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arXiv:2607.14263

AutoHF: a general Hartree-Fock solver utilizing direct energy minimization with automatic differentiation

Ryan Levy, Brandon Eskridge, Lukas Weber, Miguel A. Morales, Shiwei Zhang

Published Jul 15, 2026• 0 citations
arXivHTMLPDF

Abstract

We present autohf, a general, easy-to-use mean-field solver for quantum many-fermion Hamiltonians. It allows the user to bypass the process of deciphering the mean-field form for each many-body Hamiltonian $H$ and thus avoid setting up a tailored program for each $H$. Rather, autohf finds the optimal Slater determinant $|Ψ\rangle$, written in terms of orbital coefficients and subject to symmetry constraints, by directly minimizing the variational energy $\langle H \rangle$. By embracing this variational approach, autohf makes use of the growing power of automatic differentiation and optimization tools developed by the machine learning community.

DOI
10.48550/arxiv.2607.14263

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We present autohf, a general, easy-to-use mean-field solver for quantum many-fermion Hamiltonians. It allows the user to bypass the process of deciphering the mean-field form for each many-body Hamiltonian HHH and thus avoid setting up a tailored program for each HHH. Rather, autohf finds the optimal Slater determinant ∣Ψ⟩|Ψ\rangle∣Ψ⟩, written in terms of orbital coefficients and subject to symmetry constraints, by directly minimizing the variational energy ⟨H⟩\langle H \rangle⟨H⟩. By embracing this variational approach, autohf makes use of the growing power of automatic differentiation and optimization tools developed by the machine learning community.
Abstract Automatic differentiation (AD) is a powerful tool that allows calculating derivatives of implemented algorithms with respect to all of their parameters up to machine precision, without the need to explicitly add any additional functions. Thus, AD has great potential in quantum chemistry, where gradients are omnipresent but also difficult to obtain, and researchers typically spend a considerable amount of time finding suitable analytical forms when implementing derivatives. Here, we demonstrate that AD can be used to compute gradients with respect to any parameter throughout a complete quantum chemistry method. We present DiffiQult, a Hartree–Fock implementation, entirely differentiated with the use of AD tools. DiffiQult is a software package written in plain Python with minimal deviation from standard code which illustrates the capability of AD to save human effort and time in implementations of exact gradients in quantum chemistry. We leverage the obtained gradients to optimize the parameters of one-particle basis sets in the context of the floating Gaussian framework.
Automatic differentiation (AD) is a well-established tool that allows us to compute exact derivatives of implemented functions with respect to all their parameters. It does so without requiring any extra coding or knowledge of the analytical derivatives. Derivatives of quantum chemistry algorithms are generally very hard to obtain and often cumbersome to implement. AD shows great potential to solve this problem and to facilitate the implementation of gradients in the context of electronic structure methods. In this dissertation, we have applied AD in a broader context and demonstrate that AD tools can compute arbitrary gradients of quantum chemistry methods with respect to any input parameter and any intermediate function. To this end, we have developed the open-source code DiffiQult, a fully differentiable Hartree-Fock (HF) software package. DiffiQult is a python package that uses AD tools using the external libraries JAX and Algopy. As a proof of concept, DiffiQult is an example of an end-to-end differentiable software package of a quantum chemistry algorithm based solely on AD tools. This application has helped us understand the challenges of AD in quantum chemistry. We outline the details, benefits, and caveats of AD for the HF algorithm in this work. We have chosen HF as a starting point since it contains all the functions included in correlated electronic structure methods. In this way, it opens the door for the development of AD for more accurate algorithms. As an application, we describe DiffiQult within the framework of the fully variational HF method, where we use a gradient-based optimization method to compute system-specific basis functions. As a result, the wavefunction can capture more molecular features, and we obtain a set of basis functions for correlated methods in quantum and classical algorithms. This proves that these functions improve variationally energy while keeping a compact representation in correlated methods by interfacing with the quantum chemistry library PySCF, and with Tequila, a platform for rapid development of quantum algorithms. First, we present optimization results of minimal basis sets with respect to all parameters of the molecules H2O, CH4, NH3, HF. Later we include optimization of the potential energy surface of two isomerization mechanisms of diazene and hydrogens chains. Here, we use a minimal optimized basis set to compute FCI when possible with PySCF and VQE energies of hydrogen chains with Tequila. We obtain similar results compared to a bigger basis set with considerably fewer configurations. Also, we show how this approach improves the estimation of the thermodynamic limit in the hydrogen chain at the HF level. Finally, we apply this approach to the optimization of all atomic orbitals of benzene and compute electronic correlations restricting the active space to the 6 π orbitals of benzene using the UpCCGSD ansatz implemented in Tequila.
For a system of N fermions with Hamiltonian scrH we evaluate matrix elements of the T matrix between arbitrary Slater determinants \ensuremath{\Vert}\ensuremath{\chi}〉,\ensuremath{\Vert}\ensuremath{\chi}'〉. The resulting variational equations are driven Hartree-Fock-like equations, with source terms as right-hand sides, due to the ``channels'' \ensuremath{\Vert}\ensuremath{\chi}〉,\ensuremath{\Vert}\ensuremath{\chi}'〉. The validity of this mean-field approximation is illustrated by a numerical example.
Learning compact, interpretable descriptions of quantum many-body states is an important task in quantum science. We study the task of learning the Slater determinant with maximum fidelity to an arbitrary fermionic many-body state, with motivation from both Hartree-Fock methods and agnostic tomography. Given an nnn-fermion wavefunction built from mmm fermionic modes, we provide classical and quantum algorithms returning a Slater determinant with fidelity within ε\varepsilonε of maximal in time mpoly(n,1/ε)m^{\text{poly}(n,1/\varepsilon)}mpoly(n,1/ε). We prove matching hardness lower bounds, assuming standard complexity conjectures, along some parameter axes. Given access to quantum copies, we prove this can be accomplished with poly(m,n,1/ε)\text{poly}(m,n,1/\varepsilon)poly(m,n,1/ε) copies of ρρρ. We also show that above a fidelity of 2/32/32/3 any stationary point is the unique global maximum while below 2/32/32/3 the optimization landscape can have spurious stationary points, and hence 2/32/32/3 marks a transition point in the optimization landscape for this problem. We apply the algorithm to the Fermi-Hubbard model, extracting the closest Slater determinant from neural quantum state solutions. Together, our results provide algorithmic tools with provable guarantees in understanding fermionic many-body systems with classical or quantum simulation.
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 introduce an extension to the PySCF package, which makes it automatically differentiable. The implementation strategy is discussed, and example applications are presented to demonstrate the automatic differentiation framework for quantum chemistry methodology development. These include orbital optimization, properties, excited-state energies, and derivative couplings, at the mean-field level and beyond, in both molecules and solids. We also discuss some current limitations and directions for future work.