c. ruth le sueuer et al- on the use of variational wavefunctions in calculating vibrational band...

8/3/2019 C. Ruth Le Sueuer et al- On the use of variational wavefunctions in calculating vibrational band intensities

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MOLECULAR PHYSICS, 1992, VOL. 76, No. 5, 1147-1156

O n t h e u s e o f v a r i a t i o n a l w a v e f u n c t i o n s i n c a l c u l a t i n g v i b r a t i o n a lb a n d i n t e n s i t i e sBy C. RUTH LE SUEUR, STEVEN MILLER, JONATHAN TENNYSON

Department of Physics and Astronomy, University College London, Gower St,London WC1E 6BT, UK

and BRIAN T. SUTCLIFFEDepartment of Chemistry, University of York, York YO1 5DD, UK

(Received 19 December 1991; accepted 6 February 1992)It is shown that vibrational band intensities calculated using variationalwavefunctionsand dipole surfaces give results which depend on how the Cartesianaxes of the dipole surface are defined. It is suggested that the most consistentdefinition of these axes uses the rules proposed by Eckart for separating ro-vibrational motion. The consequences of this choice of axis system for thecalculated band intensities of H2S, LiNC and H~-, and the apparent validity ofHrnl-London factors are discussed. Computed band intensitiesare presented forH2S, HDS and D2S which correct previous literature values.

1 . I n t r o d u c t i o nThe intensity of a series of ro-vibrafional transitions (V 'J ' -V"J") are often

characterized in terms of a single parameter which is a constant for the vibrationaltransition (V'-V") under consideration. This parameter is known as the vibrationalband intensity. The intensity of each ro-vibrational line in this band is then charac-terized by a purely algebraic factor usually called a H6nl -London factor [1]. Althoughthe assumptions underlying this method of characterizing intensities sometimes breakdown, for instance when there is intensity stealing between nearby ro-vibrationallevels, vibrational band intensities are o f great use as they allow a lot of data to berepresented by a single parameter. They are thus a common currency which is widelyused to assess whether a particular band may be intense enough to be observed or toestimate the lifetime of a vibrationally excited state.

Within the harmonic oscillator approximation, vibrational band intensities can becalculated from the derivative of the dipole at equilibrium [2]. Recently, however, anumber o f ro-vibrational methods have been developed which go considerably beyondthe harmonic oscillator approximation. These methods use internal coordinates,Hamiltonians which are often exact within the Born-Oppenheimer approximation,and the variational principle to derive accurate representations of ro-vibrationalwavefunctions. The internal coordinate system used usually dictates the choice of axisembedding for these calculations. Since one is not constrained to produce a goodseparation between vibrational and rotational motion, the coordinate system (andhence the axis embedding) tend, however, to be chosen with regard to the chemistryof the molecule rather than to produce a good separation of vibrational and rotationalmotion. Such methods can be extended to calculate the intensity of individualro-vibrational transitions [3-5] but for obvious reasons of compactness, a number of

0026-8976/92 $3.00 9 1992 Taylor & Francis L td


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1148 C.R. Le Sueur et a l .authors have chosen to use rotationless (J = 0) wavefunctions from variationalcalculations and internal coordinate representations of the dipole surface to computevibrational band intensities directly; see for example studies on H 2S [6-8] and HCN[9, 10].As far as we can tell, none of these previous studies has considered explicitly theorientation of the dipole axes when computing vibrational band intensities. In thiswork we show that the results obtained depend on the orientation chosen. Sinceobtaining accurate band intensities is dependent on a good separation of rotationaland vibrational motion, we suggest that a consistent treatment should use axes givenby the Eckart conditions [11]. We demonstrate the effect of this by performing a seriesof calculations on HzS, LiNC and H~. We present vibrational band intensities forH2 S and its isotopomers which correct previously published values.

2. TheoryThe probabili ty of the transition ~u, ~ ~u" is given byl( ~'l/~Sl ~u")12 if the tran-

sition is actuated by the influence of the electric dipole/~s.In general a body-fixed wavefunction ~ ' ( J M V ) for an N-atomic molecule is given

by\{ 2J -8~ 1 -)112~F (JM V) = ~ ~k v (as~k(q)lJMk>. (1)

In other words, t t t ( J M V ) is given as a linear combination of products of angularmomentum eigenfunctions and vibrational basis functions 4~k expressed with respectto 3N - 6 internal coordinates, q. In most variational procedures the q~ are them-selves further expanded as products of functions in one or more internal coordinates,and the angular momentum functions are written in terms of Wigner ~-functions.The exact choice of how I J M k > relates to ~,k(.Q) may vary between workers--wechoose to use I J M k > = ( - 1)k~*_k--but in any case this only leads to differencesin phase factors in the following treatment. In (1), k is the projection of J, therotational angular momentum, along the body-fixed z axis, and needs to be con-sidered as indicated if a smooth transition between linear and strongly bent geo-metries is desired.

