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On the Pairing Theorem and Its Extension
Author(s) -
Löwdin PerOlov
Publication year - 1991
Publication title -
israel journal of chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.908
H-Index - 54
eISSN - 1869-5868
pISSN - 0021-2148
DOI - 10.1002/ijch.199100034
Subject(s) - molecular orbital , slater type orbital , pairing , atomic orbital , molecular orbital theory , chemistry , eigenvalues and eigenvectors , coulomb , extension (predicate logic) , quantum mechanics , unitary state , physics , molecule , superconductivity , computer science , political science , law , programming language , electron
The mirror theorem in matrix theory is briefly reviewed. It says that if R and S are matrices of the order m × n and nxm , then the two products RS and SR have the same nonvanishing eigenvalues with the same multiplicities and canonical structures. The mirror theorem is used to show that if a = {a k } and b = {b l } are two sets of orbitals of order m and n , respectively, then there exist two unitary transformations U and V such that the orbitals a ′ = aU and b ′ = bV have the special property < a k ′ | b l ′> = λ k δ kl , i.e., they are all orthogonal to each other unless they belong to the same pair having k = l. This pairing theorem is of fundamental importance in treating the Coulomb correlation in molecular calculations by means of the so‐called Alternant Molecular Orbital (AMO) method, which uses for the occupied orbitals different orbitals for different spins. A simple extension of this approach to the virtual orbitals is also shown.