Editing Slater-type orbital

{{Short description|Function used in quantum chemistry}}
'''Slater-type orbitals''' ('''STOs''') or '''Slater-type functions (STFs)''' are functions used as [[atomic orbital]]s in the [[linear combination of atomic orbitals molecular orbital method]]. They are named after the physicist [[John C. Slater]], who introduced them in 1930.<ref>
{{cite journal
 |last1=Slater |first1=J. C.
 |year=1930
 |title=Atomic Shielding Constants
 |journal=[[Physical Review]]
 |volume=36 |issue= 1|page=57
 |bibcode=1930PhRv...36...57S
 |doi=10.1103/PhysRev.36.57
|url=http://elib.bsu.by/handle/123456789/154383
 |url-access=subscription
 }}</ref>

They possess exponential decay at long range and [[Kato theorem|Kato's cusp condition]] at short range (when combined as [[hydrogen-like atom]] functions, i.e. the analytical solutions of the stationary Schrödinger equation for one electron atoms). Unlike the hydrogen-like ("hydrogenic") Schrödinger orbitals, STOs have no radial nodes (neither do [[Gaussian orbitals|Gaussian-type orbitals]]).

==Definition==
STOs have the following radial part:

: <math>R(r) = N r^{n-1} e^{-\zeta r}\,</math>
where
* {{mvar|n}} is a [[natural number]] that plays the role of [[principal quantum number]], {{mvar|n}} = 1,2,...,
* {{mvar|N}} is a [[normalizing constant]],
* {{mvar|r}} is the distance of the electron from the [[atomic nucleus]], and
* <math>\zeta</math> is a constant related to the effective [[electric charge|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 [[gamma function|the integral]]
:<math> \int_0^\infty x^n e^{-\alpha x} \, \mathrm dx = \frac{n!}{~\alpha^{n+1}\,}~. </math>
Hence
:<math>N^2 \int_0^\infty \left(r^{n-1} e^{-\zeta r}\right)^2 \, \mathrm dr = 1 \Longrightarrow N = (2\zeta)^n \sqrt{\frac{2\zeta}{(2n)!}}~. </math>

It is common to use the [[spherical harmonics]] <math>Y_l^m(\mathbf{r})</math> depending on the polar coordinates
of the position vector <math>\mathbf{r}</math> as the angular part of the Slater orbital.

==Derivatives==
The first radial derivative of the radial part of a Slater-type orbital is
:<math> {\partial R(r)\over \partial r} = \left[\frac{(n - 1)}{r} - \zeta\right] R(r) </math>
The radial Laplace operator is split in two differential operators
:<math> \nabla^2 = {1 \over r^2}{\partial \over \partial r}\left(r^2 {\partial \over \partial r}\right) </math>
The first differential operator of the Laplace operator yields
:<math> \left(r^2 {\partial\over \partial r} \right) R(r) = \left[(n - 1) r - \zeta r^2 \right] R(r) </math>
The total Laplace operator yields after applying the second differential operator
:<math> \nabla^2 R(r) = \left({1 \over r^2} {\partial\over \partial r} \right) \left[(n - 1) r - \zeta r^2 \right] R(r) </math>
the result
:<math>\nabla^2 R(r) = \left[{n (n - 1) \over r^2} - {2 n \zeta \over r} + \zeta^2 \right] R(r) </math>
Angular dependent derivatives of the spherical harmonics don't depend on the radial function and have to be evaluated separately.

==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 {{mvar|N}}, the representation of the orbitals below is
:<math>\chi_{n \ell m}({\mathbf{r}}) =
 r^{n-1}~e^{-\zeta\,r}~Y_\ell^m({\mathbf{r}})~.</math>

The [[Fourier transform]] is<ref>
{{cite journal
 |last1=Belkic |first1=D.
 |last2=Taylor |first2=H. S.
 |year=1989
 |title=A unified formula for the Fourier transform of Slater-type orbitals
 |journal=[[Physica Scripta]]
 |volume=39 |issue=2 |pages=226–229
 |bibcode=1989PhyS...39..226B
 |doi=10.1088/0031-8949/39/2/004
|s2cid=250815940
 }}</ref>
:<math>\begin{align}
\chi_{n \ell m}({\mathbf{k}}) &= \int e^{i{\mathbf{k}}\cdot {\mathbf{r}}}~\chi_{n \ell m}({\mathbf{r}})~\mathrm{d}^3 r \\
&=4\pi~(n-\ell)!~(2\zeta)^n~(i k/\zeta)^\ell~Y_\ell^m({\mathbf{k}}) \sum_{s=0}^{\lfloor(n-\ell)/2\rfloor} \frac{\omega_s^{n \ell}}{~(k^2+\zeta^2)^{n+1-s}},
\end{align}</math>
where the <math>\omega</math> are defined by
:<math>\omega_s^{n \ell} \equiv \left( -\frac{1}{4\zeta^2} \right)^s\,\frac{(n-s)!}{~s!~(n-\ell-2s)!~}.</math>

The overlap integral is
:<math> \int \chi^*_{n \ell m}(r)~\chi_{n'\ell'm'}(r)~\mathrm{d}^3 r = \delta_{\ell \ell'}\,\delta_{mm'}\, \frac{(n+n')!}{~(\zeta+\zeta')^{n+n'+1}}</math>
of which the normalization integral is a special case. The superscript star denotes [[Complex conjugate|complex-conjugation]].

