real basis functions derives from the fact that there are four basis functions in each
integral and (rs|tu) is eightfold degenerate (Eq.5.109); this approximates the
maximum number of these integrals as
Nmax¼m^4 = 8 ð 5 : 160 ÞIn the above calculations the symmetry of water (C2v) and hydrogen peroxide
(C2h) plays an important role in reducing the number of integrals which must actually
be calculated, and modern ab initio programs recognize and utilize symmetry where
it can be used (most molecules lack symmetry, but the small molecules of particular
theoretical interest usually possess it), and are also able to recognize and avoid
calculating integrals below a threshold size. Nevertheless the rapid rise in the number
of two-electron integrals with molecular and basis set size portends problems for ab
initio calculations. An ab initio calculation on aspirin, a fairly small (C 9 H 8 O 4 , 13
heavy atoms) molecule of practical interest, using the 3–21G basis set (Section 5.3.3),
which is the smallest that is usually used, requires 133 basis functions, which from
Eq.5.160could invoke up to 39 million (133^4 /8) two-electron integrals. Clearly, a
modest ab initio calculation could require tens of millions of integrals. Information on
molecular size, symmetry, basis sets and number of integrals is summarized in
Table5.2(the 3–21G basis set is explained inSection 5.3.3). Note that for those
molecules with no symmetry (C 1 ), the number of two-electron integrals calculated
from Eq.5.160is about the same as that actually calculated by Gaussian 92.
There are two problems with so many two-electron integrals: the time needed to
calculate them, and where to store them. Solutions to the first problem are, as
explained, to use Gaussian functions, to utilize symmetry where possible, and to
ignore those integrals that a preliminary check reveals are “vanishing”. The other
problem can be dealt with by storing the integrals in the RAM (the random access
memory, i.e. the electronic memory), storing the integrals on the hard drive, or not
storing them at all, but rather calculating them as they are required. Calculating all
the integrals at the outset and storing them somewhere is calledconventional scf,
being the earlier-used procedure. The latter procedure of calculating only those
two-electron integrals needed at the moment, and recalculating them again when
necessary, is calleddirect scf(presumably using “direct” in the sense of “just now”
or “at the moment”). Calculating all the two-electron the integrals and storing them
in the RAM is the fastest approach, since it requires them to be calculated only
Table 5.2 Molecular size, number of basis functions, and number of two-electron integrals
Basis functions Two-electron integrals
STO-3G 3–21G(*) Fromm^4 /8 From G92a Fromm^4 /8 From G92
HHe+ C 1 v 2 4 2 6 32 55
H 2 OC2v 7 13 300 144 3,570 1,314
H 2 O 2 C2h 12 22 2,592 738 29,282 7,713
H 2 O 2 C$ 1 12 22 2,592 2,774 29,282 28,791
H 2 O 3 C2v 17 31 10,440 3,421 115,440 31475
H 2 O 3 C 1 17 31 10,440 11,046 115,440 107,869
aThe coordinates of one of the atoms was altered slightly to get this unnatural symmetry
5.3 Basis Sets 237