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Can anyone give me a practical demonstration of the Schweinler-Wigner Orthogonalization procedure? The steps of performing it or possibly a code snippet.

The Schweinler-Wigner Orthogonalization procedure is a method for orthogonalizing the columns of a matrix in a symmetrical way.

Details can be found here https://shodhganga.inflibnet.ac.in/bitstream/10603/104432/9/09_chapter%202.pdf

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  • $\begingroup$ Hello sonaam1234. It would be helpful to add a quick description of the procedure you are mentioning, and the step(s) for which you are seeking a demonstration. Otherwise, I am afraid the question will remain unanswered or be closed. $\endgroup$
    – byouness
    Nov 26, 2019 at 13:17
  • $\begingroup$ It seems like a very specialized subject. This question may have a better chance of being answered on a Mathematics (or Mathematical Physics) forum... $\endgroup$
    – Alex C
    Nov 27, 2019 at 2:34
  • $\begingroup$ added a description $\endgroup$
    – sonaam1234
    Nov 27, 2019 at 7:59

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The paper you presented gave a thorough description. Although it actually didn't refer to Schweinler-Wignler Orthogonalisation, it referred to Lowdin Orthogonalisation, Symmetric Orthogonalisation and Canonical Orthogonalisation and the determination of the Schweinler-Wigner Matrix.

The paper can be summarised as:

You assume an initial set of linearly independent (columnwise) vectors: $V$.

You can apply a transformation to a new basis: $Z=VA$, which is orthonormal iff $Z^TZ = I$.

Letting $A = (V^TV)^{-1/2}B$, where $B$ is a unitary matrix gives the general solution. Letting $B=I$ is termed Symmetric Orthogonalisation.

The Schweinler-Wigner Matrix is calculated as $(V^TZ)^2$ where the square is element-wise, and it seems to me to be used only to prove the property claimed in the paper that symmertic orthogonalisation returns the set of orthonormal vectors "nearest" to the original set of vectors in a nearest neigbour sense with an $l2$ norm.

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  • $\begingroup$ From what I recall there is a proof in the original paper of lowdin that the orthogonalised vectors are nearest. Its a standard technique used in molecular orbital construction in quantum chemistry (ie look here in subroutine lowdin). $\endgroup$
    – will
    Dec 18, 2019 at 20:15

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