# Are any other orthonormalizations schemes besides Lowdin normally used when solving SCF equations?

When solving equations of the form:

$$F(C)C = SC\epsilon$$

(e.g. Hartree-Fock-Roothan or Kohn-Sham-Roothan) where $$F$$ is the fock matrix, $$C$$ is the coefficient matrix and $$S$$ the overlap matrix, the usual way to proceed is to first perform a change of basis and orthonormalize the AOs (atomic orbitals) by performing a Lowdin (Symmetric) orthogonalization. I know at least one case (The Psi4 program) where a different orthogonalization scheme is used (Canonical) but I haven't been able to find any information of other alternative orthogonalization schemes implemented in any programs. Are any other schemes used? If so, where and why?

• The Wikipedia article for Gram-Schmidt orthogonalization is widely used, but the source is behind a paywall. Possibly modified Gram-Schmidt, as it is more numerically stable. – Tyberius Nov 4 '18 at 0:35
• I was under the impression that Gram-Schmidt couldn't be written as a linear transformation and that a linear transformation was needed in order to transform the SCF equations into an eigenvalue problem. Is Gram-Schmidt used in this specific case or some other? – Ignacio Nov 4 '18 at 16:55
• I know that modified Gram-Schmidt is used in the Davidson procedure for CIS for instance, but that is not what I was asking about. – Ignacio Nov 4 '18 at 16:56
• As far as I know, regardless of how you make the orbitals orthogonal, doing so will put the SCF equations into standard eigenvalue form since it will turn the overlap matrix into the unit matrix. I don't have it with me at the moment, but I believe in Szabo and Ostlund they use Gram-Schmidt for one of the examples. @Ignacio – Tyberius Nov 4 '18 at 17:10