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.
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.
[3] [4] The Gaussian basis functions obey the usual radial-angular decomposition where
being a normalization constant, for Gaussian primitives the radial part is where
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
In Cartesian coordinates, Gaussian-type orbitals can be written in terms of exponential factors in the
, then the orbital has spherical symmetry and is considered an s-type GTO.
, there are six possible GTOs that may be constructed; this is one more than the five canonical d orbital functions for a given angular quantum number.
To address this, a linear combination of two d-type GTOs can be used to reproduce a canonical d function.
Similarly, there exist 10 f-type GTOs, but only 7 canonical f orbital functions; this pattern continues for higher angular quantum numbers.
[5] Taketa et al. (1966) presented the necessary mathematical equations for obtaining matrix elements in the Gaussian basis.
[6] Since then much work has been done to speed up the evaluation of these integrals which are the slowest part of many quantum chemical calculations.
Živković and Maksić (1968) suggested using Hermite Gaussian functions,[7] as this simplifies the equations.
McMurchie and Davidson (1978) introduced recursion relations,[8] which greatly reduces the amount of calculations.
Pople and Hehre (1978) developed a local coordinate method.
Gill and Pople (1990) introduced a 'PRISM' algorithm which allowed efficient use of 20 different calculation paths.
[11] The POLYATOM System[12] was the first package for ab initio calculations using Gaussian orbitals that was applied to a wide variety of molecules.
[13] It was developed in Slater's Solid State and Molecular Theory Group (SSMTG) at MIT using the resources of the Cooperative Computing Laboratory.
The mathematical infrastructure and operational software were developed by Imre Csizmadia,[14] Malcolm Harrison,[15] Jules Moskowitz[16] and Brian Sutcliffe.