Essentials of Computational Chemistry

6.3 KEY TECHNICAL AND PRACTICAL POINTS OF HARTREE–FOCK THEORY 191
some point the linear scaling algorithm will become more efficient given a system of large
enough size. In any case, because there are several features present in electronic structure
programs that allow some control over the efficiency of the calculation, we discuss here the
most common ones, recapitulating a few that have already been mentioned above.
First, there is the issue of how to go about computing the four-index integrals, which
are responsible for the formal N 4 scaling. One might imagine that the most straightforward
approach is to compute every single one and, as it is computed, write it to storage – then, as
the Fock matrix is assembled element by element, call back the computed values whenever
they are required (most of the integrals are required several times). In practice, however, this
approach is only useful when the time required to write to and read from storage is very, very
fast. Otherwise, modern processors can actually recompute the integral from scratch faster
than modern hardware can recover the previously computed value from, say, disk storage.
The process of computing each integral as it is needed rather than trying to store them all
is called ‘direct SCF’. Only when the storage of all of the integrals can be accomplished in
memory itself (i.e., not on an external storage device) is the access time sufficiently fast that
the ‘traditional’ method is to be preferred over direct SCF.
As the size of the system increases, it becomes possible to take advantage of other features
of the electronic structure that further improve the efficiency of direct SCF. For instance, it
is possible to estimate upper bounds for four-index integrals reasonably efficiently, and if
the upper bound is so small that the integral can make no significant contribution, there is
no point evaluating it more accurately than assigning it to be zero. Such small integrals are
legion in large systems, since if each of the four basis functions is distantly separated from
all of the others simple overlap arguments make it clear that the integral cannot be very
large.
With very, very large systems, fast-multipole methods analogous to those described in
Section 2.4.2 can be used to reduce the scaling of Coulomb integral evaluation to linear
(see, for instance, Strain, Scuseria, and Frisch 1996; Challacombe and Schwegler 1997), and
linear methods to evaluate the exchange integrals have also been promulgated (Ochensfeld,
White, and Head-Gordon 1998). At this point, the bottleneck in HF calculations becomes
diagonalization of the Fock matrix (a step having formal N 3 scaling), and early efforts to
reduce the scaling of this step have also appeared (Millam and Scuseria 1997).
As already described above, efficiency in converging the SCF for systems with large
basis sets can be enhanced by using as an initial guess the converged wave function from a
different calculation, one using either a smaller basis set or a less negative charge. This same
philosophy can be applied to geometry optimization, which can be quite time-consuming for
very large calculations. It is often very helpful to optimize the geometry first at a more
efficient level of theory. This is true not just because the geometry optimized with the lower
level is probably a good place to start for the higher level, but also because typically one
can compute the force constants at the lower level and use them as an initial guess for the
higher level Hessian matrix that will be much better than the typical guess generated by
the optimizing algorithm. As described in Section 2.4.1, the availability of a good Hessian
matrix can make an enormous amount of difference in how quickly a geometry optimization
can be induced to converge.