3. Fourier transform of the basis functions

We define the Fourier transform of a basis function $\chi_{n \ell m}^{\alpha}({\bf r})$ by

$${\tilde \chi}_{n \ell m}^{\alpha}({\bf k}) = \int_{\mathrm{all~space}} \mathrm{d}^{3}r~\mathrm{e}^{{\mathrm{i}}{\bf k} \cdot {\bf r}}~\chi_{n \ell m}^{\alpha}({\bf r})$$

$$= \int_{0}^{r_{\alpha}} \mathrm{d}r ~ r^2~j_{\ell}(q_{n \ell} r) \i... ...ega ~ \mathrm{e}^{{\mathrm{i}}{\bf k} \cdot {\bf r}}~\bar{Y}_{\ell m}(\Omega) .$$ (7)

The angular integral is performed by using the expansion of $\mathrm{e}^{{\mathrm{i}} {\bf k} \cdot {\bf r}}$ into spherical-waves (42, Appendix) leaving the radial integral

$${\tilde \chi}_{n \ell m}^{\alpha}({\bf k}) = 4 \pi {\mathrm{... ...lpha}} \mathrm{d}r~r^2~ j_{\ell}(q_{n \ell} r)~j_{\ell}(k r) .$$ (8)

The radial integral can now be calculated using equations (43,44) given in the Appendix and the boundary conditions (that the basis functions are finite at $r = 0$ and vanish at $r = r_{\alpha}$) for the cases when $k \not= q_{n \ell}$ and $k = q_{n \ell}$ respectively. The final result for the Fourier transform of a basis function is then

$${\tilde \chi}_{n \ell m}^{\alpha}({\bf k}) = 4 \pi {\mathrm{... ...ll} r_{\alpha} ) , & k = q_{n \ell} . & (b) \end{array}\right.$$ (9)

Equation (9b) is in fact a limiting case of (9a) which can therefore always be substituted for ${\tilde \chi}_{n \ell m}^{\alpha}({\bf k})$ in an integral over reciprocal-space.