1s Slater-type function

A normalized 1s Slater-type function is a function which has the form

:$psi\_\{1s\}(zeta,\; mathbf\{r\; -\; R\})\; =\; left(frac\{zeta^3\}\{pi\}\; ight)^\{1\; over\; 2\}\; ,\; e^\{-zeta\; |mathbf\{r\; -\; R\}.$ [cite book
last = Attila Szabo and Neil S. Ostlund
first =
title = Modern Quantum Chemistry - Introduction to Advanced Electronic Structure Theory
publisher = Dover Publications Inc.
year = 1996
pages = 153
isbn = 0486691861 ]

The parameter $zeta$ is called the Slater orbital exponent. Related sets of functions can be used to construct STO-nG basis sets which are used in quantum chemistry.

Applications for hydrogen-like atomic systems

A hydrogen-like atom or a hydrogenic atom is an atom with one electron. Except for the hydrogen atom itself (which is neutral) these atoms carry positive charge $e(mathbf\; Z-1)$, where $mathbf\; Z$ is the atomic number of the atom. Because hydrogen-like atoms are two-particle systems with an interaction depending only on the distance between the two particles, their (non-relativistic) Schrödinger equation can be exactly solved in analytic form. The solutions are one-electron functions and are referred to as "hydrogen-like atomic orbitals". [In quantum chemistry an orbital is synonymous with "one-electron function", i.e., a function of "x", "y", and "z".] The electonic Hamiltonian (in atomic units) of a Hydrogenic system is given by
$mathbf\; hat\{H\}\_e\; =\; -\; frac\{\; abla^2\}\{2\}\; -\; frac\{mathbf\; Z\}\{r\}$, where $mathbf\; Z$ is the nuclear charge of the hydrogenic atomic system. The 1s electron of a hydrogenic systems can be accurately described by the corresponding Slater orbital:
$mathbf\; psi\_\{1s\}\; =\; left\; (frac\{zeta^3\}\{pi\}\; ight\; )\; ^\{0.50\}e^\{-zeta\; r\}$, where $mathbf\; zeta$ is the Slater exponent.

Exact energy of a hydrogen-like atom

The energy of a hydrogenic system can be exactly calculated analytically as follows :
, where


. Using the expression for Slater orbital, $mathbf\; psi\_\{1s\}\; =\; left\; (frac\{zeta^3\}\{pi\}\; ight\; )\; ^\{0.50\}e^\{-zeta\; r\}$ the integrals can be exactly solved. Thus,
Integrals needed $int\_0^infty\; e^\{-alpha\; r^2\}r^n,dr\; =\; frac\{(n-1)!!\}\{2^$n/2} +1} alpha^{n/2sqrt frac{pi}{alpha} when 'n' is even. $int\_0^infty\; e^\{-alpha\; r^2\}r^n,dr\; =\; frac\{(frac\{n-1\}\{2\})!\}\{2\; alpha^$(n+1)}/2 when 'n' is odd.
$mathbf\; E\_\{1s\}\; =\; frac\{zeta^2\}\{2\}-zeta\; mathbf\; Z.$

The optimum value for $mathbf\; zeta$ is obtained by equating the differential of the energy with respect to $mathbf\; zeta$ as zero.
$frac\{dmathbf\; E\_\{1s\{dzeta\}=zeta-mathbf\; Z=0$. Thus $mathbf\; zeta=mathbf\; Z.$

Non relativistic energy

The following energy values are thus calculated by using the expressions for energy and for the Slater exponent.

Hydrogen : H
$mathbf\; Z=1$ and $mathbf\; zeta=1$
$mathbf\; E\_\{1s\}=$−0.5 E_h
$mathbf\; E\_\{1s\}=$−13.60569850 eV
$mathbf\; E\_\{1s\}=$−313.75450000 kcal/mol

Gold : Au(78+)
$mathbf\; Z=79$ and $mathbf\; zeta=79$
$mathbf\; E\_\{1s\}=$−3120.5 E_h
$mathbf\; E\_\{1s\}=$−84913.16433850 eV
$mathbf\; E\_\{1s\}=$−1958141.8345 kcal/mol.

Relativistic energy of Hydrogenic atomic systems

Hydrogenic atomic systems are suitable models to demonstrate the relativistic effects in atomic systems in a simple way. The energy expectation value can calculated by using the Slater orbitals with or without considering the relativistic correction for the Slater exponent $mathbf\; zeta$. The relativistically corrected Slater exponent $mathbf\; zeta\_\{rel\}$ is given as
$mathbf\; zeta\_\{rel\}=\; frac\{mathbf\; Z\}\{sqrt\; \{1-mathbf\; Z^2/c^2$.
The relativistic energy of an electron in 1s orbital of a hydrogenic atomic systems is obtained by solving the Dirac equation.
$mathbf\; E\_\{1s\}^\{rel\}\; =\; -(c^2+mathbf\; Zzeta)+sqrt\{c^4+mathbf\; Z^2zeta^2\}$.
Following table illustrates the relativistic corrections in energy and it can be seen how the relativistic correction scales with the atomic number of the system.

Atomic system	$mathbf\; Z$	$mathbf\; zeta\_\{non\; rel\}$	$mathbf\; zeta\_\{rel\}$	$mathbf\; E\_\{1s\}^\{non\; rel\}$	$mathbf\; E\_\{1s\}^\{rel\}$using $mathbf\; zeta\_\{non\; rel\}$	$mathbf\; E\_\{1s\}^\{rel\}$using $mathbf\; zeta\_\{rel\}$
H	1	1.00000000	1.00002663	−0.50000000 Eh	−0.50000666 Eh	−0.50000666 Eh
				−13.60569850 eV	−13.60587963 eV	−13.60587964 eV
				−313.75450000 kcal/mol	−313.75867685 kcal/mol	−313.75867708 kcal/mol
Au(78+)	79	79.00000000	96.68296596	−3120.50000000 Eh	−3343.96438929 Eh	−3434.58676969 Eh
				−84913.16433850 eV	−90993.94255075 eV	−93459.90412098 eV
				−1958141.83450000 kcal/mol	−2098367.74995699 kcal/mol	−2155234.10926142 kcal/mol

Notes