i v e t h a n t h e r o v i b r a t i o n a l c a l c u l a t io n s w h i c h u s e i t .

APPENDIX A

Three-angle embedding

I n § § 2 a n d 3 we h a v e u s e d a r o t a t i o n o v e r tw o a n g le s , ~ a n d f l, t o d e f in e t h e

b o d y - f i x e d f r a m e w i t h r e s p e c t to th e s p a c e - f i x e d o ne . T h i s l e a v e s u s fr e e t o

d e f i n e a t h i r d a n g l e o f r o t a t i o n y t o f i x c o m p l e t e l y t h e f r a m e i n t h e s y s t e m . 4 B .



1038 G . Brock s e t a l .

W e c a n c h o o s e t h a t a n g l e b y e m b e d d i n g o n e o f th e p a r ti c le s , s a y t h e l as t p a rt ic l e

o f m o n o m e r A w i t h c o o r d i n a t e s z N . _ l a i n t h e o r i gi n a l ( p a r t l y ) b o d y - f i x e d f ra m e

of § 2 , in th e x z p l a n e o f t h e n e w (c o m p l e t e l y ) b o d y - f i x e d f r a m e . I f t h e p o l a r

ang les o f th i s pa r t i c le ZNA_I = r a re (0 , ~ ) , the ro ta t ion an g le Y m us t be eq ua l to4 , s o t h a t

w i t h

r ' = C ' T r = 0

CO S

/ c o s ¢ - s in ¢ i )C ' = / s i n 6 c o s $ .

\o 0

( A 1 )

( A 2 )

T h e c o o r d i n a t e s o f t h e o t h e r p a r ti c le s i n t h e n e w b o d y - f i x e d f r a m e a r e

z i ' a = C ' ~ ' z i a , f o r i = l , .. ., N a - 2 , ]

a n d l ( A 3 )z i B = C 'T z ~B, f o r i = l , . . . , N B - 1 .

T h e t r a n s fo r m a t i o n o f t h e a n g u l a r m o m e n t u m o p e r a to r s p r o c e e ds a lo n g th e

s a m e l in e s a s i n § 2 . T h e g r a d i e n t s a n d a n g u l a r m o m e n t a o f a ll p a r t ic l e s r e f e r r e dt o b y ( A 3 ) r e t a i n t h e s a m e e x p r e s s io n s i n t h e n e w c o o r d i n a t e s, z~ a a n d z i B .I t i s c o n v e n ie n t t o d e f i n e a n e w a n g u l a r m o m e n t u m o p e r a t o r

NA -- 2 NB -- 1

J ' = E i ( z i ' A ) + E J (Z i B ) ( A 4 )i A = I i = 1

w h i c h i s d i f f e r e n t f r o m j i n t h a t t h e f i r s t s u m m a t i o n i s r e s t ri c t e d t o ( N A - 2 )p a r t i c l e s . o n A . U s i n g t h i s d e f i n i t i o n , we o b t a i n

j(z i) = C ' i (z~ ')

w i t h t h e f o l l o w i n g e x p r e s s i o n s h o l d i n g f o r t h e c o m p o n e n t s o f J i n t h e n e w

c o o r d i n a t e s z~ '

j ~ = j , x + c o t O ( j , z t ~ )i '

, , h a ( A 5 )

h ~J o = ] - ~ #

T h e t r a n s fo r m a t i o n f o r t h e t ot a l a n g ul a r m o m e n t u m o p e r a t o r is

a (z ) = c ' J ( z / )



Q u a n t u m d y n a m ic s o [ v a n d e r W a a l s d im e r s 1 0 3 9

w i t h i n th e n e w f r a m e

sin 8 # - cosec/~ cos ¢ ~ + cot/~ cos ¢ ~ ,

d u = ~ . ( c o s ~ ~ + c o s e c fl s i n ~ ~ - c o t f l s i n ¢ ~ ) , ( A 6 )

j _ h b

i 8 ¢ = j ' '

