Linear representations of an empirical correlation energy correction for transition-metal ions

The empirical correlation energy correction (CEC) introduced by Pueyo and Richardson (J. Chem. Phys. 67, 3577 (1977)) to improve the theoretical calculation of the d-d spectra of transition-metal ions in ionic crystals has been analyzed. The contribution of relaxation energies and basis effects to this correction has been discussed by using SCF and frozen-orbital values computed with different basis sets. The CEC is not always a uniformly increasing function of the atomic number of the ion due to the presence of factors other than correlation energy in this correction. A search has been conducted to examine the possible relations between the CEC and the total quantum numbers of the multiplets. Linear expression in these numbers, analogous to those in atomic spectroscopy, describe the CEC with rather large deviations. The more accurate relation of this type is that involving the orbital and spin quantum numbers and Racah’s seniority number. As expected, this quantity increases with the number of the d electrons, being linear in the number of pairs. The CEC has been, then, represented by means of linear equations in pair-correlation terms. Two models are proposed for obtaining these energies. Results of this calculation suggest that the pair energy associated to the distributions with equal values of the mi quantum numbers could be the more significant part of the CEC