Kohn–Sham DFT and ligand-field theory — Is there a synergy?

In forming electronic states of the partially filled shell of transition-metal atomic and molecular systems, real, symmetry-based, fixed, Kohn–Sham eigenorbitals can be used to bridge KS-states with strong-field, ligand-field states. Thereby, DFT computations, restrained by the use of these frozen orbitals of the so-called average-of-configuration type, allow a central-field modeling of the partially filled shell whose Hamiltonian matrix consists of mutually orthogonal diagonal and non-diagonal parts, of which only the former can be computed. Mutually orthogonal operators of ligand-field theory are particularly suited to parameterize the energy “data” obtained from the bridges between molecular Kohn–Sham DFT states and ligand-field states. With the d 2 configuration as the simplest example encompassing both ligand-field and interelectronic repulsion, each one-electron parameter, though defined by energy differences of perturbed d orbitals, is associated with a 45 × 45, diagonal, theoretical, strong-field-type coefficient matrix of the ligand field repulsion model (LFR), which is mapped in a one-to-one fashion onto a likewise diagonal KS-DFT computational energy matrix. For sets of mutually orthogonal operators, the mapping determines the value of any such ligand-field parameter as a scalar product between the DFT matrix and the coefficient matrix of the associated ligand-field operator. Each and every two-electron parameter of LFR is in the same strong-field function basis associated with a 45 × 45 coefficient matrix that includes a non-diagonal part. This matrix, nevertheless, by the formation of a scalar product with the appropriate diagonal, computational DFT matrix, provides the value of the two-electron parameter. In spite of the lacking non-diagonal DFT information, its non-diagonal elements of the two-electron interelectronic repulsion matrices are indirectly accessible through the parameterization based upon the computed diagonal DFT matrices combined with the mapping of the DFT energy results onto the parametric LFR. In this way, LFR delivers back to DFT a quantification of the deviation of the systems’ eigenbasis from the DFT-computed states, which are defined by having unit occupation numbers. This work focuses firstly on using the LFR model for forming a full DFT energy matrix and dissecting it into mutually orthogonal one- and two-electron parts and secondly on the use of the two-electron parts to obtain a complete ligand-field image of a nephelauxetic, molecular atom, intrinsic of the chemical system.

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