Abstract
Enthalpies of formation of twenty-nine compounds of aluminium were calculated employing various density functional theory (DFT) (B3LYP, B2PLYP, LC-wPBE, PBE1PBE, BMK, M06 and M06-2X) and composite methods (EnAt1, EnAt2, G3X-CEP, G3X(CCSD)-CEP and G4), using atomization. The best agreement with experimental data was achieved by using Gn and Gn-CEP multilevel techniques. It was found that the best performance among DFT methods within the atomization approach demonstrated the long range corrected M06-2X level theory. The mean absolute error calculated for EnAt1 and EnAt2, which has recently been reported as a very accurate method for calculating enthalpies of formation, for 248 compounds from the G3/05 test set presented the best results when compared with functional DFT.
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1 Introduction
Aluminium is the most common metal in the world and the second most used metal. It is constantly found associated with other atoms, such as chlorine [1], oxygen [2], hydrogen [3] and fluorine [4]. Aluminium reacts so quickly with oxygen that it is never naturally find in its pure form. Compounds of aluminium are found in large quantities in the Earth’s crust as an ore called bauxite [5].
Aluminium is found with other metals in many different ways as composites (hybrid materials), such as aluminium matrix composites. Metal matrix composites (MMCs) [6] consist of metal alloys mounted with fibres, particulates or whiskers. Alloys of numerous metals (i.e. aluminium-titanium [7] and aluminium-copper [8]) have been used as matrices to date. In the NASA Space Shuttle [9], for example, many materials are produced from aluminium mounted with boron fibres [10].
Enthalpy of formation (298.15 K) of compounds of aluminium, calculated with composite methods, presented better results when compared with ab initio methods. Curtiss et al. [11, 12] calculated that Al2Cl6 presented a mean absolute error (MAE) of 7.8 and 5.2 kcal mol−1 for G3X and G4, respectively, whereas for the density functional theory with 6-311 + G(3df,2p), the MAE was −29.1 kcal mol−1, −31.9 kcal mol−1 and 30.0 kcal mol−1 for B3LYP, O3LYP and VSXC [11], respectively.
A recent endeavour to reduce the CPU time of calculations suggested the Gn theory started by Pereira et al. [13, 14], which associates a compact effective potential (CEP) [15] with the G3 theory, resulting in a method known as G3CEP [13, 14]. After this first successful generalization by combining pseudopotential with the G3 theory, other proposals were expanded, such as G3(MP2)//B3LYP-CEP [16], G3(MP2)-CEP [17], G4-CEP [18], G3(MP2)//B3-SBK [19], G3X-CEP [20] and G3X(CCSD)-CEP [20]. Enforcement of these methods manifested a significant decrease in CPU time, preserving an outstanding accuracy when compared with the original all-electron versions. These methods were applied to transition metals [21] and pKa [22], for example.
Silva and Custodio [20] revealed the chance to improve the calculated enthalpies of formation with respect to accurate experimental data by scaling the atomization energies. This sequence of steps, along with the G3X and G3X(CCSD) methods and pseudopotential, generated EnAt1 [20] and EnAt2 [20], which provided results as accurate as the combination of G4CEP [18] and G3CEP [13] theories.
For 248 enthalpies of formation from the G3/05 test set, the MAE of the calculations with EnAt1 (1.02 kcal mol−1) and EnAt2 (0.98 kcal mol−1) is significantly better than the G3X-CEP results (1.16 kcal mol−1) and G3X(CCSD)-CEP (1.12 kcal mol−1) with the original experimental atomization energies. Silva and Custodio [20] calculated that Al2Cl6 presented a MAE of 6.7 and 7.6 kcal mol−1 for G3X-CEP and G3X(CCSD)-CEP, respectively, whereas the MAE was 0.0 kcal mol−1 for both EnAt1 and EnAt2 methods.
EnAt1 and EnAt2 had the scaling of the experimental atomization energies for the calculation of the enthalpies of formation. The experimental atomization energy of every element of the periodic table is multiplied by an adaptable parameter that decreases the MAE between the theoretical enthalpies of formation and the experimental one. Every parameter is set initially equal to 1.0 and optimized using the simplex method [23]. At the end of this optimization, it is complemented with the re-optimization of the higher-level correction (HLC) parameters.
