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Concept# Slater-type orbital

Summary

Slater-type orbitals (STOs) are functions used as atomic orbitals in the linear combination of atomic orbitals molecular orbital method. They are named after the physicist John C. Slater, who introduced them in 1930.
They possess exponential decay at long range and Kato's cusp condition at short range (when combined as hydrogen-like atom functions, i.e. the analytical solutions of the stationary Schrödinger equation for one electron atoms). Unlike the hydrogen-like ("hydrogenic") Schrödinger orbitals, STOs have no radial nodes (neither do Gaussian-type orbitals).
STOs have the following radial part:
where
n is a natural number that plays the role of principal quantum number, n = 1,2,...,
N is a normalizing constant,
r is the distance of the electron from the atomic nucleus, and
is a constant related to the effective charge of the nucleus, the nuclear charge being partly shielded by electrons. Historically, the effective nuclear charge was estimated by Slater's rules.
The normalization constant is computed from the integral
Hence
It is common to use the spherical harmonics depending on the polar coordinates
of the position vector as the angular part of the Slater orbital.
The first radial derivative of the radial part of a Slater-type orbital is
The radial Laplace operator is split in two differential operators
The first differential operator of the Laplace operator yields
The total Laplace operator yields after applying the second differential operator
the result
Angular dependent derivatives of the spherical harmonics don't depend on the radial function and have to be evaluated separately.
The fundamental mathematical properties are those associated with the kinetic energy, nuclear attraction and Coulomb repulsion integrals for placement of the orbital at the center of a single nucleus. Dropping the normalization factor N, the representation of the orbitals below is
The Fourier transform is
where the are defined by
The overlap integral is
of which the normalization integral is a special case.

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