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arXiv:2411.18493 · Physical Review Letters

Advancing Natural Orbital Functional Calculations through Deep Learning-Inspired Techniques for Large-Scale Strongly Correlated Electron Systems

Juan Felipe Huan Lew-Yee, Jorge M. del Campo, Mario Piris

Published May 20, 2025• 8 citations
arXivHTMLPDF

Abstract

Natural orbital functional (NOF) theory provides a valuable framework for studying strongly correlated systems at an affordable computational cost, with an accuracy comparable to highly demanding wave-function-based methods. However, its widespread adoption in cases involving a large number of correlated electrons has been limited by the extensive iterations required for convergence. In this Letter, we present an approach that integrates the techniques used for optimization in deep learning into NOF calculations, enabling a substantial expansion in the scale of accessible systems. The proposed procedure employs an adaptive momentum-based technique for orbital optimization, alternated with the optimization of occupation numbers, significantly improving the computational feasibility of challenging calculations. We illustrate the capabilities of our approach through three challenging test cases: (i) the symmetric dissociation of a large hydrogen cluster with 1000 electrons, (ii) an analysis of occupancy distributions in fullerenes, and (iii) a study of the singlet-triplet energy gap in linear acenes. These examples demonstrate the method's applicability to large-scale systems and strongly correlated electron phenomena, extending the reach of NOF theory to increasingly complex systems.

