Evaluation of Uncertainty of Ideal-Gas Entropy and Heat Capacity

Article pubs.acs.org/jced

Evaluation of Uncertainty of Ideal-Gas Entropy and Heat Capacity Calculations by Density Functional Theory (DFT) for Molecules Containing Symmetrical Internal Rotors Ctirad Č ervinka,† Michal Fulem,*,†,‡ and Květoslav Růzǐ čka† †

Department of Physical Chemistry, Institute of Chemical Technology, Prague, Technická 5, CZ-166 28 Prague 6, Czech Republic Department of Semiconductors, Institute of Physics, Academy of Sciences of the Czech Republic, v. v. i., Cukrovarnická 10, CZ-162 00 Prague 6, Czech Republic

‡

S Supporting Information *

ABSTRACT: The uncertainty of thermophysical data is indispensable information when reporting both experimental and calculated values. In this paper, we present an evaluation of the uncertainty of the ideal-gas entropy and heat capacity calculations by density functional theory (DFT) for molecules containing symmetrical internal rotors. The rigid-rotor harmonic oscillator (RRHO) and one-dimensional hindered rotor (1-DHR) models are compared as well as the effect of the scale factors employed. The calculations of the standard ideal-gas entropy (Sg0) are performed for a selected set of 33 molecules for which reliable reference data were found in the literature. The RRHO model provides Sg0 with the absolute average percentage deviations (σr) about 2 % from the reference data. Scaling the frequencies does not lead to any improvement when using the RRHO model. A significant improvement is achieved when the 1-DHR model and scale factors for low and high frequencies are applied simultaneously (σr less than 0.3 %). The ideal-gas heat capacity (Cg0 p ) calculations were tested on a set of 72 molecules. The RRHO model yields Cg0 p values with σr up to 3 % at 300 K and 1 % at 1000 K while using the 1-DHR model coupled with a pair of scale factors lowers σr to less than 1.5 % and 0.5 % at 300 K and 1000 K, respectively.

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INTRODUCTION Knowledge of ideal-gas thermodynamic properties is indispensable for the calculation of fundamental thermodynamic functions of ideal gas which are useful approximations to the properties of real gases at low pressures. Furthermore, if knowledge of the state behavior of the fluid and its ideas-gas thermodynamic properties is available, the thermodynamic properties of real fluids can be obtained by basic thermodynamic relationships. Ideal-gas heat capacities also find application in studying solute−solvent interactions through the solvation heat capacities, in temperature adjustments of enthalpies of vaporization or sublimation, and in thermodynamic correlations such as the multiproperty simultaneous correlation of vapor pressure and related thermal data1,2 used in our laboratory. This correlation requires ideal-gas heat capacities with an uncertainty about 1 % chiefly at ambient and subambient temperatures. Therefore, we decided to explore the current possibilities of quantum chemistry methods coupled with statistical thermodynamics to achieve this goal. Several previous works dealing with the calculations of idealgas thermodynamic properties by combining the methods of statistical thermodynamics with quantum chemistry calculations and evaluation of their uncertainties can be found in the literature.3−11 Guthrie4 calculated ideal-gas entropies at 298.15 K for a set of 128 molecules using the rigid rotor−harmonic oscillator © 2013 American Chemical Society

(RRHO) approximation at the B3LYP/6-31G(d,p) level of theory. The RRHO values were corrected for the number of low lying conformations by adding an approximation for entropy of mixing term. The training set consisted of relatively large molecules with up to 10 carbon atoms. The calculated data were compared with the compilation by Stull et al.,12 and an overall standard deviation of 5.4 J·K−1·mol−1 (≈ 2 %) was obtained. Marriott and White5 performed the calculation of ideal-gas heat capacities for 27 molecules in the RRHO approximation using low-level quantum chemical methods (up to B3LYP/6-31G(d) level of theory) and compared them with various group contribution methods for the estimation of ideal-gas heat capacities.13 Agreement is reasonable for rather rigid molecular structures and less satisfactory for flexible molecules. DeTar6 presented a study concerning 18 hydrocarbons with various molecular structures. The ideal-gas entropies and heat capacities were calculated in the RRHO approximation for all significant conformers of a given molecule. The final values were computed as the sum of fractional values based on the Boltzmann fractions corrected for hindered rotation of methyl groups using the tables of Pitzer Received: February 15, 2013 Accepted: April 10, 2013 Published: April 23, 2013 1382

dx.doi.org/10.1021/je4001558 | J. Chem. Eng. Data 2013, 58, 1382−1390

Journal of Chemical & Engineering Data

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Figure 1. Contribution of the lowest frequency corresponding to internal rotation to the ideal-gas thermodynamic properties. (a) Contribution to standard ideal-gas entropy Sg0; (b) contribution to ideal-gas heat capacity Cg0 p . Solid lines, 1,1,1-trichloro-2,2,2-trifluoroethane (energy barrier to internal rotation V0 = 31.7 kJ·mol−1, wavenumber ν = 80 cm−1, reduced moment of inertia of internal rotation Ir = 1.2 × 10−45 kg·m2; values calculated at the B3LYP/6-311+G(d,p) level of theory); dashed lines; ethane (V0 = 11.3 kJ·mol−1, ν = 308 cm−1, Ir = 2.6 × 10−47 kg·m2); dotted lines, 2-methylpyrrol (V0 = 1.4 kJ·mol−1, ν = 109 cm−1, Ir = 4.9 × 10−47 kg·m2). Black lines represent the FR treatment, blue lines the 1-DHR treatment, and red lines the HO approximation.

