The variational approach to the two−body density matrix

A variational method for the two−body density matrix is developed for practical calculations of the properties of many−fermion systems with two−body interactions. In this method the energy E = 𝒥Hijkl ρijkl is minimized using the two−body density matrix elements ρijkl = 〈ψ‖a+ja+iakal‖ψ〉 as variational parameters. The approximation consists in satisfying only a subset of necessary conditions—the nonnegativity of the following matrices: the two−body density matrix, the ’’two−hole matrix’’ Qijkl = 〈Ψ‖ajaia+ka+l‖Ψ〉 and the particle−hole matrix Gijkl = 〈Ψ‖ (a+iaj−ρij)+ (a+kal−ρk) ‖Ψ〉. The idea of the method was introduced earlier; here some further physical interpretation is given and a numerical procedure for calculations within a small single−particle model space is described. The method is illustrated on the ground state of Be atom using 1s, 2s, 2p orbitals.