Analysis of Uncertainty in the Calculation of Ideal-Gas Thermodynamic

Article pubs.acs.org/jced

Analysis of Uncertainty in the Calculation of Ideal-Gas Thermodynamic Properties Using the One-Dimensional Hindered Rotor (1-DHR) Model Ctirad Č ervinka,* Michal Fulem, Vojtěch Štejfa, and Květoslav Růzǐ čka Department of Physical Chemistry, University of Chemistry and Technology, Prague, Technická 5, CZ-166 28 Prague 6, Czech Republic S Supporting Information *

ABSTRACT: The uncertainty analysis of the calculation of ideal-gas thermodynamic properties using the one-dimensional hindered rotor (1-DHR), which is designed for quantum chemical treatment of internal rotations, is presented. The sensitivity of ideal-gas heat capacities and entropies to the input parameters of the 1-DHR model, that is, the reduced moments of inertia and barriers to internal rotation, is analyzed and evaluated. The calculations of energy barriers to internal rotations using the B3-LYP functional coupled with six basis sets and the empirical D3 correction accounting for dispersion interactions for a set of 60 molecular structures containing internal rotation motions were also performed, compared with available experimental values, and statistically evaluated to estimate the impact of uncertainty associated with the calculated barriers to internal rotation (15% on average, underestimated in two-thirds of cases) on ideal-gas thermodynamic properties.

1. INTRODUCTION Knowledge of ideal-gas thermodynamic properties is indispensable in many applications including thermochemistry, temperature adjustments of enthalpies of vaporization or sublimation, studies of solute−solvent interactions through the solvation heat capacities, equations of states, and thermodynamic correlations such as the multiproperty simultaneous correlation of vapor pressure and related thermal data1,2 used in our laboratory, etc. Direct (calorimetric) or indirect measurements (for example, speed-of-sound measurements) of ideal-gas heat capacities as well as ideal-gas entropies derived from experimental data on heat capacities of condensed phases, vapor pressure measurements, and a description or estimate of state behavior of fluid (often called third-law entropies in the literature), are nowadays very scarce, and most of the data are obtained by the methods of statistical thermodynamics using either experimental or calculated spectroscopic data. Statistical thermodynamic calculations of ideal-gas thermodynamic properties using molecular parameters, fundamental vibrations, and energy barriers to internal rotations obtained by quantum chemistry methods have become prevalent over the last years. Unfortunately, such calculations are often reported without their associated uncertainties, making them incomplete statements. The uncertainty of calculated ideal-gas thermodynamic properties arises from several sources, often occurring together3,4(i) uncertainty in calculated molecular parameters and vibrational frequencies and neglecting the vibrational anharmonicity, (ii) treatment of internal large amplitude motions (e.g., hindered © XXXX American Chemical Society

rotation, ring puckering) as harmonic vibrations, (iii) coupling of different types of molecular motions, and/or (iv) neglecting the existence of different molecular conformations. In our previous work,5 we evaluated the uncertainty of calculations of ideal-gas thermodynamic properties by the density functional theory (DFT) with the B3LYP functional for a large set of rigid molecules without internal rotations around a single bond and other large amplitude motions for which reliable reference data in the ideal gaseous state were available in the literature. For rigid molecules, the use of the rigid rotor− harmonic oscillator (RRHO) approximation for the calculation of ideal-gas thermodynamic properties is appropriate, and the main source of uncertainty remains the uncertainty in the calculated harmonic vibrations. A common practice to minimize this uncertainty is to bring the calculated harmonic frequencies closer to the experimental fundamental frequencies by applying an empirical scaling factor6 or better a pair of frequency dependent scale factors. It was concluded in ref 5 that relative uncertainties of calculated ideal-gas heat capacities g,0 Cg,0 for rigid molecules can be lowered by p and entropies S using a pair of frequency-dependent scale factors below 1% and 0.5%, respectively. For molecules containing internal rotations, the treatment of internal rotations using the harmonic oscillator approximation is known to be inappropriate except for internal rotations with Received: August 25, 2016 Accepted: November 16, 2016

