Multidimensional derivative-free optimization. A case study on minimization of Hartree-Fock-Roothaan energy functionals
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911349832941568 |
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| author | Bagci, A. |
| author_facet | Bagci, A. |
| contents | 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. This work presents the first systematic investigation of derivative-free optimization methods for Hartree-Fock$-$Roothaan energy minimization with non-integer Slater orbitals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24509 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multidimensional derivative-free optimization. A case study on minimization of Hartree-Fock-Roothaan energy functionals Bagci, A. Quantum Physics Mathematical Physics 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. This work presents the first systematic investigation of derivative-free optimization methods for Hartree-Fock$-$Roothaan energy minimization with non-integer Slater orbitals. |
| title | Multidimensional derivative-free optimization. A case study on minimization of Hartree-Fock-Roothaan energy functionals |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2512.24509 |