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# 1s Slater-type function

A normalized 1s Slater-type function is a function which has the form $\psi_{1s}(\zeta, \mathbf{r - R}) = \left(\frac{\zeta^3}{\pi}\right)^{1 \over 2} \, e^{-\zeta |\mathbf{r - R}|}.$

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. 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 $\mathbf \zeta_{rel}$ 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

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.