Circular Dichroic Intensities in the Vibronic Transitions of Chiral Metal

3

Circular D i c h r o i c Intensities i n the V i b r o n i c Transitions

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of C h i r a l M e t a l Complexes

FREDERICK S. RICHARDSON Department of Chemistry, University of Virginia, Charlottesville, VA 22901

The chiroptical properties of optically active transition metal complexes have played an enormously influential role in the stereochemical and electronic structural characterization of metal coordination compounds. Werner's early work (1) in resol ving the optical isomers of bis- and tris-chelated transition metal complexes containing achiral ligands established the octa hedral structure of hexa-coordinated complexes and posed the pro blems of molecular stereochemistry and absolute configuration of metal coordination compounds. In the 1930's, the optical rota tory properties of Werner's complexes were studied extensively by Jaeger (2), Mathieu (3), and Kuhn (4, 5, 6) with the objective of relating these properties to specific stereochemical features (most notably, the absolute configuration) of the systems. At tempts were also made to construct a theory of optical activity in transition metal complexes which would permit systematic cor relation of the observed optical rotatory (and circular dichroic) properties with absolute configuration (4, 5). These latter at tempts employed a classical representation of the metal ion and ligand electronic structure (and electronic transitions), and they were restricted to treating only those optical rotatory pro perties associated with absorptions occurring in the visible re gion of the spectrum. The first definitive determination of the absolute confi guration of a chiral metal complex was reported by Saito and coworkers (7) in 1955 using the anomalous x-ray scattering method. Saito and coworkers (7) found that the tris(ethylenediamine)cobalt(III) isomer which is dextrorotatory at the sodium D-line, (+)-[Co(en) ] , has the A-configuration (8). This finding was contrary to the configurational assignment predicted according to the Kuhn and Bein (4, 5) classical coupled oscillator model for tris-chelated Co(III) complexes. Moffitt (9) introduced the first quantum mechanical theory of optical activity in chiral transition metal complexes. He 3+

3

Q-8412-0538-8/80/47-119-043$07.50/0 © 1980 American Chemical Society In Stereochemistry of Optically Active Transition Metal Compounds; Douglas, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1980.

STEREOCHEMISTRY OF TRANSITION METALS

44

adopted a c r y s t a l f i e l d model f o r r e p r e s e n t i n g t h e s p e c t r o s c o p i c s t a t e s o f the m e t a l i o n d^-electrons, and used t h e " o n e - e l e c t r o n " t h e o r y o f m o l e c u l a r o p t i c a l a c t i v i t y p r o p o s e d b y Condon, A l t e r , and E y r i n g ( 1 0 ) t o d e v e l o p e x p r e s s i o n s f o r t h e r o t a t o r y s t r e n g t h s o f t h e m e t a l i o n cl-d ( l i g a n d f i e l d ) t r a n s i t i o n s . M o f f i t t ' s w o r k m a r k e d t h e a d v e n t o f "modern" d e v e l o p m e n t s i n t h e t h e o r y o f o p t i c a l a c t i v i t y i n t r a n s i t i o n m e t a l c o m p l e x e s . Many o f t h e t h e o r e t i c a l developments i n t r a n s i t i o n m e t a l complex o p t i c a l a c t i v i t y d u r i n g t h e I 9 6 0 s a n d e a r l y 1 9 7 0 s were b a s e d on t h e f u n d a m e n t a l a s p e c t s o f M o f f i t t ' s m o d e l . However, s e v e r a l t h e o r i e s and m o d e l s have b e e n d e v e l o p e d a n d p r o p o s e d w h i c h r e p r e s e n t m a j o r departures from M o f f i t t ' s simple o n e - e l e c t r o n c r y s t a l f i e l d m o d e l ( 1 1 , 1 2 ) . These i n c l u d e t h e o r i e s w h i c h r e p r e s e n t t h e wave f u n c t i o n s o f t h e s p e c t r o s c o p i c s t a t e s i n terms o f m o l e c u l a r o r b i t a l models ( o f v a r y i n g degrees o f s o p h i s t i c a t i o n and c o m p l e t e n e s s ) , a n d i n d e p e n d e n t systems/perturbâtion m o d e l s w h i c h i n c l u d e b o t h s t a t i c ( p o i n t charge c r y s t a l f i e l d ) and dynamic ( l i g a n d polarization) metal ion-ligand interactions. Furthermore; t h e o r e t i c a l t r e a t m e n t s have b e e n e x p a n d e d t o i n c l u d e c o n s i d e r a t i o n o f t h e o p t i c a l a c t i v i t y a s s o c i a t e d w i t h metal^->ligand charget r a n s f e r t r a n s i t i o n s and i n t r a - l i g a n d t r a n s i t i o n s , a s w e l l a s w i t h t h e m e t a l d-d t r a n s i t i o n s ( 1 1 , 1 2 ) .


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The p r i m a r y o b j e c t i v e s o f n e a r l y a l l o f t h e t h e o r e t i c a l s t u d i e s c a r r i e d o u t on t r a n s i t i o n m e t a l c o m p l e x o p t i c a l a c t ' i v i t y have b e e n t o c a l c u l a t e e l e c t r o n i c r o t a t o r y s t r e n g t h s a n d t o c o r r e l a t e t h e s i g n s and magnitudes o f t h e s e e l e c t r o n i c rotatory s t r e n g t h s w i t h s p e c i f i c s t e r e o c h e m i c a l and e l e c t r o n i c s t r u c t u r a l f e a t u r e s o f a g i v e n s y s t e m . The s t e o r e o c h e m i c a l f e a t u r e s o f p r i mary i n t e r e s t h a v e been a b s o l u t e c o n f i g u r a t i o n , conformational f e a t u r e s i n t h e l i g a n d environment ( o r i n c h e l a t e r i n g s ) , and l o c a l d i s t o r t i o n s w i t h i n t h e m e t a l i o n - d o n o r atom c o o r d i n a t i o n cluster. The e l e c t r o n i c s t r u c t u r a l f e a t u r e s o f p r i m a r y i n t e r e s t have been i d e n t i t i e s ( a s s i g n m e n t s ) o f e l e c t r o n i c t r a n s i t i o n s and t h e i r magnetic dipole versus e l e c t r i c dipole character. The r o t a t o r y s t r e n g t h a s s o c i a t e d w i t h a n e l e c t r o n i c t r a n s i t i o n ο -> m may be w r i t t e n a s R

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and N ( T ) i s a n o r m a l i z e d B o l t z m a n n w e i g h t i n g f a c t o r r e f l e c t i n g the p o p u l a t i o n o f t h e v - t h v i b r a t i o n a l l e v e l o f t h e ground e l e c t r o n i c s t a t e ( o ) a t t e m p e r a t u r e T. I n E q . ( 2 ) , μ_ a n d m a r e t h e e l e c t r i c and m a g n e t i c d i p o l e moment o p e r a t o r s , r e s p e c t i v e l y , φ and φ ^ a r e v i b r a t i o n a l wave f u n c t i o n s , and ψ a n d ty a r e e l e c t r o n i c wave f u n c t i o n s . I n w r i t i n g E q . ( 2 ) , we h a v e assumed t h e B o r n - O p p e n h e i m e r a d i a b a t i c a p p r o x i m a t i o n a n d we t a k e ψ a n d ψ V

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In Stereochemistry of Optically Active Transition Metal Compounds; Douglas, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1980.

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Intensities

to be e i g e n f u n c t i o n s o f t h e e l e c t r o n i c H a m i l t o n i a n o f t h e system. I n t h e Born-Oppenheimer a d i a b a t i c a p p r o x i m a t i o n , e l e c t r o n i c mo t i o n i s f u l l y c o r r e l a t e d w i t h the nuclear p o s i t i o n s (instantaneous s t a t i c c o n f i g u r a t i o n s ) , b u t i s n o t c o r r e l a t e d w i t h n u c l e a r motion. I n t h i s a p p r o x i m a t i o n , t h e v i b r o n i c wave f u n c t i o n s h a v e w e l l d e f i n e d e l e c t r o n i c a n d v i b r a t i o n a l quantum numbers. I n c a s e s where t h e a d i a b a t i c Born-Oppenheimer a p p r o x i m a t i o n b r e a k s down, E q s . (1) and (2) a r e n o l o n g e r a p p r o p r i a t e s i n c e t h e system cannot e x i s t i n s t a t i o n a r y s t a t e s w i t h w e l l - d e f i n e d e l e c t r o n i c quantum numbers. I n t h e s e c a s e s , t h e s y s t e m c a n be v i e w e d as s a m p l i n g d i f f e r e n t p a r t s of t h e e l e c t r o n i c c o n f i g u r a t i o n a l space as t h e n u c l e i v i b r a t e . These c a s e s c a n b e e x p e c t e d t o o c c u r when electron-nuclear v i b r a t i o n a l coupling (vibronic coupling) i s large and when t h e r e a r e d e g e n e r a t e o r n e a r l y d e g e n e r a t e e l e c t r o n i c s t a t e s i n t h e s y s t e m . A b r e a k d o w n i n t h e a d i a b a t i c Born-Oppenheimer approximation f o r degenerate s t a t e s i s r e f e r r e d to as the Jahn-Teller (JT) e f f e c t , and a breakdown o f t h i s a p p r o x i m a t i o n i n t h e p r e s e n c e o f n e a r l y d e g e n e r a t e s t a t e s i s commonly r e f e r r e d t o a s t h e pseudo Jahn-Teller(PJT) e f f e c t . In the nonadiabatic approximation, t h e v i b r o n i c wave f u n c t i o n s o f t h e s y s t e m must b e e x p r e s s e d a s Ψ. = Σ Σ C. ψ φ (3) j j,nvVnv η ν * where, now, t h e c o m p o s i t e v i b r o n i c quantum number j i s t h e o n l y "good" quantum number f o r d e s i g n a t i n g m o l e c u l a r s t a t e s , a n d t h e ίψηΦην^ t h e a d i a b a t i c B o r n - O p p e n h e i m e r v i b r o n i c wave f u n c tions. The e x p a n s i o n c o e f f i c i e n t s , C j v , r e f l e c t e l e c t r o n nuclear v i b r a t i o n a l coupling. I n c a s e s where t h e a d i a b a t i c a p p r o x i m a t i o n b r e a k s down, t h e only well-defined t r a n s i t i o n s are vibronic t r a n s i t i o n s with r o t a t o r y s t r e n g t h s g i v e n by J

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(for the v i b r o n i c t r a n s i t i o n i j ) . I fstrong v i b r o n i c coupling e x i s t s only w i t h i n a s m a l l , w e l l - d e f i n e d subset of molecular e l e c t r o n i c s t a t e s , t h e n i t may be p o s s i b l e t o c a l c u l a t e a n e t e l e c t r o n i c r o t a t o r y s t r e n g t h f o r t r a n s i t i o n s t o t h e s e s t a t e s by summing over a l l the v i b r o n i c r o t a t o r y strengths a s s o c i a t e d w i t h v i b r o n i c l e v e l s f a l l i n g w i t h i n the manifold of coupled e l e c t r o n i c s t a t e s . Only i n t h i s c o n t e x t does t h e term, e l e c t r o n i c r o t a t o r y s t r e n g t h , have any m e a n i n g i n t h e p r e s e n c e o f v e r y s t r o n g v i b r o n i c c o u p l i n g (and a c o n s e q u e n t breakdown o f t h e a d i a b a t i c Born-Oppenheimer approximation). This excursion into e l e c t r o n i c versus vibronic rotatory s t r e n g t h s and t h e a d i a b a t i c v e r s u s t h e n o n a d i a b a t i c a p p r o x i m a t i o n may be h i g h l y r e l e v a n t t o t h e d e t a i l e d i n t e r p r e t a t i o n o f t r a n s i t i o n m e t a l complex c h i r o p t i c a l s p e c t r a . I n most o p t i c a l l y a c t i v e t r a n s i t i o n m e t a l c o m p l e x e s , t h e symmetry o f t h e m e t a l i o n - d o n o r atom c l u s t e r r e m a i n s r a t h e r h i g h ( n e a r l y 0 o r n e a r l y Di* ) and, a s a r e s u l t , i t i s common t o f i n d many n e a r - d e g e n e r a c i e s ( o r e v e n e x a c t d e g e n e r a c i e s ) among t h e s p e c t r o s c o p i c s t a t e s o f i n t e r e s t . Even i n n

