Theoret. China. Acta (Berl.) 43,261-271 (1977) THEORETICA CHIMICA ACTA 9 by Springer-Verlag 1977 On the Calculation of Multiplet Energies by the Hartree-Fock-Slater Method Tom Ziegler and Arvi Rank Department of Chemistry, University of Calgary, Calgary, Alberta, Canada T2N 1N4 Evert J. Baerends Seheikundig Laboratorium Der Vrije Universiteit, De Boelelaan 1083, Amsterdam, The Netherlands It is shown that a consistent application of the pi/3 approximation of the Hartree- Fock-Slater method requires the use of one specific procedure, the sum method, for the calculation of the energy Es 1 of singlet excited states of closed shell mole- cules. Further, Es 1 is found to be in reasonable agreement with experiment for a number of molecules, contrary to the energy Es 2 obtained according to another method discussed in the literature. The calculation of other multiplet splittings than singlet-triplet in the Hartree-Fock-Slater method is also considered. Key words: Multiplet energies - Calculation by the Hartree-Fock-Slater method 1. Introduction The Hartree-Fock-Slater Method [1 ] has proved itself to be a powerful tool for calcula- tions of excitation energies [2, 3]. Bagus and Bennett [4] have recently shown from numerical results that when singlet-triplet splittings are calculated by the Xa method, using two independent expressions derivable from one-electron theory, the results differ by a large amount. These authors expressed caution at the quantitative value of the splittings and made no mention of the absolute values of the singlet excitation energies. This communication begins by an analysis of the statistical exchange expression [ 1] in Sect. 3, where it is shown that the statistical exchange approximation only in general leads to the well-known statistical energy expression [ 1] for single determinantal wave functions. The results by Bagus and Bennett are discussed in Sect. 4, where it is shown that only one of the two values for the triplet-singlet splitting is consistent with the statistical exchange approximation. Finally, the conditions under which one can calcu- late multiplet energies by the Hartree-Fock-Slater method are discussed in Sect. 5.