The [[Kinetic energy#Quantum mechanical kinetic energy of rigid bodies|kinetic energy]] integral is
<math display="block">\begin{align}&
\int \chi^*_{n \ell m}(r)~\left(-\tfrac{1}{2} \nabla^2\right)\,\chi_{n'\ell'm'}(r)~\mathrm{d}^3 r \\&=
\frac{1}{2}\delta_{\ell\ell'}\,\delta_{mm'}\,
\int_0^\infty e^{-(\zeta+\zeta')\,r}
\left[
[\ell'(\ell'+1) - n'(n'-1)]\,r^{n+n'-2} + 2\zeta'n'\,r^{n+n'-1} - \zeta'^2\,r^{n+n'}
\right]~ \mathrm dr~,
\end{align}</math>
a sum over three overlap integrals already computed above.

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

:<math>
\chi^*_{n \ell m}({\mathbf{r}}) = \int \frac{~e^{i \mathbf{k} \cdot  \mathbf{r}}~}{(2\pi)^3}~
\chi^*_{n \ell m}({\mathbf{k}})~\mathrm{d}^3 k
</math>

which yields
<math display="block">\begin{align}
\int \chi^*_{n \ell m}( \mathbf{r} ) \frac{1}{
\left| \mathbf{r} - \mathbf{r}' \right|}~\chi_{n'\ell'm'}( \mathbf{r}')~ \mathrm{d}^3 r
&=
4\pi
\int
\frac{1}{(2\pi)^3}~
\chi^*_{n \ell m}( \mathbf{k} ) ~\frac{1}{k^2}~\chi_{n'\ell'm'}( \mathbf{k} ) ~\mathrm{d}^3 k
\\
&=
8\,\delta_{\ell \ell'}\,
\delta_{mm'}~
(n-\ell)!~
(n'-\ell)!~
\frac{\,(2\zeta)^n\,}{\zeta^\ell}
\frac{\,(2\zeta')^{n'}\,}{\zeta'^\ell}
\int_0^\infty k^{2\ell} \left[
\sum_{s=0}^{\lfloor (n-\ell)/2\rfloor}
\frac{\omega_s^{n \ell}}{(k^2+\zeta^2)^{n+1-s}}
\sum_{s'=0}^{\lfloor (n'-\ell)/2\rfloor}
\frac{\omega_{s'}^{n'\ell'}}{~~(k^2+\zeta'^2)^{n'+1-s'}~} \right]
\mathrm dk
\end{align}</math>
These are either individually calculated with the [[Methods of contour integration|law of residues]] or recursively as proposed by Cruz ''et al''. (1978).<ref>
{{cite journal
 |last1=Cruz |first1=S. A.
 |last2=Cisneros |first2=C.
 |last3=Alvarez |first3=I.
 |year=1978
 |title=Individual orbit contribution to the electron stopping cross section in the low-velocity region
 |journal=[[Physical Review A]]
 |volume=17 |issue=1 |pages=132–140
 |bibcode=1978PhRvA..17..132C
 |doi=10.1103/PhysRevA.17.132
}}</ref>

==STO software==
Some quantum chemistry software uses sets of [[1s Slater-type function|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.<ref>
{{cite journal
 |last1=Guseinov |first1=I. I.
 |year=2002
 |title=New complete orthonormal sets of exponential-type orbitals and their application to translation of Slater Orbitals
 |journal=[[International Journal of Quantum Chemistry]]
 |volume=90 |issue=1 |pages=114–118
 |doi=10.1002/qua.927
}}</ref>

Analytical ab initio software for polyatomic molecules has been developed, e.g., STOP: a Slater Type Orbital Package in 1996.<ref>
{{cite journal
 |last1=Bouferguene |first1=A.
 |last2=Fares |first2=M.
 |last3=Hoggan |first3=P. E.
 |year=1996
 |title=STOP: Slater Type Orbital Package for general molecular electronic structure calculations
 |journal=[[International Journal of Quantum Chemistry]]
 |volume=57 |issue=4 |pages=801–810
 |doi=10.1002/(SICI)1097-461X(1996)57:4<801::AID-QUA27>3.0.CO;2-0
}}</ref>

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.

After the work of [[John Pople]], [[Warren. J. Hehre]] and [[Robert F. Stewart]], a least squares representation of the Slater atomic orbitals as a sum of Gaussian-type orbitals is used. In their 1969 paper, the fundamentals of this principle are discussed and then further improved and used in the [[GAUSSIAN]] DFT code. <ref>{{Cite journal|last1=Hehre|first1=W. J.|last2=Stewart|first2=R. F.|last3=Pople|first3=J. A.|date=1969-09-15|title=Self‐Consistent Molecular‐Orbital Methods. I. Use of Gaussian Expansions of Slater‐Type Atomic Orbitals|journal=The Journal of Chemical Physics|language=en|volume=51|issue=6|pages=2657–2664|doi=10.1063/1.1672392|bibcode=1969JChPh..51.2657H|issn=0021-9606}}</ref>

==See also==
*[[Basis set (chemistry)|Basis sets used in computational chemistry]]

==References==
{{Reflist}}
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 |volume=45 |issue=1 |pages=116
 |bibcode=1966JChPh..45..116H
 |doi=10.1063/1.1727293
}}
*{{cite journal
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}}
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{{DEFAULTSORT:Slater-Type Orbital}}
[[Category:Quantum chemistry]]
[[Category:Computational chemistry]]