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46

the p r e s e n c e of low-symmetry l i g a n d f i e l d s , t h e r e remains u n c e r t a i n t y regarding the r e l a t i v e p e r t u r b a t i v e strengths of v i b r o n i c c o u p l i n g v e r s u s low-symmetry l i g a n d f i e l d p o t e n t i a l s i n i n f l u e n c i n g the d e t a i l e d n a t u r e of the s p e c t r o s c o p i c s t a t e s i n a m e t a l complex. I n f a c t , i n some c a s e s s t r o n g v i b r o n i c c o u p l i n g c a n s e r v e t o q u e n c h o r m o d e r a t e l i g a n d f i e l d e f f e c t s , and i n o t h e r c a s e s i t c a n l e a d to a m p l i f i c a t i o n of l i g a n d f i e l d e f f e c t s (13). Clearly, in d e a l i n g w i t h t h e c h i r o p t i c a l s p e c t r a o f t r a n s i t i o n m e t a l com p l e x e s i t i s i m p o r t a n t t o be c o g n i z a n t o f t h e p o s s i b l e i n f l u e n c e s o f v i b r o n i c c o u p l i n g upon t h e s p e c t r a l d e t a i l s . In p a r t i c u l a r , g r e a t c a r e must be e x e r c i s e d i n m a k i n g s p e c t r a - s t r u c t u r e c o r r e l a t i o n s b a s e d on t h e a s s i g n m e n t o f s p e c i f i c s p e c t r a l f e a t u r e s t o well-defined (pure)electronic transitions. Concern about the p o s s i b l e i n f l u e n c e of v i b r o n i c i n t e r a c t i o n s upon t h e CD s p e c t r a o f t r a n s i t i o n m e t a l c o m p l e x e s was f i r s t e x p r e s s e d by R.G. D e n n i n g ( 1 4 ) . D e n n i n g p r o p o s e d t h a t t h e * Τ excited state of C o ( e n ) undergoes a s t r o n g ( t e t r a g o n a l ) J a h n - T e l l e r d i s t o r t i o n v i a c o u p l i n g t o an eg v i b r a t i o n a l mode o f t h e C o N c l u s t e r . This s t r o n g t e t r a g o n a l JT d i s t o r t i o n was t h e n presumed t o be e f f e c t i v e i n "quenching" the c r y s t a l f i e l d induced t r i g o n a l s p l i t t i n g of t h e T x g s t a t e ( a m a n i f e s t a t i o n o f t h e s o - c a l l e d Ham e f f e c t ) i n C o ( e n ) 3 + . D e n n i n g (14) f u r t h e r s u g g e s t e d t h a t t h e two CD b a n d s observed i n the r e g i o n of the A •> T t r a n s i t i o n i n Co(en)^"** a r i s e f r o m two d i f f e r e n t JT v i b r o n i c s t a t e s d e r i v e d f r o m ^ l g - e g c o u p l i n g , r a t h e r t h a n f r o m t h e two t r i g o n a l components (*Ε and A) of the T e l e c t r o n i c s t a t e . The i n f l u e n c e o f J a h n - T e l l e r and pseudo J a h n - T e l l e r i n t e r a c t i o n s upon t h e CD s p e c t r a o f t h e d-d t r a n s i t i o n s i n t r a n s i t i o n m e t a l c o m p l e x e s has been s t u d i e d i n c o n s i d e r a b l e d e t a i l ( t h e o r e t i c a l l y ) by R i c h a r d s o n and c o w o r k e r s ( 1 5 , 16, 17, 18, 1 9 ) . These l a t t e r s t u d i e s i n c l u d e d c o n s i d e r a t i o n of m e t a l complexes belonging to t r i g o n a l l y symmetric s t r u c t u r a l c l a s s e s ( 1 6 ) , as w e l l a s m e t a l c o m p l e x e s o f pseudo-tetragonal symmetry ( 1 5 , 17, 1 8 ) . The m a i n c o n c l u s i o n o f t h e s e s t u d i e s was t h a t w h e r e a s v i b r o n i c i n t e r a c t i o n s o f t h e JT and PJT t y p e s ( w i t h i n t h e m a n i f o l d o f d-d e x c i t e d s t a t e s ) w i l l n o t , i n g e n e r a l , a l t e r t h e n e t ( o r t o t a l ) d-d_ r o t a t o r y s t r e n g t h for a given sys tem, t h e y c a n p l a y a d o m i n a n t r o l e i n d e t e r m i n g how CD i n t e n s i t y i s d i s t r i b u t e d t h r o u g h o u t t h e d.-d t r a n s i t i o n r e g i o n . I t was f o u n d t h a t i n t h e p r e s e n c e o f s t r o n g JT and PJT i n t e r a c t i o n s among t h e d-d_ s t a t e s , i t becomes i m p o s s i b l e ( o r m e a n i n g l e s s ) t o a s s i g n s p e c i f i c f e a t u r e s i n t h e CD s p e c t r a t o s p e c i f i c ( i - d e l e c tronic transitions. The i n d i v i d u a l CD b a n d s , i n s u c h c a s e s , w i l l g e n e r a l l y r e f l e c t "mixed" e l e c t r o n i c parentage. 1 β

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I n t h o s e c a s e s where t h e a d i a b a t i c a p p r o x i m a t i o n can be a s sumed t o h o l d , t h e i n f l u e n c e o f v i b r o n i c c o u p l i n g on t h e s p e c t r o s c o p i c p r o p e r t i e s o f a s y s t e m c a n be t r e a t e d w i t h i n t h e H e r z b e r g T e l l e r ( p e r t u r b a t i v e ) f o r m a l i s m (20) f o r v i b r o n i c i n t e r a c t i o n s . T h i s f o r m a l i s m i s a p p l i c a b l e when t h e v i b r o n i c i n t e r a c t i o n e n e r g i e s a r e s m a l l compared t o t h e e n e r g y s p a c i n g s between t h e



3.

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Dichroic

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coupled e l e c t r o n i c s t a t e s . V i b r o n i c r o t a t o r y strengths and c i r c u l a r dichroism spectra, considered w i t h i n the a d i a b a t i c a p p r o x i mation and the Herzberg-Teller(HT) formalism, have been treated i n considerable d e t a i l by Weigang and coworkers ( 2 1 , 2 2 , 2 3 ) . Weigang concentrated p r i m a r i l y on the CD of organic chromophores i n h i s a p p l i c a t i o n s of the theory. V i b r o n i c a l l y induced coupling of the d_-d spectroscopic s t a t e s t o odd-parity (ungerade) e l e c t r o n i c states of t r a n s i t i o n metal complexes plays a s i g n i f i c a n t (and sometimes dominant) r o l e i n determining the observed d i p o l e strengths and absorption i n t e n s i t i e s of d-d t r a n s i t i o n s . The poss i b l e i n f l u e n c e of these v i b r o n i c i n t e r a c t i o n s upon d-d r o t a t o r y strengths has been considered q u a l i t a t i v e l y by M.J. Harding ( 2 4 ) using Weigang s theory. However, no d e t a i l e d or q u a n t i t a t i v e studies have been reported on t h i s problem. Bray, Ferguson, and Hawkins (25) have a p p l i e d the v i b r o n i c coupling model o f P e r r i n and Gouterman (26) i n t h e i r i n t e r p r e t a t i o n o f the CD spectra produced by the coupled ligand->ligand (molecular exciton) t r a n s i t i o n s i n t r i s complexes of 1,10-phenant h r o l i n e and 2 , 2 - b i p y r i d i n e with Zn(II) and N i ( I I ) . This problem i n v o l v i n g a trimer comprised of three t r i g o n a l l y disposed i n t e r a c t i n g monomer u n i t s ( l i g a n d chromophores) i s formally analogous t o the problem i n v o l v i n g a t r i g o n a l l y perturbed T i g s t a t e (treated by Richardson, e t a l . , (16) f o r Co(en)3+). In the weak ( v i b r o n i c ) coupling l i m i t , e l e c t r o n i c r o t a t o r y strengths are well-defined; whereas i n the strong ( v i b r o n i c ) coupling l i m i t , one can only speak of v i b r o n i c r o t a t o r y strengths of mixed e l e c t r o n i c composition. In the present study, we s h a l l re-examine the theory of v i b r o n i c coupling i n c h i r a l t r a n s i t i o n metal complexes as i t pert a i n s t o r o t a t o r y strength c a l c u l a t i o n s and to the i n t e r p r e t a t i o n of the observed CD spectra f o r these systems. We s h a l l focus p r i m a r i l y on the d-d ( l i g a n d - f i e l d ) t r a n s i t i o n s , and s h a l l cons i d e r v i b r o n i c coupling both w i t h i n the manifold of and Q6(t2 )« a t r i g o n a l l y d i s t o r t e d (D ) MLe c l u s t e r , ten normal coordinates are required to describe the 15 v i b r a t i o n a l degrees-of-freedom. We s h a l l denote the normal coordinates o f the n

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e

a

a

n

d

Q3e( >> Q ^ ( 2 ) , Qi* (> Q5 ( >> Q6 < l)> Qee(e). If the t r i g o n a l d i s t o r t i o n i s assumed small, then we can expect the t r i g o n a l modes to r e f l e c t strong octahedral parentage. Thus, f o r example, the Q 3 ( a ) and Q 3 ( e ) t r i g o n a l modes are expected to c o r r e l a t e s t r o n g l y with the Q 3 ( t ] ) octahedral mode. We s h a l l w r i t e the v i b r a t i o n a l - e l e c t r o n i c ( v i b r o n i c ) Hamiltonian of the system as a

e

a

a

e

2

a

e


u

H(r,Q) = H (r,Q) + T ( Q ) , e

(5)

V

where T ( Q ) i s the k i n e t i c energy operator f o r n u c l e a r v i b r a t i o n a l motion w i t h i n the MLg c l u s t e r , and H (r,Q) i s the e l e c t r o n i c Hamiltonian operator d e f i n e d by V

£

H (r,Q) = T ( r ) + V(r,Q). e

(6)

e

In Eq. ( 6 ) , T ( r ) i s the k i n e t i c energy operator f o r the e l e c trons and V(r,Q) i s the t o t a l p o t e n t i a l energy operator f o r the system. The c o l l e c t i o n of e l e c t r o n coordinates i s denoted by {r} and the c o l l e c t i o n of normal coordinates f o r the MLg c l u s t e r i s denoted by {Q}. Expanding V(r,Q) i n the normal coordinates {Q} about the e q u i l i b r i u m nuclear c o n f i g u r a t i o n of the MLg c l u s t e r , we may w r i t e e

V(r,Q) = V°(r) + Σ V^Q

a

+ ft) Σ Σ V£ Q Qg+ a

a

,

(7)

a 3 2

where V£ - [3V(r,Q)/3Q ] , VJJ = [8 V(r,Q)/3Q 9Qg] , and α and 3 l a b e l normal coordinates of the MLg system. The term V ( r ) represents the p o t e n t i a l energy of the complete system with the n u c l e i clamped i n t h e i r e q u i l i b r i u m p o s i t i o n s . The operator V°(r) has t r i g o n a l d i h e d r a l ( D 3 ) symmetry and may be w r i t t e n as a

Q

V°(r) = V° + V°,

3

a

Q

(8)

where V§ i s the octahedral (Oh) p a r t of V°(r) and v£ i s the t r i g o n a l d i h e d r a l ( D 3 ) part of V°(r). The ungerade components of VÇ r e f l e c t the c h i r a l i t y of the system. The second and t h i r d terms i n Eq. ( 7 ) are v i b r o n i c c o u p l i n g terms and t h e i r i n f l u e n c e on the e l e c t r o n i c p r o p e r t i e s of*the system w i l l be t r e a t e d by p e r t u r b a t i o n techniques. In our perturbat i o n treatment we define the zeroth-order e l e c t r o n i c Hamiltonian to be H

0 ( ) = T ( r ) + V°(r), r

£

(9)

with e i g e n f u n c t i o n s ψη obtained as s o l u t i o n s to the Schrodinger equation Η°(Γ)Ψ°(Γ) ε η

= EηVη( r ) .

(10)


3.

