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 Schr\"odinger 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 Schr\"odinger 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 H₂O, CH₂, NH₄⁺, HF, and N₂, and potential energy and dipole-moment surfaces are computed for boron hydride in a polarized double-ζ basis set. The computed 2-RDM's very closely satisfy known N-representability conditions.