Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The Journal of Physical Chemistry Letters

Direct Variational Calculation of Two-Electron Reduced Density Matrices via Semidefinite Machine Learning

Luis H. Delgado-Granados, David A. Mazziotti

Published Jun 18, 2026• 1 citation
Publisher

Abstract

Abstract We introduce a data-driven framework for approximating the convex set of N-representable two-electron reduced density matrices (2-RDMs). Traditional approaches characterize this set through linear matrix inequalities that define its supporting hyperplanes. Here, we instead learn a vertex-based approximation to its boundary from molecular data and use this information to improve the set defined by low-order positivity constraints, without explicitly constructing higher order conditions. The resulting semidefinite machine learning approach, combining an input convex neural network with semidefinite programming, drives a direct variational calculation of the 2-RDM with enhanced accuracy at a computational cost comparable to two-positivity calculations. Applications to the potential energy curves of C22–, N2, O22+, CO, NO+, and CN– demonstrate these systematic improvements as well as close agreement with complete active space configuration interaction results. Overall, semidefinite machine learning interweaves data-driven boundary information with semidefinite positivity constraints to yield more accurate energies and 2-RDMs without explicit higher order positivity conditions.

DOI
10.1021/acs.jpclett.6c00615

Related papers

Direct Variational Calculation of Two-Electron Reduced Density Matrices via Semidefinite Machine Learning

Luis H. Delgado-Granados, David A. Mazziotti

We introduce a data-driven framework for approximating the convex set of $N$-representable two-electron reduced density matrices (2-RDMs). Traditional approaches characterize this set through linear matrix inequalities that define its supporting hyperplanes. Here, we instead learn a vertex-based approximation to its boundary from molecular data and use this information to improve the set defined by low-order positivity constraints, without explicitly constructing higher-order conditions. The resulting semidefinite machine learning approach -- combining an input convex neural network with semidefinite programming -- drives a direct variational calculation of the 2-RDM with enhanced accuracy at computational cost comparable to two-positivity calculations. Applications to the potential energy curves of ${\rm C}_2^{2-}$, ${\rm N}_2$, and ${\rm O}_2^{2+}$ demonstrate these systematic improvements as well as close agreement with complete active space configuration interaction results. Overall, semidefinite machine learning interweaves data-driven boundary information with semidefinite positivity constraints to yield more accurate energies and 2-RDMs without explicit higher-order positivity conditions.

physics.chem-phCross-listedarXiv:2603.05524•Feb 25, 2026•0 citations

Machine Learning of Two-Electron Reduced Density Matrices for Many-Body Problems

Luis H. Delgado-Granados, LeeAnn M. Sager-Smith, Kristina Trifonova +1 more

We present a novel machine learning algorithm for the many-electron problem, predicting the convex combination of two-electron reduced density matrices (2-RDMs)─obtained from upper- and lower-bound energy calculations─that closely approximates the exact energy. In contrast to other recently developed approaches based on the wave function or one-electron density, our 2-RDM machine-learning approach predicts energies and properties without steep scaling or functional approximation. As conjectured by Preskill and co-workers, a small amount of data in a physics-based machine learning algorithm─in this case, information about the RDMs and their violation of selected higher-order N -representability conditions─yields highly accurate electronic energies that capture both dynamic and static correlation. We demonstrate the method by predicting the potential energy curves for BH and N 2 within a few millihartrees of results from exact diagonalization. This machine learning algorithm provides a general framework for improving electronic structure calculations, with the potential for wide-reaching applications to both moderately and strongly correlated molecular systems.

