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Asymptotic behavior of electron densities and computation of one‐electron properties
Author(s) -
Núñez Marco A.
Publication year - 1996
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/(sici)1097-461x(1996)57:6<1077::aid-qua6>3.0.co;2-q
Subject(s) - bounded function , electron , hamiltonian (control theory) , mathematics , hilbert space , limiting , sequence (biology) , physics , mathematical physics , mathematical analysis , quantum mechanics , chemistry , mechanical engineering , mathematical optimization , biochemistry , engineering
The role of the asymptotic behavior of approximating sequences of electron densities ρ n ( r ) in the calculation of one‐electron properties is studied. Rigorous mathematical results in the frame of Hilbert spaces are used to prove the following facts: (i) Both the L 2 convergence of wave functions ψ n and the E convergence of the corresponding energies E n guarantee the correctness of the limiting procedure lim n → x ∫ Ω s ((overline)x(/overline)|ψ n | 2 d (overline)x(/overline) = ∫ Ω s ((overline)x(/overline))|ψ| 2 d (overline)x(/overline) for the most frequently used operators s ( x ), Ω being any bounded region of the n ‐particle configuration space R 3 N ; and (ii) the uniform boundedness of the sequence {ρ n } together with both the L 2 and E convergencies is sufficient to guarantee the correctness of the limiting procedure lim n →x ∫ ∞ 0 s ( r )ρ n r 2 dr = ∫ x 0 s ( r )ρ r 2 dr for most one‐electron operators s ( r ) including the power moment operators r k which, for large k , are representative of the class of operators not relatively form‐bounded by the Hamiltonian. The mathematical concept of uniform boundedness is used to give a characterization of the capability of {ρ n } to reproduce the asymptotic behavior of the true electron density ρ and it is shown by means of numerical examples how a sequence {ρ n } that does not reproduce the correct asymptotic behavior is not uniformly bounded and can give divergent expectation values of one‐electron operators s ( r ) not relatively form‐bounded by the Hamiltonian. © 1996 John Wiley & Sons, Inc.