On Löwdin orthogonalization

It is shown that the Löwdin orthogonalization gives the unique minimum for the functional ϕ measuring the least squares distance between the given orbitals and the orthogonalized orbitals. Furthermore a much stronger result is obtained, namely that ϕ has only one local minimum, which is attained at the Löwdin orthogonalization and which is global. This justifies certain computer programs that compute Löwdin orthogonalization via minimization procedures. Finally there is a discussion of replacing the least squares metric by other metrics. The Löwdin orthogonalization turns out to be optimal with respect to all the commonly encountered norms.