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MOLECULAR ORBITAL THEORY CALCULATIONS 205

8.2.5. Basis Set
Both ab initio and DFT methods use sets of mathematical functions to represent
the atomic orbitals. These are called the basis set. These mathematical func-
tions are themselves made from a combination of simpler mathematical functions
called primitives. Increasing the number of primitive functions and including
contributions from valence orbitals imposes less restriction on the location of
the electron and therefore more accurately models the exact molecular orbitals,
but correspondingly increases the computational cost. The molecular orbitals are
approximated as linear combinations of the basis functions. In the ab initio meth-
ods, a Gaussian-type atomic function is used as the basis function, which has the
general form:
n m l −αr
g(α, r) = cx y z e 2 (8.5)

r
where vector is the position of the electron, which is composed of coordinates
x, y,and z,and α is a constant that determines the size (radial extent) of the
function. The Gaussian function e −αr 2 is multiplied by powers (possibly zero) of
x, y,and z and is normalized by constant c,sothat:

2
g = 1 (8.6)
allspace
Linear combinations of primitive Gaussian functions shown in Eq. 8.5 are used
to form the basis functions.
A set of standard basis sets has been devised to increase the comparability
between researchers and to simplify the nomenclature when describing the model
chemistry: STO-3G, 3-21G, 6-311G, 6-311G(d,p), 6-311 + G(d,p)... in the
order of increasingly large basis sets. The smallest of these, STO-3G, is the
abbreviated name for three Gaussian primitives (3G) to model a Slater-type orbital
(STO). When each orbital is represented by two or more sizes of basis functions,
we have split-valence basis set. The 3-21G uses two sizes of Gaussian primitives
to represent each orbital, and a 6-311G uses three sizes of Gaussian primitives
to represent each orbital. To allow orbitals to change shape, we can include
contributions from orbitals of higher angular momentum, for instance, include
one of more d orbitals on carbon. The notation for this involves adding a letter
for the orbital type to the end of the abbreviation, for example, 6-31G(2d,p) adds
two d orbitals to heavy atoms and one p orbital to hydrogen atoms. A further
development in basis has been to add very large versions of orbitals to the basis
set, called diffuse functions. This is denoted by adding a + to the abbreviation
before the G, so 6-31+G adds diffuse functions to heavy atoms and 6-31++G
adds diffuse functions to both heavy and hydrogen atoms. The best choice of
basis set is largely dependent on the chemistry being studied.

8.2.6. Effective Core Potentials
Relativistic effects must be considered in the applications of ab initio molecular
orbital calculations for the heavier elements; they have a significant influence on