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Physical Review

N -Representability Problem for Fermion Density Matrices. II. The First-Order Density Matrix with N Even

Darwin W. Smith

Published Jul 29, 1966• 67 citations
Publisher

Abstract

For a quantum-mechanical system of $N$ identical fermions, the $N$-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 $N$-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, $N$, is even, the necessary and sufficient condition that an approximate first-order density matrix corresponding to a nondegenerate energy eigenstate be $N$-representable is that its natural spin-orbital occupation numbers be equal in pairs.

DOI
10.1103/physrev.147.896

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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 \ensuremathΓ(p)(12\ensuremath⋯p∣1\ensuremath′2\ensuremath′\ensuremath⋯p\ensuremath′){\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})\ensuremathΓ(p)(12\ensuremath⋯p∣1\ensuremath′2\ensuremath′\ensuremath⋯p\ensuremath′), there exists an antisymmetric NNN-particle wave function \ensuremathΨ(12\ensuremath⋯N)\ensuremath{\Psi}(12\ensuremath{\cdots}N)\ensuremathΨ(12\ensuremath⋯N) such that \ensuremathΓ(p)(12\ensuremath⋯p∣1\ensuremath′2\ensuremath′\ensuremath⋯p\ensuremath′)=(Np)\ensuremath∫\ensuremathΨ(12\ensuremath⋯N)\ifmmode×\else\texttimes\fi\ensuremathΨ∗(1\ensuremath′2\ensuremath′\ensuremath⋯p\ensuremath′,p+1\ensuremath⋯N)d\ensuremathτp+1\ensuremath⋯d\ensuremathτN{\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}}\ensuremathΓ(p)(12\ensuremath⋯p∣1\ensuremath′2\ensuremath′\ensuremath⋯p\ensuremath′)=(Np)\ensuremath∫\ensuremathΨ(12\ensuremath⋯N)\ifmmode×\else\texttimes\fi\ensuremathΨ∗(1\ensuremath′2\ensuremath′\ensuremath⋯p\ensuremath′,p+1\ensuremath⋯N)d\ensuremathτp+1​​\ensuremath⋯d\ensuremathτN​​. 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 proof of Smith, which gives the solution to the N-representability problem of the first-order density matrix for the case of N even and a nondegenerate energy eigenstate, is also shown to be valid for certain linear combinations of degenerate states. A connection is shown between these proper linear combinations and the eigenfuctions of a perturbed Hamiltonian for which all of the degeneracies have been lifted. For the case of S=0, all the eigenfunctions of the pth-order density matrix (natural p functions) can be taken to be eigenfunctions of total spin (Σ lim i=1pSi)2,and if orbital angular momentum is a constant of the motion, these natural p functions are eigenfunctions of the total orbital angular momentum for L=0. Simple wavefunctions are presented for which the natural 2 functions are not all eigenfunctions of the total pair spin. It is then shown that for MS=0 (any integral spin) or for L=1, ML=0, the natural 2 functions are eigenfunctions of (S1+S2)2 and (L1+L2)2, respectively.
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 NNN-representability problem is the problem of determining whether there exists NNN-particle states with some prescribed property. Here we report an affirmative solution to the fermion NNN-representability problem when both the density and the paramagnetic current density are prescribed. This problem arises in current-density functional theory and is a generalization of the well-studied corresponding problem (only the density prescribed) in density functional theory. Given any density and paramagnetic current density satisfying a minimal regularity condition (essentially that a von Weiz\"acker--like canonical kinetic energy density is locally integrable), we prove that there exists a corresponding NNN-particle state. We prove this by constructing an explicit one-particle reduced density matrix in the form of a position-space kernel, i.e., a function of two continuous-position variables. In order to make minimal assumptions, we also address mathematical subtleties regarding the diagonal of, and how to rigorously extract paramagnetic current densities from, one-particle reduced density matrices in kernel form.
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.