In the absence of external magnetic fields all space-fixed directions are equivalentand so states ~ ( J M V ) with differing M are degenerate. The line strength S~ for agiven transition is given by summing the transition probability over all M' and M"states. The equivalence of all space-fixed axes also means that the total probability isthree times the transition probability actuated by a dipole acting only in the space-fixed z direction. The line strength can therefore be written as

(2J' + 1)(2J" + 1) 1 l~+ k" . . , '~v '* . . , "k ' v" /~s ' . ~J '~ . ,s lo~s~ . .~r~ ' . fS, 3 ( 8 2 ) 2 I E ( - 1 . ] l '' ,M"k ' , v " k ~ , v "

( 2 )where we have substituted for I J M k ) as described above.It is advantageous, when evaluating (2), to express /~s in terms of the dipolereferred to the body-fixed axes (/~, #ya, /~a). This may be done quite simply bysubstituting

#s = ~ , . # ~, (3)1


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11 50 C . R . L e S u e u r e t a L( se e , f o r e x a m p l e , e q u a t i o n ( 6 .1 2 9 ) o f r e f e r e n c e [1 5 ]) . A l t h o u g h t h i s f a c t o r i z a t i o n i sn o t s t r i c tl y p o s s i b l e u n l e ss f o r s o m e r e a s o n t h e s u m o v e r l i s r e s t r i c te d t o o n e v a l u e ,i t is c o m m o n l y a s s u m e d . I n t h i s c a s e t h e v i b r a t i o n a l b a n d i n t e n s i ty i s d e f i n e d as

s V i bv . v , = I (V ' I ~ B IV " ) I 2 Z I ( V ' l ~ l V " ) l 2 . ( 1 2 )tF o r s y m m e t r i c t o p a n d l i n e a r m o l e c u le s , s y m m e t r y c o n s i d e r a t i o n s d i c ta t e t h a t

o n l y o n e c o m p o n e n t o f t h e t r a n s i t i o n d i p o l e i s e v e r n o n - z e r o a n d s o t h i s f a c t o r i z a t i o ni s e x a ct . F u r t h e r m o r e , s i n ce k i s t h e n a g o o d q u a n t u m n u m b e r , t h e S ~ 3 c a n b ec a l c u l a te d a lg e b r a i c al l y a n d a r e fr e q u e n t l y t a b u l a t e d a s H r n l - L o n d o n f a c t o r s [1 , 1 5 ].L e a v i n g a s id e t h e q u e s t i o n o f e x a c t ly w h a t s h o u l d b e u s e d a s t h e r o t a t i o n a l f a c t o r s f o ra g e n e r a l a s y m m e t r i c t o p , i t i s c l e a r t h a t b y u s i n g t h i s a p p r o x i m a t i o n t o o b t a i nv i b r a t i o n a l b a n d d i p o l e s a n d h e n c e i n t e n s i t i e s , i t i s p o s s i b l e t o o b t a i n a f a i r i d e a o fh o w i n t en s e t h e t r a n s i t i o n s w i t h i n a v i b r a t i o n a l b a n d w i ll be .T h e i d e n t i f i ca t i o n o f a n d s e p a r a t i o n b e t w e e n v i b r a t io n a l a n d r o t a t i o n a l m o t i o n sis o f c o u r s e f u n d a m e n t a l i n t h e u s e o f v i b r a t io n a l b a n d i n te n s it ie s a n d a s s o c ia t e dr o t a t i o n a l f a c to r s . W h a t a p p e a r s t o h a v e b e e n g e n e r a ll y i g n o r e d i n p r e v io u s w o r kw h i c h u s e d v a r i a t i o n a l w a v e f u n c t i o n s a n d d i p o l e s u rf a c es t o c a l c u l a t e ( V ' I ~ B I V " ) ist h a t i d e n t i f y i n g a x e s f o r t h e b o d y - f i x e d d i p o l e , /~ B, i m p l i c i t ly a l s o i d e n t if i e s th eb o d y - f i x e d a x e s fo r t h e r o t a t i o n a l m o t i o n . I f th e s e a xe s a re i n a p p r o p r i a t e t h e n a p o o rs e p a r a t i o n b e t w e e n v i b r a t i o n a l a n d r o t a t i o n a l m o t i o n s w i l l r e s u l t a n d p a r t i t i o n i n gb e t w e e n t h e s e m o t i o n s w i ll n o l o n g e r b e v a l id . T h i s i s t r u e e v e n t h o u g h v i b r a t i o n a lw a v e f u n c t i o n s a r e g e n e r a ll y o b t a i n e d f r o m r o t a t io n l e s s ( J = 0 ) c a l c u l a ti o n s w h i c ha r e i n d e p e n d e n t o f a x is e m b e d d i n g s [ 1 4 ]. T h i s i s b e c a u s e t h e r e i s n o r e a s o n w h yd i f f e r e n t a x i s e m b e d d i n g s s h o u l d g i v e th e s a m e v a l u e s o f I ( V ' l t ~ B I V " ) 1 2 . A s m e n t i o n e da b o v e , t h e m e t h o d o f o b t a i n i n g t h e b e st s e p a r a t io n o f v i b r a ti o n a l a n d r o t a t i o n a lm o t i o n h a s p r e v i o u s l y b e e n o u t l i n e d b y E c k a r t . B y c h o o s i n g a s e t o f b o d y f i x ed a x e st h a t o b e y h i s c o n d i t i o n s

E mir~ ri = 0 , E miri = 0 , ( 1 3 )i iwh ere m~ i s the ma ss an d r~ the po s i t ion (re. a t equ i l ib r ium ) o f pa r t i c l e i, we ho pe too b t a i n m o r e a c c u r a t e b a n d i n t e n s i t i e s .