A

DOI: 10.1021/acs.jced.6b00757 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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different approximations to calculate Ir, one can obtain values differing by up to 1 order of magnitude or even meaningless negative values. Also, East and Radom9 concluded, based on solving the Schrö dinger equation using the tables by Pitzer,10,11,18,19 that to reach an accuracy in Sg,0 acceptable from the thermodynamical point of view, it is advisible to lower the uncertainty in Ir and Vb below 8·10−49 kg·m2, and below 0.8 kJ·mol−1, respectively. In the overview of the 1-DHR model,7 several aspects influencing the resulting thermochemistry data are discussed. Pfaendtner et al.7 compared the performance of HF, MP2, and B3LYP methods with three different basis sets to predict potential energy profiles for backbone and methyl group rotations and the impact of the obtained differences on Sg,0. The authors studied also the effect of the angle step during the calculations of the V(φ) profile which can be important in the case of highly asymmetric profiles, while for symmetric rotors (such as methyl), it is sufficient to know the Vb and to approximate the profile by a cosine function. Pfaendtner et al.7 created their tool to solving the Schrö dinger equation numerically using the Fourier grid Hamiltonian (FGH) method20 and noticed that they observed a comparable sensitivity of resulting Sg,0 as East and Radom.9 In the literature, there are also works discussing and comparing various ways of description of the hindered internal rotations and corresponding differences from the harmonic oscillator model.21,22 Furthermore, there are several works studying the effect of coupling of individual internal rotations which use a multidimensional Schrö dinger equation,23−26 which, however, exceeds the scope of the present work. The phenomenon of the potential energy of the internal rotations has been studied for decades.27,28 The existence of a nontrivial V(φ) profile is determined by an interplay of steric hindrance, hyperconjugation effects, and dispersion and electrostatic forces.29 Recently, Pereira et al.30 published a broad study on the theoretical calculations of Vb for 43 organic molecules comparing several modifications of ab initio techniques, such as G3, MP2, and QCISD(T), including a comparison of Vb with experimental values and results of Vb calculations using different types of basis sets and higher-order correlation terms. In accord with some other works31−38 analyzing only a limited number of model compounds, Pereira et al.30 conclude that the uncertainty of their Vb calculations usually does not exceed the 4.2 kJ·mol−1 level (chemical accuracy). In this work, the impact of the individual sources of g,0 caused by the uncertainties of the contributions to Cg,0 p and S input parameters (mainly Ir and Vb) of the 1-DHR model is g,0 on the Ir evaluated. An analysis of the sensitivity of Cg,0 p and S and Vb, both varying within their expected physically meaningful ranges, is presented along with a broad statistical study on the accuracy of DFT calculations of Vb, which enabled us to draw some generalizing conclusions and to quantify the g,0 uncertainty of contributions of internal rotations to Cg,0 p and S within the 1-DHR model. Also, the results of several programs17,39,40 solving the one-dimensional Schrö dinger equation are compared and commented.

high energy barriers. One of the most successful methods for treating internal rotation is the one-dimensional hindered rotor (1-DHR) model, reviewed by Pfaendtner et al.7 More elaborated treatments of internal rotations can be found in the literature8 (an overview can also be found in refs 7 and 9), but given their high computational cost and complexity, they are rarely used for calculations of ideal-gas thermodynamic properties of organic species. Following the 1-DHR computational scheme, the harmonic vibrational frequency is excluded from the construction of the vibrational partition function, and a one-dimensional Schrödinger equation is solved instead for a given internal degree of freedom −

ℏ2 d2 Ψ(φ) + V (φ)Ψ(φ) = ε Ψ(φ) 2Ir dφ

(1)

where ℏ is the Planck constant h divided by 2π, Ψ is the wave function, ε is the energy, Ir is the reduced moment of inertia, and V(φ) is the potential energy profile of the internal rotation, and φ is the torsion angle. The energy levels obtained by solving eq 1 are used to calculate the partition function of the hindered rotation mode, required to calculate the contributions g,0 of the hindered rotation mode to Cg,0 p and S . As can be seen from eq 1, the 1-DHR model requires knowledge of the reduced moment of inertia of internal rotations Ir and potential energy profiles of internal rotations V(φ). Ir is calculated from molecular geometrical parameters; however, for asymmetric or multiple rotors its values are only approximate and dependent on the approximation employed. Several approximations of different complexity can be found in the literature.10−12 V(φ) can be obtained by quantum chemistry calculations by performing the potential energy scan along considered dihedral angle φ, usually with a step ranging from 5° to 30°. The V(φ) profiles are usually qualitatively well predicted, but the energy barrier heights are often not obtained with a sufficiently low uncertainty. The uncertainty in both properties Ir and V(φ) are simultaneously propagated throughout the 1-DHR model in the resulting ideal-gas thermodynamic properties. Finally, the Schrödinger equation, eq 1, for internal rotations has to be solved numerically. This step requires a reliable numerical technique to grant the correctness of obtained energy levels of such a motion and to avoid importing further uncertainty into the calculations of thermodynamic properties. Also, coupling of several internal rotation modes may play a significant role, especially when a rotating top (alkyl chain) consists of several more tops (methyl, ethyl, etc.). The 1-DHR is commonly employed for the calculation of ideal-gas thermodynamic properties of molecules with large amplitude motions13−15 and can provide the results of similar and often lower uncertainty than more complicated models.15,16 A few studies on the accuracy and appropriateness of the 1-DHR model can be found in the literature. In our recent work,17 we tested and statistically evaluated its performance for a large set of molecules containing one or two symmetrical rotors attached to a rigid frame concluding that the average absolute percentage deviation from the reference data amounted to 0.3% for Sg,0 and 1.5% and 0.5% for Cg,0 p at 300 and 1000 K, respectively. Such uncertainties can be regarded as satisfactory for many applications requiring the ideal-gas thermodynamic properties. East and Radom9 compared the particular approximations to Ir and concluded that their results do not differ by more than a few percent for symmetric tops. However, for highly asymmetric or coupled tops, employing

2. COMPUTATIONAL METHODS Since it is not possible to express the contribution of the g,0 analytically, a direct hindered rotation modes to Cg,0 p and S numerical summation of the first M terms of the partition function has to be used B



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Table 1. List of Studied Molecules, Corresponding Rotating Tops, and Energy Barriers to Internal Rotations Vb (in kJ·mol−1) molecule acetaldehyde acetyl chloride acetone allylacetate biacetyl buta-1,3-diene butyne 1-chloroethane 2-chloropropane diethyl amine 1,1-difluoroethane dimethyl ether ethane ethanol 1-fluoroethane glyoxal hexafluoroethane isopropenyl acetate isopropyl thiol methyl chlorodifluoroacetate 4-methyl benzaldehyde methoxy acetonitrile methyl amine 4-methyl isothiazole anti-o-cresol cis-propanal propane-2-ol propylene oxide styrene t-butyl isocyanate 1,1,1-trifluoroethane

top

a Vexp b

CH3 CH3 CH3 CH3 C2H3O C2H3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CHO CF3 CH3ac CH3 CH3 CHO OCH3 CH3 CH3 CH3 CH3 OH CH3 C2H3 NCO CH3

