1 MULTIPLET ENERGIES

The energies of atomic and molecular multiplet states that correspond to a given electron configuration can be calculated approximately with the method suggested in ref. [1]. There it is indicated that it would not be justified to take an arbitrary configuration-state function (CSF), defined in general as a linear combination of determinants that has specific spin and space symmetry properties, and use the corresponding alpha and beta spin densities in a DFT energy expression. The same holds true for the densities corresponding to the average-of-configuration (see section 1.1, the 'procedure' notes).

Therefore, we restrict ourselves to just computing the DFT energies of single-determinant wavefunctions. Usually (but not always) this is sufficient information to obtain the multiplet energies. The procedure, which is explained in [1], requires knowledge of the CSFs belonging to a given configuration. This means that a multiplet state with specific L, M_L and S, M_S values has to be written as a linear combination of the determinant wavefunctions that belong to the given configuration. With the auxiliary program ASF (Adf Single-determinants Fribourg, developed by Claude Daul in Fribourg, Switzerland) all the CSFs can be obtained, printed as linear combinations of the determinants [2]. The inverse transformation yields the determinants written as linear combinations of the CSFs.

It is often advantageous to search for CSFs that consist of one determinant only, since the energy of this determinant should correspond directly to the multiplet energy. Sometimes there is redundancy in the information and there may even be some inconsistency: two determinants may exist that both are CSFs belonging to the same multiplet state but yield somewhat different energies. We will illustrate this for the Carbon atom example treated below.

The discrepancies are a measure of 'error bars' associated with the theoretical multiplet energies. As a matter of fact, there are certain symmetry relations between the energies of the determinants of a configuration, calculated as the expectation value of the determinant for the full many-electron Hamiltonian. An example is the equal energy for the determinants of a p¹ configuration, whether the electron is placed in the p₀ (=p_z) orbital or in the p₊₁ (=(p_x+ip_y)/√2) orbital. This equality is not obtained with present-day density functionals, leaving an ambiguity ('error bar') in the determination of the energy. A more complete treatment of the symmetry relations between determinant energies is given in [2].

The auxiliary program ASF, that for finite point groups finds the CSFs as linear combinations of determinants, performs also a symmetry analysis of all the two-electron integrals for a configuration, reducing them to a minimum number of non-redundant ones. ASF expresses the energies of the multiplets in the non-redundant two-electron integrals. However (WARNING!), there have occasionally been found inconsistent results. A comparison to the results obtained by the procedure outlined in [1] may show significant differences and the latter seem more accurate and consistent.