T h i s o p e r a t o r h a s n o w o b t a i n e d i t s f a m i l i a r f o r m i n a c o m p l e t e l y b o d y - f i x e d

f r a m e [3 , 4 ] a n d , i n c o n t r a s t w i t h (2 1 ), i ts c o m p o n e n t s s at is f y t h e c o m m u t a t i o n

r e l a t i o n s u s u a l l y c a l l e d anom alous (~, '1 , ~ = x , y o r z )

h

T h e y a r e d i f f e r e n t f r o m t h e c o m m u t a t i o n r e l a t i o n s o f t h e s p a c e - f i x e d c o m p o n -

e n t s o n l y b y t h e m i n u s s i gn . A l s o , w e re c o v e r t h e re l a t io n

c o n t r a r y to (2 5 ) i n t h e p a r t ly b o d y - f i x e d fr a m e . O n t h e o t h e r h a n d , w e f i n d

t h a t

a ( z 3 = c ' j ( : / ) , c ' j ( z ( )

= L ~+ y ,? + L 2+ c ' 0 c ' r ) , j

= j ~ + c o s e c O ju s i n Oju+j~2 ( A 9 )a n d , a n a l o g o u s ly

j . a = y ~ d ~ + c o s ec 0 j u s i n O J u + j e J ~. ( A 1 0 )

F i n a l ly , w e l o o k a t t h e t ra n s f o r m a t i o n o f t h e h a m i l t o n i a n . F i r s t, w e c o n s i d e r

t h e m o n o m e r t e r m s / ~ F i n th e k i n e ti c e n e r g y o p e ra t o r (6 ), w h i c h o b t a i n t h e s a m e

e x p r e s s i o n s ( 2 3 ) i n t h e p a r t l y b o d y - f i x e d c o o r d i n a t e s z i F a s th e y h a d i n t h e s p a c e -

f i xe d c o o r d i n a t e s t i F . I f w e u s e t h e s e e x p r e ss i o n s , t h e n t h e r e s u l t f o r m o n o m e r

.4 w i ll b e c o m e v e r y m e s s y in t h e c o m p l e t e l y b o d y - f i x e d f r a m e . T h e r e a s o n i s

t h a t a l l t h e ( n o n - d i a g o n a l ) V ( Z i F ) . V ( z F ) t e r m s i n (2 3 ) w h i c h i n v o l v e t h e

p a r t i c l e z ~.~ _l a w i l l h a v e t o b e r e p l a c e d b y c o m p l i c a t e d c h a i n r u l e e x p r e s s i o n s .

I f , i n s t e a d o f t h e c o o r d i n a t e s t i d e f i n e d b y ( 4 ), w e d e f i n e J a c o b i c o o r d i n a t e s t i ' ,w h i c h m a k e s G ij d i a g o n a l a s i n (7 ' ), t h e n a ll m o n o m e r .4 t e r m s w i t h i = 1 . . . ,

N a - 2 r e m a i n f o rm - i n v a r ia n t , as w e ll a s t h e m o n o m e r B t e r m s i = 1 . . . , N B - 1,

a n d w e h a v e t o u s e t h e c h a in r u le o n ly f o r t h e t e r m w i t h i = N a - 1. F o r th i s

t e r m i n / ~ a w e ca n w r it e

i 2V 2 ( z ~ . a _ l a ) = V 2 ( r ) = r - 2 [ ~ r Z O r ~ r + ( ~ ) j 2 ( r) ] ( A l l )

a n d a p p l y s o m e o f t h e r e s u lt s j u s t o b t a i n e d f o r t h e a n g u l a r m o m e n t a

j ( r ) = i (Z l v ~ _ x a ) = j - - j ' / ( A 1 2 )

¢ 2 ( r = f l + f 2 _ 2 j ' . i Jw i t h j 2 g i v e n b y ( A 9 ) .



1 0 4 0 G . Br o c k s e t a l .

S o , l e t u s a s s u m e a t t h i s p o in t t h a t a l l t h e t r a n s f o r m e d c o o r d in a t e s z i r i n ( A 3 )

a n d r i n ( A 1 ) a c t u a l ly o r i g i n a t e f r o m J a c o b i c o o r d i n a t e s t i ' , where t~ r i s the mass

o f t h e r e d u c e d p a r t ic l e t 'N a _ l A = C Z N A _ I A . F o r t h e o t h e r k i n e t i c e n e r g y t e r m

i n t h e h a m i l t o n ia n , / (I N T , s ee ( 2 6 ), w e c a n d i r e c t l y u s e t h e a n g u l a r m o m e n t u mr e s u lt s , ( A 8 ) to ( A 1 0 ), a n d t h e f i n a l e x p r e s s i o n b e c o m e s , i n c o m p l e t e l y b o d v -

f i x e d c o o r d i n a t e s

N A - 2 h 2 N ~ - 1 h 2

I t = i = I E -2/~---] V 2( z" A )+ i:,X -2tz--- V~(z; B)

