Computational Chemistry 4.26 - Orthogonalization
Short lecture on diagonalizing the overlap matrix in Hartree-Fock theory.
The overlap matrix (S) in Hartree-Fock theory is a K x K matrix with elements of the overlap integrals of all K atomic orbital basis functions. We may take any linear combination of these basis functions as we like, and transform any representation into another by using a transformation matrix. The overlap matrix may be diagonalized performing such a unitary transformation. The inverse square root of the overlap matrix is used in symmetric orthogonalization. Symmetric orthogonalization is not always possible due to linear dependencies in the overlap matrix (at least two basis function pairs are nearly identical), making the inverse matrix undefined due to a zero determinant. In such a case canonical orthogonalization may be used which truncates matrix rows whose eigenvalue is sufficiently small, effectively reducing the number of basis functions to K-m. The orthogonalization matrix may then be used to transform the Fock matrix and coefficient matrix as part of the Hartree-Fock procedure.
Notes Slide: https://i.imgur.com/fNR1yVF.png
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