Essentials of Computational Chemistry

7.3 CONFIGURATION INTERACTION 211
clearly, if one is considering a reaction coordinate or a series of isomers, the active space
must be balanced, so any orbital contributing significantly in one calculation should probably
be used in all calculations. While methods to include dynamical correlation after an MCSCF
calculation can help to make up for a less than optimal choice of active space, it is best not
to rely on this phenomenon.
7.2.3 Full Configuration Interaction
Having discussed ways to reduce the scope of the MCSCF problem, it is appropriate to
consider the other limiting case. What if we carry out a CASSCF calculation for all electrons
including all orbitals in the complete active space? Such a calculation is called ‘full
configuration interaction’ or ‘full CI’. Within the choice of basis set, it is the best possible
calculation that can be done, because it considers the contribution of every possible CSF.
Thus, a full CI with an infinite basis set is an ‘exact’ solution of the (non-relativistic,
Born–Oppenheimer, time-independent) Schrödinger equation.
Note that no reoptimization of HF orbitals is required, since the set of all possible CSFs
is ‘complete’. However, that is not much help in a computational efficiency sense, since the
number of CSFs in a full CI can be staggeringly large. The trouble is not the number of
electrons, which is a constant, but the number of basis functions. Returning to our methanol
example above, if we were to use the 6-31G(d) basis set, the total number of basis functions
would be 38. Using Eq. (7.9) to determine the number of CSFs in our (14,38) full CI we
find that we must optimize 2.4 × 10 13 expansion coefficients (!), and this is really a rather
small basis set for chemical purposes.
Thus, full CI calculations with large basis sets are usually carried out for only the smallest
of molecules (it is partly as a result of such calculations that the relative contributions to
basis-set quality of polarization functions vs. decontraction of valence functions, as discussed
in Chapter 6, were discovered). In larger systems, the practical restriction to smaller basis
sets makes full CI calculations less chemically interesting, but such calculations remain
useful to the extent that, as an optimal limit, they permit an evaluation of the quality of
other methodologies for including electron correlation using the same basis set. We turn
now to a consideration of such other methods.
7.3 Configuration Interaction
7.3.1 Single-determinant Reference
If we consider all possible excited configurations that can be generated from the HF determinant,
we have a full CI, but such a calculation is typically too demanding to accomplish.
However, just as we reduced the scope of CAS calculations by using RAS spaces, what if
we were to reduce the CI problem by allowing only a limited number of excitations? How
many should we include? To proceed in evaluating this question, it is helpful to rewrite
Eq. (7.1) using a more descriptive notation, i.e.,
occ.
vir.
= a0HF + a r i r i +
occ.
vir.
a rs
ij rs ij +··· (7.10)
i
r
i