RICHARDSON

Circular

Dichroic

49

Intensities

In the s o - c a l l e d "crude a d i a b a t i c a p p r o x i m a t i o n " , the o r d e r v i b r o n i c wave f u n c t i o n s may be w r i t t e n a s

where t h e v i b r a t i o n a l wave f u n c t i o n s φ tions to [ T ( Q ) + V°(r)

ν

(Q) a r e f o u n d a s s o l u

+ ft) Σ (£] * ( Q ) = W ^ Q ) . (12) α I n w r i t i n g Eq. (12) we h a v e assumed t h e h a r m o n i c a p p r o x i m a t i o n f o r v i b r a t i o n a l m o t i o n s o f t h e MLg n u c l e i . In o u r p e r t u r b a t i o n treatment o f v i b r o n i c i n t e r a c t i o n s t h e wave f u n c t i o n s d e f i n e d b y Eq. (11) c o m p r i s e o u r z e r o t h - o r d e r b a s i s s e t , and the p e r t u r b a t i o n H a m i l t o n i a n i s g i v e n by v


η

zeroth-

n v

H'(r,Q) = H ( r , Q ) - H°(r)

(13)

£

or, H'(r,Q) = Σ ν ; Q + ft) Σ Σ Q Q (14) a a 3 i n c l u d i n g terms l i n e a r and q u a d r a t i c i n the normal c o o r d i n a t e s {Q}. The " p e r t u r b e d " v i b r o n i c wave f u n c t i o n s may be e x p r e s s e d a s a

V J

Σ

a

*

η ν

( 1 5 ) J

'

where t h e e x p a n s i o n c o e f f i c i e n t s , C J , a r e t o b ef o u n d b y d i a g o n a l i z i n g (H9 + H ) i n t h e b a s i s s e t {ψ£ ( r ) φ (Q) }. j I 1 V

T

η ν

B. C o ( I I I ) S y s t e m s . The m o d e l d e s c r i b e d a b o v e ( i n S e c t i o n II.A.) i s a p p l i c a b l e t o any s i x - c o o r d i n a t e system o f t r i g o n a l d i h e d r a l ( D 3 ) symmetry. To i l l u s t r a t e t h e a p p l i c a t i o n s o f t h e m o d e l , we c o n s i d e r h e r e s i x - c o o r d i n a t e C o ( I I I ) c o m p l e x e s o f D3 symmetry i n w h i c h t h e C o ( I I I ) i o n r e s i d e s i n a " s t r o n g " c r y s t a l field. I n t h i s c a s e , the ground s t a t e o f the complex i s nond e g e n e r a t e w i t h A i ( A i g ) symmetry a n d t h e ^ - e l e c t r o n ( s i n g l e t ) e x c i t e d s t a t e s a r e o f symmetry t y p e s A ( T i g ) , E ( T i ) , Α χ ( T ) a n d E(T ). A schematic energy l e v e l diagram i s given i n F i g u r e 1 f o r t h e s e d - e l e c t r o n s t a t e s a n d f o r two a d d i t i o n a l s t a t e s o f T i o c t a h e d r a l p a r e n t a g e , A ( T i ) a n d E ( T i ) . The l a t t e r s t a t e s may be assumed t o b e d e r i v e d e i t h e r f r o m metal«-*\Ligand c h a r g e - t r a n s f e r e x c i t a t i o n s o r f r o m a d ^ m e t a l i o n c o n f i g u r a t i o n . The e l e c t r i c d i p o l e (ED) a n d m a g n e t i c d i p o l e (MD) s e l e c t i o n r u l e s g o v e r n i n g t r a n s i t i o n s between t h e A i ( A i g ) g r o u n d s t a t e and t h e e x c i t e d s t a t e s shown i n F i g u r e 1 a r e summarized a s f o l l o w s : 2

g

2 g

2 g

u

2

u

u

5




Octahedral (0 ) h

Figure 1.

Trigonal

Dihedral

3

Schematic energy-level diagram for a Co(III) complex


3.

RICHARDSON

Circular

Excited

Dichroic

State

2

D3

Ok

A (Ti ) Ε (TxJ) Ai(T ) Ε (Tog) A (Ti ) Ε (T ) g

51

Intensities

MD MD

MD(z), ED(z) MD(x,y), ED(x,y)

ED ED

MD(x,y), ED(x,y) MD(z), E D ( z ) MD(x,y), ED(x,y)

2 g

2

u


l u

where t h e p o l a r i z a t i o n d e s i g n a t i o n s ( x , y , z ) r e f e r t o a t r i g o n a l c o o r d i n a t e s y s t e m ( w i t h t h e ζ-axis c o i n c i d e n t w i t h t h e C3 sym metry a x i s o f t h e system). Assuming s t r o n g o c t a h e d r a l parentage, t h e A i ( A i g ) - > A 2 ( T i ) a n d E ( T i ) t r a n s i t i o n s may be e x p e c t e d t o e x h i b i t r e l a t i v e l y s t r o n g o p t i c a l a c t i v i t y a n d o n l y weak a b s o r p t i v i t y , w h e r e a s t h e k\ ( A i ) - > A 2 ( T i ) a n d E ( T i ) t r a n s i t i o n s may be e x p e c t e d t o e x h i b i t s t r o n g o p t i c a l a c t i v i t y a n d s t r o n g a b s o r p tivity. The Αχ(Aig)->Ai(T2 ) a n d E ( T ) t r a n s i t i o n s w o u l d b e e x p e c t e d t o e x h i b i t r e l a t i v e l y weak o p t i c a l a c t i v i t y a n d weak a b sorptivity. These e x p e c t a t i o n s b a s e d on symmetry s e l e c t i o n r u l e s r e f l e c t i n g s t r o n g o c t a h e d r a l parentage a r e g e n e r a l l y borne o u t by e x p e r i m e n t a l o b s e r v a t i o n . I n o u r a n a l y s i s o f t h e m o d e l C o ( I I I ) s y s t e m s we s h a l l assume s t r o n g o c t a h e d r a l parentage f o r t h e e l e c t r o n i c s t a t e s and f o r t h e CoLe v i b r a t i o n a l modes. We s h a l l f u r t h e r assume t h a t t h e g r o u n d e l e c t r o n i c s t a t e A i ( A i g ) remains u n a f f e c t e d by v i b r o n i c i n t e r a c tions. Given these assumptions, t h e p r i n c i p a l i n f l u e n c e s o f t h e Q 2 ( e ) a n d Q s ( t 2 ) v i b r a t i o n a l modes a r e t o ( 1 ) c a u s e J a h n - T e l l e r (JT) and pseudo j a h n - T e l l e r (PJT) d i s t o r t i o n s w i t h i n t h e T i , T , and T i e x c i t e d s t a t e s , a n d ( 2 ) i n d u c e m i x i n g between t h e T i and T2ç e x c i t e d s t a t e s . These e f f e c t s w i l l b e m a n i f e s t e d i n t h e i n t e n s i t y d i s t r i b u t i o n s w i t h i n theAi -*Ti a n d A i - * T 2 g d.-d t r a n s i t i o n s , b u tthey w i l l n o t s i g n i f i c a n t l y a l t e r t h e n e t (or t o t a l ) CD a n d a b s o r p t i o n i n t e n s i t i e s o f t h e s e t r a n s i t i o n s . On t h e o t h e r hand, t h e Q 3 ( t ) , Q i * ( t i ) , a n d Q e ( t 2 u ) v i b r a t i o n a l modes w i l l b e e f f e c t i v e i n m i x i n g t h e T i e x c i t e d s t a t e w i t h t h e T^g a n d T?_g excited states. This w i l l lead t o a r e d i s t r i b u t i o n of e l e c t r i c d i p o l e i n t e n s i t y o u t o f t h e Aig->-Ti t r a n s i t i o n a n d i n t o t h e Aig-KTig a n d Aig->T2g d_-d t r a n s i t i o n s . I n what f o l l o w s , we s h a l l f i r s t e x a m i n e t h e i n f l u e n c e o f T * ( t g + e g ) c o u p l i n g on t h e CD s p e c t r u m o f t h e C o ( I I I ) A - > T t r a n s i t i o n (neglecting a l l other v i b r o n i c i n t e r a c t i o n s ) . Secondl y , we s h a l l e x a m i n e t h e i n f l u e n c e o f t h e t 2 ( Q s ) a n d e ( Q 2 ) v i b r a t i o n a l modes o n t h e A i - > T 2 CD s p e c t r u m v i a v i b r o n i c a l l y i n duced Tig-T2g m i x i n g s . F i n a l l y , we s h a l l c o n s i d e r T - T a n d 2 g ~ l u mixings under t h e i n f l u e n c e o f v i b r o n i c i n t e r a c t i o n s w i t h t h e t i ( Q 3 a n d Qi+) a n d t 2 ( Q 6 ) v i b r a t i o n a l modes. g

g

g

u

u

g

g

2 g

g

g

u

2 g

g

g

l u

g

u

u

u

l g

2

lg

g

g

g

g

l g

T

lg

l u

T

u

U

C Tig*(t?g+eg) Coupling. To t e r m s l i n e a r i n Q , t h e v i b r o n i c H a m i l t o n i a n f o r t h i s c a s e may be w r i t t e n a s a


52

STEREOCHEMISTRY

H ( r , Q ) = H°(r) + ( h V

+ 5e Q e +

5

2

+ h

5

+ Vj _Q

+

e

e

5 e

+ h

)

5 a

+ (V£

_ + V| Q e

5 a

+

OF TRANSITION METALS

Q

+ V _Q _

2 +

2

2

),

(16)

w h e r e H°(r) i s d e f i n e d by E q . ( 9 ) , h i s the harmonic o s c i l l a t o r H a m i l t o n i a n f o r t h e α-th t r i g o n a l v i b r a t i o n a l mode, and V^C^ i s t h e l i n e a r v i b r o n i c c o u p l i n g t e r m f o r t h e Q mode. The + and s u b s c r i p t s d e n o t e components o f t h e d o u b l y d e g e n e r a t e v i b r a t i o n a l modes, Q and Q5 . I t i s understood that a l l of the operators a p p e a r i n g i n Eq. (16) a r e d e f i n e d f o r t h e t r i g o n a l l y d i s t o r t e d T i g ( A + E) e l e c t r o n i c s t a t e o f t h e C o ( I I I ) m o d e l s y s t e m . Having d e f i n e d our v i b r a t i o n a l c o o r d i n a t e s w i t h respect t o t h e t r i g o n a l l y d i s t o r t e d e q u i l i b r i u m g e o m e t r y o f t h e CoLg c l u s t e r , V £ = 0 a n d t h e l i n e a r c o u p l i n g t e r m V"5 Q v a n i s h e s i n E q . ( 1 6 ) . We d e n o t e t h e t r i g o n a l components o f t h e T^g e l e c t r o n i c s t a t e by ψ ° ( Α ) , ψ°_(Ε), and ψ°(Ε). These f u n c t i o n s a r e e i g e n f u n c t i o n s o f t h e o p e r a t o r H ° ( r ) , a n d t h e y have t h e symmetry p r o p e r t i e s C * g = i|/g, 0 ψ°_ = ωψ°_, and 0 ψ ° = ω*ψ°_, where ω = e x p ( 2 i r i / 3 ) and C i s t h e t h r e e f o l d r o t a t i o n o p e r a t o r o f t h e D p o i n t group. The d o u b l y d e g e n e r a t e v i b r a t i o n a l modes may be c o n v e n i e n t l y expressed i n p o l a r c o o r d i n a t e form as a


a

2

e

2

a

a

5a

2

3

3

3

3

Q

3

= p exp(i), Q -

2 +

2

2

= p exp(-i(|>) 2

and, Q5e+

=

e x

f

P5e P(^ )»

=

Q5 -

P5e

e

e x

!

P(-^ )

with C Q = o)Q +, C Q _ = ω*(} _, a n d s i m i l a r l y f o r Q s + a n d Qse-The e f f e c t s o f t h e Q and Q s v i b r a t i o n a l modes w i l l be t o r e n d e r t h e Ε(ψ°_,ψ°_) e l e c t r o n i c s t a t e J a h n - T e l l e r u n s t a b l e and t o c o u p l e t h e Ε(φ°_,ψ9:) a n d Α ( ψ ° ) s t a t e s v i a a p s e u d o J a h n - T e l l e r mechanism. The v i b r o n i c wave f u n c t i o n f o r t h e l g * ( 2 g g ) s y s t e m may be e x p r e s s e d a s 3

2 +

2

3

2

2

2

e

e

2

T

* = + + X ^

}

t

+

+

xs?

