Hermitian matrix reduced-density

For a system of N identical fermions in a state ij/ there is associated a reduced density matrix (RDM) of order p for each integer p, 1 Hermitian operator DP, which we call a reduced density operator (RDO) acting on a space of antisymmetric functions of p particles. The case p = 2 is of particular interest for chemists and physicists who seldom consider... 

M. Rosina, (a) Direct variational calculation of the two-body density matrix (b) On the unique representation of the two-body density matrices corresponding to the AGP wave function (c) The characterization of the exposed points of a convex set bounded by matrix nonnegativity conditions (d) Hermitian operator method for calculations within the particle-hole space in Reduced Density Operators with Applications to Physical and Chemical Systems—II (R. M. Erdahl, ed.), Queen s Papers in Pure and Applied Mathematics No. 40, Queen s University, Kingston, Ontario, 1974, (a) p. 40, (b) p. 50, (c) p. 57, (d) p. 126. 

The preceding is a rather comprehensive—but not exhaustive— review of N-representability constraints for diagonal elements of reduced density matrices. The most general and most powerful V-representability conditions seem to take the form of linear inequalities, wherein one states that the expectation value of some positive semidefinite linear Hermitian operator is greater than or equal to zero, Tr [PnTn] > 0. If Pn depends only on 2-body operators, then it can be reduced into a g-electron reduced operator, Pq, and Tr[Pg vrg] > 0 provides a constraint for the V-representability of the g-electron reduced density matrix, or 2-matrix. Requiring that Tr[Pg Arrg] > 0 for every 2-body positive semidefinite linear operator is necessary and sufficient for the V-representability of the 2-matrix [22]. 

The concept of the molecular orbital and their occupation is, however, not restricted to the HF model. It has much wider relevance and is applicable also for more accurate wave functions. For each wave function we can form the first-order reduced density matrix. This matrix is Hermitian and can be diagonalized. The basis for this diagonal form of the density matrix are the Natural Orbitals first introduced in quantum chemistry by Per-Olof Lowdin [4]. 

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