Accurate evaluation of combustion enthalpy by ab-intio computations

The details of the computed molecular enthalpies at the different levels of theory employed in this work are reported in Table 1, together with the experimental values.

Using the QM-evaluated enthalpies corrected for phase change enthalpies of water and reactants, the predicted combustion enthalpies yielded AAD, MUE% and correlation coefficient of 11.94 kJ / mol, 0.40%, and 0.99999, respectively, for the CCSD (T) -F12b computations, and 13.29 kJ / mol, 0.44%, and 0.99998, respectively, for the DSD-PBEB86 computations. A comparison of the computed and reference combustion enthalpies are depicted in Fig. 1.

Figure 1

Comparison of theoretically predicted and experimentally determined combustion enthalpies. The data shown are from the CCSD (T) -F12 computations, because the DSD-PBEB86 values ​​are visually indistinguishable on the plotted range of enthalpy values.

These results, which are directly obtained by ab-initio computation without any empirical correction, show a remarkable improvement compared to results reported in previous studies. For example, the theoretically calculated combustion enthalpies reported by Mazzuca et al.22 yielded a MUE of roughly 3%, even after applying an empirical scaling. According to the results, taking into account the high pressure impacts the combustion enthalpy only marginally, and yields an improved AAD of predicted results at only 0.01 kJ / mol. Similarly, the hindered rotor correction improves the AAD of the predicted combustion enthalpies at only 0.088 kJ / mol.

The results reported in Table 1 show that the accuracy of the employed level of theory plays a key role. To further demonstrate the importance of the applied level of theory, we also computed the combustion enthalpies at the B3LYP / 6-311 + G (2d, p) level of theory for the same set of molecules. The B3LYP / 6-311 + G (2d, p) computations yielded AAD, MUE%, and correlation coefficient of 104.35 kJ / mol, 3.93%, and 0.9988, respectively, which are roughly one order of magnitude less accurate than those obtained with DSD-PBEP86-D3 / def2-QZVP or CCSD (T) -F12b / def2-QZVP.

Analyzing the computed energies shows that molecular thermal energies, ie the kinetic energy due to rotation / translation and vibrational energies, contribute on average only 0.625% and 0.541% to the computed combustion enthalpies, and the changes in ground state electronic energies of reactants and products are the main contributions to the heat released by combustion. Thus, the accuracy of the employed level of theory in reproducing the ground state electronic energy plays the key role for the accuracy of the obtained results. For the electronic energies evaluated with DSD-PBEP86-D3 / def2-QZVP and B3LYP / 6-311 + G (2d, p), we observed an AAD of 148.425 kJ / mol between the ground state electronic energies obtained with these two DFT methods , while for thermal energies the AAD was only 0.781 kJ / mol. These results also reveal why the accuracy of theoretical methods for combustion reactions is so different from the benchmark results obtained for other case studies. The reason is that the large amount of energy released by combustion reactions is mainly due to electronic energies, which implies substantial differences between electronic energies of reactants and products.

The DSD-PBEP86-D3 / def2-QZVP level of theory used in the present study supersedes most of the conventionally accepted functionalals in studying thermochemistry28. In comparison to the computations by the computationally much more demanding CCSD (T) -F12 computations, which are considered to be a gold standard in theoretical chemistry37DSD-PBEP86-D3 yields only marginally (by 0.04%) lower accuracy in the predicted combustion enthalpies, and therefore provides an excellent cost-efficiency ratio.

Next to the accuracy of the employed QM level of theory, another important source of inaccuracy in theoretically evaluated combustion enthalpies can arise from the use of high-energy conformers instead of the global minimum-energy structure. As for almost all poly-atomic molecules, several local minima exist on the potential energy surface, and thus geometry optimizations started from different initial structures can result in quite diverse conformers and energies, and consequently in different computed combustion enthalpies. As an example, our theoretical computations on the two locally optimized structures of acetic acid, corresponding to different orientations of the hydroxyl proton relative to the second oxygen atom of the carboxylate group (inward- versus outward-pointing) yield quite different combustion enthalpies. While the low-energy structure yields a combustion enthalpy with 8.64 kJ / mol absolute error, the same computation for the higher-energy structure deviates from the experimental value at 29.76 kJ / mol. Inaccuracies from such high-energy conformers can be avoided by employing efficient general global optimization algorithms or rotamer searches38 or, for small molecules, using a systematic conformer search via multi-start optimization, as was done in the present study.

