1s Slater type function

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 ( Z − 1 ) , where 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 electronic Hamiltonian (in atomic units) of a Hydrogenic system is given by
H ^ e = − ∇ 2 2 − Z r , where 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:
ψ 1 s = ( ζ 3 π ) 0.50 e − ζ r , where ζ 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 :
E 1 s = < ψ 1 s | H ^ e | ψ 1 s > < ψ 1 s | ψ 1 s > , where < ψ 1 s | ψ 1 s > = 1
E 1 s =< ψ 1 s | − ∇ 2 2 − Z r | ψ 1 s >
E 1 s =< ψ 1 s | − ∇ 2 2 | ψ 1 s > + < ψ 1 s | − Z r | ψ 1 s >
E 1 s =< ψ 1 s | − 1 2 r 2 ∂ ∂ r ( r 2 ∂ ∂ r ) | ψ 1 s > + < ψ 1 s | − Z r | ψ 1 s > . Using the expression for Slater orbital, ψ 1 s = ( ζ 3 π ) 0.50 e − ζ r the integrals can be exactly solved. Thus,
E 1 s =< ( ζ 3 π ) 0.50 e − ζ r | − ( ζ 3 π ) 0.50 e − ζ r [ − 2 r ζ + r 2 ζ 2 2 r 2 ] > + < ψ 1 s | − Z r | ψ 1 s >
E 1 s = ζ 2 2 − ζ Z .

The optimum value for ζ is obtained by equating the differential of the energy with respect to ζ as zero.
d E 1 s d ζ = ζ − Z = 0 . Thus ζ = Z .

Non relativistic energy

The following energy values are thus calculated by using the expressions for energy and for the Slater exponent.

Hydrogen : H
Z = 1 and ζ = 1
E 1 s = −0.5 E_h
E 1 s = −13.60569850 eV
E 1 s = −313.75450000 kcal/mol

Gold : Au(78+)
Z = 79 and ζ = 79
E 1 s = −3120.5 E_h
E 1 s = −84913.16433850 eV
E 1 s = −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 ζ . The relativistically corrected Slater exponent ζ r e l is given as
ζ r e l = Z 1 − 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.
E 1 s r e l = − ( c 2 + Z ζ ) + c 4 + Z 2 ζ 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.