DMM on nonsymmetrical coordination compounds

Any Fock operator can be represented as a sum of the symmetric one and of a perturbation which includes both the dependence of the matrix elements on nuclear shifts from the equilibrium positions and the transition to a less symmetric environment due to the substitution. To pursue this, we first introduce some notations. Let hi be the supervector of the first derivatives of the matrix of the Fock operator with respect to nuclear shifts Sq counted from a symmetrical equilibrium configuration. By a supervector, we understand here a vector whose components numbered by the nuclear Cartesian shifts are themselves 10 x 10 matrices of the first derivatives of the Fock operator, with respect to the latter. Then the scalar product of the vector of all nuclear shifts 6q j and of the supervector hi yields a 10 x 10 matrix of the corrections to the Fockian linear in the nuclear shifts  

Supplying this with the 10 x 10 matrix of the substitution operator 

This does not form the entire ( dressed ) perturbation because, in case the electron density changes to the first order in the above perturbation, the Fock operator acquires additional perturbation through the variation of its self-energy part, which leads to the self-consistent perturbation. Thus the perturbed Fock operator can be written as  

Here AP stands for the correction to the unperturbed projection operator P0 to the occupied MOs, which in the case of the octahedral complexes is given by eq. (4.30). This serves as a prerequisite for performing the two remaining steps of the prescription of Section 3.1 of constructing a DMM description of CCs of arbitrary (low) symmetry and of the linear response theory based on it and leading to a strictly mechanistic description of this class of molecules. 

To proceed further, we look at the perturbed density matrix. It was assumed to have the form 

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