5.3 Fourier transform of the basis functions

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

$${\tilde \chi}_{\alpha , n \ell m}({\bf k}) = \int_{\mathrm{all~space}} {\mathrm d}{\bf r}~\exp[{\mathrm{i}}{\bf k} \cdot {\bf r}]~\chi_{\alpha , n \ell m}({\bf r})$$

$$= \exp{\left[ {\mathrm i} {\bf k} \cdot {\bf R}_{\alpha} \right]} \... ...r}~ \exp[{\mathrm{i}}{\bf k} \cdot {\bf r}]~\bar{Y}_{\ell m} (\Omega_{\bf r}) .$$ (5.7)

The angular integral is performed by using the expansion of $\exp[{\mathrm{i}} {\bf k} \cdot {\bf r}]$ into spherical-waves (A.3, appendix A) leaving the radial integral

$${\tilde \chi}_{\alpha , n \ell m}({\bf k}) = 4 \pi {\mathrm{... ...pha}} {\mathrm d}r~r^2~ j_{\ell}(q_{n \ell} r)~j_{\ell}(k r) .$$ (5.8)

The radial integral can now be calculated using equations A.4 and A.5 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

$${\tilde \chi}_{\alpha , n \ell m}({\bf k}) = 4 \pi {\mathrm{... ...ll} r_{\alpha} ) , & k = q_{n \ell} . & (b) \end{array}\right.$$ (5.9)

Equation 5.9b is in fact a limiting case of (5.9a) which can therefore always be substituted for ${\tilde \chi}_{\alpha , n \ell m}({\bf k})$ in an integral over reciprocal-space.