F o r t h e p a r t i c u l a r c a se o f a t r i a t o m i c , t h i s c o n v e r s i o n i s e a s y t o m a k e . T a k i n g ,w i t h o u t l o s s o f g e n e r a li t y , a n e m b e d d i n g w h i c h f o rc e s t h e m o l e c u l e t o l ie i n t h eb o d y - f i x e d x z p l a n e , s a ti s fi e s t h r e e o f th e s i x E c k a r t c o n d i t i o n s im m e d i a t e l y . T h e o t h e rt h r e e a r e s a t i sf i e d b y f i r s t r e f e r r i n g e v e r y t h i n g t o t h e c e n t r e o f m a s s a n d t h e n t o a n e ws e t o f a x e s X Z r e l a te d t o t h e o r i g in a l b y a r o t a t i o n t h r o u g h 0 , g i v en b y

Ei3=l m , ( x ~ z i - - z e x i )t a n 0 = Z ~ = l mi( x ~ x i + z ~ z i ) " (14)

T h e s e n s e o f t h i s r o t a t i o n i s g i v e n b y t h e f i g u r e.O u r a p p r o a c h f o r c a lc u l a t in g v i b r a ti o n a l b a n d i n te n s it ie s i s n o w c le a r. I f J = 0w a v e f u n c t i o n s a r e c h o s e n , a l l t h a t n e e d s t o b e d o n e i s t o r e f e r t o t h e d i p o l e t o t h e

' E c k a r t ' i n t e r n a l a x e s a s t h e w a v e f u n c t i o n t h e m s e lv e s h a v e n o a x is d e p e n d e n c e .

3 . R e s u l t sT o d e m o n s t r a t e t h e d e p e n d e n c e o f v i b r a t i o n a l b a n d i n te n s it ie s o n t h e e m b e d d i n g

c h o s e n f o r t h e b o d y f ix e d a x is s y s t e m , w e h a v e c o m p u t e d v i b r a t io n a l b a n d i n te n s it ie s


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V a r i a t i o n a l w a v e f u n c t i o n s i n c a l c u l a ti n g b a n d i n t e n s i ti e s 1151Z

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i i

T h e E c k a r t a x e s X Z a r e r e l a t e d t o t h e o r i g i n a l i n t e r n a l a x i s s y s t em x z b y a r o t a t i o n 0 .u s in g t h e s a m e J = 0 w a v e f u n c t i o n s f o r a n u m b e r o f a x is e m b e d d i n g s fo r H e , H 2 Sa n d L i N C . T h e s e m o l e c u l e s a r e , re s p e c t i v e ly , a s y m m e t r i c to p , a n a s y m m e t r i c t o p a n da l i n e a r m o l e c u l e .

W e p e r f o r m e d c a l c u l a t i o n s o n H f a s a f u n c t i o n o f a x is e m b e d d i n g s , u s i n g th ep o t e n t i a l e n e rg y a n d d i p o l e s u r f a c e s o f M e y e r e t a l . ( M B B ) [ 16 ] a n d t h e J = 0w a v e f u n c t i o n s o f M i l l e r e t a l . [ 17 ]. H ~ h a s t w o v i b r a t i o n a l f u n d a m e n t a l s : a n A ~s y m m e t r i c s t r e t c h , v~ , a n d a d e g e n e r a t e , E b e n d , v2.

T h e s y m m e t r y o f H f m e a n s t h a t t h e v i b r a t i o n a l b a n d i n te n s i t i e s o f t h is io np r o v i d e a p a r t i c u l a r l y s t r i n g e n t t e s t o f a n y m e t h o d . W e w o u l d e x p e c t m a n y v i b r a t i o n a lb a n d s t o h a v e z e r o i n t e n s i t y ( a l l A ~ A t r a n s i t io n s ) , a n d E ~ E t r a n s i t i o n s to h a v et w o i d e n t ic a l l y i n te n s e c o m p o n e n t s . ( O u r m e t h o d d o e s n o t w o r k w i t h t h e f u ll s y m -m e t r y o f H ~- , a n d E s t a t e c o m p o n e n t s a r e c a l c u l a t e d i n s e p a r a t e e v e n ( e) a n d o d d ( o )p a r i t y c a l c u l a t i o n s [ 1 8 ] . )

C a l c u l a t i o n s w e r e p e r f o r m e d o n H ~ - f o r a n u m b e r o f a x i s e m b e d d i n g s . T h e r e su l ts ,s ee t a b l e 1 , o b t a i n e d f o r t h e s c a t te r i n g c o o r d i n a t e e m b e d d i n g , w h e r e z w a s t a k e nT a b l e 1 . V i b r a t i o n a l b a n d d i p o l e s f o r H ~ c a l c u l a t e d u s i n g a s c a t t e r i n g c o o r d i n a t e e m b e d d i n g

a n d t h e E c k a r t e m b e d d i n g . B a n d d i p o l e s c a lc u l a t e d b y M i l le r et a l . [ 1 7 ] f r o m t h eJ = 1 , -- J = 0 l i n e s t r e n g t h s ( s e e t e x t ) a r e a l s o i n c l u d e d f o r c o m p a r i s o n . T h e d i p o l ec o m p o n e n t s a r e l a b e ll e d b y w h e t h e r t h e b r a a n d k e t v i b r a t i o n a l w a v e f u n c t io n s a r e o fe v e n ( e) o r o d d ( o ) p a r i t y .S c a t t e ri n g e m b e d d i n g E c k a r t e m b e d d i n g [17 ]

