# Caculating SAB overlap of two Kohn-Sham determinants

1. Sep 4, 2014

### limorsj

Hello,

I would like to implement SAB=<psiA|psiB>which is the overlap of two Kohn-Sham determinants (psiA and psiB are two matrices containing each the molecular orbitals coefficients). Can anybody help me with this calculation? For case of SAA and SBB it is required to get the value 1 (perfect overlap for the same matrix...which is actually the probability of finding the electron somewhere),

Thank you!

Last edited: Sep 4, 2014
2. Sep 4, 2014

### cgk

As far as I am aware of, the most straightforward algorithm for computing matrix elements between non-orthogonal determinants was given in
Some background on the algorithms is described in
If you are only interested in overlap matrix elements between the determinants (instead of more complicated matrix elements, e.g., of the Hamiltonian), then my guess is that the above algorithms would turn into something along the following lines:
1. Compute the overlap matrix of the occupied orbitals $\mathbf{S}_\mathrm{occ} = \mathbf{C}_\mathrm{A}^T \mathbf{S}_\mathrm{AO} \mathbf{C}_B$, where $\mathbf{C}_\mathrm{A}$ is the coefficient matrix of A's occupied orbitals, $\mathbf{S}_\mathrm{AO}$ is the atomic orbital basis overlap matrix (overlap between basis functions in terms of which the occupied orbitals are expanded) and $\mathbf{C}_B$ is the coefficient matrix of B's occupied orbitals.
2. Compute a singular value decomposition (SVD) of the occupied orbital overlap $\mathbf{S}_\mathrm{occ}$
3. The overlap between the determinants A and B is the product of the singular values from step 2.

Last edited by a moderator: May 6, 2017