Self-Consistent-Field $X\alpha$ Cluster Method for Polyatomic Molecules and Solids

This paper describes a practical self-consistent-field (SCF) method of calculating electronic energy levels and eigenfunctions, adapted for polyatomic molecules and solids. The one-electron Schrödinger equation is set up for a so-called "muffin-tin" approximation to the true potential, spherically symmetrical within spheres surrounding the various nuclei, constant in the region between the spheres, spherically symmetrical outside a sphere surrounding the molecule. The method of solving this equation is a multiple-scattering method, equivalent to the Korringa-Kohn-Rostoker (KKR) method often used for crystals. Once the eigenfunctions and eigenvalues of this problem are determined, one assumes that the orbitals of lowest eigenvalue are occupied, up to a Fermi level. From the resulting charge densities, one can compute a total energy, using a statistical approximation for the exchange correlation. This approximation has an undetermined factor $\alpha$ (whence the name $X\alpha$ method). The spin orbitals and occupation numbers are varied to minimize this total energy, resulting in one-electron equations. The value of $\alpha$ for an isolated atom is determined by requiring that the total energy, using the statistical approximation, should equal the precise Hartree-Fock energy. This leads to very accurate spin orbitals. In a molecule or crystal, one uses the $\alpha \text{'}\mathrm{s}$ characteristic of the various atoms within the atomic spheres, and a suitable average in the region between. The computer programs for making these self-consistent calculations, for such radicals and polyatomic molecules as S${\mathrm{O}}_4^{-2\mathrm{}}$, Cl${\mathrm{O}}_4^{-}$, Mn${\mathrm{O}}_4^{-}$, and S${\mathrm{F}}_6$ have been worked out and calculations made. They are more than 100 times as fast as comparable programs using the LCAO (linear-combination-of-atomic-orbitals) method, and the results appear to be in better agreement with experiment than such LCAO results. For calculating the frequencies of optical transitions, one must make a self-consistent calculation, not for the initial or final state, but for what we call the transition state, in which occupation numbers are halfway between the initial and final states. Then it can be proved that the differences of eigenvalues of the $X\alpha$ method are more accurate than Hartree-Fock energy values, in that they take account of the modification or relaxation of the orbitals in going from the initial to the final states. These transition states, for a crystal, involve a localized perturbation at the site of the excited atom. The multiple-scattering method is adapted to the use of such perturbed crystals, as well as to isolated molecules, and to perfect crystals. It results, in such problems as x-ray absorption, in the use of localized orbitals rather than bandlike functions. The method is adapted to the calculation of magnetic problems, by use of a spin-polarized version of the method. The method can also be used for calculations of cohesive energy of crystals, and has been used successfully for several types of metals. Unlike most other SCF methods, long-range correlation is automatically included, so that the energy of a system as a function of internuclear positions automatically reduces to the proper values at infinite internuclear distances.