Essentials of Computational Chemistry

366 10 THERMODYNAMIC PROPERTIES
the less tractable vibrational partition functions restrict this choice to only the most ambitious
of calculations.)
In practice, then, it is fairly straightforward to convert the potential energy determined from
an electronic structure calculation into a wealth of thermodynamic data – all that is required
is an optimized structure with its associated vibrational frequencies. Given the many levels
of electronic structure theory for which analytic second derivatives are available, it is usually
worth the effort required to compute the frequencies and then the thermodynamic variables,
especially since experimental data are typically measured in this form. For one such quantity,
the absolute entropy So , which is computed as the sum of Eqs. (10.13), (10.18), (10.24) (for
non-linear molecules), and (10.30), theory and experiment are directly comparable. Hout,
Levi, and Hehre (1982) computed absolute entropies at 300 K for a large number of small
molecules at the MP2/6-31G(d) level and obtained agreement with experiment within 0.1 e.u.
for many cases. Absolute heat capacities at constant volume can also be computed using the
thermodynamic definition
∂U
CV =
(10.31)
∂T V
and the various equations for components of U above.
Absolute internal energies, enthalpies, and free energies, on the other hand, are somewhat
less straightforward. From a theoretical standpoint, using the electronic energy as something
to which thermodynamic components are added is equivalent to setting the absolute zero
of energy as corresponding to all nuclei and electrons infinitely separated one from another
and at rest. In the laboratory, this is a very inconvenient zero, since the relevant elementary
particles are not easily handled. The alternative conventions in common use for reporting
H and G as determined from experiment, and the steps which must be taken so that theory
and experiment may be consistently compared, are addressed next.
10.4 Standard-state Heats and Free Energies of Formation
and Reaction
The experimental convention for assigning a zero to an enthalpy or free-energy scale is
that this is the value that corresponds to the heat or free energy of formation associated
with every element in its most stable, pure form under standard conditions (273 K, 1 atm).
Thus, for instance, the elemental standard states for the first few elements are hydrogen
gas (diatomic), helium gas (monatomic), solid lithium, solid beryllium, solid boron, solid
carbon as its graphite allostere, nitrogen gas (diatomic), oxygen gas (diatomic), fluorine
gas (diatomic), and neon gas (monatomic). Following this convention, the meaning of an
experimental heat of formation for a molecule is that it is the (molar) enthalpy change
associated with removing each of the atoms in the molecule from its elemental standard
state and assembling them into the molecule.
Put in this manner, it is easy to imagine this as a two-step procedure. There is first an
enthalpy cost to pull each atom out of its elemental standard state – always a non-negative
quantity, since the elemental standard states are chosen to be the most stable forms. This