Symmetric orthogonalisation

The pairwise overlap, symmetrical orthogonalisation and Mulliken approximation together validate the NDO approximation — the orbital product in the orthogonalised GHO basis vanishes to the extent that the Mulliken approximation is realistic. This conclusion obviously has enormous consequences for any NDO approximation schemes. 

In order to be able to carry through the implementation of the solution of the SCF equations the ability to transform the equations to a basis of orthogonal functions is required. The simplest way to carry this out is the so-called symmetrical or L0wdin orthogonalisation method. 

Using the symmetric orthogonalisation method outlined earlier, the matrix which orthogonalises this (smaller) basis is ... 

Thus the relationship of the final, symmetrically orthogonalised, cubic harmonic basis to the original, non-orthogonal monomial basis is... 

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