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

4.7 5.3 3.5 1.2 31.8 24.6 12.3 15.1 18.1 12.6 13.9 11.0 12.1 13.9 14.0 24.8 15.5 1.6 18.1 4.4 34.1 19.0 8.6 1.3 4.4 9.2 7.0 11.3 13.8 0.5 13.0

0.5 1.5 0.3 0.0e 0.8 0.2 2.0 0.6 0.5 0.0h 0.4 0.2 0.2 2.0 0.1 1.0 1.5 0.0i 2.0 0.2 2.0k 1.5 0.5 0.0m 0.2 0.9 1.5 1.7 0.2 0.0n 0.3

ref. 51 53 55 57 59 61 63 65 67 69 71 55 73 74 76 78 79 81 82 84 86 88 89 91 93 95 96 98 100 102 104

b Vcalc b

4.9 4.7 2.5 0.6 23.3 29.9 13.0 15.0 15.8 13.2 12.2 10.1 11.6 13.4 13.1 25.1 14.0 1.3 14.5 3.5 39.0 16.1 8.1 1.3 4.2 8.8 5.3 10.0 17.0 0.5 11.8

molecule acetamide acetic acid acrolein benzaldehyde biphenyl butane 1-chlorobut-2-ene 2-chloro toluene 1,2-dichloro ethane diethyl ketone dimethyl amine dimethyl sulfide ethanol ferrocene glycine hexafluoro acetone hydrazine isopropenyl acetate isoprene methyl acetate methyl ketene 2-methyl propene methanol c-N-methyl formamide phenol propane propene pyridinne-4-aldehyde 1,1,1-trichloro ethane trifluoromethyl isocyanate trimethyl amine

top

a Vexp b

CH3 CH3 C2H3 CHO C6H5 C2H5 CClH2 CH3 CClH2 CH3 CH3 CH3 OH C5H5 CNH4 CF3 NH2 CH3ip C2H3 CH3es CH3 CH3 CH3 CH3 OH CH3 CH3 CHO CH3 CF3 CH3

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.3 2.0 31.1 33.5 6.5 16.5 10.0 5.6 45.3 9.2 12.7 8.7 5.4 3.8 14.2 12.3 11.2 8.5 24.2 5.1 4.9 8.9 4.5 3.6 14.1 14.0 8.3 26.4 21.8 0.6 18.3

c

0.0 0.0d 6.0 2.0f 2.0 1.0 1.0 0.3 4.0g 0.4 0.7 0.7 0.7 1.3 1.3g 1.0 0.1 0.0j 0.5 0.7 0.2 0.2 0.0l 0.1 0.5 0.7 0.1 1.0k 1.3 0.0o 4.0

ref.

b Vcalc b

52 54 56 58 60 62 64 66 68 70 72 55 74 75 77 67 80 81 83 85 87 55 90 92 94 72 97 99 101 103 67

0.3 1.6 35.5 36.9 9.6 12.4 4.1 5.3 31.9 9.2 12.7 8.1 5.2 2.2 13.5 6.1 13.4 8.3 25.4 4.1 4.8 8.8 4.3 3.8 14.8 12.9 8.2 30.3 22.7 0.9 16.7

a exp Vb

= experimental values. They are given with their estimated expanded uncertainty (k = 2) and references to the primary literature sources. bVcalc b −1 d = calculated at the B3-LYP-D3/6-311+G(2df,p) level of theory, neglecting ZPVE based terms. cEstimated Uc(Vexp Reported b ) = 0.01 kJ·mol exp exp −1 e −1 f Uc(Vb ) = 0.005 kJ·mol Reported Uc(Vb ) = 0.01 kJ·mol Based on the polemic in ref 105, the values reported in refs 106 and 107 were h exp exp −1 i rejected. gUc(Vexp b ) estimated as the mean expanded uncertainty for the whole molecular set. Reported Uc(Vb ) = 0.02 kJ·mol . Reported Uc(Vb ) 105 108 l exp −1 k ) = 0.04 kJ·mol . Based on the polemic in ref, the value reported in ref was rejected. Reported U (V = 0.0001 kJ·mol−1. jReported Uc(Vexp b c b ) exp exp exp −1 m −1 n −1 o −1 = 0.0003 kJ·mol . Reported Uc(Vb ) = 0.001 kJ·mol . Reported Uc(Vb ) = 0.0004 kJ·mol . Reported Uc(Vb ) = 0.0001 kJ·mol .