and Gwinn.14 A performed error analysis resulted in the rootmean-square deviation range of 1.1 J·K−1·mol−1 for the calculated ideal-gas heat capacities and 1.5 J·K−1·mol−1 for the calculated ideal-gas entropies when using this computational approach. East and Radom3 presented three procedures of different complexity designated E1, E2, and E3 for calculating the ideal-gas entropies and applied them to a comprehensive set of small molecules. The procedure E1 treated all vibrational degrees of freedom in the harmonic oscillator (HO) approximation and errors up to 1.8 J·K−1·mol−1 per internal rotation were observed. The procedure E2 treated the individual internal rotations by a simple cosine potential and the contributions of internal rotations to entropy were obtained from the tables of Pitzer and Gwinn.14 The uncertainty of the calculated ideal-gas entropies were less than 1 J·K−1·mol−1 for molecules with zero or one internal rotation and less than 2 J·K−1·mol−1 for molecules with two internal rotation modes. The procedure E3 took into account the coupling of internal rotational modes which improved the accuracy of the calculated entropies for molecules with two internal rotations to approximately 1 J·K−1·mol−1. Vansteenkiste et al.8 applied the one-dimensional hindered rotor (1-DHR) model15 for the treatment of internal rotations to calculate the ideal-gas entropies and heat capacities of n-alkanes ranging from ethane to n-decane. While the standard deviation of the calculated ideal-gas entropies from experimental values remained constant and less than 1 % up to n-decane, the relative deviations of the calculated ideal-gas heat capacities increased with the number of carbon atoms and amounted up to 16 % in the case of n-decane suggesting that the 1-DHR model, which treats the internal rotations as uncoupled, is not appropriate for longchain molecules with several internal rotations. Vansteenkiste et al. also used 1-DHR approach to calculate the ideal-gas thermodynamic properties with heteroatoms O or S10 and fourmembered ring molecules.9 In both studies, the agreement of the calculated ideal-gas entropies with the reference values was more satisfactory than in the case of the calculated ideal-gas heat capacities. For four-membered ring molecules, the 1-DHR model provided the ideal-gas heat capacities which were in worse agreement with the reference data than the values obtained by the HO approximation. Another approach to calculate the ideal-gas thermodynamic properties of molecules

containing internal rotations was recently presented by Santos et al.7 who used various isodesmic reaction schemes to establish the ideal-gas heat capacity data for benzoic acid. In our previous work,11 we statistically evaluated the uncertainty in the calculated ideal-gas heat capacities and entropies for rigid molecules without internal rotations around a single bond and other large amplitude motions. In this work, we performed an analogous study for selected molecular structures containing one or two symmetrical internal rotors for which reliable reference data in the ideal gaseous state were available in the literature. The calculations were carried out using the 1-DHR model, which is expected to be appropriate for this type of molecular structures, and the easy-to-use RRHO model for comparison. We also studied the effect of employed scale factors for scaling of fundamental vibrations. As a significant portion of the studied species were aromatic molecules with atoms predominantly in the sp2 hybridization, we could compare the performance of the specialized scale factors for the sp2 structures with the widely used scale factors suggested by Merrick et al.16

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THEORETICAL SECTION The basic approach to calculate the thermodynamic properties in the ideal-gas state by statistical thermodynamics is based on the rigid rotor−harmonic oscillator (RRHO) model. One assumes here the harmonicity of all vibrational modes and free rotation of the molecule which remains rigid during the rotationbond lengths and angles are not deformed, and all atoms of the molecule rotate with the same angular velocity and in the same sense. The mathematical expressions for the computations of ideal-gas entropy and heat capacity using the RRHO model (requiring the principal moments of inertia, molecular vibrations and molar mass as inputs) are described in many textbooks,17−20 and therefore they are not repeated here. The RRHO is appropriate for molecules which do not contain internal rotations and/or other low frequency motions such as for example ring puckering. For these molecules, the main source of uncertainty remains the uncertainties in fundamental vibrational frequencies and their treatment as harmonic oscillations when calculating ideal-gas thermodynamic properties. Treatment of internal rotations introduces another uncertainty in the calculated values of ideal-gas thermodynamic 1383



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Calculation of ideal-gas thermodynamic properties using the 1-DHR model starts with the electronic structure calculations to obtain optimized geometries and corresponding harmonic vibrational frequencies. This step is followed by identifying the low frequency modes corresponding to torsional motions which are subsequently removed from the vibrational partition function and replaced by the contributions of internal rotations to the partition function Qint,rot. The 1-DHR model treats the internal rotations as uncoupled and requires additional input quantitiesthe reduced moments of inertia Ir and the potential energy scans of internal rotations V(φ). The contributions of internal rotations to the partition function are calculated using

Table 1. Scale Factors Used in This Work sp2 structurea

B3LYP/6-31G(d) B3LYP/6-311+G(d,p)

below 2000 cm−1

above 2000 cm−1

various structureb

0.9749 0.9808

0.9551 0.9618

0.9613 0.9688

a

The scale factors developed in our previous work11 for scaling the fundamental vibrations of molecules containing atoms predominantly in the sp2 hybridization. bThe scale factors suggested by Merrick et al.16 The scale factor for a given level of theory was developed based on the comparison of the calculated vibrational frequencies (1066 in total) with experimental values for a variety of predominantly small chemical species with variety of elements (see the SI of ref 16).

