$\bf{k}\cdot\bf{p}$ formalism within FLAPW method

We provide $\bf{k}\cdot\bf{p}$ formalism within the full-potential linearized augmented plane wave (FLAPW) method. Unlike the pure plane waves, the LAPW functions do not behave trivially in moving from $\bf{k}$ to $\bf{k}+\bf{q}$ and their incompleteness as a basis set should be taken into account. Derivatives of the sphere matching coefficients play the key role, for which we find a simple formula. Concrete formula for the $\bf{k}\cdot\bf{p}$ matrix elements is derived and numerically tested. Generalized second-order perturbation theory allowing for a degenerate case is presented and the literally-exact electronic band gradients and curvatures are accessible.