Q ir =

1 σint

M

contain 62 internal rotation modes and whose experimental data on given rotation barrier heights were found in the literature. The CCCBDB database41 by NIST was used as the primary tool for searching the experimental data which were then refined using the original literature sources. Conservative estimates of the experimental uncertainty were made based on the uncertainties reported in the primary sources and the scatter of available literature experimental data. The Vb values were computed using software Gaussian 03,42 the B3-LYP functional within the DFT framework and the same set of six valence split basis sets43 as when determining the scale factors in our previous paper:5 6-31G(d), 6-31+G(d,p), 6-31G(2df,p), 6-311+G(d,p), 6-311+G(2df,p), and 6-311++G(3df,3pd). As can be seen, this set of basis sets contains various combinations of basis set size,44 polarization,45 and diffuse46 functions. As long as the DFT in its pure form is not capable of accounting for the dispersion interactions, which can affect the V(φ) profile significantly, especially for long or bulky flexible molecules, the calculations of Vb using the empirical D3 correction for the dispersion interactions parametrized by Grimme et al.,47 coupled with the 6-31G(d), and 6-311+G(2df,p) basis sets, as

∑ e−ε /kT j

j=1

(2)

where σint is the internal symmetry number of rotating top, k is the Boltzmann constant, T is the temperature, and εj are the energy levels obtained by solving the Schrödinger equation, eq 1. To study the sensitivity of the results of the 1-DHR model on the uncertainty of its input parameters, we evaluated the g,0 contributions of model hindered rotors to Cg,0 p and S , with either fixed Ir or Vb while the other parameter was varied. Using this procedure, we isolated the resulting uncertainty in final Cg,0 p and Sg,0 introduced by a given uncertainty in Ir and Vb. To solve the one-dimensional Schrödinger equation, eq 1, we used three distinct programs: our code17 and the program tool FGH-1D by Johnson,40 both performing the FGH method,20 and the code StatTD by Diky39 which is based on a variational procedure expanding the hindered rotor wave function in the set of basis functions of a free rotor. To evaluate the accuracy of theoretical calculations of Vb, a set of 60 organic molecules of various structures (including aliphatic and aromatic hydrocarbons, their derivates containing oxygen, nitrogen, sulfur, or halogen atoms; see Table 1) which C



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implemented in Gaussian 09,48 were also included in the current study. The Vb were evaluated as either (i) the difference of the total electronic energies of the ground state (a stable conformer) and of the saddle point on the hyper-surface of the molecular potential energy corresponding to a maximum of the onedimensional V(φ) profile, or (ii) the difference of the sums of total electronic and zero-point vibrational energies (ZPVE) of both structures (the ZPVE was evaluated within the harmonic approximation excluding the vibrational frequency of the torsional mode). The results of these two approaches were compared and statistically evaluated.

performed solving the following modified Schrö d inger equation: −

Cpir(X ) − Cpir(X 0) Cpir(X 0)

σS = 100

(4)

where Ieq r corresponds to the reduced moment of inertia of the most stable conformation, while a real varying Ir(φ) profile is used to scale the V(φ) profile to account for the variation of Ir during the internal rotation. A testing case of an ethyl rotor in butane molecule was selected to illustrate the effect of neglecting the variation of Ir on the resulting uncertainty of Cirp and Sir. Results of this analysis are summarized in Figure 1 where it can be seen that Vb can be modified by tens of percent leading to an uncertainty in Cirp and Sir of the order of units of percent.

3. RESULTS AND DISCUSSION 3.1. Sensitivity Analysis of the 1-DHR Model to the Input Parameters. In this section, the evaluation of the impact of uncertainties in the input quantities of the 1-DHR model, the reduced moments of inertia of internal rotation Ir, and energy barriers to internal rotation Vb, to internal rotation contributions to ideal-gas heat capacity Cirp and entropy Sir is presented. The impacts of uncertainties in Ir and Vb are separated by fixing one of these quantities and varying the other throughout the whole range of values typical for common chemical species. The introduced uncertainties to Cirp and Sir are expressed as relative percentage deviations σC = 100

I (φ ) ℏ2 d2 Ψ′(φ) + r eq V (φ)Ψ′(φ) = ε′Ψ′(φ) eq 2Ir dφ Ir

S ir(X ) − S ir(X 0) S ir(X 0) (3)

where X is either Ir or Vb and X0 denotes its selected reference values (hereafter denoted as I0r and V0b). 3.1.1. Sensitivity of Cirp and Sir to Uncertainty in Ir. In the simplest casea molecule containing only one symmetric rotating topthe Ir value obtained by the relation of Pitzer and Gwinn is exact.10 The only uncertainty in Ir here arises from the computed optimized molecular geometry, which usually differ negligibly from the experimental structure for organic molecules on the B3-LYP level of theory. In a general case, the exact analytic value of Ir is inaccessible, so that the relations for Ir proposed by Pitzer and co-workers11,12 provide only estimates for asymmetric, multiple, or coupled rotors. Several approaches applicable to estimate Ir at different levels of approximation can be found in the literature as overviewed by East and Radom.9 Various formulas provide different values of Ir, and their scatter can reach even an order of magnitude, presenting thus a significant source of uncertainty in Cirp and Sir values, as shown by East and Radom9 and by Pfaendtner et al.7 Another uncertainty may arise from the fact that Ir varies during the internal rotation. For symmetric methyl rotors in ethane and butane, the relative variance does not exceed a few units of percent for any possible conformation. However, for Ir of asymmetric ethyl rotor in butane, a variance amounting to 40% during its rotation was observed in this work. Incorporation of a Ir(φ) dependence into eq 1 imposes a need for a more complex numerical method for its solution which is out of scope of this paper. In practical calculations, this issue can be partially overcome by using the conformation mixing models3 and evaluating Ir for all conformers.49,50 For simplicity, we assume a constant Ir during the internal rotations equal to the value in the most stable conformation throughout this paper. An estimation of the accuracy of this approximation can be

Figure 1. Illustration of the effect of variation of Ir during the internal rotation of the ethyl group in butane (φ = π corresponds to the most stable all trans conformer). Upper left panelcalculated V(φ) profile; upper right panelcalculated Ir(φ) profile; bottom left panelscaled Vsc(φ) profile accounting for Ir(φ) variation; bottom right panel comparison of contributions Cirp (red) and Sir (blue) calculated either neglecting (X) or assuming (Xsc) variation of Ir(φ).