Q int,rot =

properties. Describing the torsional motions as harmonic vibrations is known to be inappropriate except for internal rotations with high energy barriers. One of the most successful methods for treating internal rotation is the one-dimensional hindered rotor (1-DHR) model, which has recently been reviewed by Pfaendtner et al.15 More elaborated treatments of internal rotations can be found in the literature (an overview can also be found in ref 15), but given their high computational cost and complexity they are not so far of use for practical calculations of ideal-gas thermodynamic properties of organic species. The 1-DHR model is briefly described below; more details can be found in the paper by Pfaendtner et al.15

1 σint

∑ e−ε /kT j

j

(1)

where σint is the internal symmetry number of rotating top, k is the Boltzmann constant, T is the temperature, and the energy levels εj are obtained by solving a one-dimensional Schrödinger equation for hindered internal rotation −

ℏ2 d2Ψ + V (φ )Ψ = E Ψ 2Ir dφ 2

(2)

where ℏ is Planck constant h divided by 2π, Ψ is the wave function, E is the energy, and φ is the torsional angle. In this work, we evaluate the performance of the 1-DHR model with both unscaled and scaled vibrational frequencies to

Figure 2. Average absolute percentage deviation, σr (eq 1), of calculated Sg0 from the reference values.27,29,45−65 (a) B3LYP/6-31G(d) level of theory, all 33 compounds listed in Table S2 in the SI; (b) B3LYP/6-31G(d) level of theory, subset of 23 molecules for which a pair of SF for sp2 structures11 could be applied (denoted as “sp2” in Table S2); (c) B3LYP/6-311+G(d,p) level of theory, all 33 compounds listed in Table S2 in the SI; (d) B3LYP/6-311+G(d,p) level of theory, subset of 23 molecules for which a pair of SF for sp2 structures11 could be applied (denoted as “sp2” in Table S2). 1384



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Figure 3. Fraction of positive deviations of calculated Sg0 from the reference values.27,29,45−65 (a) B3LYP/6-31G(d) level of theory, all 33 compounds listed in Table S2 in the SI; (b) B3LYP/6-31G(d) level of theory, subset of 23 molecules for which a pair of SF for sp2 structures11 could be applied (denoted as “sp2” in Table S2); (c) B3LYP/6-311+G(d,p) level of theory, all 33 compounds listed in Table S2 in the SI; (d) B3LYP/6-311+G(d,p) level of theory, subset of 23 molecules for which a pair of SF for sp2 structures11 could be applied (denoted as “sp2” in Table S2).

be optimized at a given level of theory. The reduced moments of inertia Ir for internal rotations were calculated according to Pitzer14,22 from the optimized molecular parameters. Energy levels were calculated by the variation method using the program StatTD23 and our own code performing the Fourier grid Hamiltonian method.24 Both methods yielded the same results.

calculate the ideal-gas thermodynamic properties of molecules containing one or two internal rotors for which we expect the model to be appropriate. The model 1-DHR is also compared with the RRHO model to estimate the uncertainty one can expect when using the basic and easy to use RRHO model for this group of molecules. Figure 1 illustrates the difference in treating the low frequencies (torsional modes) in the HO approximation, using the 1-DHR model, and as a free rotation (FR) when calculating ideal-gas thermodynamic properties. Figure 1 shows a model situation and contribution of the lowest frequency to ideal-gas thermodynamic properties for three molecules with high, intermediate, and low energy barriers to internal rotation V0 (1,1,1-trichloro-2,2,2-trifluoroethane, V0 = 31.7 kJ·mol−1; ethane, V0 = 11.3 kJ·mol−1; 2-methylpyrrole, V0 = 1.4 kJ·mol−1; values calculated at the B3LYP/6-311+G(d,p) level of theory).

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DATA SET SELECTION The compounds selected for this study fulfill the following criteria: (i) molecular structures containing one or two symmetrical internal rotors (methyl, nitro, etc. groups), (ii) compounds for which reliable reference thermodynamic data in the ideal gaseous state are available in the literature. We accepted molecules with one or two symmetrical uncoupled tops to avoid dealing with (i) the input uncertainty in the reduced moment of inertia Ir as this value is accessible analytically only for a single symmetrical top and according to Pitzer and Gwinn14 with a good approximation for molecules with two symmetrical tops and (ii) the uncertainty arising from the coupling of internal rotation modes. Our initial intention was to use solely experimental data for both ideal-gas entropies and heat capacities as a reference for comparison with our calculated data. This turned out to be a problem particularly for heat capacities as direct (calorimetric) or indirect measurements (for example speed-of-sound measurements) are very scarce and most of the data are obtained by the methods of statistical thermodynamics using spectral data. Therefore, we extended the database of heat

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COMPUTATIONAL DETAILS All quantum chemistry calculations were performed using the program package Gaussian, version 03.21 Molecular geometry optimizations and calculations of vibrational frequencies and potential functions V(φ) were carried out using the density functional theory (DFT) with the B3LYP functional, which is recommended for thermochemistry calculations.16,18 The B3LYP functional was used with two basis sets 6-31G(d) and 6-311+G(d,p) to study the effect of the basis set size. The potential functions V(φ) of internal rotations were determined by scanning the dihedral angle φ from 0° to 360° at 10° increments and allowing all other structural parameters to 1385



