Essentials of Computational Chemistry

370 10 THERMODYNAMIC PROPERTIES
the elemental standard states are not, and thus the combinations of the last two terms on
the r.h.s.s of Eqs. (10.32) and (10.33) are not equal. It is probably simplest to see this
by considering the example of molecular hydrogen. Its translational enthalpy is given by
Eq. (10.16) as ( 3
2 )RT . So, at 0 K it has no translational enthalpy and at 298 K it has
roughly 0.9 kcal mol−1 of such enthalpy. Analogous changes are associated with rotation
and vibration. However, molecular hydrogen is the elemental standard state, so it is defined
experimentally to have a zero heat of formation at whatever temperature. Thus, when we
compute the 298 K thermal contributions to the enthalpy of two H atoms, we determine
from theory an absolute translational contribution of 3RT (again from Eq. (10.16) now
applied to two separate particles), but experimentally we would only obtain 3
2RT for this
term, since the reference elemental standard state also has increased absolute enthalpy
at 298 K.
Having discussed in detail how to go about computing heats and free energies of formation,
we should now consider how useful typical electronic-structure methods are for that purpose.
The somewhat disappointing answer is that most single levels of theory are disastrously
bad, with the problem lying primarily in the computation of E between the molecule
and its constituent atoms (the leftmost vertical line in Figure 10.1). As there is vastly more
correlation energy in a molecule, with its collection of bonded pairs of electrons, than there is
in a collection of atoms, and as practically affordable correlated electronic-structure methods
capture at best perhaps 70–90% of the correlation energy, the differential error can be very
large. Only with very, very small molecules is it possible to apply a single sufficiently high
level of theory to accurately compute heats and free energies of formation ab initio. However,
a number of different approaches employing varying degrees of semiempiricism have been
promulgated to improve on this situation.
10.4.2 Parametric Improvement
In Section 7.7, parametric methods for improving the quality of correlated electronicstructure
calculations were discussed in detail. Similarly, in Section 8.4.3, the mild
parameterization of density functional methods to give maximal accuracy was described.
Given that background, and the substantial data presented in those earlier chapters,
this section will only recapitulate in a rough categorical fashion the various approaches
whose development was motivated by a desire to compute more accurate thermochemical
quantities.
Most attention has been focused on the computation of Eelec, because even fairly modest
levels of theory can compute molecular geometries and vibrational frequencies sufficiently
accurately to give good ZPVEs and thermal contributions, particularly if the frequencies are
scaled by an appropriate factor (see Section 9.3). The simplest approach to improved Eelec
estimation is to scale it as a raw value as well, and this is the formalism implicit in the
PCI-80 and SAC methods described in Section 7.7.1.
At a higher level of complexity, correlation energies are computed assuming that effects
associated with basis-set incompleteness and, say, truncated levels of perturbation theory,