T r a n s i ti o n c % / cm - 1 # ~ / D t #o o/D # ir /D # ~ / D #o o/D # J D ~ / Dvo ~- vo 0"0 0"015 - 0 "015 0 -001 - 0 " 0 0 1 0 -000v2 ~ Vo 2521"3 0"162 - 0" 229 0"160 - 0"226 0-2 262v2(l = O) ~- v2 2255"7 0-1 37 - 0-13 7 0"139 - 0"139 0"1 312v2(1 = 2) ~- v2 247 6"1 0 "1 71 0"168 0"240 0"163 0"164 0 " 2 31 0"234v2 + vl ~ vl 2375"4 0"178 - 0-25 2 0"174 - 0"246 0"245v2 + vl ~ v~ 3032"4 0"028 0-043 0-051 0"032 0 " 0 3 1 0"045 0-053vl ~ vo 3178"3 0"004 - 0-00 4 0"000 - 0"000 0-00 02v2(l = 2) ~ v0 4997"4 0"075 - 0"106 0"0 61 - 0"086 0"0893v2(1 = 1) ~ v2 4482"2 0"044 0"079 0"090 0" 05 1 0"049 0 "0 7 1 0"0723v2(1 = + 3 ) ~ v2 4761"2 0"073 - 0"073 0"060 - 0"060 0-0613 v 2( 1 = - - 3 ) ~- v2 4971"3 - 0"096 0"096 - 0"079 0"079 0"0882v2(1 = 2) + v l ~ v l 4690"3 0"075 - 0-106 0"061 - 0"086 0-089

t l D = 3 " 3 35 6 4 1 0 -Z ~ c m .


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Tab l e 2 .C . R . L e S u e u r et al.

Vibrat ional absorpt ion intensi t ies , Sbt , for H2S obtained using a variety of axisembeddings.Ab sorpt ion in tens i t ies /a tm- t cm -2

(v i , v2, v3) @ f /cm - I B isec to r Scat ter ing R a d a u E c k a r t(0, 1, 0) 1190.4 2-65 2-72 49-50 2.64(0, 2, 0) 2372 .0 0"32 0.33 0-03 0-33(1, 0, 0) 262 0.4 1-40 1.53 1-46 1.40(0, 0, 1) 2631"0 1-54 67"51 1-09 1.08

tS b (a t 297 K ) in atm -l c m -z = 10-245 x to~ ( in cm -1) x S~ ( in D 2) [7].

p a r a ll e l t o t h e v e c t o r c o n n e c t i n g th e c e n t re o f a n H - H d i a t o m t o t h e th i r d H , w e r et y p ic a l o f t h o s e o b t a i n e d f o r n o n - E c k a r t e m b e d d i n g s . T h e r e su l ts p r e s e n t e d int a b l e 1 s h o w t h a t t h e n o i s e ( as m e a s u r e d b y h o w d i f f e re n t t h e i n te n s it ie s o f s u c h b a n d sa r e c a l c u l a t e d t o b e , a n d h o w i n t e n s e t h e b a n d s w h i c h s h o u l d h a v e z e r o i n te n s i t y a rep r e d i c t e d t o b e ) f o r t h e E c k a r t e m b e d d i n g i s m u c h s m a l l e r t h a n f o r a s t a n d a r dt r ia t o m i c e m b e d d i n g . T h e H r n l - L o n d o n f a c t o r f o r t h e J = 1 * - J = 0 l in e o f as y m m e t r i c t o p w i t h i n a v i b r a t i o n a l b a n d i s 1 ; t h i s w a s u s e d b y M i l l e r et al. [17] toc o m p u t e e f fe c ti v e v i b r a t i o n a l b a n d d i p o l e s b y c o n s i d e r i n g J - - l ~ - J = 0 t r a n s i t i o nd i p o l e s #01 ( se e c o l u m n 9 o f t a b l e 1 ). T h e s e c a n b e c o m p a r e d w i t h ~ if f r o m t h e E c k a r te m b e d d i n g t o s e e h o w v e r y a c c u r a t e t h e b a n d i n te n s it ie s r e s u lt in g f r o m t h is e m b e d -d i n g a r e .A n u m b e r o f th e o r e ti c a l s tu d ie s h a v e b e e n p e r f o r m e d o n t h e s p e c tr a o f H 2 S a n di ts is o t o p o m e r s . W e h a v e f o l l o w e d t h e m o s t r e c e n t a n d a c c u r a t e o f t h e se a n d u s e d t h eab initio p o t e n t i a l e n e r g y a n d d i p o l e s u r fa c e o f S e n e k o w i t s c h et aL [7 ] an d t he J = 0R a d a u c o o r d i n a t e v i b r a ti o n a l w a v e f u n c t i o n s o f M i ll e r et aL [ 8 ] . W e a r e o n l y c o n -c e r n e d w i t h J = 0 w a v e f u n c t i o n s , t h u s w e e n c o u n t e r e d n o n e o f t h e d i ff ic u lt ie s e x p e r i -e n c e d b y M i l l e r et aL i n s y m m e t r i z i n g t h e i r v i b r a t i o n a l w a v e f u n c t i o n s f o r H 2 S a n dD 2 S .T a b l e 2 s u m m a r i z e s o u r r e s u l ts f o r H 2 S a n d s h o w s t h a t t h e a n s w e r s d e p e n d o nh o w t h e a x e s a r e d e f i n e d . I t i s i n t e r e s t i n g t o n o t e t h a t c a l c u l a t i o n s i n w h i c h t h e z a x isis o r i e n t e d a l o n g t h e H S H b i s e c t o r ( c o l u m n 3 ) a g r e e w e ll w i t h t h o s e i n w h i c h t h e z ax isi s o r i e n t e d a l o n g t h e s c a t t e r i n g c o o r d i n a t e ( c o l u m n 4 ) f o r t h e b e n d a n d s y m m e t r i cs t re t c h , b u t m u c h le ss w e l l f o r t h e a s y m m e t r i c s tr e t c h w h e r e t h e t w o a x i s e m b e d d i n g sb e g i n t o d i f f e r s i g n i fi c a n t ly . E q u a l l y , t h e R a d a u e m b e d d i n g , i n w h i c h t h e z a x is ist a k e n a l o n g o n e o f th e R a d a u c o o r d i n a t e s ( a n d h e n c e n e a r ly p a ra l le l t o o n e o f t h e H - Sb o n d s ) , g e n e r a l l y gi v es v e r y d i ff e r e n t e s t im a t e s f o r t h e b a n d i n te n s it ie s . W e n o t e t h a tS e n e k o w i t s c h et al. [7 ] u s e d a d i p o l e o r i e n t e d s o t h a t t h e z a x i s b i s e c t e d t h e H S H a n g l e .M i l l e r et al. c o p i e d t h is a p p r o a c h . O u r c a l c u l a ti o n s , in c o l u m n 3 o f t a b l e 2, r e p r o d u c et h e r e s u l t s o f t h e s e w o r k e r s . T h e f i n a l c o l u m n o f t h i s t a b l e g i v e s r e s u l t s c a l c u l a t e du s i n g t h e E c k a r t e m b e d d i n g a s o u t l i n e d a b o v e . W h i l e b o t h t h e b i s e c t o r a n d s c a t te r i n ge m b e d d i n g s s e e m t o g i v e a c c u r a t e r e s u l ts f o r t h e t w o b e n d s t at e s g i v e n , i t is c le a r t h a tf o r o t h e r e x c i t e d st a te s , i n p a r t i c u l a r t h e a s y m m e t r i c s t re t c h , c a l c u l a ti o n s u s i n g t h e s ee m b e d d i n g s c o u l d b e m i s le a d in g . T h e R a d a u e m b e d d i n g , o n t h e o t h e r h a n d , g iv e s ag o o d r e s u lt f o r t h is s t a te , b u t i s i n a c c u r a t e f o r t h e m o r e s y m m e t r i c s ta t es .