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17 Figure 4. Average absolute percentage deviation, σr (eq 1), of calculated Cg0 p from the reference values. (a) B3LYP/6-31G(d) level of theory, all 72 compounds listed in Table S3 in the SI; (b) B3LYP/6-31G(d) level of theory, subset of 38 molecules for which a pair of SF for sp2 structures11 could be applied (denoted as “sp2” in Table S3); (c) B3LYP/6-311+G(d,p) level of theory, all 72 compounds listed in Table S4 in the SI; B3LYP/631G(d) level of theory; (d) B3LYP/6-311+G(d,p) level of theory, subset of 38 molecules for which a pair of SF for sp2 structures11 could be applied (denoted as “sp2” in Table S4).

and “ref” stand for the calculated and reference property, respectively. The calculations of ideal-gas thermodynamic properties were performed with the following models: (a) RRHO model with unscaled vibrational frequencies (RRHO), (b) RRHO model with vibrational frequencies scaled by the scaling factor suggested by Merrick et al.16 (RRHO + 1SF), (c) RRHO model with a pair of scaling factors (only for structures containing atoms predominantly in sp2 hybridization) suggested for low and high vibrational frequencies, respectively, in our previous work11 (RRHO + 2SF), (d) 1-DHR model with unscaled vibrational frequencies (1-DHR), (e) 1-DHR model with vibrational frequencies scaled by a scaling factor16 (1-DHR + 1SF), and (f) 1-DHR model with vibrational frequencies scaled by the two scaling factors11 (1-DHR + 2SF). The used scale factors are summarized in Table 1. Standard Ideal-Gas Entropies Sg0. A set of 33 chemical species fulfilling our criteria was selected. Fifteen of these molecules contain one internal rotor; the remaining 18 molecules contain two internal rotors. A subset of compounds contains predominantly atoms in the sp2 hybridization, and therefore the pairs of scale factors for sp2 systems11 could be used for this subset. The comparison of calculated Sg0 with the reference values is represented by the average absolute percentage deviation σr (eq 1) in Figure 2; the distribution of positive and negative deviations is shown in Figure 3. Table S2 in the SI contains the relative deviations of the calculated data

capacity reference data by selected data from TRC database17 obtained by statistical thermodynamics using reliable experimental spectroscopic parameters. Table S1 in the Supporting Information (SI) indicates which ideal-gas heat capacity data from the TRC database can be verified with the available experimental values25−44 along with the experimental techniques used, temperature ranges of experimental determinations, uncertainty of experimental determinations, and deviations of the experimental values from the data listed in the TRC database. The overall agreement of experimental data25−44 with the values from the TRC database17 was within 1 %, in most cases within 0.5 %. The reference third-law entropies, derived from experimental data on heat capacities of condensed phases, vapor pressure measurements, and a description or estimate of state behavior of fluid, were culled from various literature sources.27,29,45−65

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RESULTS

The deviations of the calculated ideal-gas thermodynamic properties from the reference data are represented by the average absolute percentage deviation σr defined as n

σr = 100/n ∑ (|Y calc − Y ref | /Y ref )i i=1

(3)

where Y is either ideal-gas entropy or heat capacity, n is the number of data points in the data set, and superscripts “calc” 1386



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17 Figure 5. Fraction of positive deviations of calculated Cg0 p from the reference values. (a) B3LYP/6-31G(d) level of theory, all 72 compounds listed in Table S3 in the SI; (b) B3LYP/6-31G(d) level of theory, subset of 38 molecules for which a pair of SF for sp2 structures11 could be applied (denoted as “sp2” in Table S3); (c) B3LYP/6-311+G(d,p) level of theory, all 72 compounds listed in Table S4 in the SI; B3LYP/6-31G(d) level of theory; (d) B3LYP/6-311+G(d,p) level of theory, subset of 38 molecules for which a pair of SF for sp2 structures11 could be applied (denoted as “sp2” in Table S4).

Messerly et al.68,69 For higher temperatures, the standard absolute uncertainties are higher, about (2 to 3) J·K−1·mol−1. The uncertainty of Sg0 values calculated by 1-DHR model is in the order of tenths of percent which is comparable with that of experimental measurements. The performance of 1-DHR for the calculation of Sg0 for the molecules with one or two symmetrical tops can thus be considered satisfactory. Ideal-Gas Heat Capacities Cg0 p . A set of 72 molecules containing one internal rotor (42 molecules) and two internal rotors (30 molecules) was selected for comparison of the calculated values with the reference data reported in TRC database.17 The pair of SF developed in our previous work11 could be applied for 38 molecules from this set as the atoms in the sp2 hybridization were prevailing in these structures; SF by Merrick et al.16 could be applied for all of the molecules from the data set. The results are summarized in Figures 4 and 5; more detailed statistical results can be found in Table S3 (B3LYP/6-31G(d) level of theory) and Table S4 (B3LYP/6-311+G(d,p) level of theory) in the SI. The calculation with both basis sets give the same trends among different modelsthe larger basis set 6-311+G(d,p) provides lower σr than the smaller basis set 6-31G(d). However, this difference amounts to only few tenths of percent. The RRHO model without employing SF leads to σr about 3 % and 1 % at low and high temperatures, respectively. These data exhibit an unbiased distribution of relative deviations (see Figure 5). The application of any SF to the RRHO model leads to greater σr, showing an inappropriateness

