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arXiv:2512.24509

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

A. Bagci

quant-phmath-phSubmitted Dec 30, 2025• Announced Jan 1, 2026
v1 · Dec 30, 2025v2 · Jan 1, 2026
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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.

Journal reference
2026
DOI
10.55730/1300-0101.2810

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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.
A tractable direct optimization algorithm is developed to optimize orbitals in the Slater determinants (S-dets) in a resonating Hartree-Fock (Res HF) wave function. We reduce the variation space using the orbitals to put the first order energy variation in the steepest descent direction. The orbitals in the next iteration are determined so as to minimize the energy functional including up to the second order variation. This algorithm is applied to the one dimensional Hubbard model of half-filling. The optimized S-dets much deviate from the trial S-dets prepared from the HF calculations. The Res HF ground state generated with a few S-dets explains from 99.9 to 95.0% of the ground state correlation energy in all the correlation regimes. We have spin correlation functions with the correct short and long range behaviors and the lowest triplet and singlet spin excitations with correct dispersions, suggesting that the optimization of orbitals incorporates long range spin fluctuations and their mode-mode couplings.
We present a Riemannian optimization framework for Hartree-Fock theory formulated directly in the Sobolev space H1H^1H1. The orthonormality constraints are interpreted geometrically via infinite-dimensional Stiefel and Grassmann manifolds endowed with the embedded H1H^1H1 metric. Explicit expressions for Euclidean and Riemannian gradients, tangent-space projections, and retractions are derived using resolvent operators, avoiding distributional formulations. The resulting algorithms include Riemannian steepest descent and a preconditioned nonlinear conjugate gradient method equipped with Armijo backtracking and Powell-type restarts. Particular attention is given to physically motivated preconditioning based on inversion of the kinetic energy operator. The framework is naturally compatible with adaptive multiwavelet discretizations, where Coulomb-type convolutions can be evaluated efficiently. Numerical experiments demonstrate robust convergence and competitive performance compared to conventional SCF-DIIS schemes. In addition, for small molecules the gradient descent method converges from random initial guesses. The proposed formulation provides a geometrically consistent and discretization-independent perspective on electronic structure optimization and offers a foundation for further developments in infinite-dimensional Riemannian methods for quantum chemistry.
In quantum chemistry, self-consistent field (SCF) algorithms define a nonlinear optimization problem, with both continuous and discrete components. In this work, we derive Hartree–Fock-inspired SCF algorithms that can be exactly written as a sequence of Quadratic Unconstrained Spin/Binary Optimization problems (QUSO/QUBO). We reformulate the optimization problem as a series of MaxCut graph problems, which can be efficiently solved using semidefinite programming techniques. This procedure provides performance guarantees at each SCF step, irrespective of the complexity of the optimization landscape. We numerically demonstrate the QUBO-SCF and MaxCut-SCF methods by studying the hydroxide anion OH – and molecular Nitrogen N 2 . The largest problem addressed in this study involves a system comprised of 220 qubits (equivalently, spin–orbitals). Our results show that QUBO-SCF and MaxCut-SCF suffer much less from internal instabilities compared with conventional SCF calculations. Additionally, we show that the new SCF algorithms can enhance single-reference methods, such as configuration interaction. Finally, we explore how quantum algorithms for optimization can be applied to the QUSO problems arising from the Hartree–Fock method. Four distinct hybrid-quantum classical approaches are introduced: GAS-SCF, QAOA-SCF, QA-SCF and DQI-SCF.
In this thesis, we develop, analyze, and implement effective and efficient methods for constrained stochastic and derivative-free optimization. For each method, we prove the convergence result under general assumptions. We also conduct numerical experiments to demonstrate the efficiency and efficacy of our proposed methods. In Chapter 2, we develop a model-based derivative-free optimization method. A key component of variational quantum algorithms (VQAs) is the choice of a classical optimizer to update the ansatz parameterization. Quantum algorithms, however, are destined to run on noisy hardware with limited fidelity, leading to objective function evaluations–such as those in the quantum approximate optimization algorithm (QAOA) or the variational quantum eigensolver (VQE)–that are compromised by both stochastic errors and hardware-induced noise. To address these challenges, we adapt recent developments from the noise-aware numerical optimization literature to these commonly used derivative-free model-based methods. We introduce the key defining characteristics of these novel noise-aware derivative-free model-based methods that separate them from standard model-based methods. We study an implementation of such noise-aware derivative-free model-based methods and compare its performance on demonstrative VQA simulations to classical solvers packaged in scikit-quant. Moreover, we propose a series of sequential quadratic programming (SQP) methods tailored for stochastic equality constrained optimization. In Chapter 3, we develop a stochastic method for solving equality constrained problems that utilizes predictive variance reduction within the SQP paradigm, and prove that a measure of first-order stationarity converges to zero in expectation under reasonable assumptions, with practical performance demonstrated on constrained binary classification problems. In Chapter 4, we propose a line search SQP method for general nonlinear equality constrained optimization that employs a modified line search strategy using second-order information to mitigate the Maratos effect, thereby achieving global convergence and local superlinear convergence. We further extend this approach to settings where the objective function is stochastic or represented as a finite sum, and design a practical inexact matrix-free variant and demonstrate the numerical performance empirically. Finally, in Chapter 5, we propose, analyze, and implement an SQP algorithm for minimizing a nonlinear smooth function subject to smooth equality constraints with noisy function and derivative estimates, addressing the challenges posed by bounded noise in the objective and constraint functions as well as potential rank-deficiency in the constraint Jacobians. This algorithm employs a decomposition approach to compute inexact search directions and utilizes adaptive step size selection schemes, which guarantee global convergence of infeasible stationarity error to a neighborhood of zero that is proportional to the noise level.
Slater determinants have underpinned quantum chemistry for nearly a century, yet their full potential has remained challenging to exploit. In this work, we show that a variational wavefunction composed of a few hundred optimized non-orthogonal determinants can achieve energy accuracies comparable to the state of the art. This is obtained by introducing an optimization method that leverages the quadratic dependence of the variational energy on the orbitals of each determinant, enabling an exact iterative optimization, and uses an efficient tensor-contraction algorithm to evaluate the effective Hamiltonian with a computational cost that scales as the fourth power of the number of basis functions. We benchmark the accuracy of the proposed method with exact full-configuration interaction results where available, and we achieve lower variational energies than coupled cluster (CCSD(T)) for several molecules in the double-zeta basis. Accurate description of electronic wave functions typically requires large numbers of Slater determinants. Here, the authors develop an optimization method that achieves state-of-the-art accuracy for small molecules with only few hundred determinants.
This work involves studying and developing new algorithms for molecular numerical integration used for density functional theory and new algorithms for Hartree-Fock method. -- New insight about molecular numerical integration is presented through a detailed study of the performance of some of the well known grids in addition to our implementation of the most recently developed MultiExp grid. A comprehensive study of numerical integration was conducted by evaluating several molecular properties: number of electrons, dipole moment, potential energy, and Coulomb repulsion energy using fifteen grids including a large benchmark grid. The standard grid (SG-1) and a slightly modified version of the Treutler and Alhrichs (TA) grid performed reasonably well. The MultiExp grid, which is more efficient, was studied as well and found to be less accurate. -- Studying large molecules using Hartree-Fock method is a challenge both in terms of CPU time and memory requirements. However, there is a high demand to perform quantum chemical calculations for large molecules. Projection from a smaller basis set to a larger basis set was studied in detail. It was found that projection from the STO-3G basis set to the 6-31G basis set performed well. Projection was used to develop a new version of a divide and conquer algorithm. Our divide and conquer algorithm was used to calculate the protonation energy for a series of peptides. Algorithms to skip calculating two-electron integrals of zero or negligible values are presented in addition to an algorithm to generate a better initial guess.