Anti-Hermitian Contracted Schr{\"o}dinger Equation for the Determination of Ground-state Energies and Two-electron Reduced-density-matrices without Wavefunctions

A recent advance in the theory of the contracted Schr{\"o}dinger 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-RDMs) that yield 95-100\% of the correlation energy of atoms and molecules [Mazziotti, Phys. Rev. Lett. {\bf 97}, 143002 (2006)]. Here we discuss in detail the anti-Hermitian contracted Schr{\"o}dinger equation (ACSE) and its comparison to the CSE with regard to cumulant reconstruction of RDMs, the role of Nakatsuji's theorem, and the structure of the wavefunction. 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. The computed 2-RDMs very closely satisfy known $N$-representability conditions.