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.
A detailed study is made of the properties of the ``particle-hole matrix''Gabcd≡〈g| (a†b−ρab)†(c†d−ρcd) |g〉,where |g> is a many-fermion ground state and ρab is the 1-body density matrix. It is shown that the zero eigenvalues of the particle-hole matrix are intimately and simply related to the one-body symmetries of the ground state. It is also shown that the large eigenvalues of Gabcd are closely related to the collective features of the ground state.
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