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Concept
Gaussian orbital
Summary
In computational chemistry and molecular physics, Gaussian orbitals (also known as Gaussian type orbitals, GTOs or Gaussians) are functions used as atomic orbitals in the LCAO method for the representation of electron orbitals in molecules and numerous properties that depend on these.
The use of Gaussian orbitals in electronic structure theory (instead of the more physical Slater-type orbitals) was first proposed by Boys in 1950. The principal reason for the use of Gaussian basis functions in molecular quantum chemical calculations is the 'Gaussian Product Theorem', which guarantees that the product of two GTOs centered on two different atoms is a finite sum of Gaussians centered on a point along the axis connecting them. In this manner, four-center integrals can be reduced to finite sums of two-center integrals, and in a next step to finite sums of one-center integrals. The speedup by 4-5 orders of magnitude compared to Slater orbitals outweighs the extra cost entailed by the larger number of basis functions generally required in a Gaussian calculation.
For reasons of convenience, many quantum chemistry programs work in a basis of Cartesian Gaussians even when spherical Gaussians are requested, as integral evaluation is much easier in the cartesian basis, and the spherical functions can be simply expressed using the cartesian functions.
The Gaussian basis functions obey the usual radial-angular decomposition
where is a spherical harmonic, and are the angular momentum and its component, and are spherical coordinates.
While for Slater orbitals the radial part is
being a normalization constant, for Gaussian primitives the radial part is
where is the normalization constant corresponding to the Gaussian.
The normalization condition which determines or is
which in general does not impose orthogonality in .
Because an individual primitive Gaussian function gives a rather poor description for the electronic wave function near the nucleus, Gaussian basis sets are almost always contracted:
where is the contraction coefficient for the primitive with exponent .
Official source
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