Below, the impact of uncertainty in fixed Ir to Cirp and Sir is evaluated. The internal rotation barrier was successively fixed at several levels, and Cirp and Sir were calculated for a range of Ir values from (0.1 to 10)·10−46 kg·m2 for each barrier height. I0r equal to 1.0·10−46 kg·m2 was chosen as this value lies roughly in the middle of the range of physically meaningful values. A comparison of Cirp and Sir obtained with altered Ir with those obtained with the reference value I0r is represented by relative percentage deviations σC and σS which are plotted as a function of Ir in Figures 2 and 3, respectively. The analysis was performed for temperature of 300 K. As can be seen from Figure 2, an overestimation of Ir leads to positive deviations σC (eq 3), while an underestimation of Ir has the opposite effect. Furthermore, Figure 2 reveals that σC strongly depends on the assumed Vb. In general, higher values of Vb result in higher sensitivity of Cirp to Ir. This is in accord with the fact that the contribution of a free rotor to ideal-gas heat capacity is a universal value of 1/2 R (R = 8.3144598 J· K−1·mol−1) independent of the Ir value. It can be easily deduced D



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uncertainty in Ir up to a few tens of percent is still acceptable for the same uncertainty target for the calculations of Cirp . 3.1.2. Sensitivity of Cirp and Sir to the Uncertainty in Vb. An analogous procedure as in the previous section was followed to obtain data on the sensitivity of Cirp and Sir to the uncertainty in Vb. Ir was successively fixed at several values, and Cirp and Sir were evaluated for a range of Vb. The V(φ) profile was modeled as a cosine function with a 3-fold symmetry as for the most common methyl top. A reference value for Vb was chosen as V0b = 12 kJ·mol−1. This value corresponds approximately to Vb in ethane which can be considered as a medium size value suitable for treating the internal rotation by the 1-DHR model. The deviations σC and σS obtained at 300 K are displayed in Figures 4 and 5.

Figure 2. Sensitivity of Cirp (Ir) on the uncertainty in Ir, expressed as σC (see eq 3 for exact definition) as a function of Ir at 300 K. Legend of particular Vb: purple ■, 1 kJ·mol−1; blue ⬟, 8 kJ·mol−1; gray ●, 12 kJ· mol−1; red ▲, 16 kJ·mol−1; green ▼, 20 kJ·mol−1 and black ◆, 30 kJ· mol−1.

Figure 4. Sensitivity of Cirp (Vb) on the uncertainty in Vb, expressed as σC (see eq 3 for exact definition) as a function of Vb at 300 K. Legend of particular Ir: black ■, 1·10−47 kg·m2; blue ●, 2·10−47 kg·m2; gray ▲, 5·10−47 kg·m2; red ▼, 1·10−46 kg·m2 and green ◆, 5·10−46 kg·m2.

Figure 3. Sensitivity of Sir(Ir) on the uncertainty in Ir, expressed as σS (see eq 3 for exact definition) as a function of Ir at 300 K. Legend of particular Vb: purple ■, 1 kJ·mol−1; blue ★, 8 kJ·mol−1; gray ●, 12 kJ· mol−1; red ▲, 16 kJ·mol−1; green ▼, 20 kJ·mol−1 and black ◆, 30 kJ· mol−1.

that, with increasing temperature, the sensitivity shall decrease as the higher thermal energy of the molecule enables the internal rotation to take place with less hindrance. Concerning the magnitude of σC deviations, applying Ir = 2 I0r leads to an overestimation of Cirp by several units of percent while an underestimation of Ir by a factor of 2 yields a several times higher uncertainty of Cirp , showing a nonlinear behavior. Figure 3 illustrates the sensitivity of Sir to the uncertainty in Ir represented by the deviations σS (eq 3). It is clear that the overestimation of Ir leads again to positive σS and vice versa. The Ir sensitivity increases with increasing Vb as well. However, two differences compared to the Ir sensitivity of Cirp can be noted. Sir exhibits a larger Ir-sensitivity with the deviations σS amounting to several tens of percent even when the Ir differs from I0r only by a factor of 2. Moreover, substantial deviations σS occur already for a free rotation. To conclude, when calculating the ideal-gas thermodynamic properties using the 1-DHR model, stricter requirements on the accuracy of Ir arise with increasing Vb, and in the case of Sg,0, rather than Cg,0 p . For example, to reach σS lower than 3% the uncertainty in Ir should be lower than 4−8%, while the

Figure 5. Sensitivity of Sir(Vb) on the uncertainty in Vb, expressed as σS (see eq 3 for exact definition) as a function of Vb at 300 K. Legend of particular Ir: black ■, 1·10−47 kg·m2; blue ●, 2·10−47 kg·m2; gray ▲, 5· 10−47 kg·m2; red ▼, 1·10−46 kg·m2 and green ◆, 5·10−46 kg·m2.