from the reference data for all compounds in the data set as well as numerical values for σr and bias distribution. As can be seen, the RRHO model provides results with σr about 2 %. Applying any scaling factors (SF) leads to larger σr for both used basis sets and the smaller basis set gives lower σr. The distribution of deviations of the results obtained by RRHO model without SF is almost unbiased while the results including frequency scaling are overestimated in more than 75 % of the cases, as shown in Figure 3. Employing the 1-DHR model leads to significantly lower σr, which amounts to approximately 0.6 % for unscaled frequencies. Applying the single SF by Merrick et al.16 worsens the results, while employing SF pairs11 leads to Sg0 values with σr about 0.3 % (see Figure 2). Both basis sets provide values of comparable σr. Sg0 values obtained by the 1-DHR model without scaling the vibrational frequencies are in almost 90 % of the cases underestimated for both basis sets, while the results obtained using the single SF16 are overestimated. When applying the pair of SF,11 the smaller basis set gives underestimated results, while the larger basis set provides overestimated Sg0 values (see Figure 3). When comparing the calculated data with experimental values, one should also take into account the uncertainties of experimental determinations. The standard uncertainties of the experimental third-law entropies at temperatures near 300 K amount to (0.3 to 0.6) J·K−1·mol−1 in the case of the recent studies by Chirico et al.66,67 (around 0.1 % in relative numbers) and to (0.8 to 1.0) J·K−1·mol−1 in the case of older works of 1387



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Funding

of the harmonic oscillator model for treatment of internal rotations. Furthermore, the values so computed exhibit an overestimation in up to 95 % of the cases, as shown in Figure 5. Employing the 1-DHR model without using SF provides data of comparable quality as in the case of the RRHO model without SF and Cg0 p values are underestimated in about 90 % of the cases. A significant improvement is achieved when the 1-DHR model is coupled with the use of SF. A pair of SF11 leads to better results than the single SF,16 especially at low temperatures. When the single SF16 is applied, Cg0 p values are overestimated in about 80 % of the cases (see Figure 5). Application of pairs of SF11 gives much more unbiased data whose σr does not exceed 1.5 % for low temperatures and 0.5 % for high temperatures (see Figures 4 and 5). Such low values of σr are comparable with the uncertainty of the reference data17 (see Table S1 in the SI) and can be regarded as satisfactory.

This work is supported by the Ministry of Education of the Czech Republic under project ME10049. C.Č . acknowledges financial support from specific university research (MSMT No. 20/2013). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Vladimir Diky and Ala Bazyleva for providing the program StatTD.

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CONCLUSION The uncertainty of the ideal-gas entropy and heat capacity calculations for overall of 82 molecules containing symmetrical internal rotors was statistically evaluated. A comparison of performance of the RRHO and 1-DHR models using one11 or two scaling factors16 and two basis sets of different complexity was performed. The effect of the basis set was not significant; B3LYP/6-311+G(d,p) level of theory provided results with only a slightly lower uncertainty compared to B3LYP/6-31G(d) level of theory. The best computational procedure among the studied approaches is the 1-DHR model coupled with the specialized pairs of scale factors for the sp2 structures,11 implying that appropriate treatment of internal rotations and frequency scaling can lead to low uncertainties in the calculated values. In this case, the average absolute percentage deviation from the reference data amounted to 0.3 % for ideal-gas entropies Sg0 and 1.5 % and 0.5 % for ideal-gas heat capacities Cg0 p at 300 K and 1000 K, respectively. Such uncertainties can be regarded as satisfactory for many applications of ideal-gas thermodynamic properties including the multiproperty simultaneous correlation of vapor pressure and related thermal data1,2 used in our laboratory. When applying the easy-to-use RRHO model, the scaling of vibrational frequencies does not lead to any improvement, and one can expect the uncertainties in the order of (2 to 3) % for Sg0 and Cg0 p calculations at 300 K and in the order of (1 to 2) % for Cg0 p calculations at 500 K and 1000 K for molecules of a similar structure as those studied in this work.

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ASSOCIATED CONTENT

S Supporting Information *

Tables (.xlsx) with the list of compounds used in the ideal-gas heat capacity calculations and comparison of values from the TRC database with experimental values (Table S1), relative percentage deviations of calculated ideal-gas entropies at standard pressure (Table S2) and relative percentage deviations of calculated ideal-gas heat capacities at the B3LYP/6-31G(d) (Table S3) and B3LYP/6-3111+G(d,p) (Table S4) levels of theory. This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

