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Effective Hamiltonian Crystal Field for Magnetic Interactions in Polynuclear Transition Metal Complexes. Sequential Derivation and Exemplary Numerical Estimates

By this we extend our work of the year 1992 devoted to calculating the intrashell excitations in the d-shells of coordination compounds of the first transition metal row, which resulted in the Effective Hamiltonian Crystal Field (EHCF) method, to their polinuclear analogs in order to assure the description of several open d-shells and of magnetic interactions of the effective spins residing in these shells. This is a challenging task since it requires improving the precision of ca. 1000 cm$^{-1}$ (that of describing the excitation energies of the single d-shells by the already well successful EHCF method) to the that of ca. 10 $÷$ 100 cm$^{-1}$ characteristic for the energies required to reorient the spins i.e. eventually by two orders of magnitude. This is performed within the same paradigm as used for the EHCF method: the concerted usage of the McWeeny's group-function approximation and the Löwdin partition technique. These are applied to develop the effective description of the d-system of the polynuclear complexes, composed of several d-shells, including the working formulae for the exchange parameters between the d-shells belonging to different transition metal ions. These formulae are implemented in the package kramers and tested against a series of binuclear complexes of trivalent Cr and Fe cations featuring $μ$-oxygen superexchange paths in order to confirm the reproducibility of the trends in the series of values of exchange parameters for the compounds differing by the chemical substitutions and other details of composition and structure. The results of calculations are in a reasonable agreement with available experimental data and other theoretical methods.