1. The 1-matrix γ1(1|1′) [“1” stands for position and spin] can be diagonalized in terms of natural orbitals and corresponding natural 1-body occupancies. The cumulant partitioning of the 2-matrix γ1(1|1′, 2|2′) = γ1(1|1′)γ1(2|2′) - γ1(1|2′)γ1(2|1′) - χ2(1|1′, 2|2′) defines the cumulant 2-matrix χ2(1|1′, 2|2′). The first and second term may be referred to as Hartree and Fock term, respectively. γ1(1|1′) and χ2(1|1′, 2|2′) are size-extensively normalized, what means ∫ d1 γ1(1|1) = N, ∫ d1d2 χ2(1|1, 2|2) = cN ∝ N, to be compared with ∫ d1d2 γ2(1|1, 2|2) = N(N - 1) ∝ N2. The quantity c=$ 1 \over N $∫ d1 [γ(1|1) - ∫ d2 γ(1|2, γ2|1)], measuring the non-idempotency of the 1-marix, is called Löwdin parameter. Both γ2(1|1′, 2|2′) and χ2(1|1′, 2|2′) can be diagonalized. γ(1|1′, 2|2′) yields natural geminals (2-body wave functions) and cor-responding natural 2-body occupancies, χ2(1|1′, 2|2′) yields cumulant geminals and cumulant 2-body occu- pancies. The contraction SR ∫ d2 χ2(1|1′, 2|2) = γ(1|1′) - ∫ d2 γ(1|2′)γ(2|1′) is a quadratic equation for the 1-matrix, supposed χ(1|1′, 2|2) is known. For reduced density matrices in general cf. [2-4], for the 2-matrix and its cumulant partitioning in particular cf. e.g. [5, 6], for inequalities cf. [7-9].
2. , Reduced Density Matrices in Quantum Chemistry (Academic, New York, 1976).
3. and , Reduced Density Matrices: Coulson's Challenge (Springer, New York, 2000).
4. , Many-Electron Densities and Reduced Density Matrices (Kluwer/Plenum, New York, 2000).
5. Cumulant 2-matrix of the high-density electron gas and the density matrix functional theory