# Accurate evaluation of the enthalpy of combustion by ab-intio calculations

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

Using QM-evaluated enthalpies corrected for water and reactant phase change enthalpies, the predicted combustion enthalpies gave AAD, MUE% and a correlation coefficient of 11.94 kJ/mol, 0.40 % and 0.99999, respectively, for CCSD(T)-F12b calculations, and 13.29 kJ/mol, 0.44% and 0.99998, respectively, for DSD-PBEB86 calculations. A comparison of calculated and reference enthalpies of combustion is shown in Fig. 1.

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

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

The analysis of the calculated energies shows that the molecular thermal energies, i.e. the kinetic energy due to the energies of rotation/translation and vibration, contribute on average only for 0.625% and 0.541% to the enthalpies of combustion calculated, and changes in the ground state electronic energies of the reactants and products are the major contributors to the heat released by combustion. Thus, the accuracy of the level of theory used to reproduce the ground state electronic energy plays the key role for the accuracy of the obtained results. For the electron 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 electron 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 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 the 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 functions in the study of thermochemistry^{28}. Compared to the calculations of the much more computationally demanding CCSD(T)-F12 calculations, which are considered a gold standard in theoretical chemistry^{37}DSD-PBEP86-D3 gives only a slightly lower accuracy (by 0.04%) in predicted enthalpies of combustion, and therefore provides excellent cost-effectiveness.

In addition to the accuracy of the QM theory level used, another significant source of inaccuracy in the theoretically evaluated enthalpies of combustion can arise from the use of high-energy conformers instead of the overall minimum-energy structure. As with almost all polyatomic molecules, multiple local minima exist on the potential energy surface, and so geometric optimizations from different initial structures can result in quite diverse conformers and energies, and hence enthalpies of combustion different calculations. As an example, our theoretical calculations 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 (pointing inward versus outward ) give quite different enthalpies of combustion. While the low energy structure gives an enthalpy of combustion with an absolute error of 8.64 kJ/mol, the same calculation for the higher energy structure deviates from the experimental value of 29.76 kJ/mol. The inaccuracies of these high energy conformers can be avoided by using efficient global optimization algorithms or rotamer searches^{38} or, for small molecules, using a systematic search for conformers via multi-boot optimization, as was done in the present study.

Yet another reason for the discrepancy between optimal and QM-predicted enthalpies may be the neglect of non-ideality effects. As mentioned earlier, the increase in ambient pressure can directly influence the phase change and gas phase enthalpies, while QM enthalpies are calculated for molecules in vacuum. We investigated 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 shown in Table 1. The most significant impact of ambient pressure on gas-phase enthalpies can be attributed to the formation of molecular clusters in the gas. high pressure phase. For example, for an accurate assessment of the phase change enthalpy and saturation vapor pressure of water, it has been shown that the clustering of molecules in the gas phase must be taken into account.^{39}. Such grouping in the gas phase reduces the enthalpy in the gas phase compared to the vacuum state. Similarly, the partial condensation of water molecules^{23} as well as the dissolution of CO_{2} in the water produced during the combustion process or formation of combustion by-products other than CO_{2} may lead to other discrepancies between the ab-initio calculation and the experiment. An empirical way to account for these effects is to scale the enthalpies of H_{2}O or CO_{2} or both. As a result, we found the optimal scale factor of 0.9999857 to empirically correct the theoretically predicted enthalpy of water in the CCSD(T)-F12b calculations, which reduced the AAD (MUE%) to 5.80 kJ/mol (0.26%). Scaling the enthalpy of CO_{2} calculated by CCSD(T)-F12b of 0.999994328 even reduces the AAD (MUE%) of the predicted enthalpies of combustion to 2.64 kJ/mol (0.15%). Still, further improvement in results can be achieved by simultaneously scaling the enthalpies of H_{2}O and CO_{2} calculated by CCSD(T)-F12b as 1.000006465 and 0.999992212, giving AAD (MUE%) of 2.00 kJ/mol (0.12%). These scale factors are derived from the enthalpies evaluated at the theoretical level CCSD(T)-F12b, which could require their re-evaluation for other theoretical levels. However, we hypothesized that, at least for methods that provide results similar to CCSD(T)-F12b, the scale factors might not strongly depend on the applied level of theory. Indeed, the use of the (unchanged) scale factors obtained by CCSD(T)-F12b for the enthalpies calculated with DSD-PBEP86-D3/def2-QZVP gives similar improvements, with AAD values (MUE%) of 8.69 kJ/mol (0.33%), 5.21 kJ/mol (0.21%) and 3.70 kJ/mol (0.15%), obtained by scaling the calculated enthalpies from H_{2}O, CO_{2}and both simultaneously, respectively.

In addition to inaccuracies resulting from theoretical calculations, systematic or operational errors in experimental data can also contribute to inconsistency between theoretical and experimental reference data. For example, we observed 1.59 kJ/mol AAD in the phase change enthalpies of our studied reactants between the NIST and DIPPR databases, which results in the same discrepancy between the theoretically predicted raw enthalpies of combustion calculated at l using each of these two databases. Similar to the enthalpy of vaporization, experimentally determined combustion enthalpies from different sources also show some variation. For example, a slight inaccuracy in the measurement of the enthalpy of combustion of benzoic acid, which is used to calibrate the calorimeter^{23}, can result in a linearly distributed discrepancy (lag) between the measured enthalpies of combustion of all other compounds. This may be a potential reason for the suitability of a linear fitting to empirically correct the predicted enthalpies of combustion, proposed in several studies.^{21.22}.

In summary, in the present study, we discuss ab-initio quantum chemical approaches capable of providing highly accurate predictions of the enthalpy of combustion. To this end, the main considerations in the theoretical calculations should aim to select an appropriate level of theory for the applied quantum chemical method and to carefully identify the minimum energy conformers. To reproduce the net heat of combustion, the enthalpy of phase change of the reactants must be subtracted from the gas phase enthalpies evaluated by QM. For the gross heat of combustion, the enthalpy of vaporization of water must also be subtracted from the enthalpy of water in the gaseous phase evaluated by QM. Thus, the consideration or not of the enthalpies of phase change, as well as the experimental measurement of the enthalpy of combustion, can also contribute to inconsistencies between the enthalpies of combustion predicted theoretically and determined experimentally.