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

With the use of the density equation [H. Nakatsuji. Phys. Rev. A 14. 41 (1976)], the second-order density matrices are directly determined without any use of the wave functions. The third- and fourth-order reduced density matrices (RDM's) are decoupled into lower-order ones using the Green's function technique. This method is applied to Be, Ne, H2O. H3O+, NH3, CH4, BH4- NH4+. and CH3F, and the results are successfully compared with the full configuration interaction results. The convergence is fairly good, and the calculated second-order RDM's almost satisfy the necessary conditions of the N representability, the P, Q, and G conditions, and the first-order RDM's are exactly N representable. These results show that the present method is very promising.