V i b r a t i o n a l b a n d i n t e n s i t i e s w e r e c a l c u l a t e d u s i n g t h e d i p o l e s u r f a c e s o fS e n e k o w i t s c h et al. b u t w i t h t h e x a n d z d i r e c t i o n s d e f i n e d i n t h e m a n n e r o u t l i n e da b o v e . T a b l e 3 c o n t a i n s r e s u lt s f o r t h e l o w e s t 61 v i b r a t i o n a l b a n d s o f H 2 S a n d D 2 S .


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1 15 4 C . R . L e S u e u r e t a l .T a b l e 3. V i b r a t i o n a l b a n d o r i g i n s, a ~ , a n d v i b r a t i o n a l a b s o r p t i o n i n te n s i ti e s , S b , f r o m t h e

g r o u n d s t a te f o r H 2 S a n d i s o t o p o m e r s c o m p u t e d u s i ng t h e E c k a r t e m b e d d i n g .L o c a l N o r m a l H 2S D 2S H D Sm o d e m o d e

In,, n 2 ] ( V 2 ) (Vl, 2 , V 3 ) ~ o ~ 1 S b l C O ~ I S b l ~ & /c m - ' a t m - ' c m -~ c m -~ a t m - ~ c m - 2 c rn -~ a t m - ~ c m - 2[ 0 ,0 ] ( 1 ) ( 0 , 1 , 0 ) 1 1 9 0 . 4 2 . 6 4 ( + 0 ) 8 6 0 .4 1 . 5 9 ( + 0 ) 1 0 3 9 .2 5 . 9 5 ( + 0 )[ 0 ,0 ] ( 2 ) ( 0 , 2 , 0 ) 2 3 7 2. 0 3 . 2 9 ( - 1 ) 1 7 1 6.6 1 . 1 5 ( - 1 ) 2 0 7 1 .9 1 . 5 2 ( - 1 )[O ,O ] ( 3 ) ( 0 , 3 , 0 ) 3 5 4 3.5 4 . 1 6 ( - 2 ) 2 5 6 8.0 1 . 0 9 ( - 2 ) 3 0 9 7.3 5 . 1 7 ( - 2 )[ 0 ,0 ] ( 4 ) ( 0 , 4 , 0 ) 4 7 0 3.7 2 . 4 6 ( - 3 ) 3 4 1 4. 1 3 . 5 5 ( - 4 ) 4 1 1 4 .6 1 . 3 5 ( - 3 )[ 0 ,0 ] ( 5 ) ( 0 , 5 , 0 ) 5 8 5 1.6 1 . 2 7 ( - 4 ) 4 2 5 4.7 1 . 8 5 ( - 5 ) 5 1 2 3.0 8 . 3 7 ( - 6 )[ 0 ,0 ] ( 6 ) ( 0 , 6 , 0 ) 6 9 8 6. 3 2 . 6 5 ( - 6 ) 5 0 8 9.2 3 . 6 1 ( - 7 ) 6 1 2 2.0 4 . 5 0 ( - 7 )[ 0 ,0 ] ( 7 ) ( 0 , 7 , 0 ) 8 1 0 6. 9 3 . 4 7 ( - 8 ) 5 9 1 7.4 5 . 0 6 ( - 9 ) 7 1 1 0. 7 1 . 1 6 ( - 8 )[ 0 ,0 ] ( 8 ) ( 0 , 8 , 0 ) 9 2 1 2.5 4 . 7 1 ( - 1 0 ) 6 7 3 8.8 2 . 0 7 ( - 1 0 ) 8 0 8 8.7 5 . 1 6 ( - 1 0 )[ 0,0 ] ( 9 ) ( 0 , 9 , 0 ) 1 0 3 0 2 . 1 5 . 2 9 ( - 1 1 ) 75 5 3.2 7 . 3 9 ( - 1 2 ) 9 0 55 .9 9 . 0 8 ( - 6 )