Figure 4 reveals that, in calculations with Vb > V0b, an underestimation of Vb would lead to an underestimation of Cirp , while for Vb < V0b, an overestimation of Vb would lead to an underestimation of Cirp with roughly doubled sensitivity. This happens because Cirp reaches its maximum value at around 300 K exactly for the V0b level (when using reasonable Ir values close to or above 5·10−47 kg·m2). As can be seen in Figure 6, the maximum of Cpir moves to higher Vb with increasing temperature, making the first mentioned trend more probable to occur. Overall, using of Vb differing from the V0b by a factor of E



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FGH method as given in ref 20 without modifying the Hamiltonian discretization procedure for even values of HN. Temperature-dependent illustration of the convergence rate is given in Figure S2. It should be noted that temperaturedependent deviations of Cirp and Sir from a reference data set are always bound together by basic thermodynamic relations expressing Sir change over a temperature interval and Cirp . In particular, Sir exhibits maximum (or minimum) deviations at temperatures at which Cirp heat capacity exhibits zero deviations, as illustrated in Figure S2. 3.3. Computations of Barriers to Internal Rotations. A broad statistical study on the accuracy of the DFT calculations of Vb is presented in this section since the uncertainty in calculated Vb represents one of the largest uncertainty terms in the 1-DHR model. A set of 60 molecules was examined and obtained results analyzed and compared to relevant experimental data (obtained by high resolution rotational microwave or Raman vibrational spectroscopy) and their estimated uncertainties (see Table S1 in the Supporting Information for a complete comparison of calculated and experimental Vb data). Molecules included in this study are summarized in Table 1 along with experimental data on Vb (their uncertainties and values calculated on the B3-LYP-D3/6-311+G(2df,p) level of

Figure 6. Temperature trends of Cirp (T,Vb) term for Ir = 5·10−47 kg·m2 and selected Vb values: black, 2 kJ·mol−1; red, 5 kJ·mol−1; blue, 12 kJ· mol−1; green, 20 kJ·mol−1 and purple, 30 kJ·mol−1.

2, may cause an error in the resulting Cirp up to 30% depending on the particular combination of V0b and T. Figure 5 illustrates the sensitivity of Sir to Vb. A monotonous declining trend can be observed for the deviations of Sir obtained for Vb altered from the reference values V0b. In other words, an overestimation of Vb causes an underestimation of Sir and vice versa. The lower the value of Ir, the steeper increase of σS can be observed. In other words, the entropy sensitivity to Vb increases with decreasing values of Ir. The magnitude of sensitivity of Sir to the uncertainty of Vb is larger by a factor of two or three compared to that of Cirp as opposed to the sensitivity to Ir which exhibits an order of magnitude higher sensitivity for Sir when compared to Cirp . 3.2. Programs for the Solution of the One-Dimensional Schrödinger Equation. The numerical solution of the one-dimensional Schrödinger equation, eq 1, is an important step which can influence the overall accuracy of resulting Cirp and Sir whose evaluation is based on the energy levels of the hindered rotation. To prove the appropriateness of the codes commonly used for this purpose, we compared the results obtained by our code,17 the program tool FGH-1D by Johnson,40 and the code StatTD by Diky.39 All three codes yield practically identical results (converged contributions Cirp and Sir do not differ by more than 0.2%) for symmetric tops, for which the V(φ) profile can be approximated by a n-fold cosine function. However, the StatTD39 code yields results significantly differing (lower by a factor of 2 for large Vb) from the other two codes (which give the same results) for asymmetric tops, for which the V(φ) profile has to be constructed using both sine and cosine terms. For more details on this comparison, see Figure S1 in the Supporting Information. Besides comparing the performance of given codes, we used our FGH-based code to test the convergence of eigenvalues of eq 1 and resulting thermodynamic properties with respect to the coarseness of the Hamiltonian discretization and to the corresponding size of Hamilton matrix HN. A negative correlation of the convergence rate with the Vb value was observed; however, the convergence was generally fast. Fully satisfactorily converged Cirp and Sir for neopentane (methyl Vb ≈ 16 kJ·mol−1) were obtained for HN = 5001, corresponding to an energy cutoff Ec ≈ 1.6·105 RT at 300 K. When compared to Cirp for neopentane culminating at around 400 K obtained for HN = 10 001, the results differed by 0.03, 0.01, and 0.005 J K−1 mol−1 for HN = 201, 501, and 1001, respectively. Concerning Sir, the same differences culminating at around 200 K amount to 0.07, 0.03, and 0.015 J K−1 mol−1 for HN = 201, 501, and 1001, respectively. For species exhibiting lower Vb, even faster convergence was observed. Note that using the odd values of HN is of utmost importance for the original formulation of the

Figure 7. Mean absolute percentage deviations σabs V (see eq 5 for the exact definition) calculated using different levels of theory from corresponding experimental data. Data set legend: black, neglecting the ZPVE term; gray, including the ZPVE term.

theory). Figure 7 shows the comparison of resulting mean absolute percentage deviations σabs V , defined as σVabs =

100 N

N

∑ i=1

exp |V b,calc i − V b, i |

V b,expi

(5)

where N is the number of studied internal rotation modes (N = exp 62) and Vcalc b and Vb stand for the calculated and experimental Vb, respectively. The values of σabs V are compared for used eight levels of theory, each combined with either neglecting or including the ZPVE; see Table S2 for the precise values. The ZPVE terms were computed in the harmonic approximation using calculated fundamental frequencies of the ground state conformer and the rotation barrier state, excluding the frequency corresponding to the internal rotation mode. Among the B3-LYP calculations without the D3 dispersion corrections, the most accurate Vcalc b were obtained with the 6311+G(2df,p) and 6-311++G(3df,3pd) basis sets, both yielding σabs V equal to 16%, while the other smaller studied basis sets yielded σabs V only slightly higher, not exceeding 21%. After the inclusion of the D3 correction, the corresponding σ Vabs decreased to 19% for the 6-31G(d) and to 15% for the 6F