Yet another reason of deviation between the QM predicted and optimum enthalpies can be overlooking non-ideality effects. As discussed earlier, increasing the ambient pressure can directly influence the phase change and gas phase enthalpies, while QM enthalpies are computed for molecules in vacuo. We studied the impact of pressure on gas phase enthalpies via Eq. (5). However, this correction was found to only marginally improve the accuracy of the predicted combustion enthalpies, as can be seen in Table 1. The more significant impact of the ambient pressure on gas phase enthalpies can be attributed to the formation of molecular clusters in the gas phase at high pressures. For example, for accurate evaluation of the phase change enthalpy and the saturation vapor pressure of water, it has been shown that clustering of molecules in the gas phase should be taken into account39. Such gas phase clustering reduces the gas phase enthalpy compared to the in vacuo state. Similarly, partial condensation of water molecules23 as well as dissolution of CO2 in the water produced in the combustion process or formation of combustion side-products other than CO2 can result in further deviations between ab-initio computation and experiment. One empirical way to take such effects into account is scaling the enthalpies of H2O or CO2 or both. Accordingly, we found the optimal scaling factor of 0.9999857 for empirically correcting the theoretically predicted enthalpy of water in CCSD (T) -F12b computations, which reduced the AAD (MUE%) to 5.80 kJ / mol (0.26%). Scaling the enthalpy of CO2 computed by CCSD (T) -F12b by 0.999994328 even reduces the AAD (MUE%) of predicted combustion enthalpies to 2.64 kJ / mol (0.15%). Yet further improvement of the results can be achieved by simultaneously scaling the enthalpies of H2O and CO2 computed by CCSD (T) -F12b at 1.000006465 and 0.999992212, which yields AAD (MUE%) or 2.00 kJ / mol (0.12%). These scaling factors are derived from the enthalpies evaluated at the CCSD (T) -F12b level of theory, which might necessitate their re-evaluation for other levels of theory. However, we speculated that, at least for methods that provide results similar to CCSD (T) -F12b, the scaling factors might not strongly depend on the level of theory applied. Indeed, using the (unchanged) scaling factors obtained by CCSD (T) -F12b for the enthalpies computed with DSD-PBEP86-D3 / def2-QZVP yields similar improvements, with AAD (MUE%) values ​​of 8.69 kJ / mol (0.33% ), 5.21 kJ / mol (0.21%), and 3.70 kJ / mol (0.15%), obtained via scaling the computed enthalpies of H2O, CO2and both of them simultaneously, respectively.

In addition to the inaccuracies resulting from the theoretical computations, systematic or operational errors in experimental data can also contribute to inconsistency between the theoretical and experimental reference data. For example, we observed 1.59 kJ / mol AAD in phase change enthalpies of our studied reactants between the NIST and DIPPR databases, which results in the same deviation between the theoretically predicted gross combustion enthalpies calculated using each one of these two databases. Similar to the vaporization enthalpy, the experimentally determined combustion enthalpies from different sources also show some variations. For example, slight inaccuracy in measuring the combustion enthalpy of benzoic acid, which is used to calibrate the calorimeter23, can result in a linearly distributed deviation (offset) between measured combustion enthalpies of all other compounds. That can be a potential reason for the suitability of a linear fitting to empirically correct the predicted combustion enthalpies, proposed in several studies21.22.

In summary, in the present study, we discuss ab-initio quantum chemistry approaches capable of providing highly accurate predictions of combustion enthalpy. To that end, the main considerations in theoretical computations should be directed towards selecting an appropriate level of theory for the quantum chemistry method applied, and carefully identifying the minimum-energy conformers. For reproducing the net heat of combustion, the phase change enthalpy of the reactants should be subtracted from the QM-evaluated gas-phase enthalpies. For the gross heat of combustion, the vaporization enthalpy of water should also be subtracted from the QM-evaluated gas-phase enthalpy of water. Accordingly, taking the phase change enthalpies, as well as the experimental measurement of combustion enthalpies, into consideration or not can also contribute to inconsistencies between the theoretically predicted and experimentally determined combustion enthalpies.

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