[ 1 , 0 + ] ( 0 ) ( 1 , 0 , 0 ) 2 62 0 .4 1 . 4 0 (+ 0 ) 1 9 00 .6 6 . 6 6 ( - 1 ) 1 9 0 5.9 4 . 9 0 ( - 1 )[ 1 , 0 - ] ( 0 ) ( 0 , 0 , 1 ) 2 63 1 .0 1 .0 .8 ( + 0) 19 1 2 .0 2 . 8 4 ( - 1 ) 2 6 25 .6 1 . 3 1 ( + 0 )[ 2 , 0 + ] ( 0 ) ( 2 , 0 , 0 ) 5 1 54 .2 3 . 5 2 ( - 1 ) 3 7 60 .3 1 . 1 6 ( - 1 ) 3 7 62 .2 2 . 1 9 ( - 1 )[ 2 , 0 - ] ( 0 ) ( 1 , 0 , 1 ) 5 1 55 .5 6 . 7 7 ( - 1 ) 3 7 63 .2 3 . 1 3 ( - 1 ) 4 5 30 .2 3 . 6 5 ( - 2 )[ 1 ,1 ] ( 0 ) ( 0 , 0 , 2 ) 5 2 5 1.2 3 . 9 9 ( - 2 ) 3 8 1 4.4 4 . 5 5 ( - 3 ) 5 1 5 5.3 5 . 0 0 ( - 1 )[ 3 , 0 + ] ( 0 ) ( 3 , 0 , 0 ) 7 5 8 9.3 9 . 9 1 ( - 3 ) 5 5 6 9.2 1 . 9 6 ( - 3 ) 7 0 5 8. 8 2 . 6 5 ( - 3 )[ 3 , 0 - ] ( 0 ) ( 2 , 0 , 1 ) 7 5 89 .4 3 . 3 5 ( - 2 ) 5 5 6 9 .6 1 . 1 4 ( - 2 ) 6 38 4 .9 2 . 0 2 ( - 3 )

