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.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.

Physical ReviewJul 5, 196934 citations

N Completeness, N Representability, and Geminal Expansions

Mary Beth Ruskai

The complete solution is presented for the $N$ -completeness problem, i.e., the problem of determining when a set of $p$ -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 $N$ representability of $N$ completeness is discussed.The complete solution is presented for the $N$-completeness problem, i.e., the problem of determining when a set of $p$-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 $N$ representability of $N$ completeness is discussed.

Physical Review AMar 1, 197214 citations

Representations of symmetric states

M. Fannes, D. Testard, André F. Verbeure

Jan 1, 19820 citations

Finite-dimensional *-product and matrix algebras

Goce Chadzitaskos, J. Tolar

Czechoslovak Journal of PhysicsFeb 1, 19960 citations

On the permutation symmetry of atomic and molecular wavefunctions

Francisco Marcelo Fernández

In this paper we analyze a recently proposed approach for the construction of antisymmetric functions for atomic and molecular systems. It is based on the assumption that the main problems with Hartree-Fock wavefunctions stem from their lack of proper permutation symmetry. This alternative building approach is based on products of a space times a spin function with opposite permutation symmetry. The main argument for devising such factors is that the eigenfunctions of the non-relativistic Hamiltonian are either symmetric or antisymmetric with respect to the transposition of the variables of a pair of electrons. However, since the eigenfunctions of the non-relativistic Hamiltonian are basis for the irreducible representations of the symmetric group they are not necessarily symmetric or antisymmetric, except in the trivial case of two electrons. We carry out a simple and straightforward general analysis of the symmetry of the eigenfunctions of the non-relativistic Hamiltonian and illustrate our conclusions by means of two exactly-solvable models of $N=2$ and $N=3$ identical interacting particles.In this paper we analyze a recently proposed approach for the construction of antisymmetric functions for atomic and molecular systems. It is based on the assumption that the main problems with Hartree-Fock wavefunctions stem from their lack of proper permutation symmetry. This alternative building approach is based on products of a space times a spin function with opposite permutation symmetry. The main argument for devising such factors is that the eigenfunctions of the non-relativistic Hamiltonian are either symmetric or antisymmetric with respect to the transposition of the variables of a pair of electrons. However, since the eigenfunctions of the non-relativistic Hamiltonian are basis for the irreducible representations of the symmetric group they are not necessarily symmetric or antisymmetric, except in the trivial case of two electrons. We carry out a simple and straightforward general analysis of the symmetry of the eigenfunctions of the non-relativistic Hamiltonian and illustrate our conclusions by means of two exactly-solvable models of $N=2$ and $N=3$ identical interacting particles.

arXiv:1904.06378Apr 12, 20190 citations

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.

The complete solution is presented for the

N

-completeness problem, i.e., the problem of determining when a set of

p

-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

N

representability of

N

completeness is discussed.

N=2

and

N=3

identical interacting particles.