Dual-Cone Variational Calculation of the 2-Electron Reduced Density Matrix

Abstract

The computation of strongly correlated quantum systems is challenging because of its potentially exponential scaling in the number of electron configurations. Variational calculation of the two-electron reduced density matrix (2-RDM) without the many-electron wave function exploits the pairwise nature of the electronic Coulomb interaction to compute a lower bound on the ground-state energy with polynomial computational scaling. Recently, a dual-cone formulation of the variational 2-RDM calculation was shown to generate the ground-state energy, albeit not the 2-RDM, at a substantially reduced computational cost, especially for higher $N$-representability conditions such as the T2 constraint. Here we generalize the dual-cone variational 2-RDM method to compute not only the ground-state energy but also the 2-RDM. The central result is that we can compute the 2-RDM from a generalization of the Hellmann-Feynman theorem. Specifically, we prove that in the Lagrangian formulation of the dual-cone optimization the 2-RDM is the Lagrange multiplier. We apply the method to computing the energies and properties of strongly correlated electrons -- including atomic charges, electron densities, dipole moments, and orbital occupations -- in an illustrative hydrogen chain and the nitrogen-fixation catalyst FeMoco. The dual variational computation of the 2-RDM with T2 or higher $N$-representability conditions provides a polynomially scaling approach to strongly correlated molecules and materials with significant applications in atomic and molecular and condensed-matter chemistry and physics.

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• Reduced-density-matrix-based ab initio cavity quantum electrodynamics
• Towards a Resolution of the Static Correlation Problem in Density Functional Theory from Semidefinite Programming
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• Beginnings of Exciton Condensation in Coronene Analog of Graphene Double Layer
• Bootstrapping the Quantum Hall problem
• Quantum Simulation of Molecules without Fermionic Encoding of the Wave Function
• Determining the ensemble N-representability of Reduced Density Matrices
• Solving one-body ensemble N-representability problems with spin
• Spin adaptation of the cumulant expansions of reduced density matrices
• Refining ensemble $N$-representability of one-body density matrices from partial information
• Representability for Quantum Theory beyond Particle-Number Conservation
• Entanglement Complexity in Many-body Systems from Positivity Scaling Laws
• Correlated Purification for Restoring $N$-Representability in Quantum Simulation