) .

e

+

xS?) (17)

The v i b r o n i c e n e r g y l e v e l s and v i b r a t i o n a l a m p l i t u d e f u n c t i o n s Οία » Χ α ^ > * Χα ^ » where α = 2,5e, o r 5a) may be f o u n d by s o l v i n g t h e s e c u l a r e q u a t i o n , Eq. ( 1 8 ) . I n t h i s m a t r i x e q u a t i o n , +

and

a n (

Η°(χ)ψ? = Ε?φ2,

(19)

Η ? ( Γ ) Ψ ° = E°X

(20)

t h e i n t e r a c t i o n m a t r i x elements, E^j and T y , E

ij

=

< ψ

°

V

Q

I * + 2 + + ν£_0. _|ψ°>, 2

a r e g i v e n by (21)


RICHARDSON

+ ^ CM Χ


Ο w CM , Χ

Circular

I VCM X

Dichroic

w

Intensities

0 0 > + < l ) l < l ) O c O + c O m w m w i n w i n w i n X X X X X

I cO

m

cO

m f ο ο w ι

+

κ

+

ω m

m

ο ο

+ +

α) m

ί ο ο

CN

eg

+ ο

Ο I

ι° °



54 and, T

=

ij

22

îê+Qse-H + n _Q e-U°>e

< >

5

Synnnetry a r g u m e n t s r e q u i r e t h a t = Ε = o-

E


*>

τ

+

-o

=

= Τ = ο-

+0

E

E

-

E

=

o+

Y 2 Î ! 2 +

= T*

=

ex

i

Y2P2 P( t )»

(24)

= Y Q5e+ - Ϊ5 Ρ5ββχρ(1Φ'),

+

5e

β

(25)

k Q _ = k p exp(-i), 2

2

2

2

and,

* k5 Q5 e

=

ksePBe^Pt

e

- 1

(26)

1

* )'

where t h e l i n e a r c o u p l i n g c o n s t a n t s f o r t h e P J T i n t e r a c t i o n s , are d e f i n e d by Ύα = < Φ + Ι ( ν / 3 Ρ ) | φ ο > = < Ψ ? Κ | Ψ Ο > , « = 2 o r 5e. 3

α+

The l i n e a r c o u p l i n g c o n s t a n t d e f i n e d by

f o r the JT interactions,

k , are a

= , α Ξ 2 o r 5e.

k a - a

α

(27)

+

0

γ ,

(28)

α

The m a t r i x e q u a t i o n ( 1 8 ) i s b l o c k - d i a g o n a l ( a s shown) o n l y i f t h e c o u p l i n g modes, α = 2,5e, a n d 5 a , a r e m u t u a l l y o r t h o g o n a l . The t w o - d i m e n s i o n a l h a r m o n i c o s c i l l a t o r H a m i l t o n i a n s f o r t h e α = 2 a n d 5e modes a r e g i v e n b y h

Ι 0 / 3 ρ £ ) + U/Pa> 0/*Ρα>

ω

a

2

- (^> αΡα "

+ (1/ρ ) 0 /3φ )]. α

The

2

2

2

eigenfunctions of h Va.vA

=

ε

X

may be o b t a i n e d f r o m s o l u t i o n s t o

a

α,ν)1 a , v A

(29)

=

(v

a

+1)

K

\,vi

(

3

0

)

where t h e quantum numbers ν a n d £ c a n t a k e on t h e v a l u e s , ν = 0,1,2,··· and £ = v,v-2,···,-v. F o r α = 5 a , we w r i t e

h x

Α

α α,ν

- ε χ α,ν α , ν

= ( v + %)ηω χ α ' α α,ν

Λ

Λ

(31)

where h i s a o n e - d i m e n s i o n a l h a r m o n i c o s c i l l a t o r H a m i l t o n i a n expressed i n terms o f Q s . I t i s now c o n v e n i e n t t o e x p a n d t h e v i b r a t i o n a l a m p l i t u d e f u n c t i o n s (χ(°), χ ^ , a n d X^~0 a p p e a r i n g i n E q s . ( 1 7 ) a n d ( 1 8 ) i n terms o f b a s i s s e t s c o n s t r u c t e d from t h e e i g e n s o l u t i o n s o f Eqs. ( 3 0 ) a n d ( 3 1 ) . Upon d o i n g t h i s , t h e e i g e n s o l u t i o n s o f E q . (18) may b e e x p r e s s e d a s a

a


3.

RICHARDSON

ψ

= J

Circular

ΑΣ

A

Dichroic

(Ο

(o) X

jvil 2,v&

+

Σ

vil

° A

55

Intensities

Β^

θ )

χί

+

θ )

A

jv£ 5e,vil

Σ

c< x< o )

0 )

A

j v 5a,v

ν

V

v +ψ°(Σ A jvil 2,vil v£

+

+

+

( + )

( + )


A

Ψ°(Σ

A - y jviT2,v£ (

vil

)

(

)

Σ

vil Σ

vil

+

A

jvil 5e,v£

Σ

c

ν

+

A

jvil 5e,v£

Σ

ν

W

x
^ (where λ = o,-, or + correspond to e l e c t r o n i c states ψ§, ψ ° , and ψ°_, r e s p e c t i v e l y ) are found by s o l v i n g Eq. (18). The v i b r a t i o n a l f u n c t i o n s , X^vSL (a = 2 or 5e), a r e obtained from Eq. (30) f o r the appropriate λ(ο,-, or +) and the v i b r a t i o n a l f u n c t i o n s , X ^ ^ v ' from Eq. (31) f o r the appropriate λ. For a more complete d e s c r i p t i o n of the v i b r o n i c l e v e l s associated with the T i ( A 2 + E) e l e c t r o n i c e x c i t e d s t a t e , the wave functions »j[Eq. 132)] must be augmented to include v i b r a t i o n a l wave functions associated with a l l of the normal modes other than α = 2,5a, and 5e. Here we s h a l l r e s t r i c t our a t t e n t i o n to j u s t those v i b r o n i c l e v e l s derived from the v i b r a t i o n a l modes α = 2,5a, and 5e. Assuming no v i b r o n i c couplings i n v o l v i n g the ground e l e c t r o n i c s t a t e of our model system, the r o t a t o r y strength of a t r a n s i t i o n between the lowest v i b r a t i o n a l l e v e l o f the ground e l e c t r o n i c s t a t e and the j - t h v i b r o n i c l e v e l of the T}g*(t2g + eg) coupled s t a t e may be w r i t t e n as B

a n
- j g go'- j j '-'Vgo r

Y

1

>

(33)

where ψ? i s the e l e c t r o n i c wave f u n c t i o n f o r the Α ι ( Α ^ ) ground state o f our model system and g i s the v i b r a t i o n a l wave f u n c t i o n f o r the ground v i b r a t i o n a l l e v e l o f the ground e l e c t r o n i c state. In expanded form, Eq. (33) may be w r i t t e n as 0

β

R. - Σ Η < > [ Σ Σ λ α,ν,Α α · , ν \ 4 ' J

λ

X

D< > ^

ν Α ( α )

X

J

DÎ >* ,, * > V

(

α

Χ S^golvDS^v'A'Igo)],

W

where, R.

(e)

= Ι*·

(35)

Φ

—

—

Λ g Λ Λ g defines the purely e l e c t r o n i c r o t a t o r y strength the ψ° -> ψ° (λ = ο,-, or +) t r a n s i t i o n and S λ S (go|viQ = < ! χ > ( X )

( g )

X

( λ )

associated

with

(36)



56

i s a Franck-Condon o v e r l a p i n t e g r a l o v e r the ground s t a t e v i b r a t i o n a l f u n c t i o n XQ[ ο the e x c i t e d s t a t e v i b r a t i o n a l f u n c t i o n *a^v£ f ° a - t h ' n o r m a l mode ( a = 2,5a, o r 5e and λ = ο,-, o r +)! For α = 5a, £ = £ = o f o r a l l v a l u e s of v. The c o e f f i c i e n t s a

r

t

n

n

d

e

f


D

jv!(a)

a

D

( X )

D

( X )

r

ed

e

f

i

n

=

jv£(2)

e

d b

:

( λ )

A

Α jv£'

=

jv£(5e)

v

Β

( λ )

*jvV

and, η

(λ) jv£(5a)

β

Γ L

(λ) jv ' QA$

X

where t h e c o e f f i c i e n t s A j £ , B j ^ , and Eq.

(32).

The

complex c o n j u g a t e

a r e as d e f i n e d

of D^!,( )

i

s

a

denoted

in

by

D

gv^(a)' I f we make t h e s i m p l i f y i n g a s s u m p t i o n t h a t t h e v i b r a t i o n a l wave f u n c t i o n s o f t h e g r o u n d ( g ) and e x c i t e d ( X ) e l e c t r o n i c s t a t e s a r e i d e n t i c a l , t h e n Eq. (34) r e d u c e s t o R.

J

= R

( e )

[|Af

ο

0 )

joo'

1

|2

+ | f B

0

)

| 2 + |C^ |2]

joo

1

0 )

1

JO

1

1

^ l„ T i ( A + E) e l e c t r o n i c t r a n s i t i o n i s d i s t r i b u t e d among t h e v i b r o n i c l e v e l s d e r i v e d f r o m ( A + E ) * ( a i + 2e) c o u p l i n g s . The t o t a l (ornet) r o t a t o r y strength of the A i T i transition i s , t h e r e f o r e , given by i g

g

2

2


g

R(Aig

T

l g

) = Σ R. = R

( e )

+ RJ

e )

g

e )

+ R^ ,

(39)

e;

where R j i s d e f i n e d b y Eq. ( 3 4 ) a n d t h e R J ^ ( X = o,-, o r +) a r e d e f i n e d b y E q . ( 3 5 ) . The " s t a t i c " s t e r e o c h e m i c a l a n d s t r u c t u r a l f e a t u r e s o f t h e m e t a l complex d e t e r m i n e t h e s i g n and magnitude o f R ( A j g -> T i g ) a s w e l l a s t h e s i g n s and m a g n i t u d e s o f t h e e l e c t r o n i c r o t a t o r y s t r e n g t h s R ^ , R ^ ) , a n d R ^ ) . The c h i r a l a s p e c t s o f these " s t a t i c " s t r u c t u r a l f e a t u r e s a r e d e s c r i b e d by t h e p o t e n t i a l e n e r g y o p e r a t o r , V ^ ( r ) , o f E q . ( 8 ) . The t o t a l ( o r net) A i g T i g r o t a t o r y strength i s i n v a r i a n t t o T i * ( t + e ) vibronic coupling. Only t h e d i s t r i b u t i o n o f e l e c t r o n i c r o t a t o r y s t r e n g t h among t h e component v i b r o n i c t r a n s i t i o n s i s a f f e c t e d by the T i g * ( t g + e ) c o u p l i n g s . The v i b r o n i c c o u p l i n g m o d e l d e s c r i b e d i n t h i s s e c t i o n ( U . C . ) i s h i g h l y r e s t r i c t e d i n s o f a r a s o n l y l i n e a r c o u p l i n g terms have been i n c l u d e d e x p l i c i t l y , a n d o n l y two e - t y p e t r i g o n a l modes (those d e r i v e d from t h e t a n d e g o c t a h e d r a l CoLg c l u s t e r modes) h a v e been c o n s i d e r e d . I t i s r e l a t i v e l y easy t o extend o u r t r e a t ment t o i n c l u d e q u a d r a t i c c o u p l i n g t e r m s . However, t h e i n c l u s i o n o f a d d i t i o n a l e - t y p e t r i g o n a l modes ( e . g . , t h o s e d e r i v e d f r o m t h e t a n d t i u o c t a h e d r a l modes) w o u l d d r a s t i c a l l y c o m p l i c a t e o u r model. T h i s l a t t e r e x t e n s i o n o f t h e model s h o u l d n o t prove n e c e s s a r y so l o n g a s t h e CoLc c l u s t e r r e t a i n s v e r y n e a r l y o c t a h e d r a l symmetry ( i . e . , VÇ T i t r a n s i t i o n i n CoLg s y s t e m s i s m a g n e t i c d i p o l e a l l o w e d w h e r e a s t n e A i g -> T g t r a n s i t i o n i s m a g n e t i c dipole forbidden. U s i n g t h e n o t a t i o n o f F i g u r e 1, t h e t r i g o n a l (D3) s e l e c t i o n r u l e s s p e c i f y t h a t t h e A i ( A i ) -> A ( T i ) , E ( T i ) , and E b ( T g ) t r i g o n a l t r a n s i t i o n s a r e m a g n e t i c d i p o l e a l l o w e d , w h i l e t h e A i ( A i ) - A i ( T ) t r a n s i t i o n remains magnetic d i p o l e forbidden. I n the usual p e r t u r b a t i o n treatments of o p t i c a l a c t i v i t y i n t r i g o n a l d i h e d r a l C o ( I I I ) c o m p l e x e s , i t i s assumed t h a t t h e A i -*· E^ component o f t h e e r s t w h i l e A i -> T transition "borrows"magnetic d i p o l e c h a r a c t e r (and, t h e r e f o r e , r o t a t o r y s t r e n g t h ) f r o m t h e A i + E component o f t h e A ^ -> T i g t r a n s i t i o n . T h i s i s assumed t o o c c u r v i a T i - T m i x i n g induced by t h e g e r a d e components o f t h e t r i g o n a l f i e l d p o t e n t i a l , VS. By t h i s m o d e l , t h e s i g n a n d m a g n i t u d e o f t h e A i ( A i ) •+ E b ( T ; r o t a t o r y s t r e n g t h ( a n d CD) may b e c o r r e l a t e d d i r e c t l y t o t h a t o f t h e g

