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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.^{[1]}
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 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
The normalization constant is computed from the integral
Hence
It is common to use the spherical harmonics Y_l^m(\mathbf{r}) depending on the polar coordinates of the position vector \mathbf{r} 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
The Fourier transform is^{[2]}
where the \omega are defined by
The overlap integral is
of which the normalization integral is a special case. The starlet in the superscript denotes complex-conjugation.
The kinetic energy integral is
a sum over three overlap integrals already computed above.
The Coulomb repulsion integral can be evaluated using the Fourier representation (see above)
which yields
These are either individually calculated with the law of residues or recursively as proposed by Cruz et al. (1978).^{[3]}
Some quantum chemistry software uses sets of Slater-type functions (STF) analogous to Slater type orbitals, but with variable exponents chosen to minimize the total molecular energy (rather than by Slater's rules as above). The fact that products of two STOs on distinct atoms are more difficult to express than those of Gaussian functions (which give a displaced Gaussian) has led many to expand them in terms of Gaussians.^{[4]}
Analytical ab initio software for poly-atomic molecules has been developed e.g. STOP: a Slater Type Orbital Package in 1996.^{[5]}
SMILES uses analytical expressions when available and Gaussian expansions otherwise. It was first released in 2000.
Various grid integration schemes have been developed, sometimes after analytical work for quadrature (Scrocco). Most famously in the ADF suite of DFT codes.
Basis sets used in computational chemistry
Proton, Electron, Electric current, Electromagnetism, Glass
Quantum mechanics, Periodic table, Niels Bohr, Pauli exclusion principle, Hydrogen-like atom
University of Cambridge, University of Copenhagen, Physics, Harvard University, Mit
Quantum mechanics, Cusp (singularity), Density functional theory, Electron density, Atomic number
Hartree–Fock equation, Basis set (chemistry), Gaussian orbital, Slater-type orbital, Molecular orbital
Energy, Hartree–Fock method, Quantum Monte Carlo, Tight binding, Density functional theory
Slate, Tradesman, Seax, Axe