[ 2 , 1 + ] ( 0 ) ( 1 , 0 , 2 ) 7 7 6 8.4 5 . 8 4 ( - 3 ) 5 6 58 .1 9 . 8 0 ( - 4 ) 5 56 9 .2 6 . 5 3 ( - 3 )[ 2 , 1 - ] ( 0 ) ( 0 , 0 , 3 ) 7 78 9 .3 5 . 5 4 ( - 3 ) 5 6 79 .7 1 . 4 9 ( - 3 ) 7 5 91 .6 2 . 3 0 ( - 2 )[ 4 , 0 + ] ( 0 ) ( 4 , 0 , 0 ) 9 9 29 .0 5 . 9 8 ( - 6 ) 7 3 31 .4 a 2 . 1 8 ( - 6 ) 8 9 12 .5 2 . 3 2 ( - 7 )[ 4 , 0 - ] ( 0 ) ( 3 , 0 , 1 ) 9 9 29 .0 1 . 0 0 ( - 3 ) 7 3 31 .4 a 2 . 2 5 ( - 4 ) 8 1 89 .7 9 . 9 9 ( - 5 )[ 3 , 1 + ] ( 0 ) ( 2 , 0 , 2 ) 1 0 2 08 .3 3 . 8 9 ( - 4 ) 7 4 6 8.5 5 . 1 1 ( - 5 ) 7 3 2 6. 7 1 . 4 7 ( - 4 )[3 ,1- ](0) (1 ,0 ,3) 10212.0 4 .3 7( -4 ) 7476.1 7 .75 (-5) 9562.7 9 .53(-8)[2,2] (0) (0 ,0, 4) 10308.8 1.5 9( -5 ) 7529.6 1.05(-5 ) 10010.9 1.35(-6)[1,0+] (1) (1 ,1 ,0 ) 3794 .6 1.78(+ 0) 2752.5 6.9 1(- 1) 2934.8 1.09(+0)[ 1 , 0 - ] ( 1 ) ( 0 , 1 , 1 ) 3 7 99 .8 2 . 2 8 ( + 0 ) 2 7 61 .3 8 . 8 8 ( - 1 ) 3 6 4 5.6 1 . 5 3 ( + 0 )[ 1 , 0 - ] ( 2 ) ( 0 , 2 , 1 ) 49 6 0.0 4 . 5 3 ( - 2 ) 3 6 0 6 .4 1 . 3 1 ( - 2 ) 4 6 59 .5 1 . 8 2 ( - 2 )[ 1 , 0 + ] ( 2 ) ( 1 , 2 , 0 ) 4 9 6 0.1 6 . 3 8 ( - 3 ) 3 6 0 0 .2 7 . 6 5 ( - 4 ) 3 9 5 7.2 1 . 4 5 ( - 2 )[ 1 , 0 - ] ( 3 ) ( 0 , 3 , 1 ) 6 1 1 0.2 8 . 7 4 ( - 4 ) 4 4 4 6 .8 1 . 5 3 ( - 4 ) 5 6 6 6.2 8 . 6 1 ( - 4 )[ 1 , 0 + ] ( 3 ) ( 1 , 3 , 0 ) 6 1 1 5.6 2 . 0 8 ( - 3 ) 4 4 4 3 .3 3 . 5 2 ( - 4 ) 4 9 7 2.4 1 . 3 8 ( - 3 )[ 1 , 0 - ] ( 4 ) ( 0 , 4 , 1 ) 7 2 49 .4 4 . 9 2 ( - 5 ) 5 2 82 .0 5 . 2 9 ( - 6 ) 6 6 65 .0 4 . 3 9 ( - 5 )[ 1 , 0 + ] ( 4 ) ( 1 , 4 , 0 ) 7 2 5 9.8 5 . 5 5 ( - 5 ) 5 2 8 1 .2 5 . 0 7 ( - 6 ) 5 9 7 9.5 2 . 4 6 ( - 5 )[1 ,0- ](5) (0 ,5 , 1) 8376.5 1 .7 9(- 7) 6111.7 3 .90 (-8) 7655.2 5 .61(-7)[ 1 , 0 + ] ( 5 ) ( 1 , 5 , 0 ) 8 39 1 .6 1 . 0 8 ( - 5 ) 6 1 13 .4 1 . 0 2 ( - 6 ) 6977 . 7 3 . 3 9 ( - 7 )[ 1 , 0 - ] ( 6 ) ( 0 , 6 , 1 ) 9 49 0 .6 8 . 6 9 ( - 8 ) 6 9 35 .5 5 . 6 5 ( - 9 ) 8 63 6 .1 8 . 4 0 ( - 9 )[ 1 , 0 - t - ]( 6 ) ( 1 , 6 , 0 ) 9 5 1 0. 2 1 . 8 1 ( - 7 ) 6 9 3 9.6 1 . 9 3 ( - 8 ) 7 9 6 6 .4 9 . 7 6 ( - 9 )[ 1 , 0 - ] ( 7 ) ( 0 , 7 , 1 ) 1 0 5 90 .9 1 . 7 7 ( - 9 ) 7 7 5 2. 9 2 . 0 2 ( - 1 0 ) 9 6 0 7. 1 1 . 3 1 ( - 9 )[ 1 , 0 + ] ( 7 ) ( 1 , 7 , 0 ) 1 0 61 4 .6 9 . 4 4 ( - 9 ) 7 7 59 .4 7 . 9 5 ( - 1 0 ) 8 9 44 .9 6 . 4 1 ( - 1 0 )[ 2 , 0 + ] ( 1 ) ( 2 , 1 , 0 ) 6 3 07 .7 4 . 3 7 ( - 2 ) 4 6 02 .1 1 . 2 9 ( - 2 ) 4 7 80 .9 4 . 0 3 ( - 2 )[ 2 , 0 - ] ( 1 ) ( 1 , 1 , 1 ) 6 30 7 .7 1 . 9 4 ( - 1 ) 4 6 03 .9 5 . 2 8 ( - 2 ) 55 3 9.6 1 . 8 1 ( - 3 )[ 1 ,1 ] ( 1 ) ( 0 , 1 , 2 ) 6 4 0 3.0 1 . 6 7 ( - 4 ) 4 6 5 4.2 1 . 0 7 ( - 6 ) 6 1 5 6. 2 1 . 1 2 ( - 1 )


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Variational wavefunctions in calculating band intensitiesT a b l e 3 . (continued)

1 1 5 5

L o c a l N o r m a l H 2 S D 2 S H D Sm o d e m o d e (/)if/ ~ b/ ('Oif/ Sb/ ~O~1 & t[ ni,n 2 ]( v2 ) ( v i, v 2, v 3) c m - l a t m - t c m - 2 c m - I a t m - t c m - 2 c rn - t a t m - l c m - 2[2 , 0 - ] (2 ) (1 , 2 , 1 ) 7451 .5 3 . 27 ( -3 ) 5440 .4 5 . 7 8 ( - 4 ) 6542 .8 3 . 86 ( - -4 )[2 , 0+ ] (2 ) (2 , 2 , 0 ) 7452 .1 1 . 11 ( -5 ) 5439 .6 1 . 7 0 ( -5 ) 5793 .2 2 . 9 4 ( -4 )

[1 ,1 ] (2 ) (0 , 2 , 2 ) 7546 .9 3 . 04 ( -6 ) 5490 .1 4 . 0 6 ( - 6 ) 7151 .1 1 . 1 4 ( - 3 )[ 2 , 0 - ] ( 3 ) ( 1 , 3 , 1 ) 8 5 85 .3 8 . 5 5 ( - 5 ) 6 2 72 .4 7 . 3 5 ( - 6 ) 7 5 39 .0 2 . 4 8 ( - 5 )[ 2 , 0 + ] ( 3 ) ( 2 , 3 , 0 ) 8 5 85 .9 4 . 5 8 ( - 5 ) 6 2 72 .2 7 . 7 8 ( - 6 ) 6 7 98 .3 3 . 9 7 ( - 5 )[1 ,1 ] (3 ) (0 , 3 , 2 ) 8681 .5 2 . 25 ( -5 ) 6321 .7 2 . 0 0 ( - 6 ) 8139 .2 3 . 2 1 ( -5 )

