physical behaviour would seem to have a conceptual flaw, and in fact lack of size-
consistency also places limits on the utility of the method. For instance, in trying to
study the hydrogen-bonded water dimer we would not be able to equate the
decrease in energy (compared to twice the energy of one molecule) with stabiliza-
tion due to hydrogen bonding, and it is unclear how we could computationally turn
off hydrogen bonding and evaluate the size-consistency error separately (actually,
there is a separate problem, basis set superposition error – see below – with species
like the water dimer, but this source of error can be dealt with). It might seem that
any computational method must be size-consistent (why shouldn’t the energy of a
large-separation (M)ncome out atntimes that of M?). However, it is not hard to
show that CI is not size-consistent unless Eqs.5.168include all possible determi-
nants, i.e. unless it isfullCI. Consider a CISD calculation with a very large
(“infinite”) basis set on two helium atoms which are separated by a large (“infinite”;
say ca. 20 A ̊) distance, and are therefore non-interacting. Note that although helium
atoms do not form covalent He 2 molecules, at short distances theydointeract to
form van der Waals molecules. The wavefunction for this four-electron system will
contain, besides the HF determinant, only determinants with single and double
excitations (because we are using CISD). Lacking the triple and quadruple excita-
tions which are possible in principle for a four-electron system, it is not a full CI
calculation, and so it will not yield the exact energy of our noninteracting He–He
system, which logically must be twice that of one helium atom; instead it will yield
a higher energy. Now, a CISD calculation with an infinite basis set on asingleHe
atomwillgive the exact wavefunction, and thus the exact energy of the atom
(because only single and double promotionsare possiblefor a two-electron system,
this is a full CI calculation). Thus in this CISD calculation, the energy of the
infinitely-separated He–He system is not, as it “should” be, twice the energy of a
single He atom. This conclusion holds for any CI calculation which does not confer
full “mobility” on all the electrons.
5.4.3.2 Variational Behavior
The other factor to be discussed in connection with post-HF calculations is whether
a particular method isvariational. A method is variational (see the variation
theorem,Section 5.2.3.3) if any energy calculated from it is not less than the true
energy of the electronic state and system in question, i.e. if the calculated energy is
anupper boundto the true energy. Using a variational method, as the basis set size
is increased we get lower and lower energies, levelling off above the true energy (or
at the true energy in the unlikely case that our method treats perfectly electron
correlation, relativistic effects, and any other minor effects). Figure5.18shows that
the calculated energy of H 2 using the HF method approaches a limit (#1.133 h)
with increasingly large basis sets. The calculated energy can be lowered by using a
correlated method and an adequate basis: full CI with the very big 6–311þþG
(3df,3p2d) basis gives#1.17288 h, only 4.0 kJ mol#^1 (small compared with the
5.4 Post-Hartree–Fock Calculations: Electron Correlation 277