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.