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Orbital FT Kernels

 

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

The s spherical harmonic tex2html_wrap_inline3532 which has no angular dependence gives rise to the simplest kernel

eqnarray1965

Higher order spherical harmonics give rise to more complex kernels, for example tex2html_wrap_inline3534 gives

eqnarray1979

Here tex2html_wrap_inline3536 , and similarly we define tex2html_wrap_inline3538 and tex2html_wrap_inline3540 .

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 tex2html_wrap_inline3548 and tex2html_wrap_inline3550 kernels may be found as follows:

eqnarray2025

The kernels tex2html_wrap_inline3552 and tex2html_wrap_inline3554 may be found in the same way as above and are:

eqnarray2053

Therefore tex2html_wrap_inline3548 and tex2html_wrap_inline3550 are given by

eqnarray2064

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

eqnarray2079

where tex2html_wrap_inline3560 , tex2html_wrap_inline3562 , etc.



Matthew Segall
Wed Sep 24 12:24:18 BST 1997