DOI
10.1103/physrevlett.134.206401

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Natural orbital functional (NOF) theory provides a valuable framework for studying strongly correlated systems at an affordable computational cost, with an accuracy comparable to highly demanding wave-function-based methods. However, its widespread adoption in cases involving a large number of correlated electrons has been limited by the extensive iterations required for convergence. In this Letter, we present an approach that integrates the techniques used for optimization in deep learning into NOF calculations, enabling a substantial expansion in the scale of accessible systems. The proposed procedure employs an adaptive momentum-based technique for orbital optimization, alternated with the optimization of occupation numbers, significantly improving the computational feasibility of challenging calculations. We illustrate the capabilities of our approach through three challenging test cases: (i) the symmetric dissociation of a large hydrogen cluster with 1000 electrons, (ii) an analysis of occupancy distributions in fullerenes, and (iii) a study of the singlet-triplet energy gap in linear acenes. These examples demonstrate the method's applicability to large-scale systems and strongly correlated electron phenomena, extending the reach of NOF theory to increasingly complex systems.
In the past decade, natural orbital functional (NOF) approximations have emerged as prominent tools for characterizing electron correlation. Despite their effectiveness, these approaches, which rely on natural orbitals and their associated occupation numbers, often require hybridization with other methods to fully account for all correlation effects. Recently, a global NOF (GNOF) has been proposed [Phys. Rev. Lett. 127, 233001 (2021)] to comprehensively address both dynamic and static correlations. This study evaluates the performance of GNOF on strongly correlated model systems, including comparisons with highly accurate Full Configuration Interaction (FCI) calculations for hydrogen atom clusters in one, two, and three dimensions. Additionally, the investigation extends to a BeH2 reaction, involving the insertion of a beryllium atom into a hydrogen molecule along a C2v pathway. According to the results obtained using GNOF, consistent behavior is observed across various correlation regions, encompassing a range of occupation and orbital schemes. Furthermore, distinctive features are identified when varying the dimensionality of the system.
We present a quantum-classical algorithm for electronic structure calculations that dramatically reduces the quantum measurement cost of variational quantum eigensolver (VQE) approaches. While conventional VQE methods require measurements scaling as O(M^4) with system size M, the proposed occupation-number VQE (ON-VQE) reduces this cost to O(M/2) by avoiding reduced density matrix (RDM) measurements and relying exclusively on ONs. The method exploits only the diagonal elements of the one-particle RDM in the natural orbital representation, where occupations are obtained directly from computational-basis measurement outcomes. By restricting the variational ansatz to double excitations within orbital subspaces associated with electron pairs, the required measurements can be grouped into a small number of qubit-wise commuting observables, yielding an efficient and scalable measurement strategy. The approach is validated through simulations and executions on quantum hardware for the cubic H8_88​ cluster, demonstrating the feasibility of extracting accurate ONs from quantum measurements and evaluating electronic energies within the natural orbital functional (NOF) framework. Across representative molecular systems, the extracted ONs enable accurate energy evaluation with state-of-the-art NOFs while maintaining a dramatically reduced measurement cost. These results establish a scalable route toward quantum simulation of strongly correlated electronic systems, demonstrating that accurate electronic energies can be obtained from quantum measurements of ONs alone.
Density functional theory with plane-wave basis sets is widely employed in computational materials science, including applications to isolated molecular systems. However, the inadequate description of electron correlation remains a fundamental limitation. Accurate correlation treatments based on many-body Hamiltonians require reliable representations of both occupied and virtual orbitals, yet virtual orbitals are often poorly described in conventional computational schemes, resulting in reduced accuracy. In this work, we introduce localized correlation-converged virtual orbitals (LCCVOs) as an efficient basis for constructing accurate many-body Hamiltonians in molecular systems. Using a substantially reduced number of orbitals, the LCCVO framework yields dissociation energies for singlet, doublet, and triplet molecules that are comparable to, and in many cases exceed, those obtained with high-level correlation-consistent basis sets such as cc-pVXZ (X = D, T, Q, 5). These results demonstrate the efficiency, scalability, and robustness of the LCCVO approach for high-accuracy quantum chemical calculations.
We introduce a novel energy functional for ground-state electronic-structure calculations. Its fundamental variables are the natural spin-orbitals of the implied singlet many-body wave function and their joint occupation probabilities. The functional derives from a sequence of controlled approximations to the two-particle density matrix. Algebraic scaling of computational cost with electron number is obtainable in general, and Hartree-Fock scaling in the seniority-zero version of the theory. Results obtained with the latter version for saturated small molecular systems are compared with those of highly-accurate quantum-chemical computations. The numerical results are variational, capturing most of the correlation energy from equilibrium to dissociation. Their accuracy is considerably greater than that obtainable with current density-functional theory approximations and with current functionals of the one-particle density matrix only.
We develop a new family of electronic structure methods for capturing at the same time the dynamic and nondynamic correlation effects. We combine the natural orbital functional theory (NOFT) and many-body perturbation theory (MBPT) through a canonicalization procedure applied to the natural orbitals to gain access to any MBPT approximation. We study three different scenarios: corrections based on second-order Møller–Plesset (MP2), random-phase approximation (RPA), and coupled-cluster singles doubles (CCSD). Several chemical problems involving different types of electron correlation in singlet and multiplet spin states have been considered. Our numerical tests reveal that RPA-based and CCSD-based corrections provide similar relative errors in molecular dissociation energies ( D e ) to the results obtained using a MP2 correction. With respect to the MP2 case, the CCSD-based correction improves the prediction, while the RPA-based correction reduces the computational cost.
This work deals with the problem of strongly correlated electrons in two-dimensions. We give a reduced density matrix (RDM) based tool through which the ground-state energy is given as a functional of the natural orbitals and their occupation numbers. Specifically, the Piris Natural Orbital Functional 7 (PNOF7) is used for studying the 2D Hubbard model and hydrogen square lattices. The singlet ground-state is studied, as well as the doublet mixed quantum state obtained by extracting an electron from the system. Our method satisfies two-index necessary N-representability conditions of the two-particle RDM (2RDM) and guarantees the conservation of the total spin. We show the ability of PNOF7 to describe strong correlation effects in two-dimensional (2D) systems by comparing our results with the exact diagonalization, density matrix renormalization group (DMRG), and auxiliary-field quantum Monte Carlo calculations. PNOF7 overcomes variational 2RDM methods with two- and three-index positivity N-representability conditions, reducing computational cost to mean-field scaling. Consistent results are obtained for small and large systems up to 144 electrons, weak and strong correlation regimes, and many filling situations. Unlike other methods, there is no dependence on dimensionality in the results obtained with PNOF7 and no particular difficulties have been observed to converge PNOF7 away from half-filling. Smooth double occupancy of sites is obtained, regardless of the filling. Symmetric dissociation of 2D hydrogen lattices shows that long-range nondynamic correlation dramatically affects electron detachment energies. PNOF7 compares well with DMRG along the dissociation curve.