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We define the Fourier transform of a basis function
by
The angular integral is performed by using the expansion of
into spherical-waves
(A.3, appendix A) leaving the radial integral
![\begin{displaymath}
{\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) .
\end{displaymath}](img637.gif) |
(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
and vanish at
) for the cases when
and
respectively. The final result for the
Fourier transform of a basis function is then
![\begin{displaymath}
{\tilde \chi}_{\alpha , n \ell m}({\bf k}) = 4 \pi {\mathrm{...
...ll} r_{\alpha}
) , & k = q_{n \ell} . & (b)
\end{array}\right.
\end{displaymath}](img642.gif) |
(5.9) |
Equation 5.9b is in fact a limiting case of
(5.9a) which can therefore always be substituted for
in an integral over
reciprocal-space.
Next: 5.4 Overlap matrix elements
Up: 5. Localised basis-set
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Peter Haynes