2

2 £

g

g

2

g

2

g

a

2

>

g

2 g

g

2 g

a

g

2 g

g

2 g


g

58


A

A

E

T

l( lg) a( lg) transition. The A T CD w o u l d n o t r e f l e c t any a s p e c t s o f t h e Αχ -* A component o f t h e A i g T i g t r a n s i t i o n . T h i s p i c t u r e i s a l t e r e d d r a s t i c a l l y when v i b r o n i c c o u p l i n g e f f e c t s a r e taken i n t o account. V i b r o n i c c o u p l i n g may e n t e r i n t o t h i s p r o b l e m i n s e v e r a l ways. F i r s t , b o t h t h e T i and T states are subject t o " i n t r a - s t a t e " c o u p l i n g s o f t h e types ( f o r example): T i * ( t g + e ) and T g * ( t + e ). Secondly, " i n t e r - s t a t e " c o u p l i n g o f t h e t y p e ( T i g + T ) * ( t g + e g ) may become i m p o r t a n t . We s h a l l e x a m i n e t h e p o s s i b l e i n f l u e n c e o f t h i s l a t t e r ( " i n t e r s t a t e " ) type o f c o u p l i n g here. T r i g o n a l e - t y p e modes ( d e r i v e d , f o r e x a m p l e , f r o m t h e t g o r e g o c t a h e d r a l modes) c a n e f f e c t t h e f o l l o w i n g " i n t e r - s t a t e c o u p l i n g s o f t r i g o n a l e l e c t r o n i c components: ( E + E )*e, ( E + A i ) * e , and ( A + Eb)*e. T h i s has t h e consequence t h a t v i b r o n i c components o f t h e A i ( A i g ) -> A i ( T ) t r a n s i t i o n c a n e x h i b i t a n o n v a n i s h i n g CD w h i c h h a s been " b o r r o w e d " f r o m t h e A i ( A i g ) + E ( T i g ) transition. F u r t h e r m o r e , v i b r o n i c components o f t h e A i ( A i g ) -> E ( 2g) t r a n s i t i o n may e x h i b i t CD " b o r r o w e d " f r o m b o t h t h e A i •> E and A i -* A t r i g o n a l components o f t h e A i g -> T i g t r a n s i t i o n . H e r e t o f o r e , t h e a p p e a r a n c e o f m u l t i p l e components i n t h e A i g •> T CD r e g i o n h a s a l w a y s been a t t r i b u t e d t o t h e p r e s e n c e o f m u l t i p l e s p e c i e s t y p e s o r t o a r e d u c t i o n o f symmetry ( f r o m D3 t o C 3 ) i n t h e complex under study. Strong T i - T m i x i n g under t h e i n f l u e n c e o f a n e - t y p e t r i g o n a l v i b r a t i o n a l mode i s , p e r h a p s , a more l i k e l y c a u s e f o r t h e a p p e a r a n c e o f m u l t i p l e components in the A -* T CD s p e c t r u m . Detailed studies of (Tig + T ) * ( t + e ) coupling to octa h e d r a l systems and o f ( A + E + A i + E )*e coupling i n trigonal s y s t e m s h a v e n o t y e t been c a r r i e d o u t . Such s t u d i e s i n c l u d i n g n u m e r i c a l c o m p u t a t i o n s w o u l d be e x t r e m e l y c o m p l e x . However, t h e q u a l i t a t i v e c o n s e q u e n c e s o f s u c h c o u p l i n g s o n t h e CD s p e c t r a o f c h i r a l t r i g o n a l s y s t e m s may be d i s c e r n e d r e a d i l y f o l l o w i n g t h e arguments g i v e n above. I n b r i e f , ( A + E + A i + E ) * e c o u p l i n g c a n be e x p e c t e d t o l e a d t o CD o f m i x e d s i g n and m i x e d p o l a r i z a t i o n i n t h e A i -> T transition region, reflecting the sign and p o l a r i z a t i o n p r o p e r t i e s o f t h e A i ( A i ) ->• A ( T i ) + E ( T i ) CD bands. l g

i g

2

g

g

2

g

2

2 g

g

2 g


2 g

2

2

a

D

a

2

2 g

a

D

T

a

2

2 g

g

i g

2 g

2 g

2

2

g

2

a

g

2

g

g

D

a

D

2 g

g

2

g

a

g

E. ( T i g + T i ) * ( t i u + t ) Coupling. This type o f coupling i s t h e b a s i s f o r t h e s o - c a l l e d H e r z b e r g - T e l l e r (HT) v i b r o n i c t h e o r y o f d-d i n t e n s i t i e s i n o c t a h e d r a l (Oh) t r a n s i t i o n m e t a l c o m p l e x e s . I n t h i s t h e o r y a p p l i e d t o CoL,6, t h e A i g Tigtransition i s as sumed t o g a i n e l e c t r i c d i p o l e c h a r a c t e r v i a v i b r o n i c a l l y i n d u c e d T i g - T i m i x i n g u n d e r t h e i n f l u e n c e o f an u n g e r a d e v i b r a t i o n a l mode ( o f e i t h e r t i or t symmetry). ( H e r e we s h a l l i g n o r e v i b r o n i c a l l y induced m i x i n g o f t h e A i g ground s t a t e w i t h ungerade e x c i t e d s t a t e s , and c o n s i d e r o n l y v i b r o n i c p e r t u r b a t i o n s on t h e excited state Ti .) The l o w - t e m p e r a t u r e A i + T i electric d i p o l e (absorption) spectrum i s , then, p r e d i c t e d t o c o n s i s t o f t h r e e p r o g r e s s i o n s based on f a l s e o r i g i n s l o c a t e d a t 0 - 1 ( 0 ) 3 ) , u

2 u

u

u

g

2

u

g

g


3.

RICHARDSON

Circular

Dichroic

59

Intensities

0 - 1 ( 0 ) ^ ) , a n d 0-1 (ως), where 0 ) 3 , ω^, a n d ως a r e t h e f u n d a m e n t a l f r e q u e n c i e s o f t h e Q 3 ( t i ) , Q t + ( t i ) , a n d Q6(t2u) n o r m a l modes, respectively. Strong Jahn-Teller d i s t o r t i o n s w i t h i n t h e T and/or T s t a t e s would, o f course, tend t o complicate t h i s simple t h r e e - p r o g r e s s i o n spectrum. I n t r i g o n a l d i h e d r a l c o m p l e x e s o f " n e a r " o c t a d e d r a l sym metry: T -> A + E a n d T i •+ + E (using the notation of F i g u r e 1 ) . I n these systems, t h e A (Αχ) -*· T ( A + E ) t r a n s i t i o n h a s i n h e r e n t e l e c t r i c d i p o l e s t e n g t h due t o t h e u n g e r a d e components o f t h e " s t a t i c " t r i g o n a l f i e l d p o t e n t i a l vÇ. However, a d d i t i o n a l e l e c t r i c d i p o l e c h a r a c t e r may be i n t r o d u c e d i n t o t h i s t r a n s i t i o n v i a ( A + E + A + E ' ) * e c o u p l i n g , where t h e c o u p l i n g mode i s an e - t y p e t r i g o n a l v i b r a t i o n . A complete treatment o f ( A + E + A + E')*e c o u p l i n g would r e q u i r e s i m u l taneous c o n s i d e r a t i o n o f i n t r a - s t a t e v i b r o n i c i n t e r a c t i o n s ( J a h n - T e l l e r a n d pseudo J a h n - T e l l e r i n t e r a c t i o n s w i t h i n b o t h T and T i ) a n d i n t e r - s t a t e v i b r o n i c i n t e r a c t i o n s ( o f t h e H e r z b e r g T e l l e r t y p e between T i a n d T i ) . F u r t h e r m o r e , a l l e - t y p e t r i g o n a l modes s h o u l d be c o n s i d e r e d i n d e p e n d e n t o f t h e i r o c t a h e d r a l parentage. I n t h e p r e s e n t t r e a t m e n t , we s h a l l a d o p t a much more r e s t r i c t e d m o d e l i n w h i c h o n l y i n t e r - s t a t e ( T i - T i ) i n t e r a c t i o n s a r e c o n s i d e r e d and i n which o n l y those e-type t r i g o n a l modes o f t i or t o c t a h e d r a l ancestry a r e taken i n t o account. T h i s l a t t e r r e s t r i c t i o n t o v i b r a t i o n a l modes d e r i v e d from t h e t i and t o c t a h e d r a l v i b r a t i o n s s h o u l d be a c c e p t a b l e so l o n g a s t h e ΟοΙ,ς c l u s t e r r e m a i n s v e r y n e a r l y o c t a h e d r a l a n d Ί° « V§. u

u

l g

l u

1

l g

2

a

u


L

2

2

&

a

G

L G

2

a

2

2

l g

u

g

u

g

u

u

2

2

u

u

u

Ύ

Given | A ) , and electronic We f u r t h e r 2

t h e a p p r o x i m a t i o n s c i t e d a b o v e , we t a k e | A ) , | E ) , | E ) t o be e i g e n f u n c t i o n s o f t h e t r i g o n a l l y s y m m e t r i c H a m i l t o n i a n d e f i n e d i n E q . ( 9 ) , H°(r) = T ( r ) + V°(r). define a ( l i n e a r ) v i b r o n i c i n t e r a c t i o n operator 2

a

?

£

H'(r,Q) = Σ v ; Q (40) a where Σ i s t a k e n o v e r t h e e - t y p e t r i g o n a l modes d e r i v e d f r o m t h e Q 3 ( t i ) , Q i + ( t i ) , a n d Q 6 ( t 2 ) o c t a h e d r a l modes. T h a t i s , a=3e,4e, a n d 6e ( s e e S e c t i o n I I . A . f o r n o t a t i o n ) . Restricting our c o n s i d e r a t i o n t o T i - T i i n t e r - s t a t e m i x i n g s and u s i n g f i r s t - o r d e r t i m e - i n d e p e n d e n t p e r t u r b a t i o n t h e o r y , we may e x p r e s s t h e p e r t u r b e d wave f u n c t i o n s o f i n t e r e s t a s f o l l o w s : a

α

u

u

u

g

u

|A >

= | A ) + Σ λ (Ε' , A ) | E ' ) Q a

I V

- |E ) + Σ À ( E \ E ) | E ' ) Q a

2

2

a

α

a

2

a

a

(41)

a

+ Σ X (A»,E )|A£)Q a a

a

a

(42)

where, (43)



60

i s a perturbation expansion c o e f f i c i e n t . (Note t h a t t h e p e r t u r b e d wave f u n c t i o n s a r e d e n o t e d b y p o i n t e d k e t s !>, w h e r e a s t h e u n p e r t u r b e d wave f u n c t i o n s a r e d e n o t e d b y r o u n d e d k e t s I ) ) . Having r e s t r i c t e d our p e r t u r b a t i o n treatment t o T i - T i i n t e r s t a t e i n t e r a c t i o n s , i t i s e a s y t o s e e t h a t t h e wave f u n c t i o n s e x p r e s s e d by E q s . ( 4 1 ) a n d (42) a r e a c t u a l l y good t o f i r s t - a n d second-order i n H . We s h a l l i g n o r e a l l v i b r o n i c p e r t u r b a t i o n s on t h e g r o u n d e l e c t r o n i c s t a t e o f o u r C o ( I I I ) m o d e l s y s t e m , so that: g

u


f

|A > = I Α χ ) .

(44)

X

U t i l i z i n g E q s . ( 4 1 ) , ( 4 2 ) , a n d ( 4 4 ) , t h e e l e c t r i c a n d magne t i c d i p o l e t r a n s i t i o n moments f o r t h e Αχ A and E t r a n s i t i o n s may be e x p r e s s e d ( t o s e c o n d - o r d e r ) a s 2

1

2

,

= (k \0\A ) l

a

f

+ Σ λ (E ,A )Q ^(A |0|E ) α

2

2

r

(45)

1

and = ( A i | 0 | E ) + Σ X ( E ' , E ) Q ( A | 0 | E ' ) 1

a

a

a

a

a

1

+ Σ X (A|,E )Q (A |Ô|Aj), a

Λ

a

a

(46)

1

a

Λ

where 0 = μ o r m. The r o t a t o r y s t r e n g t h s e x p r e s s e d t o s e c o n d - o r d e r , a r e g i v e n by R(A

X

-* A ) = R

R °Ul (

+

( o )

2

f o r these t r a n s i t i o n s ,

( A ! -> A ) 2

+ Ε')Σ Σ

α α

λ (E\A )X 2

Ut

•

OC

(E\A )Q 2

Ut

Q .