(1) Růzǐ čka, K.; Majer, V. Simultaneous treatment of vapor pressures and related thermal data between the triple and normal boiling temperatures for n-alkanes C5-C20. J. Phys. Chem. Ref. Data 1994, 23, 1−39. (2) Růzǐ čka, K.; Majer, V. Simple and controlled extrapolation of vapor pressures toward the triple point. AIChE J. 1996, 42, 1723− 1740. (3) East, A. L. L.; Radom, L. Ab initio statistical thermodynamical models for the computation of third-law entropies. J. Chem. Phys. 1997, 106, 6655−6674. (4) Guthrie, J. P. Use of DFT methods for the calculation of the entropy of gas phase organic molecules: An examination of the quality of results from a simple approach. J. Phys. Chem. A 2001, 105, 8495− 8499. (5) Marriott, R. A.; White, M. A. Comparison of ab initio and group additive ideal gas heat capacities. AIChE J. 2005, 51, 292−297. (6) DeTar, D. F. Calculation of entropy and heat capacity of organic compounds in the gas phase. Evaluation of a consistent method without adjustable parameters. Applications to hydrocarbons. J. Phys. Chem. A 2007, 111, 4464−4477. (7) Santos, L. S. M. N. B. F.; Rocha, M. A. A.; Gomes, L. G. R.; Schröder, B.; Coutinho, J. O. A. P. Gaseous phase heat capacity of benzoic acid. J. Chem. Eng. Data 2010, 55, 2799−2808. (8) Vansteenkiste, P.; Van Speybroeck, V.; Marin, G. B.; Waroquier, M. Ab initio calculation of entropy and heat capacity of gas-phase nalkanes using internal rotations. J. Phys. Chem. A 2003, 107, 3139− 3145. (9) Vansteenkiste, P.; Van Speybroeck, V.; Verniest, G.; De Kimpe, N.; Waroquier, M. Applicability of the hindered rotor scheme to the puckering mode in four-membered rings. J. Phys. Chem. A 2006, 110, 3838−3844. (10) Vansteenkiste, P.; Verstraelen, T.; Van Speybroeck, V.; Waroquier, M. Ab initio calculation of entropy and heat capacity of gas-phase n-alkanes with hetero-elements O and S: Ethers/alcohols and sulfides/thiols. Chem. Phys. 2006, 328, 251−258. (11) Č ervinka, C.; Fulem, M.; Růzǐ čka, K. Evaluation of accuracy of ideal-gas heat capacity and entropy calculations by density functional theory (DFT) for rigid molecules. J. Chem. Eng. Data 2012, 57, 227− 232. (12) Stull, D. R.; Westrum, E. F.; Sinke, G. C. The Chemical Thermodynamics of Organic Compounds; Wiley: New York, 1969. (13) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Gases and Liquids; McGraw-Hill: New York, 2001. (14) Pitzer, K. S.; Gwinn, W. D. Energy Levels and Thermodynamic Functions for Molecules with Internal Rotation I. Rigid Frame with Attached Tops. J. Chem. Phys. 1942, 10, 428−440. (15) Pfaendtner, J.; Yu, X.; Broadbelt, L. J. The 1-D hindered rotor approximation. Theor. Chem. Acc. 2007, 118, 881−898. (16) Merrick, J. P.; Moran, D.; Radom, L. An evaluation of harmonic vibrational frequency scale factors. J. Phys. Chem. A 2007, 111, 11683− 11700. (17) Frenkel, M. L.; Kabo, G. J.; Marsh, K. N.; Roganov, G. N.; Wilhoit, R. C. Thermodynamics of Organic Compounds in the Gas State;

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Thermodynamics Research Center: College Station, TX, 1994; Vols. 1 and 2. (18) Irikura, K. K.; Frurip, D. J. Computational thermochemistry: prediction and estimation of molecular thermodynamics; American Chemical Society: Washington, DC, 1998. (19) Jensen, F. Introduction to computational chemistry; John Wiley & Sons: Chichester, 2008. (20) Cramer, C. J. Essentials of computational chemistry: theories and models; Wiley: Chichester, 2004. (21) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03, Revision D.01; Gaussian, Inc.: Wallingford, CT, 2004. (22) Pitzer, K. S. Energy Levels and Thermodynamic Functions for Molecules with Internal Rotation: II. Unsymmetrical Tops Attached to a Rigid Frame. J. Chem. Phys. 1946, 14, 239−243. (23) Diky, V.; Bazyleva, A. Personal communication, 2011. (24) Marston, C. C.; Balint-Kurti, G. G. The Fourier grid Hamiltonian method for bound state eigenvalues and eigenfunctions. J. Chem. Phys. 1989, 91, 3571−3576. (25) Scott, D. W.; McCullough, J. P.; Douslin, D. R.; Todd, S. S.; Hossenlopp, I. A.; Messerly, J. F. 4-Fluorotoluene: chemical thermodynamic properties, vibrational assignment, and internal rotation. J. Chem. Phys. 1962, 37, 867−873. (26) Scott, D. W.; Hubbard, W. N.; Todd, S. S.; Messerly, J. F.; McCullough, J. P.; Good, W. D.; Douslin, D. R.; Hossenlopp, I. A. Chemical thermodynamic properties and internal rotation methylpyridines. I. 2-methylpyridine. J. Phys. Chem. 1963, 67, 680−685. (27) Pennington, R. E.; Finke, H. L.; Hubbard, W. N.; Messerly, J. F.; Frow, F. R.; Hossenlopp, I. A.; Waddington, G. The chemical thermodynamic properties of 2-methylthiophene. J. Am. Chem. Soc. 1956, 78, 2055−2060. (28) Scott, D. W.; Todd, S. S.; Good, W. D.; Guthrie, G. B.; McCullough, J. P.; Hossenlopp, I. A.; Osborn, A. G. Chemical thermodynamic properties and internal rotation of methylpyridines. II. 3-methylpyridine. J. Phys. Chem. 1963, 67, 685−689. (29) McCullough, J. P.; Sunner, S.; Finke, H. L.; Hubbard, W. N.; Gross, M. E.; Pennington, R. E.; Messerly, J. F.; Good, W. D.; Waddington, G. The chemical thermodynamic properties of 3methylthiopene from 0 to 1000 K. J. Am. Chem. Soc. 1953, 75, 5075−5081. (30) Hossenlopp, I. A.; Scott, D. W. Vapor heat capacities and enthalpies of vaporization of six organic compounds. J. Chem. Thermodyn. 1981, 13, 405−414. (31) Bier, K.; Ernst, G.; Kunze, J.; Maurer, G. Thermodynamic properties of propylene from calorimetric measurements. J. Chem. Thermodyn. 1974, 6, 1039−1052. (32) Scott, D. W.; Guthrie, G. B.; Messerly, J. F.; Todd, S. S.; Berg, W. T.; Hossenlopp, I. A.; McCullough, J. P. Toluene: thermodynamic properties, molecular vibrations, and internal rotation. J. Phys. Chem. 1962, 66, 911−914. (33) Scott, D. W.; Douslin, D. R.; Messerly, J. F.; Todd, S. S.; Hossenlopp, I. A.; Kincheloe, T. C.; McCullough, J. P. Benzotrifluoride: chemical thermodynamic properties and internal rotation. J. Am. Chem. Soc. 1959, 81, 1015−1020.

