Matrix elements

$$(H-ES)_{ij} = \langle {\bf k}+{\bf K}_i\vert H-ES\vert{\bf k... ...B_{ij}+\sum_{l=0}^{l_{max}}C_{ijl} \frac{R'_l(r_0)}{R_l(r_0)}.$$ (249)

$$A_{ij} = \frac{-4\pi r_0^2}{\Omega}\frac{j_1(\vert{\bf K}_i-{\bf K}_j\vert r_0)}{\vert{\bf K}_i-{\bf K}_j\vert}+\delta_{ij}$$ (250)

$$B_{ij} = \frac{\hbar^2}{2m} A_{ij}({\bf q}_i\cdot {\bf q}_j)$$ (251)

$$C_{ijl} = (2l+1)\frac{2\pi r_0^2}{\Omega}P_l\left(\frac{{\bf q}_i\cdot {\bf q}_j}{k_ik_j}\right)j_l(q_ir_0)j_l(q_jr_0)$$ (252)

where $\Omega$ is the volume of the unit cell, and $P_l$ is the Legendre polynomial, and ${\bf q}_i = {\bf K}_i+{\bf k}$.