Dual-cone variational calculation of the two-electron reduced density matrix

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.

Figure 1

The (a) total energy and the (b) energy errors of the potential energy curve of the ${\mathrm{H}}_4$ chain are shown for its equally spaced dissociation in a [4,4] active space in the cc-pVQZ basis set. Errors, shown with a base-10 log, are relative to CASSCF. The variational 2-RDM energies with the DQG and DQGT conditions agree with those from CASSCF to about ${10}^{-3}$ and ${10}^{-5}$ a.u., respectively.

Figure 2

The metallic character of the ${\mathrm{H}}_4$ chain as a function of the distance $R$ between the equally spaced hydrogen atoms is shown with a [4,4] active space in the cc-pVQZ basis set. The metallic character is computed from the sum of the squares of the atomic-orbital 1-RDM elements between atoms. Based on this criterion, the Hartree-Fock theory predicts that the chain remains a metal upon dissociation while the variational 2RDM method with the DQG and DQGT conditions and CASSCF predict a Mott metal-to-insulator transition.

Figure 3

(a) The structure of FeMoco and (b) the atomic charges of FeMoco are shown. The Mulliken charges are computed from the 1-RDM of the variational 2-RDM method with DQGT conditions. The red indicates positive charge while blue denotes negative charge with the magnitudes of the charges given in Table 3.

Figure 4

The fractional occupations of the natural orbitals of the lowest singlet state of FeMoco are shown visually from the active-space variational 2-RDM method with the DQGT conditions for a [30,30] active space in the DZP basis set. The blue and red lines correspond to orbitals that are occupied or unoccupied in the Hartree-Fock limit, respectively.

Figure 5

The electron density of the ${209}^{\mathrm{th}}$ natural orbital of FeMoco from the (a) Hartree-Fock method and the (b) variational 2-RDM method with DQGT conditions is displayed. The highest occupied molecular orbital of the Hartree-Fock method becomes half filled and much more localized on the Fe centers in the variational 2-RDM method.