(321g) Computing Thermophysical Properties of Aromatic Compounds: Comparison of Theory and Experiment

In this paper, we systematically
examine the practical procedure used to obtain the entropy of various compounds
in the ideal gas reference state, an important physical property required for
computing equilibrium coefficients for combustion reactions. This procedure
uses both quantum mechanics and statistical mechanics to calculate the entropy.
Using ab initio calculations, the electronic structure of the molecule
is optimized. Normal vibrational frequencies, energetic barriers to internal
rotations and moments of inertia are calculated from the optimized structures
using quantum mechanics. Given the current state-of-the-art in the numerical
solution of the Schrödinger equation, it is necessary to apply an empirical
scaling factor to the normal vibrational frequencies computed by ab initio
means [1]. Scaling factors are well tabulated for a variety of quantum methods
and basis sets [2]. The scaling factor reduces the level of theory
approximation error between experimental and calculated frequencies. Finally,
taking the empirically scaled results from quantum mechanical calculations,
statistical mechanics is used to calculate translational, vibrational,
rotational and internal rotation contributions to the entropy. This procedure
is well established [3-6].

We present a systematic evaluation
of the combined quantum mechanical and statistical mechanical procedure for
generating reference entropies.  We vary the choice of quantum mechanical
method and basis set for a set of 15 aromatic molecules. Our standard, by which
the procedure is evaluated, is a set of highly precise experimental
measurements of the reference entropies for these compounds [7-12]. The
compounds include benzene, toluene, the three xylene isomers and the ten
dimethyl naphthalene isomers. We examined the effect of both method,
Hartree-Fock (HF) and B3LYP, and basis set size, 6-31G(d), 6-31++G(d,p) and
6-311++G(3df,2pd).  

We first examined the reliability
of the published empirical scaling factors for the vibrational
frequencies.  For some of the compounds?benzene, toluene, and the three
xylene isomers, very accurate experimental frequencies are available in the
literature.[7, 8] Therefore, for these compounds, we computed an empirical
scaling factor directly. This allowed us to verify the published scaling
factors as well as to examine the variation in an average scaling factor across
various compounds. For the dimethyl naphthalenes, the same quality of data does
not exist for the vibrational frequencies.  As a result, we rely on an
average scaling factor. After applying scaling factors, the average percentage
error between experimental and calculated frequencies for benzene, toluene and
xylene isomers is below 2.5%.

Before applying the scaling factors
to the vibrational frequencies, the more sophisticated B3LYP method shows much
better agreement in general with experimental frequencies (roughly 2% error)
than the Hartree-Fock (HF) method (10% error). There is a slight computational
cost of performing B3LYP over HF. For both B3LYP and HF, we varied the size of
the basis set.  This results in  at most nominal improvement in the
error (after empirical correction), while a heavy computational penalty is paid
for the larger basis sets. Therefore, strictly on the basis of vibrational
frequencies, it is suggested to use higher accuracy methods like B3LYP rather
than HF with a small basis set.

We next used the scaled vibrational
frequencies, the energetic barriers to internal rotation of methyl groups, and
the moments of inertia from the quantum mechanical calculations as input in a
statistical mechanical model. The translational, vibrational, rotational and
internal rotation contributions to entropy were calculated for all molecules
and levels of theory across a range of 250K to 540K (where experimental data
was available). We compared these entropies to experimental determined
reference entropies [9-12].  Again, it is found that there is an across
the board improvement in the error between theory and experiment is smaller for
B3LYP than for HF, but the improvement due to increased basis set size is not
guaranteed.

For all molecules that had no
internal rotation (benzene) or essentially free internal rotation (toluene,
meta- and para-xylene, as well as 7 of the 10 dimethyl naphthalene isomers),
the entropy difference between all calculations and available experimental
values is less than 0.5% (for B3LYP and small basis set) across a range from
250 K to 540 K.  For orthoxylene, in which the two methyl groups are situated
on adjacent carbons and therefore experience hindered rotation, the error was
less than 1% across the temperature range.  The fact that the internal
rotation can double the error in the quantum calculations can be attributed to
the fact that the vibrational contribution has an empirical correction in it,
whereas the smaller contribution, that of internal rotation, does not have an
empirical correction. 

Reference:

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