Essentials of Computational Chemistry

182 6 AB INITIO HARTREE–FOCK MO THEORY
usually easier to converge a small-basis-set HF calculation than a larger one. This suggests
a method for bootstrapping one’s way to the convergence of a large-basis-set calculation:
First, obtain a wave function from a minimal basis set (e.g., STO-3G), then use that as an
initial guess for a calculation with a small split-valence basis set (e.g., 3-21G), and repeat
this process with increasingly larger basis sets until the target is reached. Because of the
exponential scaling, the early calculations typically represent a negligible time investment,
especially if they are saving steps in a slowly converging SCF for the full-sized basis set by
providing a more accurate initial guess.
The above process has another possible utility that is associated with the molecular geometry.
Often when an SCF is difficult to converge, the problem is that the molecular structure
is very bad. If that is the case, there can be a very small separation between the highest occupied
MO (HOMO) and the lowest unoccupied MO (LUMO). Such small separations wreak
havoc on the SCF process, because it is possible that occupation of either orbital could lead
to HF eigenfunctions of similar energy. In that case, the characters of the two orbitals are very
sensitive to all the remaining occupied orbitals, which generate the static potential felt by the
highest energy electrons, and their coefficients can undergo large changes that fail to converge
(an issue of non-dynamical electron correlation, see Section 7.1). Optimizing the geometry
at a low level of theory, where the wave function can be coaxed to converge, is typically an
efficient way to overcome this problem. Some care must be exercised, however, in systems
where the lowest levels of theory may not be reliable for molecular geometries. As a general
rule, however, visualization of the structure, and some thoughtful analysis of it by comparison
to whatever analogs or prior calculations may be available, is nearly always worth the effort.
Very complete basis sets, or those with many diffuse functions, pose some of the worst
problems for SCF convergence because of near-linear dependencies amongst the basis
functions. That is, some basis functions may be fairly well described as linear combinations
of other basis functions. This is most readily appreciated by considering two very diffuse s
orbitals on adjacent atoms; if they have maxima in their radial density at 40 ˚A but the two
atoms are only 1.5 ˚A apart, the two basis functions are really almost indistinguishable from
one another throughout most of space. If a basis set has a true linear dependence, then it is
necessarily impossible to assure orthogonality of all of the MOs (a division by zero occurs at
a particular point of the SCF process), so very near-linear dependence can lead to numerical
instabilities. Thus, it is again important to have a good guess. In a case like this, sometimes
it is useful not only to carry out bootstrap calculations in terms of basis sets, but in terms of
electrons. Thus, if one is interested in an anion, for instance, one can first try to converge
a large-basis-set wave function for the neutral (or the cation), to get a good estimate of the
more compact MOs, and then import that wave function as a guess for the anionic system,
trying thereby to reduce the impact of possible numerical instabilities.
6.3.2 Symmetry
The presence of symmetry in a molecule can be used to great advantage in electronic structure
calculations, although some care is required to avoid possible pitfalls that are simultaneously
introduced (Appendix B provides a brief overview of nomenclature (e.g., the term “irrep”,