In the present work, the thermochemical values of aluminium compounds considered have been determined using the EnAt1 and EnAt2 methods, which presented the best results.
2 Computational methods
Heats of formation at 298.15 K and 1 atm were computed from total atomization energies. This requires an accurate and balanced energetic description of the molecule and its constituent atoms, which places stringent requirements on the quantum methods employed.
The results obtained atomization using the density functional theory (DFT) with different exchange and correlation: B3LYP [24, 25], B2PLYP [26], BMK [27], M06 [28], M06-2X [28], LC-wPBE [29, 30] and PBE1PBE [29, 31]. The 6-311G(d,p) [32, 33], cc-pVDZ [34], cc-pVTZ [35], aug-cc-pVTZ [34, 35] basis sets were employed in the DFT calculation. Besides the optimized geometries at the B3LYP/6-311G(d,p) level, different strategies have been adopted by the composite methods, such as the following: (a) B3LYP/6-311G(d,p) was used for the geometry optimization and with a scaling factor of λ = 0.99 for vibrational frequencies as used by B3LYP/cc-pVDZ, B3LYP/cc-pVTZ, B3LYP/extrapolation, B2PLYP/cc-pVDZ, B2PLYP/cc-pVTZ, B2PLYP/extrapolation, MP2/cc-pVDZ, MP2/cc-pVTZ, MP2/extrapolation, BMK/aug-cc-pVTZ, M06/aug-cc-pVTZ, M06-2x/aug-cc-pVTZ, LC-wPBE/aug-cc-pVTZ and PBE1PBE/aug-cc-pVTZ; (b) B3LYP/6-31G(2df,p) was used for geometry optimization and frequencies with λ = 0.9854, used by G3X-CEP, G3X(CCSD)-CEP, G4, EnAt1 and EnAt2; (c) extrapolation is described in G4 theory [12].
In summary, we have adapted the G3X and G3X(CCSD) methods together with the CEP pseudopotential to create two methods labelled as: G3X-CEP and G3X(CCSD)-CEP [20]. The G3X reference energy is calculated at the QCISD(T)/6-31G(d) level, for as much as the G3X(CCSD) regards the CCSD(T)/6-31G(d) level of theory [20]. The general expression for G3X contains corrections for diffuse (\(\Delta E_{ + }\)) and polarization (\(\Delta E_{2df,p}\)) functions, electron correlation effects (\(\Delta E_{QCISD\left( T \right)}\)), effects from the size of the basis functions (\(\Delta E_{{G3{\text{large}}}}\)), and improvement in the G3large basis set (ΔEHF), spin–orbit correction (\(E_{SO}\)) from the literature, zero point energy and thermal effects (\(E_{ZPE}\)), and an empirical higher level correction (\(E_{HLC}\)). The combination of all these effects is summarized in the following expression [20]:
where the mathematical definition of each correction is:
The molecular equilibrium geometries were obtained from B3LYP/6-31G(2df,p). The zero point energy (ZPE) is also obtained from the harmonic approximation at the B3LYP/6-31G(2df,p) level, and the frequencies are scaled by a factor of 0.9854 [20].
In this work, the behaviour of the composite techniques EnAt1 and EnAt2 is evaluated for a test set of enthalpies of formation of 29 representative molecules containing second-row and third-row periodic table elements. The main criteria for selecting these 29 molecules were an experimental uncertainty of less than 1 kcal mol−1. This test set also included six molecules with larger uncertainties.
All calculations were performed by using the Gaussian 09 package of programs [36].
3 Results and discussion
Table 1 shows the enthalpy of formation (298.15 K) for compounds of aluminium, calculated using G3X-CEP, G3X(CCSD)-CEP, G4, EnAt1 and EnAt2 and compared to experimental data. The resulting MAE, standard deviation (Std. Dev.), largest negative deviation (lar. neg. dev) and largest positive deviation (lar. pos. dev.) are also reported in Table 1. Comparing the MAEs, EnAt1 and EnAt2 result in MAEs of 3.23 and 3.09 kcal mol−1, respectively, while G3X-CEP, G3X(CCSD)-CEP and G4 result in MAEs of 3.79, 3.58 and 3.47 kcal mol−1, respectively. The difference between the general performance of calculations with EnAt2 and G4 is 0.38 kcal mol−1.
