Orthonormality of Kohn-Sham orbitals

Question

I was wondering if Kohn-Sham orbitals corresponding to a different Bloch wavevector should be orthogonal? I know that we should have $$\int d \boldsymbol{r}\phi_i(\boldsymbol{r}) \phi_j^*(\boldsymbol{r}) = \delta_{ij}\tag{1}$$ but I was wondering whether, if we also considered the $\boldsymbol{k}$ dependence, we should also have $$\int d \boldsymbol{r}\phi_i(\boldsymbol{r}, \boldsymbol{k}) \phi_j^*(\boldsymbol{r},\boldsymbol{k}' ) = \delta_{ij} \delta(\boldsymbol{k} - \boldsymbol{k}')\tag{2}$$

My feeling is that this should be the case and so the appropriate Lagrange multipliers used in the solution of the KS equations would have to be sought for each $\boldsymbol{k}$ point?

Answer 1

You are correct that orbitals from different $k$-points should be orthogonal. $k$-points are irreps of the translation group and, similar to the irreps of point groups for molecules, integrals of two functions with different irreps will always result in $0$.

More generally, we can actually separately solve the SCF equations for each $k$-point, since the Fock matrix can be split into blocks for each $k$-point: $$F^kC^k=S^kC^k\epsilon^k$$ Some details about efficiently solving for the energy and forces of periodic systems using Gaussian type orbitals and transformations back-and-forth from real space to $k$-space are given in 1.

1. Konstantin N. Kudin and Gustavo E. Scuseria Phys. Rev. B 61, 16440 DOI: 10.1103/PhysRevB.61.16440