The kernels of the Orbital FTs as defined in Equation may be calculated analytically as an integral over the angular polar coordinates.

The s spherical harmonic which has no angular dependence gives rise to the simplest kernel

Higher order spherical harmonics give rise to more complex kernels, for example gives

Here , and similarly we define and .

The kernels of spherical harmonics with non-zero *m* must be
constructed by finding the kernels corresponding to linear
combinations of spherical harmonics with equal *l* and *m* of equal
magnitude, but opposite sign. These combinations are constructed such
that the expressions generated are either purely imaginary or
real. The kernels of these linear combinations may then be combined to
generate the kernels corresponding to the original spherical harmonic.
For example the and kernels may be found as
follows:

The kernels and may be found in the same way as above and are:

Therefore and are given by

The kernels for the d orbitals may be calculated in a similar way and are;

where , , etc.

Wed Sep 24 12:24:18 BST 1997