Direct determination of the quantum-mechanical density matrix using the density equation. II.

Abstract

With the use of the density equation [Phys. Rev. A 14, 41 (1976)], second-order density matrices of atoms and molecules are calculated directly without any use of the wave function. The decoupling of the third- and fourth-order density matrices is done using the diagrammatic procedure of the Green's-function method, including terms up to second order in electron correlation. In addition to the results given in a previous communication [Phys. Rev. Lett. 76, 1039 (1996)], we give more detailed formulations, discussions, and additional results for ${\mathrm{H}}_{2}\mathrm{O},$ ${\mathrm{NH}}_{3},$ ${\mathrm{CH}}_{4},$ HF, ${\mathrm{N}}_{2},$ CO, and acetylene using double-\ensuremath{\zeta} basis sets, and for ${\mathrm{CH}}_{3}\mathrm{OH},$ ${\mathrm{CH}}_{3}{\mathrm{NH}}_{2},$ and ${\mathrm{C}}_{2}{\mathrm{H}}_{6}$ (staggered and eclipsed) using minimal basis sets. The present density-equation method gave energies as accurate as, and reduced density matrices (RDM's) more accurate than the single and double excitation configuration interaction method. The present method seems to give better quality results as the system becomes large. The convergence was fairly good and the calculated second-order RDM's almost satisfied some necessary conditions of the $N$ representability, the so-called $P,$ $Q,$ and $G$ conditions, while the first-order RDM's were exactly $N$ representable. The variety of the molecules calculated and the quality of the calculated results show that the density-equation method can be a promising alternative to the wave-function approach in quantum mechanics.