Löwdin orthogonalization as a minimum energy perturbation

Let f1,...,fn be the atomic orbitals of a molecule in a molecular orbital calculation. There are advantages to using an orthonormal set e1,...,en instead of f1,...,fn in the computation. Of all the possible choices for the ei, we show that the Löwdin orthogonalization is the one which moves the fi the least. More precisely the sum of the squares of the lengths of the difference vectors ei − fi attains a minimum when the ei are obtained from the fi by the Löwdin orthogonalization.