Equation for the direct determination of the density matrix

A nonvariational equation, called the density equation, is proposed for the direct determination of the density matrix without using a wave function. It is connected with the Schrödinger equation by a necessary and sufficient theorem. The equation for the lowest order depends explicitly only on the fourth-order density matrix ${\Gamma }^{\mathrm{}\left(4\right)\mathrm{}}$ (or ${\Gamma }^{\mathrm{}\left(3\right)\mathrm{}}$ in a special case) and not on the higher-order density matrices. The equation always gives the density matrix and the associated energy which coincide with those obtained indirectly from the Schrödinger equation. This is true even if we solve the equation only with the known and tractable $N$-representability conditions, although in such a case some unphysical solution may also occur in the non-$N$-representable space. The equation is applicable to both fermion and boson systems, and to both ground and excited states. In contrast to the Schrödinger equation, the labor of solving the equation does not increase when the number of particles of the system is increased. When we have the Hartree-Fock solution, the equation is transformed such that the correlated density matrix and the correlation energy are the direct solution. The correlated density equation thus obtained is suitable for the study of electron correlations.