(34) Counsell, J. F.; Hales, J. L.; Lees, E. B.; Martin, J. F. Thermodynamic properties of fluorine compounds. Part VIII. The heat capacity and entropy of 2,3,4,5,6-pentafluorotoluene. J. Chem. Soc. A 1968, 0, 2994−2996. (35) Pitzer, K. S.; Scott, D. W. The thermodynamics and molecular structure of benzene and its methyl derivatives. J. Am. Chem. Soc. 1943, 65, 803−829. (36) Hossenlopp, I. A.; Scott, D. W. Vapor heat capacities and enthalpies of vaporization of four aromatic and/or cycloalkane hydrocarbons. J. Chem. Thermodyn. 1981, 13, 423−428. (37) Kistiakowsky, G. B.; Rice, W. W. Gaseous heat capacities. III. J. Chem. Phys. 1940, 8, 618−622. (38) Ernst, G.; Büsser, J. Ideal and real gas state heat capacities Cp of C3H8, i-C4H10, CH2ClCF3, CF2ClCFCl2, and CHF2Cl. J. Chem. Thermodyn. 1970, 2, 787−791. (39) Estrada-Alexanders, A. F.; Trusler, J. P. M. The speed of sound and derived thermodynamic properties of ethane at temperatures between 220 and 450 K and pressures up to 10.5 MPa. J. Chem. Thermodyn. 1997, 29, 991−1015. (40) Coleman, C. F.; Vries, T. D. The heat capacity of organic vapors. V. Acetaldehyde. J. Am. Chem. Soc. 1949, 71, 2839−2841. (41) Boyes, S. J.; Ewing, M. B.; Goodwin, A. R. H. Heat capacities and second virial coefficients for gaseous methanol determined from speed-of-sound measurements at temperatures between 280 and 360 K and pressures from 1.03 to 80.5 kPa. J. Chem. Thermodyn. 1992, 24, 1151−1166. (42) McCullough, J. P.; Scott, D. W.; Pennington, R. E.; Hossenlopp, I. A.; Waddington, G. Nitromethane: the vapor heat capacity, heat of vaporization, vapor pressure, and gas imperfection; the chemical thermodynamic properties from 0 to 1500 K. J. Am. Chem. Soc. 1954, 4791−4796. (43) Beekermann, W.; Kohler, F. Acoustic determination of ideal-gas heat capacity and second virial coefficients of some refrigerants between 250 and 420 K. Int. J. Thermophys. 1995, 16, 455−464. (44) Pennington, R. E.; Kobe, K. A. The thermodynamic properties of acetone. J. Am. Chem. Soc. 1957, 79, 300−305. (45) Scott, D. W.; McCullough, J. P.; Douslin, D. R.; Todd, S. S.; Hossenlopp, I. A.; Messerly, J. F. 4-Fluorotoluene - Chemical Thermodynamic Properties, Vibrational Assignment, and Internal Rotation. J. Chem. Phys. 1962, 37, 867−873. (46) Messerly, J. F.; Finke, H. L.; Good, W. D.; Gammon, B. E. Condensed-Phase Heat-Capacities and Derived Thermodynamic Properties for 1,4-Dimethylbenzene, 1,2-Diphenylethane, and 2,3Dimethylnaphthalene. J. Chem. Thermodyn. 1988, 20, 485−501. (47) Messerly, J. F.; Todd, S. S.; Finke, H. L.; Good, W. D.; Gammon, B. E. Condensed-Phase Heat-Capacity Studies and Derived Thermodynamic Properties for Six Cyclic Nitrogen-Compounds. J. Chem. Thermodyn. 1988, 20, 209−224. (48) Chirico, R. D.; Knipmeyer, S. E.; Nguyen, A.; Cowell, A. B.; Reynolds, J. W.; Steele, W. V. Thermodynamic equilibria in xylene isomerization. 3. The thermodynamic properties of o-xylene. J. Chem. Eng. Data 1997, 42, 758−771. (49) Chirico, R. D.; Knipmeyer, S. E.; Nguyen, A.; Reynolds, J. W.; Steele, W. V. Thermodynamic equilibria in xylene isomerization. 2. The thermodynamic properties of m-xylene. J. Chem. Eng. Data 1997, 42, 475−487. (50) Chirico, R. D.; Knipmeyer, S. E.; Nguyen, A.; Steele, W. V. Thermodynamic equilibria in xylene isomerization. 1. The thermodynamic properties of p-xylene. J. Chem. Eng. Data 1997, 42, 248−261. (51) Chirico, R. D.; Klots, T. D.; Knipmeyer, S. E.; Nguyen, A.; Steele, W. V. Reconciliation of calorimetrically and spectroscopically derived standard entropies for the six dimethylpyridines between the temperatures 250 and 650 K: a stringent test of thermodynamic consistency. J. Chem. Thermodyn. 1998, 30, 535−556. (52) Chirico, R. D.; Knipmeyer, S. E.; Nguyen, A.; Steele, W. V. Thermodynamic properties of the methylpyridines. Part 2. Vapor pressures, heat capacities, critical properties, derived thermodynamic functions between the temperatures 250 and 560 K, and equilibrium 1389