The Journal of Physical Chemistry Letters•Feb 21, 2025•6 citations

First-order semidefinite programming for the direct determination of two-electron reduced density matrices with application to many-electron atoms and molecules

David A. Mazziotti

Direct variational calculation of two-electron reduced density matrices (2-RDMs) for many-electron atoms and molecules in nonminimal basis sets has recently been achieved through the use of first-order semidefinite programming [D. A. Mazziotti, Phys. Rev. Lett. (in press)]. With semidefinite programming, the electronic ground-state energy of a molecule is minimized with respect to the 2-RDM subject to N-representability constraints known as positivity conditions. Here we present a detailed account of the first-order algorithm for semidefinite programming and its comparison with the primal-dual interior-point algorithms employed in earlier variational 2-RDM calculations. The first-order semidefinite-programming algorithm, computations show, offers an orders-of-magnitude reduction in floating-point operations and storage in comparison with previous implementations. We also examine the ability of the positivity conditions to treat strong correlation and multireference effects through an analysis of the Hamiltonians for which the conditions are exact. Calculations are performed in nonminimal basis sets for a variety of atoms and molecules and the potential-energy curves for CO and H(2)O.

The Journal of Chemical Physics•Nov 30, 2004•87 citations

Variational reduced-density-matrix method using three-particle N -representability conditions with application to many-electron molecules

David A. Mazziotti

Molecular two-electron reduced density matrices (2-RDMs) are computed variationally without the many-electron wave function by constraining the 2-RDM with a set of three-particle $N$-representability conditions known as three-positivity conditions. These constraints restrict four distinct three-particle probability distributions, which can be defined for any $N$-particle system, to be nonnegative. The variational calculation of the 2-RDM with full three-positivity conditions is implemented with the first-order semidefinite programming algorithm [D.A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. We also derive and implement a generalization of the ${T}_{2}$ representability condition, which is a subset of the three-positivity conditions. Energies and 2-RDMs are computed for several molecules as well as the nitrogen molecule at both equilibrium and nonequilibrium geometries. The ground-state energies for nitrogen are within $0.0015\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$ of the values from full configuration interaction at all internuclear distances.

Physical Review A•Sep 1, 2006•123 citations

Variational reduced density matrix method in the doubly occupied configuration interaction space using three-particle N-representability conditions

Diego R. Alcoba, P. Capuzzi, Álvaro Rubio-García +5 more

Ground-state energies and two-particle reduced density matrices (2-RDMs) corresponding to N-particle systems are computed variationally within the doubly occupied configuration interaction (DOCI) space by constraining the 2-RDM to satisfy a complete set of three-particle N-representability conditions known as three-positivity conditions. These conditions are derived and implemented in the variational calculation of the 2-RDM with standard semidefinite programming algorithms. Ground state energies and 2-RDMs are computed for N2, CO, CN−, and NO+ molecules at both equilibrium and nonequilibrium geometries as well as for pairing models at different repulsive interaction strengths. The results from the full three-positivity conditions are compared with those from the exact DOCI method and with approximated 2-RDM variational ones obtained within two-positivity and two-positivity plus a subset of three-positivity conditions, as recently reported [D. R. Alcoba et al., J. Chem. Phys. 148, 024105 (2018) and A. Rubio-García et al., J. Chem. Theory Comput. 14, 4183 (2018)]. The accuracy of these numerical determinations and their low computational cost demonstrate the usefulness of the three-particle variational constraints within the DOCI framework.

The Journal of Chemical Physics•Nov 20, 2018•20 citations

Rank restriction for the variational calculation of two-electron reduced density matrices of many-electron atoms and molecules