[ 2 , 0 - ] ( 4 ) ( 1 , 4 , 1 ) 9 70 7.9 9 . 5 4 ( - 6 ) 7 0 9 9 . 1 6 . 0 8 ( - 7 ) 8 52 7.4 4 . 2 8 ( - 7 )[2 , 0+ ] (4 ) (2 , 4 , 0 ) 9708 .0 1 . 42 ( -6 ) 7099 .3 7 . 8 0 ( - 8 ) 7795 .4 8 . 9 9 ( -7 )[1 ,1 ] (4 ) (0 , 4 , 2 ) 9805 .6 6 . 10 ( -7 ) 7148A 2 . 4 8 ( - 8 ) 9119 .9 3 . 4 8 ( - 6 )[ 2 , 0 + ] ( 5 ) ( 2 , 5 , 0 ) 1 0 8 1 7 . 4 5 . 9 1 ( - 7 ) 7 9 20 .6 4 . 7 4 ( - 8 ) 8 7 83 .7 7 . 1 1 ( - 9 )[ 2 , 0 - ] ( 5 ) ( 1 , 5 , 1 ) 1 0 81 8 .2 6 . 4 6 ( - 1 0 ) 79 2 0.3 2 . 6 5 ( - 9 ) 9 5 07 .0 3 . 4 7 ( - 8 )[1 ,1 ] (5 ) (0 , 5 , 2 ) 10917. 9 3 . 6 8 ( - 7 ) 7969 .9 1 . 54 ( -8 ) 10093. 2 2 . 1 6 ( -8 )

[ 3 , 0 - ] ( 1 ) ( 2 , 1 , 1 ) 8 7 23 .0 7 . 7 7 ( - 3 ) 6 4 00 .7 1 . 5 4 ( - 3 ) 7 3 84 .0 2 . 2 9 ( - 5 )[ 3 , 0 + ] ( 1 ) ( 3 , 1 , 0 ) 8 7 23 .1 1 . 8 8 ( - 3 ) 6 4 00 .6 3 . 7 2 ( - 4 ) 8 0 49 .0 1 . 3 3 ( - 5 )[2 , 1+ ] (1 ) (1 , 1 , 2 ) 8906 .9 1 . 15 ( -5 ) 6491 .0 3 . 0 5 ( - 6 ) 6578 .0 1 . 1 3 ( -3 )[ 2 , 1 - ] ( 1 ) ( 0 , 1 , 3 ) 8 9 17 .4 2 . 4 6 ( - 6 ) 6 5 07 .8 3 . 6 2 ( - 7 ) 8 5 78 .1 5 . 0 0 ( - 3 )[ 3 , 0 - ] ( 2 ) ( 2 , 2 , 1 ) 9 8 47 .8 2 . 3 2 ( - 4 ) 7 2 27 .8 2 . 4 3 ( - 5 ) 8 3 76 .9 1 . 5 4 ( - 6 )[3 , 0+ ] (2 ) (3 , 2 , 0 ) 9848 .0 3 . 5 4 ( - 7 ) 7227 .8 1 . 96 ( -7 ) 9033 .2 1 . 1 7 ( - 5 )[2 , 1+ ] (2 ) (1 , 2 , 2 ) 10037. 4 4 . 0 0 ( - 6 ) 7320 .2 1 . 4 1 ( -6 ) 7580 .3 1 . 1 0 ( - 5 )[ 2 , 1 - ] ( 2 ) ( 0 , 2 , 3 ) 1 0 03 7 .4 1 . 4 0 ( - 6 ) 7 3 31 .9 4 . 6 1 ( - 5 ) 9 5 62 .1 7 . 4 8 ( - 5 )

a T h e s e b a n d o r ig i n s a r e o n l y a c c u r a t e t o I c m - ' .T ab l e 4 . V i b ra t i ona l ab so rp t i on i n tensi ti es , S b , fo r t he m os t in t ense bands , f r equency ~% ,o f L i N C ca l cu l a t ed u s i ng va r i ous ax i s em bedd i ngs fo r t he d i po l e su r f ace . E f f ec t i veband in t ens i t i es ca l cu la t ed f rom the J = 1 ~- J = 0 t r ans i t i ons a r e a l so p rov i ded fo rc o m p a r i s o n .

A bso rp t io n in tens it ies /a tm-~ cm -2( V S , V b ) ( D i f / C l T 1 - 1 Scat t e r ing Ec kar t J = 1 ~ J = 0 (oif/cm -!ca lcu la t ion(0, 2) 247"2 5"53 5-53 6-83 248" 1(1, 0) 753"7 796"49 796-49 788"53 754"6(2, 0) 1497"0 5"43 5-43 5-41 1497"9

R e f e r e n c e s[1] HERZBERG,G ., 1960, Infrared and Raman Spectra of Polyatomic Molecules V a n N o s t ra n d

R e i nho l d ) .[2] WILSON,E. B . JR., DECIUS,J. C. , and CROSS,P. C. , 1980, Molecular Vibrations: The Theoryof Infrared and Raman Vibrational Spectra (D o ve r P u b l ica t ions ) .[3] JENSEN,P. , and SPIRKO, V. , 1986 , J . molec. Spectrosc., 118, 208.[4] M ILLE R, S., TENNYSON, J., an d SUTCLIFFE,B. T . , 1989 , Molec. Phys., 66, 429.[5] C ARTE R, S., SENEKOWITSCH, ., HA NDY , N . C ., an d ROSMUS, P., 19 88 , Molec. Phys., 6 5 ,143.[6] BOTSCI-rWINA, ., ZILCH , A ., W ERNER,H -J. , ROSMUS,P. , a nd E-A . , REtNSCH, 1985, J . chem.Phys., 85, 5107.


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c. ruth le sueuer et al- on the use of variational wavefunctions in calculating vibrational band...

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