Article

311+G(2df,p) basis set (the decrease is of the order of one or two percentage points). The differences among σVabs for individual levels of theory are rather insignificant, and the inclusion of the D3 correction does not lead to any dramatic on average. The more improvement of the quality of Vcalc b appropriate approach of evaluating Vcalc b , which takes into account also the ZPVE-based terms, does not necessarily have to yield lower σabs V , as can be seen for the smallest 6-31G(d) basis set and both studied D3-corrected levels of theory. For other examined levels of theory, the inclusion of the ZPVE based terms lowered the σabs V by one or two percentage points, while the most significant improvement was observed for the 631+G(d,p) basis set (by five percentage points). Except the results obtained by the 6-31G(d) basis set, all σabs V for the calculations including the ZPVE terms amount to 16% or less and do not exhibit any significant differences among the studied basis sets (the lowest σabs V = 15% was obtained for the 6311+G(2df,p) basis set). The worsening of the quality of Vcalc b for D3-corrected ZPVE including calculations (by one or two percentage points) is a consequence of some coincidental compensation of errors caused by neglecting the ZPVE and the error in the calculated electronic energies. Moreover, the vibrational frequencies exhibited by both conformations, corresponding to an energetic minimum and maximum, usually differ by a few tens of cm−1 in the case of the deformation modes (the structures of higher energy exhibit usually higher frequencies) but differ by no more than several units of cm−1 in the case of the high frequency bond stretching modes (the differences are rather unbiased). For the 6-311+G(2df,p)-D3 level of theory, the ZPVE based correction to Vcalc b was positive in roughly two-thirds of cases. Since the dominant contribution to the ZPVE is determined by the high frequency modes, the ZPVE based terms play rather a minor role compared to the difference of the two conformations in the electronic energy. Despite it is not fully correct to neglect the ZPVE effect on Vcalc b , it is commonly used in the literature. This fact can be partially justified by the analysis given above. For both studied D3 corrected levels of theory, the effect of including the thermal contributions to molecular enthalpy was also studied. For this purpose, the difference of enthalpic terms was calculated from the unscaled vibrational frequencies (excluding the value for the torsional mode) of both structures using the harmonic oscillator model. Due to the mentioned prevailing trends of the frequency changes connected with the rotation between the conformations of minimum and are maximum energy, these thermal contributions to Vcalc b negative in almost all cases, which further worsens the with Vexp agreement of the already underestimated Vcalc b b . abs , σ Namely, after including the thermal contributions to Vcalc b V increased by two and eight percentage points for the 6-31G(d)D3 and 6-311+G(2df,p)-D3 levels of theory, respectively. Taking into account the results of the current study, the computational cost and the average accuracy of given calculations, the 6-311+G(2df,p)-D3 level of theory can be recommended for fast, but relatively accurate calculations of the Vcalc b of organic molecules. Figure 8 shows the bias of the mean percentage deviations σV, defined as σV =

100 N

N

∑ i=1

Figure 8. Bias ± of percentage deviations σV (see eq 6 for exact definition) of calculated Vb from corresponding experimental data calculated using different levels of theory. Data set legend: black, neglecting ZPVE term; gray, including ZPVE term.

The obtained σV exhibit an almost unbiased behavior for the 6-31G(d) basis set, regardless of the D3 correction. The larger studied basis sets yield σV overestimated in roughly one-third of the cases, while including the ZPVE terms increases the fraction of the overestimated Vcalc b by up to ten percentage points (the only decrease by a few percentage points was observed for the 6-311+G(2df,p) basis set). Still, the σV are negative in 55−70% of cases for basis sets larger than 6-31G(d). Several features of the agreement of Vcalc with Vexp b b , always common for similar molecular types, can be concluded from analyzing the results obtained at the B3-LYP/6-311+G(2df,p) level of theory. The Vb are generally very low for compounds containing methyl rotors attached to a carbonyl group (usually fairly below 5 kJ·mol−1). The Vcalc of saturated halogen b derivatives of alkanes are underestimated by roughly 10%. The Vcalc of methyl rotors are underestimated for all studied b compounds, when ZPVE and thermal terms are neglected. On the other hand, in a case of high Vb, fairly exceeding 20 kJ· mol−1, which occurs typically for unsaturated tops (carbonyl or vinyl groups) forming conjugated π-systems with the rest of the molecule (typically an aromatic frame), the Vcalc are overb estimated by about 15%, which points to an overestimation of the effect of delocalization of the conjugated electronic πsystems within planar sp2 structures on the B3-LYP level of theory. Knowing the mean accuracy of the Vcalc calculations and b being equipped with the results of the Vb sensitivity study, the uncertainty of calculated contributions of the internal rotations to the thermodynamic properties can be estimated. It can be concluded from these analyses that a mean uncertainty in Vcalc b of 15% causes an uncertainty in Cirp and Sir amounting up to 5% and 20%, respectively, depending on the particular Ir value. Since Cirp and Sir are not the dominant terms, the uncertainty in these contributions gets diluted in the total value of the thermodynamic properties. For Cg,0 p , the vibrational term is the dominant one (with exception of very low temperatures), while for entropy the translation term is significant as well. Therefore, the uncertainty Cirp and Sir in order of tens of percent yields a g,0 total uncertainty of Cg,0 p and S in order of a few units or rather tenths of percent which is comparable to the uncertainty caused by the calculated vibrational frequencies.

