1s Slater-type function explained

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 _{1s} (\zeta, r-R ) = \left( (\zeta ^{3})/(\pi) \right) ^{1} e ^{-\zeta} . ^[1]

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( 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.^[2] The electronic Hamiltonian (in atomic units) of a Hydrogenic system is given by
\hat{H }_e = - \frac - \frac, 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:
\psi _{1s} = \left ( (\zeta ^{3})/(\pi) \right ) ^{0.50} e ^{-\zeta} , where \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 :

E _{1s} = (\langle\psi _{1s} | \hat{H)/(_{e|\psi}) _{1s} \rangle}{\langle\psi _{1s} |\psi _{1s} \rangle} , where \langle\psi _{1s} |\psi _{1s} \rangle = 1
E _{1s} = \langle\psi _{1s} | - (\nabla ^{2})/(2) - (Z)/(r) |\psi _{1s} \rangle
E _{1s} = \langle\psi _{1s} | - (\nabla ^{2})/(2) |\psi _{1s} \rangle + \langle\psi _{1s} | - (Z)/(r) |\psi _{1s} \rangle
E _{1s} = \langle\psi _{1s} | - (1)/(2 r ^{2}) (\partial)/(\partial r) \left (r ^{2} (\partial)/(\partial r) \right )|\psi _{1s} \rangle + \langle\psi _{1s} | - (Z)/(r) |\psi _{1s} \rangle . Using the expression for Slater orbital, \psi _{1s} = \left ( (\zeta ^{3})/(\pi) \right ) ^{0.50} e ^{-\zeta} the integrals can be exactly solved. Thus,
E _{1s} = \left\langle \left( (\zeta ^{3})/(\pi) \right) ^{0.50} e ^{-\zeta} \right|\left. - \left( (\zeta ^{3})/(\pi) \right) ^{0.50} e ^{-\zeta} \left[ (-2 r\zeta + r ^{2\zeta} ^{2})/(2 r ^{2}) \right]\right\rangle + \langle\psi _{1s} | - (Z)/(r) |\psi _{1s} \rangle
E _{1s} = (\zeta ^{2})/(2) - \zeta Z.

The optimum value for

\zeta is obtained by equating the differential of the energy with respect to \zeta as zero.
(d E _{1s})/(d\zeta) = \zeta - Z=0 . Thus \zeta= 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 \zeta=1
E _{1s} = −0.5 E_h
E _{1s} = −13.60569850 eV
E _{1s} = −313.75450000 kcal/mol

Gold : Au(78+)

Z=79 and \zeta=79
E _{1s} = −3120.5 E_h
E _{1s} = −84913.16433850 eV
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

\zeta . The relativistically corrected Slater exponent \zeta _{rel} is given as
\zeta _{rel} = (Z)/(\sqrt { 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 ^{rel} _{1 s} = - (c ^{2+Z\zeta)+\sqrt{c} ^{4+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	Z	\zeta _{non}	\zeta _{rel}	E ^{non rel} _{1 s}	E ^{rel} _{1 s} using \zeta _{non}	E ^{rel} _{1 s} using \zeta _{rel}
H	1	1.00000000	1.00002663	−0.50000000 Eh	−0.50000666 Eh	−0.50000666 Eh
				−13.60569850 eV	−13.60587963 eV	−13.60587964 eV
				−313.75450000 kcal/mol	−313.75867685 kcal/mol	−313.75867708 kcal/mol
Au(78+)	79	79.00000000	96.68296596	−3120.50000000 Eh	−3343.96438929 Eh	−3434.58676969 Eh
				−84913.16433850 eV	−90993.94255075 eV	−93459.90412098 eV
				−1958141.83450000 kcal/mol	−2098367.74995699 kcal/mol	−2155234.10926142 kcal/mol

Notes and References

1. Book: Attila Szabo . Neil S. Ostlund . amp. Modern Quantum Chemistry - Introduction to Advanced Electronic Structure Theory . limited . . 1996 . 153 . 0-486-69186-1.
2. In quantum chemistry an orbital is synonymous with "one-electron function", i.e., a function of x, y, and z.