(47)

ΙΛ

and,

R(A - E ) = R °Ul + E ) (

X

a

a

+

R °Ul

+ E f ) Z Σ λα(Ε',Ε&)λαΙ (Ef ,Ea)QaQa,

(

+ R ^ U l + Α^)Σ Σ X ( A ^ , E ) X ( A ^ , E ) Q Q a a + I m [ ( A | î | E ) - ( E |m|A!) a

a

a f

a

a

a î

1

l

a

a

,

,

+ (A!|y|E )-(E |m|A )]ZX (E ,E )Q , α a

1

a

a

a

(48)

where R °U (

-v j ) = Ι*(ψ°|ί|ψ°)·(ψ°|η|ψ°).

(49)

Now l e t u s c o n s i d e r a s p e c i f i c v i b r o n i c t r a n s i t i o n l e a d i n g from t h e ground v i b r a t i o n a l l e v e l o f t h e ground e l e c t r o n i c s t a t e (go) t o t h e ν (β) v i b r a t i o n a l l e v e l o f t h e e x c i t e d e l e c t r o n i c


3.

RICHARDSON

Circular

Dichroic

Intensities

61

state (ev(3)). I n t h e p r e s e n t c o n t e x t , g r e f e r s t o t h e Αχ e l e c t r o n i c g r o u n d s t a t e and e may be e i t h e r t h e A o r E e l e c t r o n i c e x c i t e d s t a t e o f our C o ( I I I ) model system. 3 denotes t h e v i b r a t i o n a l mode c o n t a i n i n g t h e ν q u a n t a i n t h e v i b r o n i c l e v e l ev(3). D e n o t i n g t h e i n i t i a l a n d f i n a l v i b r a t i o n a l wave f u n c t i o n s b y Φ^ a n d Φ ( β ) , r e s p e c t i v e l y , we may w r i t e f o r t h e vibronic rotatory strengths 2


0

R

o,v(3Î

A l

β ν

* *> A

=

r

° *

(

A

-

i

2 Ι

Α

< Φ

)

8

+R^°Ui ·> Ε')Σ Σ λ (Ε',Α )λ

Φ

| 2

2

01

α α g o I Qa 'Ι e v /( 3^ ) e v/( 3 J) Qα Ι Ι go > , x

| Φ

,(Ε',Α )

2

ι Χ

a

Φ

Φ

(

»

0

5

0

>

and, R

, (Αι -> Ε ) = R ^ A i o,v(3; a 0

+

/

R °Ul (

+

Χ

1

I Q go' ^ a Ι e v/( 3\ ) 1

>Ι ν

| Φ

|

>|2

5 a e

,

(53)

F o r t h e c a s e 3=a, we

>

βν(α) ! α

>|^

$ E V ( E )

$ e v ( B )

- Ε')|λ (Ε·,Α )|2|.

(60)

0

I n t h e s p e c i a l c a s e where R ( U l

A ) = - R < ° U l -»• Ε a) y i n M o f f i t t ' s t h e o r y o f d-d o p t i c a l a c t i v i t y i n t r i g o n a l ( D 3 ) C o ( I I I ) complexes ( 9 , 1 2 ) , Eq. (60) r e d u c e s t o R

net χ

+

(Α ι x

8

2

+

Ti^) =

A

g'

[|λ (Ε»,Α )| α

R °Ul (

^

2

Σ { R

a ' go

α

2

(

° 1U I

+

a

s

f

E )

2

+ |λ (Ε',Ε )| ]

2

α

α

+ Ap|X (A^,E )| }. 2

a

a

(61)



64

Making t h e f u r t h e r assumption ( i n t h e s p i r i t o f M o f f i t t ' s simple model) t h a t R ( ° } A I + E ) = - R ( ° U X + A p , Eq. ( 6 1 ) f u r t h e r r e duces t o f

R

net


-

( A

V

*

lg

|λ (Ε·,Ε )| α

" 2

r

(

° ^

-

A

< $

^

o «'V l Q

g

net

( A

T

lg

- lg>

« 2'Vl ( A

- |λ (Ε',Α )|2].

β

α

+

R

net

( A

lg

2

(62)

2

N o t e t h a t t h e a s s u m p t i o n s made i n o b t a i n i n g quire that

R

[ | X

T

" l«> =

Eq. ( 6 2 ) a l s o r e

0

(

6

3

)

The s i m p l e f o r m a l a n a l y s i s o u t l i n e d a b o v e r e v e a l s several i n t e r e s t i n g a s p e c t s o f t h e A x •> T x CD p r e d i c t e d f o r c h i r a l C o ( I I I ) c o m p l e x e s o f t r i g o n a l d i h e d r a l symmetry. First, (A + E + A + E ) * e coupling can lead t o mixed p o l a r i z a t i o n w i t h i n t h e v i b r o n i c CD band e n v e l o p s o f b o t h t h e Αχ A and i E t r a n s i t i o n s . I n t h e absence o f v i b r o n i c c o u p l i n g , t h e Ai A t r a n s i t i o n would be p o l a r i z e d p a r a l l e l t o t h e t r i g o n a l a x i s o f t h e s y s t e m , w h e r e a s t h e Αχ ->• E t r a n s i t i o n w o u l d b e polarized perpendicular t o the trigonal axis. I n t h e presence o f v i b r o n i c c o u p l i n g , v i b r o n i c components o f b o t h p a r a l l e l a n d p e r p e n d i c u l a r p o l a r i z a t i o n may a p p e a r i n b o t h t h e Αχ •> A a n d Αχ •> E CD b a n d s . S e c o n d l y , f r o m E q s . ( 6 1 ) a n d ( 6 2 ) we s e e t h a t t h e n e t CD a s s o c i a t e d w i t h t h e A x T x t r a n s i t i o n need n o t v a n i s h e v e n when t h e z e r o t h - o r d e r n e t e l e c t r o n i c r o t a t o r y strength, R(°UI + A ) + R T x CD s p e c t r u m . g

g

f

2

a

2

2

A

a

2

a

2

a

g

2

g

a

2

a

f

g

g

F. Summary o f Q u a l i t a t i v e C o n c l u s i o n s . I nconsidering the i n f l u e n c e o f v i b r o n i c c o u p l i n g i n t h e d-d. CD s p e c t r a o f t r i g o n a l d i h e d r a l C o ( I I I ) s y s t e m s , we h a v e e x a m i n e d t h r e e g e n e r a l c a s e s . F i r s t we e x a m i n e d v i b r o n i c i n t e r a c t i o n s w i t h i n t h e T x ( A + E ) excited state o f Co(III). These i n t e r a c t i o n s a r e o f t h e J a h n T e l l e r a n d p s e u d o J a h n - T e l l e r t y p e s and r e q u i r e a n o n - a d i a b a t i c v i b r o n i c coupling formalism. The p r i n c i p a l e f f e c t s o f t h e s e i n t e r a c t i o n s a r e r e d i s t r i b u t i o n s o f r o t a t o r y s t r e n g t h among t h e v i b r o n i c components o f t h e Αχ A and E t r a n s i t i o n s . Secondly, we e x a m i n e d v i b r o n i c a l l y i n d u c e d m i x i n g s b e t w e e n t h e T x ( A + E ) and T ( A x + Εχ>) e x c i t e d s t a t e s . Q u a l i t a t i v e l y we showed t h a t ( A + E + Αχ + E ) * e c o u p l i n g c o u l d l e a d t o n o n v a n i s h i n g r o t a t o r y s t r e n g t h i n b o t h t h e Αχ Αχ a n d Αχ -> E v i b r o n i c a l l y p e r t u r b e d t r i g o n a l c o m p o n e n t s o f t h e A x •> T t r a n s i t i o n s . Cong

2

2

a

a

g

2

2 g

2

a

D

D

g

2 g


a

3.

RICHARDSON

Circular

Dichroic

65

Intensities

sequently, the A i g T g CD c o u l d e x h i b i t m i x e d p o l a r i z a t i o n . F i n a l l y , we e x a m i n e d t h e p o s s i b l e c o n s e q u e n c e s o f v i b r o n i c a l l y i n d u c e d T i ( A + E ) - Ti (A£ + E ) m i x i n g . H e r e we f o u n d t h a t v i b r o n i c components o f m i x e d p o l a r i z a t i o n c a n b e e x p e c t e d w i t h i n b o t h t h e A i -> A a n d Αχ E CD b a n d s a s s o c i a t e d w i t h t h e A i g •> Tig t r a n s i t i o n . I n t h i s c a s e , an a d i a b a t i c v i b r o n i c c o u p l i n g f o r m a l i s m was assumed v a l i d s i n c e t h e e n e r g y s e p a r a t i o n between t h e c o u p l e d e l e c t r o n i c s t a t e s c o u l d b e assumed t o b e l a r g e com pared t o t h e v i b r o n i c c o u p l i n g energy. I t i s , o f course, h i g h l y a r t i f i c i a l t o consider each o f t h e above-mentioned c o u p l i n g cases i n i s o l a t i o n . F o r example, i n c o n s i d e r i n g T i g - T i i n t e r a c t i o n s r i g o r o u s l y i t would be n e c e s sary t o take i n t o account simultaneously t h e l i k e l i h o o d t h a t b o t h T i ( A + E ) a n d Ti (A£ + E ) a r e b o t h J T a n d P J T d i s t o r t e d . The same w o u l d h o l d t r u e f o r v i b r o n i c a l l y i n d u c e d T i - T in teractions. Such c o n s i d e r a t i o n s would l e a d t o e x c e e d i n g l y t e d i o u s a n d c o m p l i c a t e d f o r m a l m a n i p u l a t i o n s a n d h o r r e n d o u s com p u t a t i o n a l r e q u i r e m e n t s , i f o n e ' s o b j e c t i v e was t o e x p l i c a t e quantitatively the vibronic d e t a i l s of the A i Ti + T CD s p e c t r a o f t r i g o n a l C o ( I I I ) systems. A t present i t would appear n e c e s s a r y t o be c o n t e n t w i t h t h e q u a l i t a t i v e c o n c l u s i o n s e l i c i t e d from t h e less-tban-complete treatments given i n t h i s section. The v i b r o n i c c o u p l i n g a n a l y s i s g i v e n h e r e h a s a v e r y i m portant i m p l i c a t i o n f o r spectra-structure c o r r e l a t i o n studies u s i n g CD. The u s u a l p r a c t i s e i n s u c h s t u d i e s i s t o a s s i g n s p e c i f i c f e a t u r e s i n t h e CD s p e c t r a t o s p e c i f i c e l e c t r o n i c t r a n s i t i o n s o f w e l l - d e f i n e d e l e c t r o n i c s t a t e p a r e n t a g e . The i n t e n s i t i e s o f t h e s e f e a t u r e s a r e t h e n assumed t o b e d i r e c t l y a n d s i m p l y r e l a t e d t o pure e l e c t r o n i c r o t a t o r y s t r e n g t h s , and t h e s i g n s and magnitudes o f these e l e c t r o n i c r o t a t o r y strengths a r e then c o r r e l a t e d w i t h s p e c i f i c s t e r e o c h e m i c a l f e a t u r e s o f t h e system ( e i t h e r by u s e o f v a r i o u s models and t h e o r i e s o r a c c o r d i n g t o some s e t o f e m p i r i c a l r u l e s ) . I n the presence o f n o n - n e g l i g i b l e v i b r o n i c i n t e r a c t i o n s t h e s e p r o c e d u r e s c a n , a t b e s t , be m i s l e a d i n g and, a t w o r s t , l e a d t o erroneous s t r u c t u r a l c o n c l u s i o n s . Under c o n d i t i o n s o f n o n - n e g l i g i b l e v i b r o n i c i n t e r a c t i o n s i t i s n o t p o s s i b l e , i n g e n e r a l , t o a s s i g n s p e c i f i c CD f e a t u r e s t o t r a n s i t i o n s o f w e l l - d e f i n e d e l e c t r o n i c parentage. I s i t p o s s i b l e t o make c l e a r - c u t s p e c t r a - s t r u c t u r e c o r r e l a t i o n s i n t h e absence o f a complete v i b r o n i c a n a l y s i s o f t h e s p e c t r a and o f t h e systems under study? I t seems l i k e l y t h a t t h e n e t CD i n t e n s i t y ( a n d n e t e l e c t r o n i c r o t a t o r y s t r e n g t h ) a s s o c i a t e d w i t h a l l o f t h e d-ci t r a n s i t i o n s i n a g i v e n c o m p l e x w i l l , i n l a r g e p a r t , be c o n s e r v e d i n t h e p r e s e n c e o f n o n - n e g l i g i b l e v i b r o n i c i n t e r a c t i o n s . T h i s assumes, o f c o u r s e , t h a t v i b r o n i c a l l y induced T i g - T i mixing w i l l g e n e r a l l y be s m a l l . S p e c t r a - s t r u c t u r e c o r r e l a t i o n s s h o u l d b e b a s e d , t h e n , on n e t d_-d_ r o t a t o r y s t r e n g t h s r a t h e r t h a n o n t h e r o t a t o r y s t r e n g t h s a s s o c i a t e d w i t h s p e c i f i c CD bands ( o f u n c e r t a i n o r m i x e d e l e c t r o n i c parentage). 2

1

g

2

a

u


2

a

u

1

g

2

a

u

g

g

g

2 g

2 g

u



66


The v i b r o n i c c o u p l i n g t h e o r y p r e s e n t e d i n t h i s s e c t i o n was developed s p e c i f i c a l l y f o r c h i r a l s i x - c o o r d i n a t e systems o f t r i g o n a l d i h e d r a l ( D 3 ) symmetry, a n d i t was a p p l i e d t o t h e d_-d CD s p e c t r a o f C o ( I I I ) c o m p l e x e s . I t i s , o f c o u r s e , s i m i l a r l y a p p l i c a b l e t o t r i g o n a l d i h e d r a l complexes o f any t r a n s i t i o n m e t a l i o n . The f o r m a l i s m a p p r o p r i a t e f o r a n a l y z i n g v i b r o n i c c o u p l i n g i n c h i r a l s y s t e m s o f o t h e r symmetry t y p e s i s s i m i l a r t o t h a t g i v e n h e r e i n a l l g e n e r a l a s p e c t s , a l t h o u g h t h e symmetrydetermined f e a t u r e s w i l l , o f course, be d i f f e r e n t . III.

