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Orthonormalization, polar decomposition, and transformation to an effective hamiltonian
Author(s) -
Jørgensen Flemming
Publication year - 1977
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.560120215
Subject(s) - polar decomposition , hamiltonian (control theory) , canonical transformation , eigenvalues and eigenvectors , operator (biology) , mathematical physics , simple (philosophy) , mathematics , hilbert space , characterization (materials science) , pure mathematics , polar , physics , quantum mechanics , quantum , chemistry , mathematical optimization , philosophy , epistemology , repressor , transcription factor , optics , biochemistry , gene
Klein has pointed out that des Cloizeaux's orthonormalization is just another formulation of Löwdin's symmetric orthonormalization. We demonstrate that des Cloizeaux's formulation is convenient for theoretical discussions and that it is intimately connected with polar decomposition–‐which is a generalization to operators of the fact that any complex number z can be written r exp ( i θ) where r is positive and θ real. We generalize two other properties for z and find an interpretation of the Carlson‐Keller theorem. Recently Lathouwers found that Löwdin's canonical orthonormalization leads to the eigenvectors of a positive operator. This operator is des Cloizeaux's. We discuss some variational theorems on this background and find the Courant‐Hilbert‐Löwdin “measure of linear independence” from a simple least squares consideration, closely related to Lathouwers' results. An orthonormalization due to Schweinler and Wigner (1970) is observed to be Löwdin's canonical. The Schweinler‐Wigner maximum characterization is referred to a simple general theorem, which also implies a new maximum characterization. In Sections 6 and 7 we consider transformation to an effective Hamiltonian from the point of view of polar decomposition and symmetric orthonormalization. It turns out that the polar part of Bloch's transformation is des Cloizeaux's. The characterizations of this by both Klein and Jørgensen are visualized in a simple way.