Single-centre molecular orbital theory of transition metal complexes

Full text: Most of the optical and magnetic properties of transition metal compounds are determined by the splitting pattern of the partially filled d-orbitals. Theoretical methods for calculating this splitting pattern range from simple empirical methods as ligand field theory (LFT) up to first principle approaches as molecular orbital (MO) theories. Whereas the latter solve the full multi-centre problem numerically, LFT constitutes a single-centre, essentially atomic-like problem that can, in principle, be solved analytically, albeit only by using empirically adjusted parameters. In order to bridge the gap between both levels of description and to maintain the atomic-like structure of LFT, a single-centre representation of the MO-Hamiltonian is derived based on ligand-orthogonalized d-orbitals that explicitly account for overlap and covalency effects. Neglecting the s- and p-orbitals of the metal, as well as ligand-ligand interactions, a simple representation of the 'ligand-field' is obtained that can be interpreted as a repulsive pseudopotential and is roughly proportional to the square of the overlap matrix between metal and ligand orbitals. Moreover, this pseudopotential can be factorized into radial and angular parts, thus, enabling the separate analysis of both contributions. The directly, within this scheme, calculated d-orbital energies for several CuII-fluorides are in quantitative agreement with fully numerical MO-calculations in the local density approximation based on the same model assumptions (neglect of metal s- p-orbitals and ligand-ligand interactions). Hence, these results confirm the reliability of this single-centre representation. Finally, the careful analysis of the results demonstrates that by far the largest part, viz. about 90 %, of the ligand-field splitting arises from the orbital interactions. Comparison of the single-centre MO-approach with LFT, still being widely used for the interpretation of optical spectra, shows that the repulsive pseudopotential exhibits the same angular dependence as the electrostatic ligand-field of LFT. From this result it is comprehensible that LFT very often provides the qualitatively correct splitting pattern of the d-orbitals. The essential difference, however, consists in the radial parts. The single-centre MO-approach yields basically an exponential dependence of the ligand-field splitting on the metal-ligand distance R, whereas LFT gives, for cubic systems, the well-known 1/R5-dependence. Accordingly, the widespread claim that the ligand-field splitting is proportional to 1/R5 is merely based on the fact that any exponential can be approximated locally by an appropriate 1/Rn but is devoid of any physical basis. Finally, it is demonstrated that certain limitations of the applicability of both the single-centre MO-approach and LFT arise mainly from the neglect of the metal s- and p-orbitals. Their influence may lead, in particular for lower symmetries, to interchanges between the d-orbitals that must be compensated in LFT by unreasonable parameter choices. In summary, an analytical expression has been derived that enables the easy calculation of d-orbital energies without using empirically adjusted parameters. Moreover, this scheme facilitates the understanding of the limits and of the reason why in many cases LFT yields qualitatively correct results. (author)