Day-Hush Disproportionation Model

The existence of a multitude of valence states for a given elanent has been evident for 200 years. This fact formed the background for the estabUshment of the periodic table. Missing valence states are commonplace and not considered as surprising. Why do we need a theory One reason is that part of the electronic spectra can be explained only as metal to other metal transitions, and then three consecutive valence states have to be involved. We have already seen that very strong colors appear if two valence states are mixed, for example, Fe + and Fe + in ink. There is a different reason for colorful substances if the disproportionation mechanism is realized. For that, we need a theory and such theory was provided by P. Day, K. Prassides, N. Hush, J. Reimers, and their collaborators. 

As previously for MV-2 systems we introduce two diabatic, parabolic PES (H and H22) in each of two equilibrium points, Q = -Qq and Q = Qq, but this time the PES correspond to localization of an electron pair at two different sites, which we may write as M+M and M M+. In MV-3 systans we have to consider the MM configuration also, although it may not form a stable ground state. We notice that M+M and M M+ are different from each other by an electron pair. 

The third diabatic PES (H33) corresponds to MM. It can be assumed that its energy minimum occurs when the geometry is the same on both sites, for Q = 0. We assume that the force constants are the same for all three states. H33 is placed above or below Hu and H22 at Q = 0  

The eigenvalues corresponding to the PES are determined from the following secular equation  

We may choose H,3 and Qo to fit previously calculated PES along a breathing or half-breathing mode. The vertical energy U at Q = 0, which appears in Equation 10.90 also appears in the theory of condensed matter physics therefore, to avoid unnecessary confusion, we use the notation employed there. U is the so-called Hubbard U parameter (see Chapter 17). 

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This model was originally used along with the Day-Hush disproportionation model, mentioned in Section 10.7.2, and got its name from the pioneers S. Piepho, E. Krausz, and P. N. Schatz (PKS). The PKS model is a useful and consistent model for the treatment of spectra of mixed valence systans. The model includes the vibrational levels for MV-2 and MV-3 systems. It is assumed that the coupling is small so that the harmonic approximation can be used. 

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