Essentials of Computational Chemistry

6.3 KEY TECHNICAL AND PRACTICAL POINTS OF HARTREE–FOCK THEORY 181
6.3.1 SCF Convergence
As noted in Chapter 4, there is never any guarantee that the SCF process will actually
converge to a stable solution. A fairly common problem is so-called ‘SCF oscillation’. This
occurs when a particular density matrix, call it P (a) , is used to construct a Fock matrix F (a)
(and thus the secular determinant), diagonalization of which permits the construction of an
updated density matrix P (b) ; this is a general description of any step in the SCF cycle. In
the oscillatory case, however, the diagonalization of the Fock matrix created using P (b) (i.e.,
F (b) ) gives a density matrix indistinguishable from P (a) . Thus, the SCF simply bounces back
and forth from P (a) to P (b) and never converges. This behavior can be recognized easily by
looking at the SCF energy for each step, which itself bounces back and forth between the
two discrete values associated with the two different unconverged wave functions defined
by P (a) and P (b) .
In more pathological cases, the SCF behaves even more badly, with large changes occurring
in the density matrix at every step. Again, observation of the energies associated with
each step is diagnostic for this problem; they are observed to vary widely and seemingly
randomly. Such behavior is not uncommon for the first three or four steps of a typical SCF,
but usually beyond this point there is a ‘zeroing-in’ process that leads to convergence.
In the abstract sense, converging the SCF equations is a problem in applied mathematics,
and many algorithms have been developed for this process. While the technical details are
not presented here, the process is quite analogous to the process of finding a minimum
on a PES as described in Chapter 2. In the SCF problem, instead of a space of molecular
coordinates we operate in a space of orbital coefficients (so-called ‘Fock space’), and there
are certain constraints beyond the purely energetic ones, but many of the search strategies
are analogous. Similarly analogous is the degree to which they tend to balance speed and
stability. Usually the default optimizer in a given program is the fastest one available, while
other methods (e.g., quadratically convergent methods) typically take more steps to converge
but are less likely to suffer from oscillation or other problems. Thus, one option for dealing
with a system where convergence proves difficult is simply to run through all the different
convergence schemes offered by the electronic structure package and hope that one proves
sufficiently robust.
In general, however, it is more efficient to solve the problem using chemistry rather than
mathematics. If the SCF equations are failing to converge, the problem lies in the initial guess
(this is, of course, something of a truism, for if you were to guess the proper eigenfunction,
obviously there would be no problem with convergence). Most programs use as their default
option a semiempirical method to generate a guess wave function, e.g., EHT or INDO. The
resulting wave function (remember that a wave function is simply the list of coefficients
describing how the basis functions are put together to form the occupied MOs) is then used
to construct a guess for the HF calculation by mapping coefficients from the basis set of the
semiempirical method to the basis set for the HF calculation.
When the HF basis set is minimal, this is fairly simple (there is a one-to-one correspondence
in basis functions) but when it is larger, some algorithmic choices are made about
how to carry out the mapping (e.g., always map to the tightest function or map based on
overlap between the semiempirical STO and the large-basis contracted GTO). Thus, it is