Essentials of Computational Chemistry

132 5 SEMIEMPIRICAL IMPLEMENTATIONS OF MO THEORY
of an HF calculation, in terms of computational resources, is the assembly of the twoelectron
(also called four-index) integrals, i.e., the J and K integrals appearing in the
Fock matrix elements defined by Eq. (4.54). Not only is numerical solution of the integrals
for an arbitrary basis set arduous, but there are so many of them (formally N 4 where
N is the number of basis functions). One way to save time would be to estimate their
value accurately in an apriori fashion, so that no numerical integration need be undertaken.
For which integrals is it easiest to make such an estimation? To answer that question, it
is helpful to keep in mind the intuitive meaning of the integrals. Coulomb integrals measure
the repulsion between two electrons in regions of space defined by the basis functions. It
seems clear, then, that when the basis functions in the integral for one electron are very
far from the basis functions for the other, the value of that integral will approach zero (the
same holds true for the one-electron integrals describing nuclear attraction, i.e., if the basis
functions for the electron are very far from the nucleus the attraction will go to zero, but
these integrals are much less computationally demanding to solve). In a large molecule, then,
one might be able to avoid the calculation of a very large number of integrals simply by
assuming them to be zero, and one would still have a reasonable expectation of obtaining a
Hartree–Fock energy close to that that would be obtained from a full calculation.
Such an approximation is what we might call a numerical approximation. That is, it
introduces error to the extent that values employed are not exact, but the calculation can be
converged to arbitrary accuracy by tightening the criteria for employing the approximation,
e.g., in the case of setting certain two-electron integrals to zero, the threshold could be the
average inter-basis-function distance, so that in the limit of choosing a distance of infinity, one
recovers exact HF theory. Other approximations in semiempirical theory, however, are guided
by a slightly different motivation, and these approximations might be well referred to as
‘chemically virtuous approximations’. It is important to keep in mind that HF wave functions
for systems having two or more electrons are not eigenfunctions of the corresponding nonrelativistic
Schrödinger equations. Because of the SCF approximation for how each electron
interacts with all of the others, some electronic correlation is ignored, and the HF energy is
necessarily higher than the exact energy.
How important is the correlation energy? Let us consider a very simple system: the helium
atom. The energy of this two-electron system in the HF limit (i.e., converged with respect to
basis-set size for the number of digits reported) is −2.861 68 a.u. (Clementi and Roetti 1974).
The exact energy for the helium atom, on the other hand, is −2.903 72 a.u. (Pekeris 1959).
The difference is 0.042 04 a.u., which is about 26 kcal mol −1 . Needless to say, as systems
increase in size, greater numbers of electrons give rise to considerably larger correlation
energies – hundreds or thousands of kcal mol −1 for moderately sized organic and inorganic
molecules.
At first glance, this is a terrifying observation. At room temperature (298 K), it requires a
change of 1.4 kcal mol −1 in a free energy of reaction to change an equilibrium constant by an
order of magnitude. Similarly, a change of 1.4 kcal mol −1 in a rate-determining free energy
of activation will change the rate of a chemical reaction by an order of magnitude. Thus,
chemists typically would prefer theoretical accuracies to be no worse than 1.4 kcal mol −1 ,