Direct minimization of Hartree-Fock-Roothaan energy functionals using derivative-free optimization algorithms: a case study on the use of noninteger Slater-type orbitals.

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Bibliographic Details
Title:	Direct minimization of Hartree-Fock-Roothaan energy functionals using derivative-free optimization algorithms: a case study on the use of noninteger Slater-type orbitals.
Authors:	BAĞCI, Ali¹ abagci@pau.edu.tr
Source:	Turkish Journal of Physics. 2026, Vol. 50 Issue 3, p148-169. 22p.
Subjects:	Simplex algorithm, Atomic models, Heuristic, Radial basis functions, Nonsmooth optimization
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 noninteger-order quantum numbers. 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 analysis is first carried out 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 nonconvex optimization landscape due to the nonlinear coupling of orbital parameters. Illustrative examples are provided for closed-shell atomic configurations, specifically the He and Be isoelectronic series, with calculations performed for energy functionals involving up to eight nonlinear parameters. This work presents the first systematic investigation of derivative-free optimization methods for Hartree-Fock-Roothaan energy minimization with noninteger Slater orbitals. [ABSTRACT FROM AUTHOR]
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Database:	Engineering Source

Description
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 noninteger-order quantum numbers. 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 analysis is first carried out 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 nonconvex optimization landscape due to the nonlinear coupling of orbital parameters. Illustrative examples are provided for closed-shell atomic configurations, specifically the He and Be isoelectronic series, with calculations performed for energy functionals involving up to eight nonlinear parameters. This work presents the first systematic investigation of derivative-free optimization methods for Hartree-Fock-Roothaan energy minimization with noninteger Slater orbitals. [ABSTRACT FROM AUTHOR]
ISSN:	13000101
DOI:	10.55730/1300-0101.2810