Slater's exchange parameters α for analytic and variational Xα calculations

Recently, we formulated a fully analytical and variational implementation of a subset of density functional theory using Gaussian basis sets to express orbital and the one-body effective potential. The implementation, called the Slater-Roothaan (SR) method, is an extension of Slater's X alpha method, which allows arbitrary scaling of the exchange potential around each type of atom in a heteroatomic system. The scaling parameter is Slater's exchange parameter, alpha, which can be determined for each type of atom by choosing various criteria depending on the nature of problem undertaken. Here, we determine these scaling parameters for atoms H through Cl by constraining some physical quantity obtained from the self-consistent solution of the SR method to be equal to its exact value. Thus, the sets of alpha values that reproduce the exact atomic energies have been determined for four different combinations of basis sets. A similar set of alpha values that is independent of a basis set is obtained from numerical calculations. These sets of alpha parameters are subsequently used in the SR method to compute atomization energies of the G2 set of molecules. The mean absolute error in atomization energies is about 17 kcal/mol and is smaller than that of the Hartree-Fock theory (78 kcal/mol) and the local density approximation (40 kcal/mol) but larger than that of a typical generalized gradient approximation (similar to 8 kcal/mol). A second set of alpha values is determined by matching the highest occupied eigenvalue of the SR method to the negative of the first ionization potential. Finally, the possibility of obtaining alpha values from the exact atomization energy of homonuclear diatomic molecules is explored. We find that the molecular alpha values show much larger deviation than what is observed for the atomic a values. The alpha values obtained for atoms in combination with an analytic SR method allow elemental properties to be extrapolated to heterogeneous molecules. In general, the sets of different alpha values might be useful for calculations of different properties using the analytic and variational SR method.