Structure of Fermion Density Matrices. II. Antisymmetrized Geminal Powers

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Abstract
Functions which can be expressed as an antisymmetrized power of a single two‐particle function (a geminal) occur in the BCS ansatz. They constitute a comparatively tractable generalization of a single Slater determinant. The second‐order reduced density matrix and bounds on its eigenvalues are obtained for the most general such antisymmetrized geminal power (AGP). These eigenvalues are physically of great importance, being analogs, in an analysis of the wavefunction into pair states, of the one‐particle occupation numbers of the independent‐particle model. A necessary and sufficient condition is given that an alleged first‐order reduced density matrix be derivable from an N‐particle AGP function. It is shown that only for exceptional functions (called extreme) is the generating geminal of the AGP function an eigenfunction of the 2‐matrix. The extreme 2‐matrix is diagonalized explicitly and provides a vivid example of the fact that one‐particle occupation numbers alone are unable to convey certain information of decisive significance about the wavefunction. The extreme 2‐matrix satisfies the assumption basic to current theories of superconductivity.