Examples

A number o f v i b r o n i c CD c a l c u l a t i o n s have b e e n r e p o r t e d i n the l i t e r a t u r e f o r t r a n s i t i o n metal complexes. C a l i g a and R i c h a r d s o n (15) t r e a t e d t h e g e n e r a l c a s e o f p a i r s o f n e a r l y degenerate e l e c t r o n i c t r a n s i t i o n s coupled by a s i n g l e n o n t o t a l l y s y m m e t r i c v i b r a t i o n a l mode. T h e i r v i b r o n i c i n t e r a c t i o n model was b a s e d on a n o n - a d i a b a t i c c o u p l i n g r e p r e s e n t a t i o n , a n d t h e i r c a l c u l a t i o n s i n c l u d e d a wide range o f values f o r t h e s p e c t r o s c o p i c and v i b r o n i c c o u p l i n g parameters i n h e r e n t t o t h e i r model. R i c h a r d s o n a n d c o w o r k e r s ( 1 7 , 18) t h e n e x t e n d e d t h i s m o d e l t o t r e a t d i s s y m m e t r i c pseudo t e t r a g o n a l ( n e a r l y D^h) m e t a l complexes i n which three n e a r l y degenerate e l e c t r o n i c t r a n s i t i o n s a r e c o u p l e d v i a pseudo J a h n - T e l l e r i n t e r a c t i o n s i n v o l v i n g e i t h e r two o r t h r e e v i b r a t i o n a l modes o f t h e s y s t e m . A g a i n , m o d e l CD s p e c t r a and v i b r o n i c r o t a t o r y s t r e n g t h s were c a l c u l a t e d f o r a wide v a r i e t y o f parameter s e t s r e f l e c t i n g t h e s p e c t r o s c o p i c p r o p e r t i e s and v i b r o n i c c o u p l i n g d e t a i l s o f t h e system. R i c h a r d s o n and Hilmes (19) i n c o r p o r a t e d v i b r o n i c e f f e c t s i n t o t h e i r t r e a t m e n t o f t h e C u ( I I ) E •> T CD o b s e r v e d i n s i n g l e c r y s t a l s o f ZnSeOi+'olÔ doped w i t h C u ( I I ) i o n s . I n t h e C u ( I I ) :ZnSeOi -6H 0 doped s y s t e m , t h e C u ( I I ) i o n s a r e e a c h c o o r d i n a t e d t o s i x w a t e r m o l e c u l e s w i t h t h e CuOg c l u s t e r s b e i n g v e r y n e a r l y o c t a h e d r a l ( 0 ^ ) . However, t h e e x a c t s i t e symmetry f o r e a c h C u ( I I ) i s C so t h a t t h e E g r o u n d s t a t e i s s p l i t i n t o two n o n d e g e n e r a t e o r b i t a l components a n d t h e T excited state i s s p l i t i n t o t h r e e n o n d e g e n e r a t e o r b i t a l components. I n c a l c u l a t i n g t h e t e m p e r a t u r e d e p e n d e n c e o f t h e n e t ( t o t a l ) CD i n t e n s i t y a s s o c i a t e d w i t h t h e E -> T Cu(II) t r a n s i t i o n , R i c h a r d s o n and H i l m e s ( 1 9 ) e x p l i c i t l y i n c l u d e d pseudo J a h n - T e l l e r t y p e i n t e r a c t i o n s w i t h i n t h e s p l i t E ground s t a t e assuming a s i n g l e p e r t u r b i n g v i b r a t i o n a l mode. The e x p e r i m e n t a l l y o b s e r v e d temp e r a t u r e d e p e n d e n c e o f t h e E •> T CD c o u l d be a c c o u n t e d f o r q u a n t i t a t i v e l y o n l y by t h e i n c l u s i o n o f these v i b r o n i c i n t e r a c t i o n s w i t h i n t h e E ground s t a t e . As was m e n t i o n e d p r e v i o u s l y , two c o m p u t a t i o n a l s t u d i e s o f v i b r o n i c e f f e c t s i n c h i r a l t r i g o n a l s y s t e m s h a v e been r e p o r t e d (18, 2 8 ) . B o t h o f t h e s e s t u d i e s were a d d r e s s e d t o t h e g e n e r a l p r o b l e m o f v i b r o n i c p e r t u r b a t i o n s on t h e c h i r o p t i c a l s p e c t r a o f t r i g o n a l m e t a l c o m p l e x e s . However, t h e c a l c u l a t i o n s r e p o r t e d i n 2

2

g

+

2 g

2

2

2

g

Z

2 g

2

2

g

2 g

2

g

2

2

g

2 g

2

g


3.

RICHARDSON

Circular

Dichroic

Intensities

67

e a c h study were based on parameter s e t s c l o s e l y r e l a t e d t o t h e A i g -> T i g CD o f C o ( I I I ) c o m p l e x e s s u c h a s C o ( e n ) and C o ( o x ) ~ . I n b o t h s t u d i e s c o n s i d e r a t i o n o f v i b r o n i c i n t e r a c t i o n s was c o n f i n e d t o J a h n - T e l l e r a n d pseudo J a h n - T e l l e r t y p e c o u p l i n g s w i t h i n a t r i g o n a l l y s p l i t excited state o f T i ( o r T ) octahedral paren tage. I n e a c h s t u d y o n l y a s i n g l e v i b r a t i o n a l p e r t u r b i n g mode o f e - t y p e t r i g o n a l symmetry was i n c l u d e d i n t h e c o m p u t a t i o n a l m o d e l . The g e n e r a l a s p e c t s o f t h e v i b r o n i c c o u p l i n g m o d e l s u s e d i n t h e s e two s t u d i e s were i d e n t i c a l t o t h o s e d e s c r i b e d i n S e c t i o n s II.Α., II.Β., a n d U . C . o f t h e p r e s e n t p a p e r e x c e p t t h a t o n l y a s i n g l e p e r t u r b i n g mode was c o n s i d e r e d . In the earlier study (by R i c h a r d s o n , e t a l . ( 1 6 ) ) , o n l y l i n e a r J a h n - T e l l e r and pseudo J a h n - T e l l e r c o u p l i n g s were i n c l u d e d i n t h e n u m e r i c a l calculations. I n t h e more r e c e n t s t u d y b y Z g i e r s k i a n d P a w l i k o w s k i ( 2 8 ) , q u a d r a t i c J a h n - T e l l e r c o u p l i n g s were a l s o i n cluded i n the c a l c u l a t i o n s . To d e m o n s t r a t e how t h e s i m p l e s t t y p e o f ( A +E)*e c o u p l i n g may i n f l u e n c e t h e A i ( A i ) - > T i ( A + E) CD o f a c h i r a l t r i g o n a l d i h e d r a l ( D 3 ) m e t a l c o m p l e x , we s h a l l p r e s e n t some r e s u l t s o b t a i n e d b y a c t u a l c o m p u t a t i o n s . We s h a l l assume a s i n g l e p e r t u r b i n g v i b r a t i o n a l mode o f e - t y p e t r i g o n a l symmetry a n d o f f u n d a m e n t a l f r e q u e n c y ω*ρ ( e x p r e s s e d i n c m ) . We s h a l l d e n o t e the t r i g o n a l s p l i t t i n g energy by Δ (cm" ) = ( E ^ - E ) / h c . Only the l i n e a r J T and PJT c o u p l i n g terms w i l l be i n c l u d e d i n t h e c o m p u t a t i o n a l m o d e l . The l i n e a r J T c o u p l i n g c o n s t a n t , k, w i l l be d e f i n e d a c c o r d i n g t o E q . ( 2 8 ) , a n d t h e l i n e a r P J T c o u p l i n g c o n s t a n t , γ, w i l l b e d e f i n e d a c c o r d i n g t o E q . ( 2 7 ) . The r a t i o of pure e l e c t r o n i c r o t a t o r y s t r e n g t h s w i l l be t a k e n i n a l l c a s e s t o b e R ^ : R Î : R o ^ = 1 : 1 : - l , where +

3


g

3

2 g

2

g

g

2

-1

1

0

E

e )

R^

e)

R^

e)

e )

e

= Im(Ai|y|E )-(E |m|Ai)

(64a)

= Im(Ai|y|A ).(A |m|Ai).

(64b)

+

±

and, 2

2

The v i b r o n i c wave f u n c t i o n s f o r t h e ( A + E ) * e c o u p l e d s t a t e s o f o u r m o d e l s y s t e m may be w r i t t e n a s 2

Ψ

( θ )

( θ )

= Ψ° Σ Α γ J ° ν ' Λ ^ ν

+

ψ

}

+ Ψ° +

( + )

Σ Α v ^ * V

X

ν

( + )

v

*

excited

(

6

5

)

°- γ 5 ν 1 4 > ν,£ where t h e n o t a t i o n i s s i m i l a r t o t h a t u s e d i n E q . (32) o f S e c t i o n U . C . e x c e p t t h a t h e r e we a r e c o n s i d e r i n g j u s t o n e c o u p l i n g mode ( a n e - t y p e t r i g o n a l mode). The e x p r e s s i o n s a n d p r o c e d u r e s o u t l i n e d i n R e f . _16 a n d i n S e c t i o n U . C . o f t h e p r e s e n t p a p e r may now b e u s e d t o c a l c u l a t e t h e r o t a t o r y s t r e n g t h s o f t r a n s i t i o n s o r i g i n a t i n g i n t h e ground v i b r a t i o n a l l e v e l o f t h e ground e l e c t r o n i c s t a t e ( A i ) and t e r m i n a t i n g i n the j v i b r o n i c l e v e l s J


STEREOCHEMISTRY OF

68

TRANSITION METALS

d e r i v e d from the (A + E)*e coupled e x c i t e d s t a t e s . I n a l l o f t h e c a l c u l a t i o n s r e p o r t e d h e r e we h a v e assumed i d e n t i c a l o s c i l l a t o r f r e q u e n c i e s f o r t h e p e r t u r b i n g mode i n t h e g r o u n d and e x c i t e d e l e c t r o n i c s t a t e s . Furthermore, i n con s t r u c t i n g t h e v i b r o n i c wave f u n c t i o n s g i v e n by Eq. (65) we h a v e employed a harmonic o s c i l l a t o r b a s i s s e t w h i c h i n c l u d e d a l l f u n c t i o n s w i t h ν £ 10. A s s u m i n g G a u s s i a n - s h a p e d v i b r o n i c CD b a n d s , t h e CD s p e c t r u m i n t h e v i c i n i t y o f t h e A i ( A i ) - * T i g ( A + E) t r a n s i t i o n o f o u r m o d e l s y s t e m may be s y n t h e s i z e d a c c o r d i n g t o


2

g

2

2

Δε(ϋΓ) = C Σ (ÔT.R./6) exp [-(ω j 3

e

2

- ω.) /ό ]

3

(66)

3

e

1

where Δε = L - R, uTj i s t h e t r a n s i t i o n f r e q u e n c y ( i n cm" ) for a t r a n s i t i o n t o t h e j - t h v i b r o n i c l e v e l , Rj i s t h e v i b r o n i c r o t a t o r y s t r e n g t h d e f i n e d a c c o r d i n g t o Eq. ( 3 3 ) , 6 i s a b a n d w i d t h p a r a m e t e r ( e x p r e s s e d i n cm" u n i t s ) , and C i s a c o n s t a n t f a c t o r whose n u m e r i c a l v a l u e d e p e n d s , i n p a r t , on t h e u n i t s chosen f o r R j . F o r our p r e s e n t p u r p o s e s , i t w i l l be more c o n v e n i e n t t o e x p r e s s Eq. (66) as f o l l o w s : 1

