Because of the discontinuity in the first derivatives of the basis functions at the sphere boundaries, a delta-function arises when the Laplacian operates on a basis function. This is integrated out when the matrix element is calculated and this contribution is included when transforming the real-space integral to reciprocal-space in equation (26).

The second line of equation (26) is identical to
equation (11) apart from a factor of
. The same
separation into individually regular terms can be applied here, and
the result is that we need to calculate the contour integral
(17) as before, except that the integer must be
replaced by and a numerical factor of
is
introduced. The calculation of the residues is identical to that
presented in the previous section, except that the integrand no longer
* always* has a pole at in every term.

The results for
when
are

(27) | |||

The calculation of the kinetic energy has been checked by projecting a set of wave functions expanded in the spherical-wave basis onto the plane-wave basis using equation (9a). As the kinetic energy cut-off for the plane-wave basis is increased, so the description of the wave functions becomes more accurate. The kinetic energy calculated using the results above can then be compared against the kinetic energy calculated by a plane-wave code [2].

From the asymptotic behaviour of the spherical Bessel functions,
the Fourier transform (9a) for large is

(28) |

(29) |