Application of an optimization technique to the discretized version of the Griffin–Hill–Wheeler–Hartree–Fock equations

Application of the SIMPLEX optimization method to define the mesh of the discretized version of the Griffin–Hill–Wheeler–Hartree–Fock (GHWHF ) equations was studied. Improved discretization parameters with respect to the original method were obtained for atomic systems with two or four electrons and for the H₂ molecule. For the atomic systems, the following correlations between the discretization parameters and the total energy were found: N = a · In(ΔE) + b; Ω₀ = a′ · In(ΔΩ) + a″; and In(ΔΩ) = b′ · ln(N) + b″. These equations provide a systematic procedure to reach a desired degree of accuracy in the energy for the atomic systems studied as well as to fix the basis set to be employed. These equations are similar to those found earlier for eventempered basis sets and permit the establishment of a relationship between the two methods. The even-tempered method is also an approximate solution of the GHWHF equations. The optimized integral discretized basis is more efficient in representing small basis sets for atoms and the basis for the hydrogen molecule in comparison to the even-tempered one. The optimization procedure was successfully applied to generate the universal basis for the atomic systems studied.