2~ R 2 8R ~ - + cos ec # s in 0 b---~

2 +cot 0 Y, +cot 0

+Yx +Yu + COt20Jz '2 + + COt 0 J u + cot 0 (g~ y~ +j~ y~ )

- 2(]~ ' J~ +] . ' d . + c o t s 0 ] ~ ' Y,) - 2 co t 0 ( ] ~ ' J , + ] ~' d x ) ]

h z ~ ~ 0 1 - , 2 - , ^

2 /zd" e r r ~ rr + ~ c°sec2 0 [ , ] 2 + j~ _ 2 j~ J~ ] + V. ( a 13 )

T h i s h a m i l t o n i a n r e d u c e s t o t h a t o b t a i n e d b y T e n n y s o n a n d S u t c l i ff e [ 1 0] f o r

t h e a t o m - d i a t o m p r o b l e m b y s et ti n g N a = 2 a n d N B = 1 , t h a t i s , o m i t t i n g t h e f i r s t

t w o t e r m s a n d a ll t h e te r m s w i t h j '. I n t h e a t o m - d i a t o m c a s e i t i s c o n v e n i e n t

b e c a u s e r c a n b e i d e n t i f i e d w i t h t h e d i a t o m b o n d l e n g t h a n d w e h a v e n o f u r t h e ri n t e rn a l m o n o m e r c o o r d i n a t e s . S t il l, o n e h a s to b e c a r e f u l w i t h t h e s i n g u l a ri t y

a t 0 = 0 .

F o r l a r g e r s y s t e m s i t i s c l e a r t h a t t h i s h a m i l t o n i a n ( A 1 3 ) , a l t h o u g h i t c o n t a in s

o n l y n o r m a l a n g u l a r m o m e n t u m o p e r a t o r s J a n d j ' , is n o t a s u s e f u l a s t h e o n e

o b t a i n e d f r o m t h e t w o - a n g l e e m b e d d i n g , g i v e n in § 2 . F i r s t, t h e s e p a r a t i o n

/ t = / t a + ~ B + / ~ I ~ T is d e s t ro y e d b y t ak in g o n e ( J ac o b i) c o o rd i na t e o u t o f

m o n o m e r . 4 a n d w e c a n n o l o n g e r f o l l o w t h e s o l u t i o n s t r a t e g y o u t l i n e d i n § 3 .

S e c o n d l y , th e s y m m e t r y b e t w e e n t h e t w o f r a g m e n t s ,4 a n d B , a l th o u g h f o r m a l l y

s t il l p r e s e n t , i s n o l o n g e r r e f l e c t e d b y t h e f o r m o f t h i s h a m i l t o n i a n . E s p e c i a l l y

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

s i m p l i f y i n g t h e s o l u t i o n ( s e e t h e c o n c l u s i o n s i n § 4 ) . T h i r d l y , t h e r e i s as in g u l a r i t y f o r 0 = 0, th a t i s , w h e n t h e J a c o b i c o o r d in a t e Z N A _I = r , t h a t i s c h o s e n

t o f i x t h e t h i r d e m b e d d i n g a n g l e , s e e ( A 1 ) , l i e s a l o n g t h e v e c t o r R b e t w e e n t h e

c e n t r e s o f m a s s .

APPENDIX B

A c t i o n o f p s e u d o - a n g u l a r m o m e n t u m o p e r at o rs o n b o d y - f i x e d b a s is

I n o r d e r t o d e r i v e t h e a c t i o n o f t h e c o m p o n e n t s o f t h e p s e u d o - a n g u l a r

m o m e n t u m o p e r a t o r J , d e f i n e d b y ( 2 0 ) o n t h e b a si s ( 3 7 ), w e a p p l y a s i m p l e

c o o r d i n a t e t r a n s f o r m a t i o n . I n t h e b o d y - f i x e d c o o r d i n a t e s y s t e m o f §§ 2 a n d 3

t h e o r i e n t a t i o n s o f t h e m o l e c u l e s ,4 a n d B a r e g iv e n b y t h e E u l e r a n g l e st° .4 = ( $ ~ , 0 4 , S A ) a n d t O B = ( $ B , O B, ~ bB ) r e s p e c t i v e ly , a n d t h e a n g l e s ( fl , ~ )



