Gaussian orbital

Mathematical form

The Gaussian basis functions obey the usual radial-angular decomposition

${\displaystyle \mathrm{\Phi }(\mathbf{r})=R_l(r)Y_{lm}(\theta ,\varphi )}$,

where ${\displaystyle Y_{lm}(\theta ,\varphi )}$ is a spherical harmonic, ${\displaystyle l}$ and ${\displaystyle m}$ are the angular momentum and its ${\displaystyle z}$ component, and ${\displaystyle r,\theta ,\varphi }$ are spherical coordinates.

While for Slater orbitals the radial part is

${\displaystyle R_l(r)=A(l,\alpha )r^l{e}^{-\alpha r},}$

${\displaystyle A(l,\alpha )}$ being a normalization constant, for Gaussian primitives the radial part is

${\displaystyle R_l(r)=B(l,\alpha )r^l{e}^{-\alpha r^2},}$

where ${\displaystyle B(l,\alpha )}$ is the normalization constant corresponding to the Gaussian.

The normalization condition which determines ${\displaystyle A(l,\alpha )}$ or ${\displaystyle B(l,\alpha )}$ is

${\displaystyle {\int }_0^{\mathrm{\infty }}\mathrm{d}r{r}^2{\left|R_l(r)\right|}^2=1}$

which in general does not impose orthogonality in ${\displaystyle l}$.

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:

${\displaystyle R_l(r)=r^l\sum _{p=1}^P{c}_{p}B(l,{\alpha }_p)\exp (-{\alpha }_p{r}^2)}$,

where ${\displaystyle c_p}$ is the contraction coefficient for the primitive with exponent ${\displaystyle {\alpha }_p}$. The coefficients are given with respect to normalized primitives, because coefficients for unnormalized primitives would differ by many orders of magnitude. The exponents are reported in atomic units. There is a large library of published Gaussian basis sets optimized for a variety of criteria available at the Basis Set Exchange portal.

Cartesian coordinates

In Cartesian coordinates, Gaussian-type orbitals can be written in terms of exponential factors in the ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$ directions as well as an exponential factor ${\displaystyle \alpha }$ controlling the width of the orbital. The expression for a Cartesian Gaussian-type orbital, with the appropriate normalization coefficient is

${\displaystyle \mathrm{\Phi }(x,y,z;\alpha ,i,j,k)={\left(\frac{2\alpha }{\pi }\right)}^{3/4}{\left[\frac{(8\alpha {)}^{i+j+k}i!j!k!}{(2i)!(2j)!(2k)!}\right]}^{1/2}{x}^i{y}^j{z}^k{e}^{-\alpha (x^2+y^2+z^2)}}$

In the above expression, ${\displaystyle i}$, ${\displaystyle j}$, and ${\displaystyle k}$ must be integers. If ${\displaystyle i+j+k=0}$, then the orbital has spherical symmetry and is considered an s-type GTO. If ${\displaystyle i+j+k=1}$, the GTO possesses axial symmetry along one axis and is considered a p-type GTO. When ${\displaystyle i+j+k=2}$, 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.

Molecular integrals

Taketa et al. (1966) presented the necessary mathematical equations for obtaining matrix elements in the Gaussian basis. 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, as this simplifies the equations. McMurchie and Davidson (1978) introduced recursion relations, which greatly reduces the amount of calculations. Pople and Hehre (1978) developed a local coordinate method. Obara and Saika introduced efficient recursion relations in 1985, which was followed by the development of other important recurrence relations. Gill and Pople (1990) introduced a 'PRISM' algorithm which allowed efficient use of 20 different calculation paths.

The POLYATOM System

The POLYATOM System was the first package for ab initio calculations using Gaussian orbitals that was applied to a wide variety of molecules. 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, Malcolm Harrison, Jules Moskowitz and Brian Sutcliffe.