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Inclusion of Solvent Effects in a Vibronic
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Coupling Model for Mixed-Valence Compounds Κ. Y. WONG and P. N. SCHATZ University of Virginia, Department of Chemistry, Charlottesville, VA 22901 Solvent effects are included in a previous vibronic coupling model (PKS) of mixed-valence systems by including a second totally symmetric subunit normal coordinate. The antisymmetric com bination is identified as a low frequency effec tive solvent mode. It is still possible to obtain the ground vibronic manifold of the system by direct diagonalization. Application is made to the Creutz and Taube ion. It proves possible to account for the observed intervalence band con tour, the change in bond length between oxidation states, and weak, diffuse low energy tunneling transitions. It is also shown that in the harmon ic approximation no totally symmetric vibration of the Creutz and Taube ion is expected to contribute significantly to the intervalence band width un less there is a large change in force constant be tween oxidation states. This contradicts other analyses in the literature. The shift of intervalence band position with solvent is discussed in general using both direct diagonalizations and analytical formulas. A v i b r o n i c c o u p l i n g model f o r mixed-valence systems has been developed over the l a s t few years ( 1.-5). The model, which i s e x a c t l y s o l u b l e , has been used to c a l c u l a t e i n t e r v a l e n c e band contours ( 1 , 3, 4, 5 ) , e l e c t r o n t r a n s f e r rates (4, 5, 6) and Raman spectra ( 5 , 7, 8 ) , and the r e l a t i o n of the model to e a r l i er t h e o r e t i c a l work has been discussed i n d e t a i l ( 3 - 5 ) . As f o r mulated to date, the model i s "one dimensional" (or one-mode). That i s , e f f e c t i v e l y o n l y a s i n g l e v i b r a t i o n a l coordinate i s used i n d i s c u s s i n g the complete ground v i b r o n i c manifold o f the system. This i s a severe l i m i t a t i o n which, among other t h i n g s , prevents an e x p l i c i t treatment o f s o l v e n t e f f e c t s which are 0097-6156/82/0198-0281 $06.00/0 © 1982 American Chemical Society Rorabacher and Endicott; Mechanistic Aspects of Inorganic Reactions ACS Symposium Series; American Chemical Society: Washington, DC, 1982.
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known t o be o f great importance. I t i s the purpose o f t h i s a r t i c l e t o o u t l i n e a method f o r i n c l u d i n g t h e s o l v e n t and t o i l l u s t r a t e some o f the consequences o f such a treatment. S o l u t i o n o f the S t a t i c Problem We f o l l o w c l o s e l y p r e v i o u s e x p o s i t i o n s o f the theory (4, 5) and i n c l u d e o n l y the p a r t i c u l a r f e a t u r e s needed f o r our present discussion. L e t us imagine a mixed-valence system composed o f two s u b u n i t s , A and B, which are a s s o c i a t e d w i t h formal o x i d a t i o n s t a t e s M and N, r e s p e c t i v e l y . We designate the correspond i n g e l e c t r o n i c Hamiltonian operators H ^ and H ^ , and i f t h e corresponding wave f u n c t i o n s are