4. CONCLUSIONS The sensitivity of the contributions of internal rotations to ideal-gas thermodynamic properties (Cpir and Sir) to the uncertainty in the inputs of the 1-DHR model, that is, the reduced moment of inertia Ir and energy barriers to internal

exp (V b,calc i − V b, i )

V b,expi

(6) G



Article

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) Frenkel, M. L.; Kabo, G. J.; Marsh, K. N.; Roganov, G. N.; Wilhoit, R. C. Thermodynamics of Organic Compounds in the Gas State; CRC Press: College Station, 1994; Vol. 1, p 2. (4) Irikura, K. K.; Frurip, D. J. Computational Thermochemistry: Prediction and Estimation of Molecular Thermodynamics; American Chemical Society: Washington, DC, 1998. (5) 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. (6) Merrick, J. P.; Moran, D.; Radom, L. An evaluation of harmonic vibrational frequency scale factors. J. Phys. Chem. A 2007, 111, 11683− 11700. (7) Pfaendtner, J.; Yu, X.; Broadbelt, L. J. The 1-D Hindered Rotor Approximation. Theor. Chem. Acc. 2007, 118, 881−898. (8) Vansteenkiste, P.; Van Neck, D.; Van Speybroeck, V.; Waroquier, M. An Extended Hindered-Rotor Model with Incorporation of Coriolis and Vibrational-Rotational Coupling for Calculating Partition Functions and Derived Quantities. J. Chem. Phys. 2006, 124, 044314. (9) 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. (10) 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. (11) 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. (12) Kilpatrick, J. E.; Pitzer, K. S. Energy Levels and Thermodynamic Functions for Molecules with Internal Rotation. 3. Compound Rotation. J. Chem. Phys. 1949, 17, 1064−1075. (13) Vansteenkiste, P.; Van Speybroeck, V.; Marin, G. B.; Waroquier, M. Ab Initio Calculation of Entropy and Heat Capacity of Gas-Phase n-Alkanes Using Internal Rotations. J. Phys. Chem. A 2003, 107, 3139− 3145. (14) 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. (15) 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. (16) Van Speybroeck, V.; Vansteenkiste, P.; Van Neck, D.; Waroquier, M. Why Does the Uncoupled Hindered Rotor Model Work Well for the Thermodynamics of n-Alkanes? Chem. Phys. Lett. 2005, 402, 479−484. (17) Č ervinka, C.; Fulem, M.; Růzǐ čka, K. Evaluation of Uncertainty of Ideal-Gas Entropy and Heat Capacity Calculations by Density Functional Theory (DFT) for Molecules Containing Symmetrical Internal Rotors. J. Chem. Eng. Data 2013, 58, 1382−1390. (18) Pitzer, K. S. Thermodynamic Functions for Molecules Having Restricted Internal Rotations. J. Chem. Phys. 1937, 5, 469−472. (19) Li, J. C. M.; Pitzer, K. S. Energy Levels and Thermodynamic Functions for Molecules with Internal Rotation. 4. Extended Tables for Molecules with Small Moments of Inertia. J. Phys. Chem. 1956, 60, 466−474. (20) 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. (21) Zheng, J. J.; Mielke, S. L.; Clarkson, K. L.; Truhlar, D. G. MSTor: A Program For Calculating Partition Functions, Free Energies, Enthalpies, Entropies, And Heat Capacities Of Complex

rotations Vb, was studied. Generally, the higher the Vb, the more strict requirements on Ir accuracy should be considered, which holds especially for asymmetric tops whose Ir vary considerably during the internal rotation. Sg,0 usually exhibits a higher sensitivity on Ir and Vb than Cg,0 p . A thorough study of the accuracy of DFT computations of Vb was performed for a set of 60 molecules and six basis sets, eventually coupled with the empirical D3 dispersion corrections. The B3-LYP/6311+G(2df,p) level of theory can be assessed as the most reliable (regardless of the D3 correction), at the same time requiring an affordable computational cost, yielding Vbcalc burdened with an uncertainty below 15% on average. This seems to be the maximum accuracy of calculations of Vcalc b reachable when using the B3-LYP functional coupled with split valence Pople basis sets. Including the ZPVE term to calculations of Vcalc b did not lead to any significant improvement abs of Vcalc (σ decreased by 2 percentage points in the best cases). b V To decrease the uncertainty of Vcalc b any further, a more costly and powerful ab initio level of theory would need to be used. The obtained average uncertainty of Vcalc around 15% is b expected to cause an uncertainty of 5% and 20% in Cirp and Sir, respectively. These values have to be taken as considerable and represent the dominant contribution to the uncertainty within the 1-DHR model for evaluating the thermodynamic properties.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.6b00757. (1) Illustration of the convergence rate of the FGH method; (2) list of calculated and experimental barriers to internal rotation for individual molecules; (3) comparison of performance of particular levels of theory concerning Vb calculations (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +420 220 444 116. Address: Technická 5, 166 28 Praha 6, Czech Republic. Funding

The authors acknowledge financial support from the Czech Science Foundation (GACR No. 15-07912S) and specific university research (MSMT No. 20/2016). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Vladimir Diky for providing the program StatTD. The access to computing and storage facilities owned by parties and projects contributing to the National Grid Infrastructure MetaCentrum, provided under the program “Projects of Large Infrastructure for Research, Development, and Innovations” (LM2010005), and the CERIT-SC under the program Centre CERIT Scientific Cloud, part of the Operational Program Research and Development for Innovations, Reg. no. CZ.1.05/ 3.2.00/08.0144 is highly appreciated.

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