7. Basis sets
A basis set is a function or a set of function that numerically represents molecular orbitals which in
turn combine to approximate the total electronic wavefunction. The two general categories of basis
1. Minimal basis set: such basis set contain minimum number of basis function required for each
atom in a system. Such basis sets describes only basic aspects of orbitals. Minimal basis sets
use fixed-size type atomic orbitals.
Orbitals can be numerically represented in two ways. They are called as slater type orbitals (STO)
and Gaussian type orbitals (GTO).
STO = N*e
GTO = N*e
where N is the normalisation constant, α is the orbital exponent and r is the radius.
The only difference between STO and GTO is the power of r. Though STO are seen to be more
accurate than a similar number of GTO, they are not mathematically convenient to use. Hence GTOs
are preferred over STO even though large number of functions are required. All basis set equations
in the form STO-NG (where N represents the number of GTOs combined to approximate the STO)
are considered to be minimal basis sets.
2. Extended basis set: They come up with much more detailed description of the orbitals and
consider higher orbitals of the molecule and account for the shape and size of molecular charge
distribution. There are different types of basis sets coming under the category of extended basis
set. Some of them are:
i. Split valence basis set: The first thing that can be done to increase the size of the basis
set so as to get better results is to increase the number of basis function per atom.
Double split valence basis set such as 6-31G, 3-21G have two basis function for each
valence orbital. For example H and C atom are represented as
H: 1s, 1sꞌ
C: 1s, 2s, 2sꞌ, 2px, 2py, 2pz, 2pxꞌ, 2pyꞌ, 2pzꞌ
Where the prime and unprimed orbitals differ in size.
ii. Polarized basis set: Split valence basis set allows change in orbital size but no change
can be made in its shape. Polarized basis set can modify shape as well by adding orbitals
with angular momentum more than required for the atom’s ground state description.
For instance, polarized basis set add d function to C atom, f functions to transition
metals and sometimes p functions to H atom. An example of a polarized basis set is 6-
31G(d) or written as 6-31G*. It indicates 6-31G basis set with d function added to heavy