Δε(Ω)

= C Σ (ô£ j 1

2

2

+ n . ) ( R . / 6 ) exp[-(Ω - Ω . ) / 6 ] , J J 3 - 1

(67)

where Ω (cm"" ) = ÔT - ôJg, Ω. ( c m ) = ôTj = and = (Eg- IÔ/hc. The new f r e q u e n c y v a r i a b l e ^ Ω, i s d e f i n e d t o be z e r o a t t h e resonance frequency of the unperturbed, pure e l e c t r o n i c t r a n s i t i o n Αχ •> E. I n t h e c a l c u l a t i o n s r e p o r t e d h e r e we have s e t aig = 20,000 cm" . CD s p e c t r a f o r t h e Αχ + ( A + E ) * e t r a n s i t i o n s o f o u r t r i g o n a l d i h e d r a l (D3) m o d e l s y s t e m w e r e c a l c u l a t e d f o r a number o f p a r a m e t e r s e t s i n w h i c h t h e v a l u e s o f Δ, γ, and k were v a r i e d . These s p e c t r a a r e d i s p l a y e d i n F i g u r e s 2-5. In these f i g u r e s , t h e CD i n t e n s i t y s c a l e (Δε) i s g i v e n i n a r b i t r a r y u n i t s and t h e frequency s c a l e i s e x p r e s s e d i n terms of Ω(cm" ). The c a l c u l a t e d s p e c t r a shown i n F i g u r e s 2-5 d e m o n s t r a t e t h e p r o f o u n d i n f l u e n c e w h i c h v i b r o n i c c o u p l i n g may have on b o t h t h e q u a l i t a t i v e and q u a n t i t a t i v e a s p e c t s o f t h e CD o b s e r v e d w i t h i n t h e Αχ -> A + Ε t r a n s i t i o n r e g i o n . Although the net ( t o t a l ) r o t a t o r y s t r e n g t h and CD i n t e n s i t y r e m a i n i n v a r i a n t t o t h e v a l u e s o f Δ, γ, and k, t h e d i s t r i b u t i o n o f CD i n t e n s i t y and t h e s i g n p a t t e r n s w i t h i n t h e CD s p e c t r u m a r e shown t o be v e r y s e n s i t i v e t o t h e s e p a r a m e t e r s . The v a l u e s o f Δ, γ, and k w i l l , o f c o u r s e , be d e t e r m i n e d t o some e x t e n t b y - t h o s e s t e r e o c h e m i c a l and e l e c t r o n i c s t r u c t u r a l f e a t u r e s o f a c o m p l e x w h i c h one w i s h e s t o e l u c i d a t e v i a CD s p e c t r a l s t u d i e s . However, i n n e a r l y a l l s p e c t r a - s t r u c t u r e c o r r e l a t i o n studies reported to date, the i n f l u e n c e o f v i b r o n i c c o u p l i n g on t h e CD s p e c t r a l f e a t u r e s has been i g n o r e d o r n e g l e c t e d . The s p e c t r a shown i n F i g u r e s 2-5 r e f l e c t o n l y the simplest k i n d s of v i b r o n i c i n t e r a c t i o n s ( i n t r a s t a t e JT and PJT c o u p l i n g s ) w h i c h may i n f l u e n c e t h e d e t a i l s o f 1

2

1

2



3.

RICHARDSON

Circular

Dichroic

1

1

Intensities

I

-600

I

0

I

69

I

I

600 iKcrrT )

I

1200

I

.

1800

1

Figure 2. Calculated CD spectra for the A, -» (A + E)*e transitions using the parameter sets R :R. :R = 1:1:—1 (in arbitrary units), Δ = ώ = 300 cm' , k = 0 and y = 0 ( ), y = 0.5 ώ ( ), and y = 1.0 ω ( ). Δε is ex pressed in arbitrary units and δ was taken to be 0.6 ω . 2

(e)

(e)

+

(e)

0

1

ρ

ρ

ρ

ρ

ÎMcm" ) 1

Figure 3. Calculated CD spectra for the A -> (A + E)*e transitions using the parameter sets R :R. :R = 1:1:—1 (in arbitrary units), Δ = k = ω = 300 cm , and y = 0 ( ), y = 0.5ω ( ), and y = 1.0 ω ( ). Acts expressed in arbitrary units and δ was taken to be 0.6 ω . t

(e)

+

1

(e)

2

(e)

0

ρ

p

ρ

ρ




70

Figure 4. Calculated CD spectra for the A -> (A + E)*e transitions using the parameter sets R :R_ :R/ = 1:1:—1 (in arbitrary units), Δ = y = ( ) and k = 1.25 m ( ). Δε is expressed in arbitrary units and δ was taken to be 0.6 w . t

fe;

fe,

2

e;

p

+

1

p

p

p

Figure 5. Calculated CD spectra for the A i -» (A + E)*e transitions using the parameter sets R/ ;R. :R/ = 1:1:—1 (in arbitrary units), y = k = ω = 300 cm , and Δ = 0.5 τ> ( ), Δ = 1.5 ΊΠ ( ), and Δ = 2.5 m (- · - -). Δε is expressed in arbitrary units and δ was taken to be 0.6 m . t

eJ

fe:)

eJ

ρ

1

ρ

ρ

p

p


3.

RICHARDSON

Circular

Dichroic

Intensities

71

t h e CD s p e c t r a a s s o c i a t e d w i t h c h i r a l t r a n s i t i o n m e t a l c o m p l e x e s . M u l t i p l e mode c o u p l i n g a n d i n t e r - s t a t e c o u p l i n g s w i l l , i n general, introduce a d d i t i o n a l complications.


IV.

Conclusions

Modest v i b r o n i c c o u p l i n g s t r e n g t h s i n m e t a l c o m p l e x e s may: (1) l e a d t o s i g n i f i c a n t m i x i n g s o f " f i x e d - n u c l e i " e l e c t r o n i c states; (2) a l t e r t h e e n e r g y l e v e l s p a c i n g s a n d o r d e r i n g s w i t h in spectroscopic state manifolds; (3) c a u s e s i g n i f i c a n t d i s t o r t i o n s w i t h i n degenerate o r n e a r l y degenerate spectroscopic s t a t e s ; and, (4) p r e c l u d e t h e a s s i g n m e n t o f t r a n s i t i o n s t o i n i t i a l and f i n a l s t a t e s o f w e l l - d e f i n e d e l e c t r o n i c i d e n t i t i e s (quantum n u m b e r s ) . The c o n s e q u e n c e s o f n o n - n e g l i g i b l e v i b r o n i c c o u p l i n g i n c h i r a l m e t a l c o m p l e x e s may b e t o : (1) p r o d u c e s i g n i f i c a n t a l t e r a t i o n s i n CD i n t e n s i t y d i s t r i b u t i o n s ( w i t h r e g a r d t o sign p a t t e r n s , r e l a t i v e i n t e n s i t i e s , and p o l a r i z a t i o n p r o f i l e s ) ; (2) r e q u i r e t h a t v i b r o n i c c o u p l i n g mechanisms a n d s t r e n g t h s be taken i n t o account along w i t h stereochemical s t r u c t u r a l f a c t o r s in constructing detailed spectra-structure correlation rules; and, (3) a b r o g a t e s e c t o r ( o r r e g i o n a l ) r u l e s b a s e d o n t h e CD o b s e r v e d f o r s p e c i f i c bands o r s p e c t r a l f e a t u r e s . V i b r o n i c c o u p l i n g e f f e c t s on the c h i r o p t i c a l s p e c t r a o f o p t i c a l l y a c t i v e m e t a l complexes tend t o obscure the i n h e r e n t r e l a t i o n s h i p s between t h e CD o b s e r v a b l e s a n d s t r u c t u r a l f e a t u r e s such a s a b s o l u t e c o n f i g u r a t i o n , l i g a n d c o n f o r m a t i o n , and l i g a n d spatial distributions. For t h i s reason, s p e c t r a - s t r u c t u r e r e l a t i o n s h i p s based on the " f i x e d - n u c l e i " approximation and on the a s s u m p t i o n o f w e l l d e f i n e d " e l e c t r o n i c " t r a n s i t i o n s must b e a p plied with considerable caution. A great wealth of spectrosc o p i c a l l y a n d s t r u c t u r a l l y i m p o r t a n t i n f o r m a t i o n may be o b t a i n e d f r o m d e t a i l e d v i b r o n i c a n a l y s e s o f CD s p e c t r a . However, v e r y few a n a l y s e s o f t h i s s o r t h a v e been r e p o r t e d t o d a t e . Acknowledgments T h i s w o r k was s u p p o r t e d b y t h e N a t i o n a l S c i e n c e F o u n d a t i o n ( G r a n t CHE77-02150) a n d b y t h e C a m i l l e a n d H e n r y D r e y f u s F o u n d a t i o n ( t h r o u g h a T e a c h e r - S c h o l a r Award t o F . R . ) .

Literature Cited 1. 2. 3.

"Alfred Werner, 1866-1919", Hevl. Chim. Acta, 1966, Com memoration Volume IX, ICCC, Zurich. Jaeger, F.M., "Spatical Arrangements of Atomic Systems and Optical Activity", George Fisher Baker Lectures, Vol. 7, Cornell University, McGraw-Hill, New York, 1930. Mathieu, J.P., "Les Theories Moleculaires du Pouvoir Rotatoire Naturel", Gauthier-Villars, Paris, 1946.


72 4. 5. 6. 7.


8.

9. 10. 11.

12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.


Kuhn, W. and Bein, Κ., Z. Physik. Chem., 1934, B24, 335. Kuhn, W. and Bein, Κ., Z. Anorg. Allg. Chem., 1934, 216, 321. Kuhn, W., Naturwissenschaften, 1938, 19, 289. Saito, Y.; Nakatsu, K.; Shiro, M.; and Kuraya, Η., Acta Crystallogr., 1955, 8, 729. For the accepted nomenclature and notation regarding ab solute configuration designations for coordination com pounds, see: (a) "I.U.P.A.C. Information Bulletin No. 33", 1968, p. 68; (b) Inorg. Chem., 1970, 9, 1. Moffitt, W., J . Chem. Phys., 1956, 25, 1189. Condon, E.U.; Alter, W.; and Eyring, Η., J . Chem. Phys., 1937, 5, 753. Mason, S.F. in "Fundamental Aspects and Recent Developments in Optical Rotatory Dispersion and Circular Dichroism", edited by J . Ciardelli and P. Salvadore, Heyden and Son, Ltd., New York, 1973; Chapter 3.6. Richardson, F.S., Chem. Rev., 1979, 79, 17. See, for example: (a) Ham, F.S., Phys. Rev., 1965, A138, 1727; (b) Sturge, M.D., Solid State Physics, 1967, 20, 91; (c) Stephens, P . J . , J . Chem. Phys., 1969, 51, 1995. Denning, R.G., Chem. Comm., 1967, 120. Caliga, D. and Richardson, F.S., Mol. Phys., 1974, 28, 1145. Richardson, F.S.; Caliga, D.; Hilmes, G.; and Jenkins, J., Mol. Phys., 1975, 30, 257. Richardson, F.S.; Hilmes, G.; and Jenkins, J., Theoret. Chim. Acta (Berl.), 1975, 39, 75. Hilmes, G.; Caliga. D.; and Richardson, F.S., Chem. Phys., 1976, 13, 203. Richardson, F.S. and Hilmes, G., Mol. Phys., 1975, 30, 237. Herzberg, G. and Teller, Ε., Z. Physik. Chem., 1933, B21, 410. Weigang, O.E., J . Chem. Phys., 1965, 43, 3609. Harnung, S.E.; Ong, E.C.; and Weigang, O.E., J . Chem. Phys., 1971, 55, 5711. Weigang, O.E. and Ong, E . C . , Tetrahedron, 1974, 30, 1783. Harding, M.J., J.C.S. Faraday II, 1972, 68, 234. Bray, R.G.; Ferguson, J.; and Hawkins, C . J . , Aust. J . Chem., 1969, 22, 2091. Perrin, M.H. and Gouterman, Μ., J . Chem. Phys., 1967, 46, 1019. See, for example: (a) Englman, R., "The Jahn-Teller Effect in Molecules and Crystals", Wiley-Interscience, New York, 1972, Chapter 3; (b) references 13 and 16 cited above. Zgierski, M.Z. and Pawlikowski, Μ., J . Chem. Phys., 1979, 70, 3444.

RECEIVED September 13, 1979.


Circular Dichroic Intensities in the Vibronic Transitions of Chiral Metal

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