# Orthonormality of Kohn-Sham orbitals

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?

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.