Slater-type orbitals (STOs) are functions used as atomic orbitals in the linear combination of atomic orbitals molecular orbital method. They are named after the physicist John C. Slater, who introduced them in 1930.^{[1]}
They possess exponential decay at long range and Kato's cusp condition at short range (when combined as hydrogen-like atom functions, i.e. the analytical solutions of the stationary Schrödinger for one electron atoms). Unlike the hydrogen-like ("hydrogenic") Schrödinger orbitals, STOs have no radial nodes (neither do Gaussian-type orbitals).
Definition
STOs have the following radial part:
- $R(r)\; =\; N\; r^\{n-1\}\; e^\{-\backslash zeta\; r\}\backslash ,$
where
- n is a natural number that plays the role of principal quantum number, n = 1,2,...,
- N is a normalizing constant,
- r is the distance of the electron from the atomic nucleus, and
- $\backslash zeta$ is a constant related to the effective charge of the nucleus, the nuclear charge being partly shielded by electrons. Historically, the effective nuclear charge was estimated by Slater's rules.
The normalization constant is computed from the integral
- $\backslash int\_0^\backslash infty\; x^n\; e^\{-\backslash alpha\; x\}\; dx\; =\; \backslash frac\{n!\}\{\backslash alpha^\{n+1\}\}.$
Hence
- $N^2\; \backslash int\_0^\backslash infty\; \backslash left(r^\{n-1\}e^\{-\backslash zeta\; r\}\backslash right)^2\; r^2\; dr\; =1\; \backslash Longrightarrow\; N\; =\; (2\backslash zeta)^n\; \backslash sqrt\{\backslash frac\{2\backslash zeta\}\{(2n)!\}\}.$
It is common to use the spherical harmonics $Y\_l^m(\backslash mathbf\{r\})$ depending on the polar coordinates
of the position vector $\backslash mathbf\{r\}$ as the angular part of the Slater orbital.
Differentials
The first radial derivative of the radial part of a Slater-type orbital is
- $\{\backslash partial\; R(r)\backslash over\; \backslash partial\; r\}\; =\; \backslash left[\backslash frac\{(n\; -\; 1)\}\{r\}\; -\; \backslash zeta\backslash right]\; R(r)$
The radial Laplace operator is split in two differential operators
- $\backslash nabla^2\; =\; \{1\; \backslash over\; r^2\}\{\backslash partial\; \backslash over\; \backslash partial\; r\}\backslash left(r^2\; \{\backslash partial\; \backslash over\; \backslash partial\; r\}\backslash right)$
The first differential operator of the Laplace operator yields
- $\backslash left(r^2\; \{\backslash partial\backslash over\; \backslash partial\; r\}\; \backslash right)\; R(r)\; =\; \backslash left[(n\; -\; 1)\; r\; -\; \backslash zeta\; r^2\; \backslash right]\; R(r)$
The total Laplace operator yields after applying the second differential operator
- $\backslash nabla^2\; R(r)\; =\; \backslash left(\{1\; \backslash over\; r^2\}\; \{\backslash partial\backslash over\; \backslash partial\; r\}\; \backslash right)\; \backslash left[(n\; -\; 1)\; r\; -\; \backslash zeta\; r^2\; \backslash right]\; R(r)$
the result
- $\backslash nabla^2\; R(r)\; =\; \backslash left)\; =\; r^\{n-1\}e^\{-\backslash zeta\; r\}Y\_l^m(\{\backslash mathbf\{r\}\}).$
The Fourier transform is^{[2]}
- $\backslash chi\_\{nlm\}(\{\backslash mathbf\{k\}\})=\; \backslash int\; d^3r\; e^\{i\{\backslash mathbf\{k\}\}\backslash cdot\; \{\backslash mathbf\{r\}\}\}\; \backslash chi\_\{nlm\}(\{\backslash mathbf\{\; r\}\})$
- $=4\backslash pi\; (n-l)!\; (2\backslash zeta)^n\; (ik/\backslash zeta)^l\; Y\_l^m(\{\backslash mathbf\{k\}\})\; \backslash sum\_\{s=0\}^\{\backslash lfloor(n-l)/2\backslash rfloor\}\; \backslash frac\{\backslash omega\_s^\{nl\}\}\{(k^2+\backslash zeta^2)^\{n+1-s\}\}$,
where the $\backslash omega$ are defined by
- $\backslash omega\_s^\{nl\}\backslash equiv(-\backslash frac\{1\}\{4\backslash zeta^2\})^s\backslash frac\{(n-s)!\}\{s!(n-l-2s)!\}$.
The overlap integral is
- $\backslash int\; \backslash chi^*\_\{nlm\}(r)\backslash chi\_\{n\text{'}l\text{'}m\text{'}\}(r)d^3r\; =\; \backslash delta\_\{ll\text{'}\}\backslash delta\_\{mm\text{'}\}\backslash frac\{(n+n\text{'})!\}\{(\backslash zeta+\backslash zeta\text{'})^\{n+n\text{'}+1\}\}$
of which the normalization integral is a special case. The starlet in the
superscript denotes complex-conjugation.
The kinetic energy integral is
- $$
\int \chi^*_{nlm}(r)(-\frac{\nabla^2}{2})\chi_{n'l'm'}(r)d^3r
=
\frac{1}{2}\delta_{ll'}\delta_{mm'}
\int_0^\infty dr e^{-(\zeta+\zeta')r}
\left[
[l'(l'+1)-n'(n'-1)]r^{n+n'-2}+2\zeta'n'r^{n+n'-1}-\zeta'^2r^{n+n'}
\right],
a sum over three overlap integrals already computed above.
The Coulomb repulsion integral can be evaluated using the Fourier representation
(see above)
- $$
\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).^{[3]}
STO Software
Some quantum chemistry software uses sets of Slater-type functions (STF) analogous to Slater type orbitals, but with variable exponents chosen to minimize the total molecular energy (rather than by Slater's rules as above). The fact that products of two STOs on distinct atoms are more difficult to express than those of Gaussian functions (which give a displaced Gaussian) has led many to expand them in terms of Gaussians.^{[4]}
Analytical ab initio software for poly-atomic molecules has been developed e.g. STOP: a Slater Type Orbital Package in 1996.^{[5]}
SMILES uses analytical expressions when available and Gaussian expansions otherwise. It was first released in 2000.
Various grid integration schemes have been developed, sometimes after analytical work for quadrature (Scrocco). Most famously in the ADF suite of DFT codes.
References
See also
Basis sets used in computational chemistry
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