1s Slater-type Function

A normalized 1s Slater-type function is a function which is used in the descriptions of atoms and in a broader way in the description of atoms in molecules. It is particularly important as the accurate quantum theory description of the smallest free atom, hydrogen. It has the form

:$\psi_(\zeta, \mathbf) = \left(\frac\right)^ \, e^.$

It is a particular case of a Slater-type orbital (STO) in which the principal quantum number n is 1. 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 electronic Hamiltonian (in atomic units) of a Hydrogenic system is given by

$\mathbf_e = - \frac - \frac$, 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:

$\mathbf \psi_ = \left (\frac \right ) ^e^$, where $\mathbf \zeta$ is the Slater exponent. This state, the ground state, is the only state that can be described by a Slater orbital. Slater orbitals have no radial nodes, while the excited states of the hydrogen atom have radial nodes.

Exact energy of a hydrogen-like atom

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

$\mathbf E_ = \frac$, where $\mathbf = 1$

$\mathbf E_ = \langle\psi_, \mathbf - \frac - \frac, \psi_\rangle$

$\mathbf E_ = \langle\psi_, \mathbf - \frac, \psi_\rangle+\langle\psi_, - \frac, \psi_\rangle$

$\mathbf E_ = \langle\psi_, \mathbf - \frac\frac\left (r^2 \frac\right ), \psi_\rangle+\langle\psi_, - \frac, \psi_\rangle$. Using the expression for Slater orbital, $\mathbf \psi_ = \left (\frac \right ) ^e^$ the integrals can be exactly solved. Thus,

\mathbf E_ = \left\langle \left(\frac \right)^ e^ \\left. -\left(\frac \right)^e^\left frac\rightright\rangle+\langle\psi_, - \frac, \psi_\rangle

$\mathbf E_ = \frac-\zeta \mathbf Z.$

The optimum value for $\mathbf \zeta$ is obtained by equating the differential of the energy with respect to $\mathbf \zeta$ as zero.

$\frac=\zeta-\mathbf Z=0$. 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 $\mathbf \zeta=1$

$\mathbf E_=$−0.5 E_h

$\mathbf E_=$−13.60569850 eV

$\mathbf E_=$−313.75450000 kcal/mol

Gold : Au(78+)

$\mathbf Z=79$ and $\mathbf \zeta=79$

$\mathbf E_=$−3120.5 E_h

$\mathbf E_=$−84913.16433850 eV

$\mathbf E_=$−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 $\mathbf \zeta_$ is given as

$\mathbf \zeta_= \frac$.

The relativistic energy of an electron in 1s orbital of a hydrogenic atomic systems is obtained by solving the Dirac equation.

$\mathbf E_^ = -(c^2+\mathbf Z\zeta)+\sqrt$.

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.

References

{{DEFAULTSORT:1s Slater-Type Function
Atoms
Quantum models