Premium
Block diagonalization problem for a Fockian matrix of molecule and its solution by means of noncommutative Rayleigh–Schrödinger perturbation theory
Author(s) -
Gineityte V.
Publication year - 1998
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(1998)68:2<119::aid-qua4>3.0.co;2-y
Subject(s) - noncommutative geometry , eigenvalues and eigenvectors , eigenfunction , mathematical physics , quantum mechanics , diagonalizable matrix , hamiltonian (control theory) , schrödinger equation , matrix (chemical analysis) , operator (biology) , mathematics , physics , mathematical analysis , symmetric matrix , chemistry , repressor , transcription factor , gene , chromatography , mathematical optimization , biochemistry
The block diagonalization problem for a Fockian or a Huckel‐type model Hamiltonian matrix ( H ) of molecule originating from the Brillouin theorem and determining the noncanonical molecular orbitals (NCMOs) has been studied. An alternative form of the problem, viz. the so‐called eigenblock equation for the matrix H , has been suggested, which formally resembles the usual secular equation for certain two‐dimensional matrix. The operator analog of the eigenblock equation also has been derived, and it acquired the form of the usual secular problem for an operator. However, the multidimensional eigenblocks of the matrix H , playing the role of eigenvalues in this new equation, do not commute with the respective multidimensional eigenfunctions. A noncommutative Rayleigh–Schrödinger perturbation theory (PT) has been developed for the solution of operator problems of the above‐mentioned type. It has been shown that the PT used previously when obtaining the NCMOs of saturated organic molecules on the basis of the Brillouin theorem [V. Gineityte, J. Mol. Struct. (Theochem) 343 , 183 (1995)] actually corresponds to the case of two eigenfunctions (eigenvalues) of the noncommutative Rayleigh–Schrödinger PT. On the whole, search for NCMOs of molecules is shown to be related to a nontrivial generalization of a two‐level problem, where multidimensional (noncommutative) characteristics stand for the usual ones. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 119–127, 1998