Slater-type Orbital - Integrals

Integrals

The fundamental mathematical properties are those associated with the kinetic energy, nuclear attraction and Coulomb repulsion integrals for placement of the orbital at the center of a single nucleus. Dropping the normalization factor N, the representation of the orbitals below is

.

The Fourier transform is

=4\pi (n-l)! (2\zeta)^n (ik/\zeta)^l Y_l^m({\mathbf{k}}) \sum_{s=0}^{\lfloor(n-l)/2\rfloor} \frac{\omega_s^{nl}}{(k^2+\zeta^2)^{n+1-s}},

where the are defined by

.

The overlap integral is

\int \chi^*_{nlm}(r)\chi_{n'l'm'}(r)d^3r =\delta_{ll'}\delta_{mm'}\frac{(n+n')!}{(\zeta+\zeta')^{n+n'+1}}

of which the normalization integral is a special case. The starlet in the superscript denotes complex-conjugation.

The kinetic energy integral is

\int \chi^*_{nlm}(r)(-\frac{\nabla^2}{2})\chi_{n'l'm'}(r)d^3r = \frac{1}{2}\delta_{ll'}\delta_{mm'} \int_0^\infty dr e^{-(\zeta+\zeta')r} \left[ r^{n+n'-2}+2\zeta'n'r^{n+n'-1}-\zeta'^2r^{n+n'} \right],

a sum over three overlap integrals already computed above.

The Coulomb repulsion integral can be evaluated using the Fourier representation (see above)

\chi^*_{nlm}({\mathbf{r}})=\int\frac{d^3k}{(2\pi)^3}e^{i{\mathbf{k}}\cdot {\mathbf{r}}} \chi^*_{nml}({\mathbf{k}})

which yields

\int \chi^*_{nlm}({\mathbf{r}})\frac{1}{|{\mathbf{r}}-{\mathbf{r}}'|}\chi_{n'l'm'}({\mathbf{r}}')d^3r = 4\pi \int \frac{d^3k}{(2\pi)^3} \chi^*_{nlm}({\mathbf{k}})\frac{1}{k^2}\chi_{n'l'm'}({\mathbf{k}})
= 8\delta_{ll'} \delta_{mm'} (n-l)! (n'-l)! \frac{(2\zeta)^n}{\zeta^l} \frac{(2\zeta')^{n'}}{\zeta'^l} \int_0^\infty dk k^{2l} \sum_{s=0}^{\lfloor (n-l)/2\rfloor} \frac{\omega_s^{nl}}{(k^2+\zeta^2)^{n+1-s}} \sum_{s'=0}^{\lfloor (n'-l)/2\rfloor} \frac{\omega_{s'}^{n'l'}}{(k^2+\zeta'^2)^{n'+1-s'}}

These are either individually calculated with the law of residues or recursively as proposed by Cruz et al. (1978).

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