Lower Bounds for the Eigenvalues of First-Order Density Matrices

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Abstract
Lower bounds for the first n eigenvalues of the first‐order density matrix corresponding to a quantum‐mechanical state of an n‐electron system are derived under the three assumptions: (1) that one knows a lower bound to the overlap integral of an arbitrary normalized Slater determinant with the true wavefunction. (2) that the configuration interaction expansion of the true wavefunction contains only singly, doubly, etc., up to m‐fold excited configurations with respect to a ``ground'' configuration, where m≤n. (3) That the electronic‐interaction term in the Hamiltonian is proportional to a parameter λ and that a perturbation expansion in terms of this parameter is possible. In the latter case one finds that the idempotency of the one‐particle density matrix is stable with respect to first‐order perturbations.