Basis Sets

Ab initio electronic structure computations are almost always carried out numerically using a basis set of orbitals. It is important to choose a basis set large enough to give a good description of the molecular wave function. Typically, the basis functions are centered on the atoms, and so sometimes they are called ``atomic orbitals.'' However, it is important to note that this does not imply that they are actually solutions to the electronic Schrödinger equation for the atom. In modern practice, these atom-centered basis functions are usually chosen to be Gaussian-type orbitals (GTO's), which have the form

$$\Psi_{GTO}(x,y,z) = x^l y^m z^n e^{- \zeta r^2},$$ (1)

where x,y,z are the local (atom-centered) Cartesian coordinates, l,m,n are positive integers which more or less describe the angular momentum of the orbital, and r is the radial distance to the atomic center. Spherical orbitals are usually given by l=m=n=0, a p_x orbital is given by l=1,m=n=0, a d_xy orbital is given by l=m=1, n=0, etc. Unlike hydrogen atom orbitals, GTO's do not have radial nodes; however, radial nodes can be obtained by combining different GTO's. Quite frequently, an atomic basis function is actually a fixed linear combination of GTO's; this is called a contracted Gaussian basis function.

The smallest possible basis set is called the minimal basis set, and it contains one orbital (which may be contracted) for every orbital we usually think of for an atom (including unoccupied orbitals). For example, hydrogen has just one orbital, but carbon has 5 (1s, 2s, 2p_x, 2p_y, and 2p_z) even though one of the p orbitals for carbon atom will be unoccupied. The STO-3G basis is a very well-known minimal basis set which contracts 3 Gaussian functions to approximate the more accurate (but more difficult to compute) Slater type orbitals. Although a contracted GTO might give a good approximation to an atomic orbital, it lacks any flexibility to expand or shrink in the presence of other atoms in a molecule. Hence, a minimal basis set such as STO-3G is not capable of giving highly accurate results.

The solution is to add extra basis functions beyond the minimum number required to describe each atom. Then, the Hartree-Fock procedure (below) can weight each atomic orbital basis function more or less to get a better description of the wave function. If we have twice as many basis functions as in a minimum basis, this is called a ``double zeta'' basis set (the zeta, $\zeta$, comes from the exponent in the GTO). Hence, a double-zeta basis set for hydrogen would have two functions, and a true double-zeta basis set for carbon would have 10 functions. However, sometimes people ``cheat'' and use only a single orbital for the core (1s), giving 9 functions for carbon. Such basis sets are said to be ``double-zeta in the valence'' space; they are also called ``split-valence'' basis sets. Double-zeta basis sets are often denoted DZ. Often additional flexibility is built in by adding higher-angular momentum basis functions. Since the highest angular momentum orbital for carbon is a p orbital, the ``polarization'' of the atom can be described by adding a set of d functions. A hydrogen atom would use a set of 3 p functions as polarization functions. A double-zeta plus polarization basis set might be designated DZP. The most famous example of a split-valence double-zeta plus polarization basis set is Pople's so-called 6-31G* basis. This obscure notation means that the core orbital is described by a contraction of 6 Gaussian orbitals, while the valence is described by two orbitals, one made of a contraction of 3 Gaussians, and one a single Gaussian function. Just to confuse you, the star (*) indicates polarization functions on non-hydrogen atoms. If polarization was added to hydrogen atoms also, this basis would be called 6-31G**. The confusing nature of this nomenclature has caused some chemists to start switching to slightly improved notation such as 6-31G(d,p), where the polarization functions are listed explicitly.

This lab will not investigate the nuances of basis set design, but you need to have a basic grasp of what basis sets are, since you'll be using them!