A DFT Based Ligand Field Theory

Abstract

A general and user-oriented ligand field (LF) theory – LFDFT with parameters adjusted to DFT energies of separate Slater Determinants (SD) of the partly filled dn shell [n=2(8), 3(7), 4(6) and 5] of transition metals (TM) complexes – is developed and tested using 22 well documented examples from the literature. These include Cr^III, d³ and Co^II d⁷ in octahedral and Cr^IV, Mn^V, Fe^VI d², Co^II d⁷, Mn^II d⁵, Ni^II d⁸ in tetrahedral complexes for which reliable values of d-d transition energies available from high-resolute ligand field spectra of complexes with halogenide (F, Cl, Br and I), oxide and cyanide ligands have been reported. The formalism has been implemented and consists of three steps allowing provision of geometries, ligand field Kohn-Sham orbitals and SD-energies in a way consistent with the LF phenomenology. In a fourth step LF parameters are utilized to yield multiplet energies using a full CI LF program. Comparing SD energies from DFT with those calculated using the LF parameter values, we can state for all considered cases, that the LF parameterization scheme is remarkably compatible with SD energies from DFT; standard deviations between DFT SD-energies and their LFDFT values being calculated between 0.016 and 0.124 eV. We find that, when based on the average of configuration with n/5 occupancy of each MO dominated by TM d-orbitals and on geometries with metal-ligand bond lengths from experiment, the 10Dq parameter values (cubic symmetry) are very close to the ones obtained from a fit to reported ligand field transitions. In contrast, when using common functionals such as LDA or gradient corrected ones (GGA) we find that the parameters B and C deduced from a fit to the SD energies are systematically lower than experimental. Thus spin-forbidden transitions which are particularly sensitive to B and C are calculated to be by 2000 to 3000 cm^–1 at lower energies compared to experiment. Based on DFT and experimental B and C values we propose scaling factors, which allow one to improve the agreement between DFT and experimental transition energies, or alternatively to develop a DFT theory based on effective LF functionals and/or basis sets. Using a thorough analysis of the dependence of the Kohn-Sham orbital energy on the orbital occupation numbers, following Slater theory, we propose a general LFDFT scheme allowing one to treat, within the same formalism low symmetric ligand fields as well. Test examples, which illustrate the efficiency of this approach, include Cs distorted CrO₄ ^4– and D_2d distorted MnO₄ ^3– chromophores. Finally, for cubic LF we propose a hybrid LFDFT model (HLFDFT) which leads to an improvement of the existing DFT-multiplet theories. We show, taking low-spin Co(CN)₆ ^3– as an example, that the new model yields better results as compared to time-depending DFT (TDDFT). A discussion of the LFDFT method in the context of other CI-DFT approaches is given.