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Physical Review

N -Representability Problem for an Odd Number of Fermions

Mary Beth Ruskai, John E. Harriman

Published May 5, 1968• 13 citations
Publisher

Abstract

The $N$-representability problem for the first- and second-order density matrices is considered when $N$ 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 $N$-representability of the first-order density matrix are given. The second-order density matrix is studied in detail. Special cases in which the $N$-representability problem for the second-order density matrix can be solved are considered.

DOI
10.1103/physrev.169.101

Related papers

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Physical Review•Jul 5, 1969•34 citations
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.
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 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.
This paper considers the determination of N-representability (for diagonal elements) of p-density matrices restricted to certain finite-dimensional subspaces of l2 of the configuration space of N identical antisymmetric particles. In particular, an arbitrary set of N + p spin orbitals is selected and one considers the (N+pp)-dimensional subspace generated by all possible Slater determinants of the spin orbitals being considered. Applying a combinatorial approach to the problem, a necessary and sufficient set of conditions is determined; previous work has dealt only with necessary conditions, except in the 1-matrix case. The paper concludes by presenting a probabilistic interpretation of these conditions which seems of particular interest for the 2-matrix case. The conditions presented here in combination with the Pauli principle give a probabilistic view of the expected occupation of p-tuples of spin orbitals in terms of the expected occupations of lower-order-tuples of spin orbitals.
The usual restriction on the permutational symmetry of density matrices is removed. An Nth-order density matrix of general permutational symmetry is reduced and the symmetry of the resulting pth-order density matrix is found. This gives rise to necessary conditions for the N-representability of spin-free density matrices. We indicate the importance of these results for symmetry under the groups GL(n), U(n), and SU(n). We generalize the question of N-representability to the tensor algebra, where contraction of tensors plays the role of reduction of density matrices. Symmetry properties of the tensors are studied and density matrices are considered as special cases.
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.