Rank Restriction for the Variational Calculation of Two-electron Reduced Density Matrices of Many-electron Atoms and Molecules

Abstract

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 wavefunction. Here we introduce a new 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 wavefunctions 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 non-equilibrium geometries. While more accurate, the rank-restricted $N$-representability conditions are less expensive computationally than the full-rank conditions.

References in corpus (4)

• Projected Quasiparticle Theory for Molecular Electronic Structure
• The Electronic Ground State Energy Problem: a New Reduced Density Matrix Approach
• Variational determination of the second-order density matrix for the isoelectronic series of beryllium, neon and silicon
• Constrained-Pairing Mean-Field Theory. IV. Inclusion of corresponding pair constraints and connection to unrestricted Hartree-Fock theory

Cited by in corpus (6)

• Challenges for variational reduced-density-matrix theory with three-particle N-representability conditions
• Robust formulation of Wick's theorem for computing matrix elements between Hartree-Fock-Bogoliubov wavefunctions
• Cooper-Pair Condensates with Non-Classical Long-Range Order on Quantum Devices
• Data-driven Refinement of Electronic Energies from Two-Electron Reduced-Density-Matrix Theory
• Active Space Pair 2-Electron Reduced Density Matrix Theory for Strong Correlation
• Fermion Correlation Inequalities derived from G- and P-Conditions