Kasra Naftchi-Ardebili, Nathania W. Hau, David A. Mazziotti

Variational minimization of the ground-state energy as a function of the two-electron reduced density matrix (2-RDM), constrained by necessary $N$-representability conditions, provides a polynomial-scaling approach to studying strongly correlated molecules without computing the many-electron wave function. Here we introduce a route to enhancing necessary conditions for $N$ representability through rank restriction of the 2-RDM. Rather than adding computationally more expensive $N$-representability conditions, we directly enhance the accuracy of two-particle (2-positivity) conditions through rank restriction, which removes degrees of freedom in the 2-RDM that are not sufficiently constrained. We select the rank of the particle-hole 2-RDM by deriving the ranks associated with model wave functions, including both mean-field and antisymmetrized geminal power (AGP) wave functions. Because the 2-positivity conditions are exact for quantum systems with AGP ground states, the rank of the particle-hole 2-RDM from the AGP ansatz provides a minimum for its value in variational 2-RDM calculations of general quantum systems. To implement the rank-restricted conditions, we extend a first-order algorithm for large-scale semidefinite programming. The rank-restricted conditions significantly improve the accuracy of the energies; for example, the percentages of correlation energies recovered for HF, CO, and N${}_{2}$ improve from 115.2%, 121.7%, and 121.5% without rank restriction to 97.8%, 101.1%, and 100.0% with rank restriction. Similar results are found at both equilibrium and nonequilibrium geometries. While more accurate, the rank-restricted $N$-representability conditions are less expensive computationally than the full-rank conditions.

