Large-Scale Semidefinite Programming for Many-Electron Quantum Mechanics

The energy of a many-electron quantum system can be approximated by a constrained optimization of the two-electron reduced density matrix (2-RDM) that is solvable in polynomial time by semidefinite programming (SDP). Here we develop a SDP method for computing strongly correlated 2-RDMs that is 10–20 times faster than previous methods [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. We illustrate with (i) the dissociation of ${\mathrm{N}}_2$ and (ii) the metal-to-insulator transition of ${\mathrm{H}}_{50}$. For ${\mathrm{H}}_{50}$ the SDP problem has $9.4\times {10}^6$ variables. This advance also expands the feasibility of large-scale applications in quantum information, control, statistics, and economics.

Figure 1

${\mathrm{N}}_2$ potential energy curve from the variational 2-RDM method is presented in the cc-pVTZ basis set.

Figure 2

The ${\mathrm{H}}_{50}$ potential energy curve from stretching all bonds (in Å) of the chain equally is shown. The variational 2-RDM method yields the correct dissociation energy which would require a wave function with ${10}^{28}$ determinants.

Figure 3

The ${\mathrm{H}}_{50}$ metal-to-insulator transition from stretching all bonds (in Å) of the chain equally is shown. While the Hartree-Fock method predicts a metal at all distances, the 2-RDM method predicts an insulator upon dissociation.