Reduced Density Matrices of Atoms and Molecules. II. On the N-Representability Problem

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Abstract
For the 2-matrix of the ``double occupancy configuration interaction'' (DOCI) wavefunction described in a previous paper, the N-representability problem is stated in terms of algebraic equations relating the density-matrix elements λi, Λij, λij to the wavefunction coefficients CK. An operator formalism is developed and used to derive a number of necessary N-representability conditions for both diagonal and off-diagonal elements of the DOCI 2-matrix. These conditions consist principally of certain linear inequalities for the diagonal elements and nonnegativity requirements for the eigenvalues of certain synthetic matrices built from both diagonal and off-diagonal 2-matrix elements. A hole-particle duality is noted and used to expedite the derivations. Four special cases for which all the N-representability conditions can be explicitly given are used to show that the conditions obtained are not yet sufficient for DOCI N representability. It is then proved that the DOCI diagonal N-representability conditions apply with equal necessity to the diagonal elements of any N-representable 2-matrix. These new necessary conditions for the general N-representability problem are given a simple and satisfying probability interpretation in terms of the Pauli exclusion principle for fermion pair distributions. Off-diagonal conditions are generated by requiring that the diagonal conditions be maintained under all possible unitary transformations of the orbital basis set. This latter procedure is illustrated by an application to the DOCI 2-matrix which accounts for all off-diagonal conditions previously obtained.