arXiv:1111.1978•Physical Review A•Nov 8, 2011•14 citations
Abstract We introduce a data-driven framework for approximating the convex set of N-representable two-electron reduced density matrices (2-RDMs). Traditional approaches characterize this set through linear matrix inequalities that define its supporting hyperplanes. Here, we instead learn a vertex-based approximation to its boundary from molecular data and use this information to improve the set defined by low-order positivity constraints, without explicitly constructing higher order conditions. The resulting semidefinite machine learning approach, combining an input convex neural network with semidefinite programming, drives a direct variational calculation of the 2-RDM with enhanced accuracy at a computational cost comparable to two-positivity calculations. Applications to the potential energy curves of C22–, N2, O22+, CO, NO+, and CN– demonstrate these systematic improvements as well as close agreement with complete active space configuration interaction results. Overall, semidefinite machine learning interweaves data-driven boundary information with semidefinite positivity constraints to yield more accurate energies and 2-RDMs without explicit higher order positivity conditions.
We introduce a data-driven framework for approximating the convex set of NNN-representable two-electron reduced density matrices (2-RDMs). Traditional approaches characterize this set through linear matrix inequalities that define its supporting hyperplanes. Here, we instead learn a vertex-based approximation to its boundary from molecular data and use this information to improve the set defined by low-order positivity constraints, without explicitly constructing higher-order conditions. The resulting semidefinite machine learning approach -- combining an input convex neural network with semidefinite programming -- drives a direct variational calculation of the 2-RDM with enhanced accuracy at computational cost comparable to two-positivity calculations. Applications to the potential energy curves of C22−{\rm C}_2^{2-}C22−​, N2{\rm N}_2N2​, and O22+{\rm O}_2^{2+}O22+​ demonstrate these systematic improvements as well as close agreement with complete active space configuration interaction results. Overall, semidefinite machine learning interweaves data-driven boundary information with semidefinite positivity constraints to yield more accurate energies and 2-RDMs without explicit higher-order positivity conditions.
We present a novel machine learning algorithm for the many-electron problem, predicting the convex combination of two-electron reduced density matrices (2-RDMs)─obtained from upper- and lower-bound energy calculations─that closely approximates the exact energy. In contrast to other recently developed approaches based on the wave function or one-electron density, our 2-RDM machine-learning approach predicts energies and properties without steep scaling or functional approximation. As conjectured by Preskill and co-workers, a small amount of data in a physics-based machine learning algorithm─in this case, information about the RDMs and their violation of selected higher-order N -representability conditions─yields highly accurate electronic energies that capture both dynamic and static correlation. We demonstrate the method by predicting the potential energy curves for BH and N 2 within a few millihartrees of results from exact diagonalization. This machine learning algorithm provides a general framework for improving electronic structure calculations, with the potential for wide-reaching applications to both moderately and strongly correlated molecular systems.
Direct variational calculation of two-electron reduced density matrices (2-RDMs) for many-electron atoms and molecules in nonminimal basis sets has recently been achieved through the use of first-order semidefinite programming [D. A. Mazziotti, Phys. Rev. Lett. (in press)]. With semidefinite programming, the electronic ground-state energy of a molecule is minimized with respect to the 2-RDM subject to N-representability constraints known as positivity conditions. Here we present a detailed account of the first-order algorithm for semidefinite programming and its comparison with the primal-dual interior-point algorithms employed in earlier variational 2-RDM calculations. The first-order semidefinite-programming algorithm, computations show, offers an orders-of-magnitude reduction in floating-point operations and storage in comparison with previous implementations. We also examine the ability of the positivity conditions to treat strong correlation and multireference effects through an analysis of the Hamiltonians for which the conditions are exact. Calculations are performed in nonminimal basis sets for a variety of atoms and molecules and the potential-energy curves for CO and H(2)O.
Molecular two-electron reduced density matrices (2-RDMs) are computed variationally without the many-electron wave function by constraining the 2-RDM with a set of three-particle NNN-representability conditions known as three-positivity conditions. These constraints restrict four distinct three-particle probability distributions, which can be defined for any NNN-particle system, to be nonnegative. The variational calculation of the 2-RDM with full three-positivity conditions is implemented with the first-order semidefinite programming algorithm [D.A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. We also derive and implement a generalization of the T2{T}_{2}T2​ representability condition, which is a subset of the three-positivity conditions. Energies and 2-RDMs are computed for several molecules as well as the nitrogen molecule at both equilibrium and nonequilibrium geometries. The ground-state energies for nitrogen are within 0.0015a.u.0.0015\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}0.0015a.u. of the values from full configuration interaction at all internuclear distances.
Ground-state energies and two-particle reduced density matrices (2-RDMs) corresponding to N-particle systems are computed variationally within the doubly occupied configuration interaction (DOCI) space by constraining the 2-RDM to satisfy a complete set of three-particle N-representability conditions known as three-positivity conditions. These conditions are derived and implemented in the variational calculation of the 2-RDM with standard semidefinite programming algorithms. Ground state energies and 2-RDMs are computed for N2, CO, CN−, and NO+ molecules at both equilibrium and nonequilibrium geometries as well as for pairing models at different repulsive interaction strengths. The results from the full three-positivity conditions are compared with those from the exact DOCI method and with approximated 2-RDM variational ones obtained within two-positivity and two-positivity plus a subset of three-positivity conditions, as recently reported [D. R. Alcoba et al., J. Chem. Phys. 148, 024105 (2018) and A. Rubio-García et al., J. Chem. Theory Comput. 14, 4183 (2018)]. The accuracy of these numerical determinations and their low computational cost demonstrate the usefulness of the three-particle variational constraints within the DOCI framework.
Variational minimization of the ground-state energy as a function of the two-electron reduced density matrix (2-RDM), constrained by necessary NNN-representability conditions, provides a polynomial-scaling approach to studying strongly correlated molecules without computing the many-electron wave function. Here we introduce a route to enhancing necessary conditions for NNN representability through rank restriction of the 2-RDM. Rather than adding computationally more expensive NNN-representability conditions, we directly enhance the accuracy of two-particle (2-positivity) conditions through rank restriction, which removes degrees of freedom in the 2-RDM that are not sufficiently constrained. We select the rank of the particle-hole 2-RDM by deriving the ranks associated with model wave functions, including both mean-field and antisymmetrized geminal power (AGP) wave functions. Because the 2-positivity conditions are exact for quantum systems with AGP ground states, the rank of the particle-hole 2-RDM from the AGP ansatz provides a minimum for its value in variational 2-RDM calculations of general quantum systems. To implement the rank-restricted conditions, we extend a first-order algorithm for large-scale semidefinite programming. The rank-restricted conditions significantly improve the accuracy of the energies; for example, the percentages of correlation energies recovered for HF, CO, and N2{}_{2}2​ improve from 115.2%, 121.7%, and 121.5% without rank restriction to 97.8%, 101.1%, and 100.0% with rank restriction. Similar results are found at both equilibrium and nonequilibrium geometries. While more accurate, the rank-restricted NNN-representability conditions are less expensive computationally than the full-rank conditions.