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

TL;DR

This paper introduces a dual-cone variational method to compute the 2-electron reduced density matrix (2-RDM) and ground-state energy of strongly correlated quantum systems with polynomial scaling, enabling detailed electronic property analysis.

Contribution

The authors generalize the dual-cone variational 2-RDM method to directly compute the 2-RDM, linking it to the Lagrange multiplier in the optimization, which was not previously achieved.

Findings

Successfully computed energies and properties of strongly correlated electrons.

Demonstrated polynomial scaling for complex molecules like FeMoco.

Enabled detailed analysis of electronic properties such as dipole moments and orbital occupations.

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.