Quantum dynamics o f van der Waals d imers 1041

d e s c r i b e t h e o v e ra l l r o t a t i o n s o f t h e s y s t e m . I n s t e a d o f t h e a n g le s ~ a a n d S B,w e in t ro d u ce th e n ew an g les y = S a an d $ = C B - -C a" T h e an g le "$ i s ac tu a l lya t r u e i n t e r n a l a n g l e o f t h e d i m e r A B , w h i le y i s t h e th i rd o v e ra l l ro t a t io n an g le

i f w e e m b e d t h e b o d y - f i x e d f r a m e w i t h i ts x z - p l a n e c o n t a i n i n g t h e l o ca l c - ax iso f m o n o m e r . 4. W e s h a ll n o t p e r f o r m t h e th r e e - an g l e e m b e d d i n g t r a n s fo r m a t i o no n t h e h a m i l t o n i a n , h o w e v e r , s i n c e t h i s l e a d s t o t h e p r o b l e m s o u t l i n e d i na p p e n d i x A .

I t i s e a s i l y p r o v e d t h a t t h e b a s i s ( 3 7 ) c a n b e r e w r i t t e n i n t e r m s o f t h e n e wan g les a s fo l lo w s

X DaAk.,t/A)*( , 0a , ~a )D n.k.O n)*(¢ , OB, ~bB)

x < j a O a j B O B I j ~ > D M a ( J r ( a , fl, 7 ) . (B 1)

T h e c o m p o n e n t s o f J ( 2 0) , c o n ta i n t h e o p e r a t o r

j ~ =j ~ +j . = ~ [ ~ - - ~ a + ~ - ~ B ) • ( B 2 )

U s i n g t h e c h a i n r u l e a g a in , w e f i n d t h a t

L ( B 3 )

~ . ~ . ~ v + ~ - ~ ~-~ = ~-~

s o t h a t j ~ i n t h e n e w c o o r d i n a t e s r e a d s s i m p l y

h bJ' = i ~y" (B 4)

S u b s t i t u t i n g t h i s i n t o t h e e x p r e s s io n s ( 25 ) a n d ( 2 0) w e o b s e r v e t h a t J 2 a n d J z

o b ta in th e i r u su a l fo rm s [4 2] in t e rm s o f th e b o d y - f ix ed f r am e an g les ~, fl an d 7 .T h e y o n l y o p e r a t e o n t h e l a s t fa c t o r i n t h e b a s is ( B 1 ) a n d t h e y y i e l d t h e r e g u l a rr esu l t s

J 2 D M n ( J ) * ( ° t , f l , 7 ) = h ~ J ( J + ~ x r ~ ( J * ) f o t f l , 7 ) ,")~Ma " ' ( B 5 )

Jz OM a (J )* ( a , f l , 7 ) ~ -~ r~ ( J ) " ~ t- ~ - - ~ . ( , ~ , v ) .

T h e o p e r a t o r s J + = J ' ~ -T- r d e f i n e d b y ( 4 1 ) w e c a n c a ll p s e u d o - s h i f t o p e r a t or s .

S u b s t i t u t i n g t h e r e s u l t ( B 4 ) f or j ~ a g a i n a n d c o m p a r i n g t h e m w i t h t h e r e g u l a r

e x p r e s s i o n s f o r s t e p - u p a n d s t e p - d o w n o p e r a t o r s ' + [ 4 2 ], w e o b s e r v e t h a t t h e y

a r e r e l a t e d t o t h e l a tt e r b y

J ± = e x p ( Z i y ~ J '± . (B 6 )

A c t in g w i th th o se o p e ra to r s o n th e l a s t f ac to r s in th e b as i s (B 1 )

J+ D M n (S)*(a, [3, r )= e x p ( ~ ir )h C jn ± DM f~+I J)*(oc, f l, 'y)

= ~ C s . ± D ~ ± V J ' ) ( ~ , ~ , 0 ) ex p ( i~v ) (B 7)

su b s t i tu t in g th i s r e su l t b ack in to (B 1 ) ag a in an d r e -ex p ress in g y an d ~ in ~ a an d

~ B , w e o b ta in th e d es i r ed r e su l t fo r th e ac t io n o f J± o n th e o r ig in a l b as i s (37 ) .



1 0 42 G . B r o c k s e t a l .

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