The largest positive deviation for G4, EnAt1 and EnAt2 is for AlS, with a deviation of 9.4, 5.42 and 5.3 kcal mol−1, respectively (see Table 1), while the largest negative deviation for G3X-CEP and G3X(CCSD)-CEP is for C6H15Al, with a deviation of -7.44 and -7.52 kcal mol−1, respectively (see Table 1). A total of 21% of the enthalpies of formation calculated with G4 showed a deviation between ± 2 kcal mol−1. A greater percentage (44%) was obtained for the results calculated with EnAt1 and EnAt2. Calculations using G4 also resulted in only 48% of the deviations between ± 3 kcal mol−1, while 62% and 72% of the results were observed for EnAt1 and EnAt2, respectively.
Some results from the literature are calculated at an equivalent level and can be partially compared with those obtained using the EnAt1 and EnAt2 calculations. Cobos [37] performed calculations of AlH at the B3LYP/6-311 ++G(d,p), B3LYP/6-311 ++G(3df,3pd) and B3LYP/6-311 ++G(3d2f,3pd), with a deviation of 2.1, 2.7 and 2.7 kcal mol−1, respectively, and EnAt1 and EnAt2 with a deviation of -0.64 and -0.29 kcal mol−1, respectively. These results of EnAt1 and EnAt2 offered significant increases in accuracy compared with B3LYP in a different basis set.
For most compounds, the MAE of EnAt1 and EnAt2 atomization-based estimates is lower than the corresponding G3X-CEP, G3X(CCSD)-CEP and G4 atomization-based estimates by 3.47–3.79 kcal mol−1. Estimates of the enthalpy of formation (298.15 K) value for AlS (lar. pos. dev.) from the EnAt1 and EnAt2 calculations deviated by 5.42 and 5.3 kcal mol−1, respectively. These values are still better than the PM6 and PM5 values calculated by Stewart [38], with a deviation of -5.9 and -13.0 kcal mol−1, respectively.
The following are heats of formation values (kcal mol−1) at 0 K for G3X-CEP, G3X(CCSD)-CEP, G4, B3LYP, B2PLYP, MP2, LC-wPBE, PBE1PBE, BMK, M06 and M06-2X: H (51.63 ± 0.001), B (136.2 ± 0.2), C (169.98 ± 0.1), O (58.99 ± 0.02), F (18.47 ± 0.007), Na (25.69 ± 0.17), Al (78.23 ± 1.00), S (65.66 ± 0.06) and Cl (28.59 ± 0.001). However, Silva and Custodio [20] proposed the possibility of refining the calculated enthalpies of formation with respect to accurate experimental data by scaling these values at 0 K to create EnAt1 and EnAt2. Heats of formation values (kcal mol−1) at 0 K for EnAt1 are: H (51.57), B (134.43), C (170.18), O (59.35), F (18.73), Na (26.73), Al (83.12), S (65.52) and Cl (28.41),and values at 0 K for EnAt2 are: H (51.61), B (135.08), C (170.01), O (59.17), F (18.60), Na (26.92), Al (82.76), S (66.12) and Cl (28.47).
Table 2 shows the enthalpy of formation (298.15 K) for compounds of aluminium calculated using B3LYP/6-311G(d,p), B3LYP/cc-pVDZ, B3LYP/cc-pVTZ, B3LYP/extrapolation, B2PLYP/cc-pVDZ, B2PLYP/cc-pVTZ, B2PLYP/extrapolation, MP2/cc-pVDZ, MP2/cc-pVTZ and MP2/extrapolation compared to experimental data. The lowest deviation from the experimental data was observed for aluminium compounds (29 compounds) with mean absolute error of 13.28 kcal mol−1 for MP2/extrapolation, while the MAE for B3LYP/extrapolation and B2PLYP/extrapolation was 16.98 kcal mol−1 and 19.94 kcal mol−1, respectively. Only 21% and 17% for the B3LYP/extrapolation and MP2/extrapolation calculations, respectively, show deviations between ± 2 kcal mol−1, whereas in B2PLYP/extrapolation, 3% of the results are between this limit. Most of the deviations are concentrated above ± 2 kcal mol−1.
