Multidimensional derivative-free optimization. A case study on minimization of Hartree-Fock-Roothaan energy functionals

TL;DR

The paper addresses derivative-free optimization of Hartree-Fock-Roothaan energy functionals that depend on nonlinear orbital parameters and noninteger Slater-type orbitals, comparing four DFO methods under identical conditions. It evaluates Powell’s conjugate-direction, Nelder-Mead simplex, coordinate pattern search, and a radial-basis-function surrogate on the Powell singular function and then applies them to HFR minimization for closed-shell atomic systems, up to eight nonlinear parameters. The key finding is that the Nelder-Mead simplex offers the most robust and efficient performance across dimensions, while Powell CD degrades with higher dimensionality and MB-RBF remains computationally intensive; NM simplex also yields energy values in close agreement with established Hartree-Fock benchmarks for Be and He-like ions. The study demonstrates that a derivative-free optimization framework can effectively accelerate convergence toward the HFR limit using $n^{*}$-$STOs$, with potential extension to larger atomic or molecular systems and to exponential-type orbitals while remaining implementable in languages beyond Mathematica.

Abstract

This study presents an evaluation of derivative-free optimization algorithms for the direct minimization of Hartree-Fock-Roothaan energy functionals involving nonlinear orbital parameters and quantum numbers with noninteger order. The analysis focuses on atomic calculations employing noninteger Slater-type orbitals. Analytic derivatives of the energy functional are not readily available for these orbitals. Four methods are investigated under identical numerical conditions: Powell's conjugate-direction method, the Nelder-Mead simplex algorithm, coordinate-based pattern search, and a model-based algorithm utilizing radial basis functions for surrogate-model construction. Performance benchmarking is first performed using the Powell singular function, a well-established test case exhibiting challenging properties including Hessian singularity at the global minimum. The algorithms are then applied to Hartree-Fock-Roothaan self-consistent-field energy functionals, which define a highly non-convex optimization landscape due to the nonlinear coupling of orbital parameters. Illustrative examples are provided for closed$-$shell atomic configurations, specifically the He, Be isoelectronic series, with calculations performed for energy functionals involving up to eight nonlinear parameters.

Figures (1)

• Figure 1: Convergence behaviour of orbital parameters in optimization of HFR energy functional with NM$-$Simplex algorithm.