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The Journal of Chemical Physics

N Representability of Fermion Density Matrices

Karl F. Freed

Published Nov 15, 1967• 6 citations
Publisher

Abstract

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.

DOI
10.1063/1.1701553

Related papers

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We consider the v-representability of the particle density for a noninteracting system of spinless fermions by introducing the idea of proper order of a set of energy levels. It is shown that if ${\mathit{E}}_{1}$(\ensuremath{\lambda}), ${\mathit{E}}_{2}$(\ensuremath{\lambda}), and ${\mathit{E}}_{3}$(\ensuremath{\lambda}) are three energy levels associated with some local potential ${\mathit{V}}_{\ensuremath{\lambda}}$(r) that is a continuous function of \ensuremath{\lambda}=(${\ensuremath{\lambda}}_{1}$,${\ensuremath{\lambda}}_{2}$,${\ensuremath{\lambda}}_{3}$) over all possible points \ensuremath{\lambda}, where ${\ensuremath{\lambda}}_{\mathit{i}}$ is the occupation number of the ith state and M=${\ensuremath{\lambda}}_{1}$+${\ensuremath{\lambda}}_{2}$+${\ensuremath{\lambda}}_{3}$ is the total number of particles distributed over the three levels, then there must be at least one \ensuremath{\lambda} for which the three levels are in so-called proper order, in which the levels below the highest occupied level are filled. This result provides a basis for the proof of ensemble v-representability of some N-particle density for which the ground-state degeneracy of the system is no more than three. As examples, three- and two-dimensional central systems are examined, and an N-particle central density is shown to be ensemble v-representable for small N (N\ensuremath{\le}14 and N\ensuremath{\le}9 for three- and two-dimensional cases, respectively). The implications for density-functional theory are discussed.

Physical Review A•Sep 1, 1991•34 citations
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.
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 density operator (density matrix) of a quantum mechenical system can be decomposed into operators which transform as irreducible representations of the symmetry group in coordinate and spin space. Each of these components has a physical meaning connected with the expectation values of certain operators. The reduced density matrices can be decomposed in a completely analogous way. The symmetry properties of the total wave function give rise to degeneracies of the eigenvalues of the reduced density matrices. These degeneracies can be removed by requiring that the natural spin orbitals (NSO, defined as the eigenfunctions of the first-order density matrix), as well as the natural spin geminals (NSG, the eigenfunctions of the second-order density matrix) and their spinless counterparts transform as irreducible representations of the symmetry group and are eigenfunctions of S/sup 2/ and S/sub z/. In many cases this requirement is compatible with the original definition of the NSO, the NSG, etc., e.g., when there is no spatial degeneracy of the total wave function and when the Z- component of the total spin vanishes. When these conditions are not fulfilled an alternative definition of the NSO and the NSG is proposed. (auth)
We consider the v-representability of the particle density for a noninteracting system of spinless fermions by introducing the idea of proper order of a set of energy levels. It is shown that if E1{\mathit{E}}_{1}E1​(\ensuremath{\lambda}), E2{\mathit{E}}_{2}E2​(\ensuremath{\lambda}), and E3{\mathit{E}}_{3}E3​(\ensuremath{\lambda}) are three energy levels associated with some local potential ${\mathit{V}}_{\ensuremath{\lambda}}$(r) that is a continuous function of \ensuremath{\lambda}=(${\ensuremath{\lambda}}_{1}$,${\ensuremath{\lambda}}_{2}$,${\ensuremath{\lambda}}_{3}$) over all possible points \ensuremath{\lambda}, where ${\ensuremath{\lambda}}_{\mathit{i}}$ is the occupation number of the ith state and M=${\ensuremath{\lambda}}_{1}$+${\ensuremath{\lambda}}_{2}$+${\ensuremath{\lambda}}_{3}$ is the total number of particles distributed over the three levels, then there must be at least one \ensuremath{\lambda} for which the three levels are in so-called proper order, in which the levels below the highest occupied level are filled. This result provides a basis for the proof of ensemble v-representability of some N-particle density for which the ground-state degeneracy of the system is no more than three. As examples, three- and two-dimensional central systems are examined, and an N-particle central density is shown to be ensemble v-representable for small N (N\ensuremath{\le}14 and N\ensuremath{\le}9 for three- and two-dimensional cases, respectively). The implications for density-functional theory are discussed.