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Integral–geometrical consideration of density matrices
Author(s) -
Zapol Boris P.
Publication year - 1995
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.560560511
Subject(s) - rdm , bounded function , order (exchange) , mathematics , positive definiteness , matrix (chemical analysis) , combinatorics , space (punctuation) , physics , density matrix , mathematical physics , pure mathematics , computational chemistry , chemistry , mathematical analysis , quantum mechanics , positive definite matrix , computer science , finance , economics , computer network , eigenvalues and eigenvectors , chromatography , quantum , operating system
The ensemble N ‐representability problem for the k ‐th order reduced density matrix ( k ‐ RDM ) as well as the problem of reconstruction of the N ‐particle system density matrices ( N ‐ DM ) from a given k ‐ RDM are studied. The spatial parts of the k ‐ RDM expansion in terms of spin tensorial operators Θ λ kare represented using particular values (at specially chosen ) of the Radon transform of the N ‐ DM spatial parts (or their sums) Nλ ( x ′ | x ″) (here, is a d ‐plane in the n ‐space ℝ n of x = ( x ′, x ″)), with n = 6 N , d = 3 ( N − k ), x ′ ≡ (r′ 1 , ⃛, r′ N ), x′ ≡ ( r 1 ″, ⃛, r ″ N ()). In this way, the problem is reduced to investigation of the properties of the functions . For a normalizable N – DM , it is proved that are bounded functions. The properties of implied by the N ‐ DM permutational symmetry, Hermiticity, and positive definiteness are found. A formal procedure of reconstruction of all N ‐ DM corresponding to a given k ‐ RDM is proposed. © 1995 John Wiley & Sons, Inc.