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1s_Slater-type_function

1s Slater-type function

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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.

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1 Applications for hydrogen-like atomic systems
2 Notes

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
       when 'n' is even.
       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 $\mathbf \zeta_{rel}$
H	1	1.00000000	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+)	79	79.00000000	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

Notes

^ Attila Szabo and Neil S. Ostlund (1996). Modern Quantum Chemistry - Introduction to Advanced Electronic Structure Theory. Dover Publications Inc., 153. ISBN 0486691861.
^ In quantum chemistry an orbital is synonymous with "one-electron function", i.e., a function of x, y, and z.

Category: Atoms

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