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Chemistry · Glossary

What is Basis set, minimal basis, Slater- and Gaussian-type orbitals?

Also known as: basis set · minimal basis set · Slater-type orbital · Gaussian-type orbital · contracted Gaussian function

Definition 3.8 University Chemistry — Year 3 · Chapter 3 — Computational Chemistry: Hartree–Fock and DFT

A basis set is the set of fixed functions χi\chi_i, centred on the atoms, from which molecular orbitals are built. A minimal basis set has one function per occupied atomic orbital (one 1s1s for H, five for C). A Slater-type orbital (STO) has the radial form rn−1e−ζrr^{n-1}\eu^{-\zeta r}; a Gaussian-type orbital (GTO) has the form e−αr2\eu^{-\alpha r^2} times a polynomial in xx, yy, zz. A contracted Gaussian function is a fixed combination ∑kdk e−αkr2\sum_kd_k\,\eu^{-\alpha_kr^2} of primitive Gaussians.

Examples

Example 3.9 (The STO-3G function of hydrogen)

The hydrogen 1s1s function of STO-3G is 0.15433 g(3.42525)+0.53533 g(0.62391)+0.44463 g(0.16886)0.15433\,g(3.42525) + 0.53533\, g(0.62391) + 0.44463\,g(0.16886), where g(α)g(\alpha) is the normalised Gaussian (2α/π)3/4e−αr2(2\alpha/\pi)^{3/4}\eu^{-\alpha r^2} (exponents in a0−2a_0^{-2}). It imitates a Slater function of exponent ζ=1.24\zeta = 1.24 — a hydrogen atom slightly compressed, as it is in molecules. Its overlap with the exact 1s1s function of the same exponent is 0.99980.9998; only the cusp at the nucleus and the far tail are missed (figure below).

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