Computational Chemistry

(Table5.3) as the next bigger widely-used basis, the 6–31G (below) and is much
less demanding of computer power, the 3–21G()basis set has been used as a
kind of workhorse for relatively big molecules; see for example a study using it
for geometry optimization investigations of pericyclic reactions [ 47 ]. As long ago
as 1988 the somewhat similar but now obsolete 4–21G basis was used, with the
3–21G() basis specifically on sulfur, for geometry optimization of a protein
(crambin) with 46 amino acid residues and 642 atoms. This represented 3,597
basis functions, and the job took 260 days [ 48 ]. It seems likely that now it would
be shorter by a factor of 10–20, on an inexpensive desktop machine. More recently
novel approaches, such as dividing a large molecule into fragments, have been
explored [ 49 ]. The general problem of optimizing large molecules has been
reviewed [ 50 ]. Even where geometry optimizations with larger bases are practical,
a survey of the problem with the 3–21G()basis is sometimes useful (it is HF/
3–21G(*)geometriesrather thanrelative energieswhich are good; consistently
getting good relative energies is a more challenging problem – seeSection 5.5.2).

5.3.3.3 6–31G*

This is a split valence basis set with polarization functions (these terms were
explained in connection with the 3–21G()basis set, above). The valence shell of
each atom is split into an inner part composed of three Gaussians and an outer part
composed of one Gaussian (hence “31”), while the core orbitals are each represented
by one basis function, each composed of six Gaussians (“6”). The polarization
functions () are present on “heavy atoms” – those beyond helium. Thus H and He

.

+

.

+

+

.

+

atomic

nucleus

p function d function

C 1 f 1 C 2 f 2 C 1 f 1 + C 2 f 2

Weighted sum of

p function and d function.

The p function has been

shifted (polarized) toward

the right by the d function.

Fig. 5.16 One basis function can be used to shift another in a given direction (to polarize it). In
minimizing the energy, the program adjusts the relative contributions of the two functions to shift
the electron density where it is needed to get the minimum energy.pFunctions are also commonly
used to polarize thesfunctions on hydrogen atoms, but the main use of polarization functions is the
utilization ofdfunctions on “heavy” atoms (atoms other than H and He)

246 5 Ab initio Calculations