Article

isomer distributions for all temperatures ≥ 250 K. J. Chem. Thermodyn. 1999, 31, 339−378. (53) Chirico, R. D.; Steele, W. V. Thermodynamic properties of 2methylquinoline and 8-methylquinoline. J. Chem. Eng. Data 2005, 50, 697−708. (54) Chirico, R. D.; Steele, W. V. Thermodynamic properties of diphenylmethane. J. Chem. Eng. Data 2005, 50, 1052−1059. (55) Chirico, R. D.; Johnson, R. D.; Steele, W. V. Thermodynamic properties of methylquinolines: Experimental results for 2,6dimethylquinoline and mutual validation between experiments and computational methods for methylquinolines. J. Chem. Thermodyn. 2007, 39, 698−711. (56) Aston, J. G.; Eidinoff, M. L.; Forster, W. S. The Heat Capacity and Entropy, Heats of Fusion and Vaporization and the Vapor Pressure of Dimethylamine. J. Am. Chem. Soc. 1939, 61, 1539−1543. (57) Aston, J. G.; Mastrangelo, S. V. R.; Moessen, G. W. The Thermodynamics of 1-Butyne from Calorimetric and Spectroscopic Data. J. Am. Chem. Soc. 1950, 72, 5287−5291. (58) Aston, J. G.; Siller, C. W.; Messerly, G. H. Heat Capacities and Entropies of Organic Compounds. III. Methylamine from 11.5 K. to the Boiling Point. Heat of Vaporization and Vapor Pressure. The Entropy from Molecular Data. J. Am. Chem. Soc. 1937, 59, 1743−1751. (59) Aston, J. G.; Szasz, G. J. The Thermodynamics of Butadiene-1,2 from Calorimetric and Spectroscopic Data. J. Am. Chem. Soc. 1947, 69, 3108−3114. (60) Jones, W. M.; Giauque, W. F. The Entropy of Nitromethane. Heat Capacity of Solid and Liquid. Vapor Pressure, Heats of Fusion and Vaporization. J. Am. Chem. Soc. 1947, 69, 983−987. (61) Kemp, J. D.; Egan, C. J. Hindered Rotation of the Methyl Groups in Propane. The Heat Capacity, Vapor Pressure, Heats of Fusion and Vaporization of Propane. Entropy and Density of the Gas. J. Am. Chem. Soc. 1938, 60, 1521−1525. (62) Kemp, J. D.; Pitzer, K. S. The Entropy of Ethane and the Third Law of Thermodynamics. Hindered Rotation of Methyl Groups. J. Am. Chem. Soc. 1937, 59, 276−279. (63) Kennedy, R. M.; Sagenkahn, M.; Aston, J. G. The Heat Capacity and Entropy, Heats of Fusion and Vaporization, and the Vapor Pressure of Dimethyl Ether. The Density of Gaseous Dimethyl Ether. J. Am. Chem. Soc. 1941, 63, 2267−2272. (64) Russell, H.; Golding, D. R. V.; Yost, D. M. The Heat Capacity, Heats of Transition, Fusion and Vaporization, Vapor Pressure and Entropy of 1,1,1-Trifluoroethane. J. Am. Chem. Soc. 1944, 66, 16−20. (65) Schumann, S. C.; Aston, J. G. Restricted rotation in ethyl alcohol, acetone and isopropyl alcohol. J. Am. Chem. Soc. 1938, 60, 985−986. (66) Chirico, R. D.; Kazakov, A. F.; Steele, W. V. Thermodynamic properties of three-ring aza-aromatics. 1. Experimental results for phenazine and acridine, and mutual validation of experiments and computational methods. J. Chem. Thermodyn. 2010, 42, 571−580. (67) Chirico, R. D.; Kazakov, A. F.; Steele, W. V. Thermodynamic properties of three-ring aza-aromatics. 2. Experimental results for 1,10phenanthroline, phenanthridine, and 7,8-benzoquinoline, and mutual validation of experiments and computational methods. J. Chem. Thermodyn. 2010, 42, 581−590. (68) Messerly, G. H.; Kennedy, R. M. The Heat Capacity and Entropy, Heats of Fusion and Vaporization and the Vapor Pressure of n-Pentane. J. Am. Chem. Soc. 1940, 62, 2988−2991. (69) Messerly, J. F.; Finke, H. L.; Osborn, A. G.; Douslin, D. R. LowTemperature Calorimetric and Vapor-Pressure Studies on Alkanediamines. J. Chem. Thermodyn. 1975, 7, 1029−1046.

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