Löwdin Population Analysis in StS module

Reform the density matrix which is made by a non-orthogonal basis set to a new density matrix in terms of a symmetrically orthogonalized basis set. The diagonal elements of the new density matrix are commonly used for a Löwdin population analysis.

What I do is to extract the coefficient matrix in terms of an orthogonal basis set obtained by canonical orthogonalization from the canonical molecular orbitals. The SCF procedure in Q-Chem keeps the lastest Fock matrices (which is formed by a non-orthogonal basis set) in scratch, named as FILE_TEMP_FOCK. I gonna going to hack or learn from liblas/fock2mo.F which makes a coefficients matrix in terms of a non-orthogonal basis set.

Damn it !! It's not so complicated as I think. Don't worry about the linear dependence problem due to multiplying with the root-squared eigenvalue of the overlap matrix, not dividing it.

The one-particle density matrix in discrete basis can be presented as sum of