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 It is a particular case of a Slater-type orbital in which the principal quantum number n is 1. The parameter 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 these atoms carry positive charge, where 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 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 electronic Hamiltonian of a Hydrogenic system is given by
, where 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:
, where 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 :
, where
. Using the expression for Slater orbital, the integrals can be exactly solved. Thus,
The optimum value for is obtained by equating the differential of the energy with respect to as zero.
. Thus
Non-relativistic energy
The following energy values are thus calculated by using the expressions for energy and for the Slater exponent. Hydrogen : H
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. The relativistically corrected Slater exponent is given as
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.
