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N ‐Representability conditions generated by a one‐particle grassmann factor of an antisymmetric function
Author(s) -
Grudziński Hubert
Publication year - 1985
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560270608
Subject(s) - antisymmetric relation , operator (biology) , mathematics , projection (relational algebra) , physics , eigenvalues and eigenvectors , combinatorics , mathematical physics , quantum mechanics , pure mathematics , mathematical analysis , chemistry , biochemistry , repressor , algorithm , transcription factor , gene
N ‐representability conditions for a two‐particle density operator implied by positive‐semidefiniteness of the projection operator P N +1 (ϕ 1 Λ Ψ N ) are derived and discussed. The operator P N +1 (ϕ 1 Λ Ψ N ) projects onto an ( N + 1)‐particle antisymmetric function ϕ 1 Λ Ψ N , the Grassmann product of a one‐particle factor ϕ 1 and an N ‐particle factor Ψ N . The polar subcone 2 N ( g , q ) to the set of N ‐representable two‐particle density operators 2 N which corresponds to these conditions is found. It is shown that its extreme rays belong to two orbits for the action of the unitary group of transformations in one‐particle Hilbert space. The facial structure of the convex set 2 N exposed by elements of 2 N ( g, q ) is analyzed. An example of the operator that changes the structure of its bottom eigenspace when the number of fermions N surpasses a certain value is noted. A new approach to the diagonal conditions for N ‐representability is found. It consists of the decomposition of the N ‐particle antisymmetric identity operator onto the mutually orthogonal projection operators.