N-Representability Problem: Conditions on Geminals

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.