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## Comparison: 1s-Slater-type orbital and 1s-Gauss-type orbital

We compare a normalized 1s Slater function (STO) for an H atom with$n=1$,$μ=1$ and the ball function$Y00=121π$ :

$ϕ1sSTO=1π⋅e−r$

with a normalized primitive Gaussian function (GTO) with $l=0, m=0, n=0$:

$ϕGTO=(2β1π)3/4⋅e−β1r2 ,$

being in both cases $r$ the distance from the electron to the nucleus (proton) is: $r=|re-R.H|$. The following figure shows the 1s Slater function on the left and the primitive Gaussian function on the right. The coefficient of the Gaussian function $β=0,270950$ one obtains with maximum overlap of the 1s-STO and the 1s-GTO function:$S.=∫drϕ1sSTOϕ1sGTO=Max.$

The exponent $β$ is a positive real number that determines how diffuse the Gaussian function is: a small one $β$ corresponds to a rapidly falling, i.e. not very diffuse, function; a big$β$ causes a slowly decreasing, i.e. strongly diffuse Gaussian function.

The specified Slater function represents the exact 1s function for the hydrogen atom. It shows a point. cusp) at $r=0$, while the Gaussian function has a maximum there and is flat. In addition, the Slater function works with large $r$ much slower towards zero than the Gaussian function. Slater-type orbitals describe the behavior of molecular orbitals better than Gauss-type orbitals. The reason why inquant-chemical calculations are used almost exclusively orbitals of the Gaussian type is that two-electron integrals can be calculated much more efficiently with primitive Gaussian functions than with Slater functions.

In practice, the Slater-type orbitals are approximated by Gaussian functions: in the above example the coefficient of the Gaussian function was chosen so that there was maximum overlap with the Slater function. This approach to Slater functions works better through linear combinations of several primitive Gaussian functions, i.e. when using contracted Gaussian functions (CGTOs). The following page gives an example of an approximation of a Slater function by a linear combination (contraction) of three primitive Gaussian functions.