Toward understanding nephelauxetism: interelectronic repulsion in gaseous dq ions computed by Kohn–Sham DFT

The nephelauxetic phenomenon is reviewed. So is the newest formulation of the Slater–Condon–Shortley model (SCS) for the multiplet term energies of atomic dq ions. This model is expressed as a sum of products of empirical parameters and their associated coefficient operators. The orthogonal-operators formalism is used by choosing Jørgensen''s spin-pairing energy parameter D as one of the repulsion parameters when the other parameter required by the SCS model is the Racah parameter B. By restricting the coefficient operators further so as to make them orthonormal, B changes into the new parameter E=(21/4)B. D accounts for the energy separation of eigenstates with different spin and/or seniority, E for the final separation into the spatially different eigenstates. The numerical values of D and E exhibit directly the relative contributions of D and E to the average configurational splitting. The parameter set {D, E} is equivalent to the SCS set {F2, F4} as well as to the Racah set {C, B} even though these sets are not associated with orthogonal operators so that cross products, as for instance CB, enter the expressions for the average configurational splitting. Moreover, neither conventional parameter set provides a visualizable interpretation of the energetic functioning of its parameters upon the eigenstate energy separations. The symmetry problem of the two-electron part of Kohn–Sham DFT is discussed in the framework of the SCS model. With the future perspective of mimicking ligand-field theory and, in particular, of dealing with the full nephelauxetic phenomenon non-empirically, the atomic part of the phenomenon is handled by calculation of the parameters D and E successfully for numerous gaseous ions with 3dq configurations. The Amsterdam Density Functional package (ADF) has been used in a two-step procedure. First, all the Kohn–Sham self-consistent field d-orbitals of the 3dq configurations as well as all the inner core orbitals were determined using population numbers of q/10 for all the 3d-spin–orbitals, that is, by doing the average-of-configuration calculation. Secondly, all these orbitals were frozen in all the subsequent calculations on the same system. Here density distributions of α and β electrons were defined by integer occupation numbers of d-spin–orbitals corresponding to single Slater determinants. On grounds of principle, complex orbitals were found not to be usable. The ADF energy associated with a real determinant was—apart from an additive constant—taken to correspond to the expectation value of the interelectronic repulsion operator acting on the dq part of this determinant. A characteristic property of the method is that the energy is expressed by a sum over contributions from the Coulomb and the exchange-correlation functional. It has turned out that for real d-orbitals both of these terms independently obey the symmetry restrictions as well as the special SCS restrictions of the dq assumption. However, the two functionals are not commensurable. They give different parametric results. Nevertheless, by averaging the total energy results from using a gradient-corrected (GC) set of functionals over the configuration dq, values for the SCS parameters that are better than those of Hartree–Fock calculations are obtained. In the local density approximation (LDA) the exchange-correlation functionals were found to provide the same energy for all real Slater determinants with the same value for MS, which means that the exchange-correlation part of the energy corresponds to B=E=0. However, the Coulomb functional invariably exaggerates the values of the E parameter so that LDA functionals end up giving fairly reasonable values of E, as well as D. We have obtained a complete bridge between KS-DFT in the form of ADF and the atomic part of ligand-field theory. [Copyright &y& Elsevier]