Essentials of Computational Chemistry

364 10 THERMODYNAMIC PROPERTIES
the presence of symmetry in a molecule may also require a structural degeneracy correction
to the molecular partition function for certain conformers, as described in more detail in
Appendix B.
Evaluation of the rotational components of the internal energy and entropy using the
partition function of Eq. (10.22) for more typically encountered non-linear molecules gives
and
√
πIAIBIC
Srot = R ln
σ
Urot = 3
2RT (10.23)
2 8π kBT
h2 3/2
+ 3
2
(10.24)
Again, it must be noted that evaluating the rotational components of U and S requires
relatively little in the way of molecular information. All that is required is the principal
moments of inertia, which derive only from the molecular structure. Thus, any methodology
capable of predicting accurate geometries should be useful in the construction of rotational
partition functions and the thermodynamic variables computed therefrom. Also again, the
units chosen for quantities appearing in the partition function must be consistent so as to
render q dimensionless.
10.3.6 Molecular Vibrational Partition Function
In a polyatomic molecule with many vibrations, we simplify the vibrational partition function
much as the original molecular partition function was simplified: we assume that the total
vibrational energy can be expressed as a sum of individual energies associated with each
mode, in which case, for a non-linear molecule, we have
qvib(T ) =
i
e −[ε1+ε2+···+ε3N−6]i/kBT
⎡
= ⎣
e −εj(1)/kBT
⎤ ⎡
⎦ ⎣
e −εj(2)/kBT
⎤ ⎡
⎦ ··· ⎣
j(1)
j(2)
j(3N−6)
⎤
e −εj(3N−6)/kBT ⎦ (10.25)
where the energies εk are the vibrational energy levels associated with each mode k, and
there are 3N − 6 such modes in a non-linear molecule (3N − 5 in a linear molecule) where
N is the number of atoms.
To evaluate the sums associated with each mode, we assume that the modes can be
approximated as quantum mechanical harmonic oscillators (QMHOs), in which case the
energy levels are given by Eqs. (9.47) and (9.48). In this case, we are offered a choice with
respect to convention. We may either take the zero of energy as the bottom of the potential
energy well on the PES, in which case the zeroth vibrational level has energy 1
2hω, orwe
may take the zero of energy as the energy of the equilibrium structure plus the ZPVE, in
which case the energy of the zeroth vibrational energy level is zero for every mode. Both