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Physical Review A

Conditions on the Expansion Coefficients of the Reduced Two-Particle Density Matrix

Louis J. Kijewski

Published Jan 1, 1970• 1 citation
Publisher

Abstract

An explicit reduction in the number of terms of the expansion of the reduced two-particle density matrix for atomic systems is shown to occur when the expansion functions are eigenfunctions of ${L}_{z}(1)+{L}_{z}(2)$ and ${S}_{z}(1)+{S}_{z}(2)$.

DOI
10.1103/physreva.1.217

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An explicit reduction in the number of terms of the expansion of the reduced two-particle density matrix for atomic systems is shown to occur when the expansion functions are eigenfunctions of Lz(1)+Lz(2){L}_{z}(1)+{L}_{z}(2)Lz​(1)+Lz​(2) and Sz(1)+Sz(2){S}_{z}(1)+{S}_{z}(2)Sz​(1)+Sz​(2).
Properties and NNN-representability conditions of the two-body density matrix
A summary is presented of the properties of the coefficient matrices formed by expanding the two-body reduced density matrix in a complete set of two-electron wave functions. Calculating the relationship between the many electron wave function and these so-called two-body density matrices enables the derivation of necessary NNN-representability conditions that ensure a given matrix obeys the postulates of many-body quantum mechanics.
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.
The reduction of density matrices defined with respect to a finite basis set is considered. A basis set can be introduced into the vector space of Hermitian matrices acting on functions of the coordinates of ppp particles for each value of ppp. An analogy with the construction of spin eigenfunctions is used to obtain a particular basis set called the reducing basis. Reduction of one of these basis elements is either one to one or maps into the origin. Any element in the preimage of a density matrix can thus be resolved into two components, one uniquely determined by the density matrix and the other arbitrary within a certin subset. A restatement of the NNN-representability problem is given, and two sufficient conditions and one necessary condition for NNN representability are given in terms of distances and norms.