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:  

1.         Anthony P. Scott, L.R., Harmonic Vibrational Frequencies: An Evaluation of Hartree-Fock, Moller-Plesset, Quadratic Configuration Interaction, Density Functional Theory, and Semiempirical Scale Factors. Journal of Physical Chemistry, 1996. 100(41): p. 16502-16513.

2.         NIST, Computational Chemistry Comparison and Benchmark DataBase. 2006, National Institute of Standards and Technology.

3.         Allan. L. L. East, L.R., Ab initio statistical thermodynamical models for the computation of third-law entropies. J. Chem. Phys., 1997. 106: p. 6655-6674.

4.         Wang, Z.-Y.Z., Zhi-Cai; Wang, Lian-Sheng; Chen, Jin-Long; Kikuchi, Osamu; Watanabe, Toshio, Prediction of gas phase thermodynamics function of polychlorinated dibenzo-p-dioxins using DFT. Journal of Molecular Structure (Theochem), 2004. 672: p. 97-104.

5.         P. Vansteenkiste, V.V.S., G. B. Martin, M. Waroquier, Ab Initio Calculation of Entropy and Heat Capacity of Gas-Phase n-Alkanes Using Internal Rotations. Journal of Physical Chemistry, A., 2003. 107: p. 3139-3145.

6.         R.A. Barrett, R.J.M., The calculation of molecular entropy using the semimpirical AM1 method. Journal of Molecular Structure (Theochem), 1996. 363: p. 203-209.

7.         Draeger, J.A., The methylbenzenes-I. Vapor-phase vibrational fundamentals, internal rotations and a modified valence force field. Spectrochimica Acta, 1985. 41A(4): p. 607-627.

8.         K. S. Pitzer, D.W.S., The thermodynamics and molecular structure of benzene and its methyl derivatives. J. Am. Chem. Soc., 1943. 65: p. 803-829.

9.         R. D. Chirico, S.E.K., A. Nguyen, W. V. Steele, Thermodynamic Equilibria in Xylene Isomerization. 1. The Thermodynamic Properties of p-Xylene. J. Chem. Eng. Data, 1997. 42: p. 248-261.

10.       R. D. Chirico, S.E.K., A. Nguyen, J. W. Reynolds, W. V. Steele, Thermodynamic Equilibria in Xylene Isomerization. 2. The Thermodynamic Properties of m-Xylene. J. Chem. Eng. Data, 1997. 42: p. 475-487.

11.       R. D. Chirico, S.E.K., A. Nguyen, A. B. Cowell, J. W. Reynolds,  W. V. Steele, Thermodynamic Equilibria in Xylene Isomerization. 3. The Thermodynamic Properties of o-Xylene. J. Chem. Eng. Data, 1997. 42: p. 758-771.

12.       Robert D. Chirico, W.V.S., Reconciliation of Calorimetrically and Spectroscopically Derived Thermodynamic Properties at Pressures Greater Than 0.1 MPa for Benzene and Methylbenzene: The Importance of the Third Virial Coefficient. Ind. Eng. Chem. Res., 1994. 33: p. 157-167.