Interactive matrix diagonalization

In general, no simple, consistent set of analytical expressions for the resonance condition of all intradoublet transitions and all possible rhombicities can be derived with the perturbation theory for these systems. Therefore, the rather different approach is taken to numerically compute all effective g-values using quantum mechanics and matrix diagonalization techniques (Chapters 7-9) and to tabulate the results in the form of graphs of geff,s versus the rhombicity r = E/D. This is a useful approach because it turns out that if the zero-field interaction is sufficiently dominant over... 

The sum over coulomb and exchange interactions in the Fock operator runs only over those spin-orbitals that are occupied in the trial VF. Because a unitary transformation among the orbitals that appear in F leaves the determinant unchanged (this is a property of determinants- det (UA) = det (U) det (A) = 1 det (A), if U is a unitary matrix), it is possible to choose such a unitary transformation to make the Ey matrix diagonal. Upon so doing, one is left with the so-called canonical Hartree-Fock equations ... 

The Breit interaction matrix can be treated in a similar way. The off-diagonal blocks can be written in terms of the magnetic fields using... 

Figure 11- Off-diagonal interaction parameter 92 between / = Oe and / = 2e levels of (0, V2, 0) is plotted vs. 02. Note that the actual Oe - 2e interaction matrix element is proportional to - (02+ 1)92 (see eqs. 4-6 of Ref. 5). Although 92 is nearly constant from v=2.

First, let the unitary transformation diagonalizes the interaction matrix that involves the operators of the electron repulsion, the crystal field, and the... 

The interaction occurs between half-filled shell states with identical quasispin character (AQ = 0). If the interaction is time-even then it has quasispin rank K = 1. From these equations such off diagonal interaction matrix elements must vanish. 

Since the excess electron is localized on nondegenerate orbital the excess electron vibronic interaction is limited to the totally symmetric distortions of the first coordination sphere. This interaction reflects the difference of metal-ligand distances in different oxidation states. The matrix of the electron-vibrational interaction is diagonal on the basis of localized states with the matrix elements... 

Three molecular orbitals with different energies and p-atomic-orbital contributions. Diagonalization of the no-adjacent-interaction matrix gives... 

It is now a straightforward process to calculate the diagonal and off-diagonal matrix elements. The interaction matrix for the three J = 3/2 positive parity states... 

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