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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}\right)^{1 \over 2} \, e^{-\zeta |\mathbf{r - R}|}.[1]

The parameter ζ 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.[2] The electonic Hamiltonian (in atomic units) of a Hydrogenic system is given by
\mathbf \hat{H}_e = -  \frac{\nabla^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} \right ) ^{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 :
\mathbf E_{1s} = \frac{<\psi_{1s}|\mathbf \hat{H}_e|\psi_{1s}>}{<\psi_{1s}|\psi_{1s}>}, where \mathbf <\psi_{1s}|\psi_{1s}> = 1
\mathbf E_{1s} = <\psi_{1s}|\mathbf -  \frac{\nabla^2}{2} - \frac{\mathbf Z}{r}|\psi_{1s}>
\mathbf E_{1s} = <\psi_{1s}|\mathbf -  \frac{\nabla^2}{2}|\psi_{1s}>+<\psi_{1s}| - \frac{\mathbf Z}{r}|\psi_{1s}>
\mathbf E_{1s} = <\psi_{1s}|\mathbf -  \frac{1}{2r^2}\frac{\partial}{\partial r}\left (r^2 \frac{\partial}{\partial r}\right )|\psi_{1s}>+<\psi_{1s}| - \frac{\mathbf Z}{r}|\psi_{1s}>. Using the expression for Slater orbital, \mathbf \psi_{1s} = \left (\frac{\zeta^3}{\pi} \right ) ^{0.50}e^{-\zeta r} the integrals can be exactly solved. Thus,
\mathbf E_{1s} = <\left (\frac{\zeta^3}{\pi} \right ) ^{0.50}e^{-\zeta r}|-\left (\frac{\zeta^3}{\pi} \right )^{0.50}e^{-\zeta r}\left[\frac{-2r\zeta+r^2\zeta^2}{2r^2}\right]>+<\psi_{1s}| - \frac{\mathbf Z}{r}|\psi_{1s}>

      Integrals needed
      \int_0^\infty e^{-\alpha r^2}r^n\,dr = \frac{(n-1)!!}{2^{{n/2} +1} \alpha^{n/2}}\sqrt \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{d\mathbf E_{1s}}{d\zeta}=\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 hartree
\mathbf E_{1s}=-13.60569850 e.V.
\mathbf E_{1s}=-313.75450000 kcal/mol

Gold : Au(78+)
\mathbf Z=79 and \mathbf \zeta=79
\mathbf E_{1s}=-3120.5 hartree
\mathbf E_{1s}=-84913.16433850 e.V.
\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 Z\zeta)+\sqrt{c^4+\mathbf Z^2\zeta^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}
H11.00000000 1.00002663-0.50000000 hartree-0.50000666 hartree-0.50000666 hartree
-13.60569850 e.V.-13.60587963 e.V.-13.60587964 e.V.
-313.75450000 kcal/mol-313.75867685 kcal/mol-313.75867708 kcal/mol
Au(78+)7979.00000000 96.68296596-3120.50000000 hartree-3343.96438929 hartree-3434.58676969 hartree
-84913.16433850 e.V. -90993.94255075 e.V. -93459.90412098 e.V.
-1958141.83450000 kcal/mol-2098367.74995699 kcal/mol-2155234.10926142 kcal/mol


  1. ^ Attila Szabo and Neil S. Ostlund (1996). Modern Quantum Chemistry - Introduction to Advanced Electronic Structure Theory. Dover Publications Inc., 153. ISBN 0486691861. 
  2. ^ In quantum chemistry an orbital is synonymous with "one-electron function", i.e., a function of x, y, and z.
  This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "1s_Slater-type_function". A list of authors is available in Wikipedia.
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