Anti-Hermitian part of the contracted Schrödinger equation for the direct calculation of two-electron reduced density matrices

A recent advance in the theory of the contracted Schrödinger equation (CSE), in which only the anti-Hermitian part of the equation is solved, permits the direct determination of ground-state two-electron reduced density matrices (2-RDM’s) that yield 95%–100% of the correlation energy of atoms and molecules [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. Here we discuss in detail the anti-Hermitian contracted Schrödinger equation (ACSE) and its comparison to the CSE with regard to cumulant reconstruction of RDM’s, the role of Nakatsuji’s theorem, and the structure of the wave function. The ACSE is also formulated in the Heisenberg representation and related to canonical diagonalization. The solution of the ACSE is illustrated with a variety of molecules including ${\mathrm{H}}_2\mathrm{O}$, ${\mathrm{CH}}_2$, ${{\mathrm{NH}}_4}^{+}$, $\mathrm{HF}$, and ${\mathrm{N}}_2$, and potential energy and dipole-moment surfaces are computed for boron hydride in a polarized double-$\zeta$ basis set. The computed 2-RDM’s very closely satisfy known $N$-representability conditions.

Figure 1

For ${\mathrm{N}}_2$ the ACSE energy converges to $0.3\mathrm{mH}$ above the FCI energy.

Figure 2

The potential energy curve of BH from the ACSE with NY reconstruction is compared with the curves from CCSD and FCI. In the equilibrium region the ACSE energies are an order of magnitude more accurate than those from CCSD. Calculations are performed in a correlation consistent polarized valence double-$\zeta$ (cc-pVDZ) basis set with the core orbital of boron frozen.

Figure 3

The dipole-moment surface for BH is shown as a function of the internuclear distance $R$ for HF, CCSD, the ACSE with NY reconstruction, and FCI. All three approximate methods predict the change in sign of the dipole moment between $R=1.6\mathrm{\AA }$ and $R=1.8\mathrm{\AA }$, but both the CCSD and ACSE surfaces deviate significantly at long bond lengths from the Hartree-Fock surface which remains linear. Calculations are performed in a correlation consistent polarized valence double-$\zeta$ (cc-pVDZ) basis set with the core orbital of boron frozen.