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Physical Review

N -Representability Problem: Conditions on Geminals

Mary Beth Ruskai

Published Jul 5, 1969• 34 citations
Publisher

Abstract

The $N$-representability problem is approached by considering geminal product expansions of symmetric and antisymmetric functions. The $N$-completeness problem, i.e., the problem of determining when a set of geminals is a suitable basis for expanding symmetric or antisymmetric $N$-particle functions, is considered. New necessary conditions are given for both $N$-completeness and $N$-representability. In some cases, one can also obtain sufficient conditions; examples of such cases are discussed. The circumstances under which a density matrix can be derived from two or more different functions are also treated. Finally, extensions to higher order are also mentioned.

DOI
10.1103/physrev.183.129

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The NNN-representability problem is approached by considering geminal product expansions of symmetric and antisymmetric functions. The NNN-completeness problem, i.e., the problem of determining when a set of geminals is a suitable basis for expanding symmetric or antisymmetric NNN-particle functions, is considered. New necessary conditions are given for both NNN-completeness and NNN-representability. In some cases, one can also obtain sufficient conditions; examples of such cases are discussed. The circumstances under which a density matrix can be derived from two or more different functions are also treated. Finally, extensions to higher order are also mentioned.
The complete solution is presented for the NNN-completeness problem, i.e., the problem of determining when a set of ppp-particle functions generates symmetric or antisymmetric functions without applying a symmetrizing or antisymmetrizing operator. Then it is shown that a symmetric or antisymmetric function has finite 2 rank if, and only if, it has finite 1 rank. Finally, the relation between NNN representability of NNN completeness is discussed.
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.
A method is given for obtaining the density matrix or matrices, expressible in terms of a given set of spin geminals, which are as nearly NNN representable as possible. A measure of NNN representability is introduced as the norm of the antisymmetric component of a normalized wave function leading to the density matrix. An error estimate is given for the maximum deviation of an energy calculated with the density matrix from that calculated with an antisymmetric wave function. Linear variational parameters may occur in the density matrix, and their number can be increased at the expense of NNN representability. Implementation of the method requires diagonalization of a large matrix, and the consequences of truncation are considered. The method is also related to the problem of the exact NNN representability of a given density matrix.
For a quantum-mechanical system of NNN identical fermions, the NNN-representability problem is the problem of recognizing whether, for a given pth-order reduced density matrix ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})$, there exists an antisymmetric NNN-particle wave function $\ensuremath{\Psi}(12\ensuremath{\cdots}N)$ such that ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})=({N}{p})\ensuremath{\int}\ensuremath{\Psi}(12\ensuremath{\cdots}N)\ifmmode\times\else\texttimes\fi{}{\ensuremath{\Psi}}^{*}({1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}},p+1\ensuremath{\cdots}N){d}_{{\ensuremath{\tau}}_{p+1}}\ensuremath{\cdots}{d}_{{\ensuremath{\tau}}_{N}}$. It is shown that if the Hamiltonian of a system is time-reversal invariant, and the number of particles, NNN, is even, the necessary and sufficient condition that an approximate first-order density matrix corresponding to a nondegenerate energy eigenstate be NNN-representable is that its natural spin-orbital occupation numbers be equal in pairs.
The NNN-representability problem for the first- and second-order density matrices is considered when NNN is an odd integer. Attention is restricted to those density matrices derivable from functions whose natural spin orbitals have certain paired properties. Neither pairing of spatial orbitals nor special spin properties are required in general, but they are discussed as special cases. New sufficient conditions for NNN-representability of the first-order density matrix are given. The second-order density matrix is studied in detail. Special cases in which the NNN-representability problem for the second-order density matrix can be solved are considered.
The problem considered is that of the diagonal N -representability of a p th-order reduced density matrix, p ≥ 2 p \geq 2 , for a system of N identical fermions or bosons. A finite number M of allowable single particle states is assumed. The problem is divided into three cases, namely: Case I. M = N + p M = N + p ; Case II. M > N + p M > N + p ; Case III. M > N + p M > N + p . Using the theory of polyhedral convex cones, a complete set of necessary and sufficient conditions is first found for Case I. This solution is then employed to find such conditions for Case II. For Case III, two algorithms are developed to generate solutions for the problem, and examples of the usage of these algorithms are given.