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Given two computational programs, one using a Gaussian basis, and the other using Slater basis, what are the practical differences, advantages and disadvantages for each choice ?

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As a general rule, Slater basis elements are closer to the actual solution, and therefore fewer of them are needed.

The advantage of a Gaussian basis is the Gaussian Product Theorem, which allows products of gaussians at different centers to be easily integrated. This allows for a significant speed up when attempting to converge around multiple nearby centers.

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Advantage of the Gaussian basis is that you can use the Gaussian Product Theorem to simplify the two electron integrals at different centers. However, Gaussians ($e^{-\alpha r^2}$) don't capture the cusp of the wavefunction nor the exponential decay ($e^{-r}$) naturally, so one needs more Gaussians for the given accuracy.

Advantage of the Slater type orbitals (STO) $r^n e^{-\zeta r}$ is that they capture both the cusp and the exponential decay, but the disadvantage is that it is harder to calculate the two electron integrals at different centers (for atomic calculations this isn't a problem and so many atomic codes use STO).

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From a results point of view, for most things, it shouldn't really matter which you use - basis sets are generally constructed and labeled in such a way that for example a TZVP basis set will give comparable results whether it's using GTOs or STOs.

The cases where they're more likely to give different results will be properties that depend heavily on non-valence regions, for example, NMR data, which depend heavily upon the electronic density at and around the nucleus (for more info on this, you'll want to look up Fermi contact terms), and properties which may depend on the longer-distance decay of the density. STOs model the density at the nucleus more accurately assuming a point nucleus (which some programs don't - Gaussian for example uses a Gaussian for the nucleus by default), and neither type gets long-distance behaviour exactly right, as far as I know - STOs decay too slowly, GTOs too quickly.

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