We studied the behaviour of B2PLYP for a test set. This method shows a stronger basis set dependence than B3LYP and MP2. B2PLYP is for the cc-pVTZ basis set; the MAE reaches its smallest value at 20.54 kcal mol−1. When the basis set is decreased to the cc-pVDZ level, the MAD becomes 30.44 kcal mol−1. Further tuning of the empirical parameters did not improve the B2PLYP results compared with B3LYP, even though some of the MP2 correlation is included in B2PLYP.
The accuracy of extrapolations can be compared among the calculations using the respective functionals and cc-pVDZ and cc-pVTZ basis sets. The best results were obtained using MP2/extrapolation, which yielded accuracy superior to the B3LYP/extrapolation and B2PLYP/extrapolation. Measuring the value of the energies with respect to the three functionals and the cc-pVDZ for cc-pVTZ basis sets, the errors were systematically reduced with the enlargement of the basis set. The typical performance of the extrapolation is compatible or better than the results obtained with the respective functionals and cc-pVDZ and cc-pVTZ basis sets.
Chinini and Custodio [39] calculated that Al2Cl6 presented a MAE of -24.73 kcal mol−1 for B3LYP/extrapolation using a different extrapolation formula (cc-pVDZ for cc-pVTZ basis sets) including corrections for electron number. In this situation, the MAE was -27.81 kcal mol−1 for B3LYP/extrapolation.
We have also examined five density functional methods for the 29-molecule set. These results are given in Table 3. The density functional methods tested include: LC-wPBE, PBE1PBE, BMK, M06 and M06-2X. The results indicate that M06-2X performs the best with a MAD of 16.14 kcal mol−1. The PBE1PBE performed much worse, with MAD of 26.88 kcal mol−1. Density functional methods, such those in Table 3, performed much worse than the EnAt1 and EnAt2 methods for the 29 enthalpies. Thus, they are much less reliable than the EnAt1 and EnAt2 methods and are not recommended for use on compounds of aluminium.
The 29 enthalpies calculated and shown in Table 3 show a tendency towards larger dispersion of the results for M06-2X when compared to the EnAt1 and EnAt2 methods. The distribution indicates that 3% of the presented cases correspond to 1 molecule accuracy in the range of ± 2 kcal mol−1 and 17% of the set test calculated with PBE1PBE showed a deviation between ± 2 kcal mol−1.
The discrepancy in the results is due to the molecules of C3H9Al and C6H15Al. The enthalpy of formation for compounds of aluminium without C3H9Al and C6H15Al presented results for MAE of LC-wPBE (8.30 kcal mol−1), PBE1PBE (15.96 kcal mol−1), BMK (14.68 kcal mol−1), M06 (13.61 kcal mol−1) and M06-2X (13.39 kcal mol−1), which are worse than MAE for EnAt1 (3.23 kcal mol−1) and EnAt2 (3.09 kcal mol−1) for C3H9Al and C6H15Al.
In agreement with the reported results of Curtiss et al. [11, 12], the VSXC/6-311 + G(3df,2p) and TPSS/6-311 + G(3df,2p) methods calculated deviations of 3.9 and -2.8 kcal mol−1 for AlF, respectively. In this work, we found results corresponding to Curtiss for LC-wPBE (− 2.13 kcal mol−1), PBE1PBE (− 5.58 kcal mol−1), BMK (0.18 kcal mol−1), M06 (2.44 kcal mol−1) and M06-2X (2.25 kcal mol−1). Zhao and Truhlar [28] showed M06-2X and M06 (these two hybrid meta-GGA exchange–correlation functionals) are the best functionals for thermochemistry calculations when compared to other functionals (LC-wPBE, PBE1PBE, BMK). This fact was again demonstrated in this paper.
Some of the most extreme values for enthalpies of aluminium compounds are not exclusive for functional. These values represent pathological systems where functionals and basis sets are not sufficient to describe subtle structural and/or electronic effects. Some of these anomalous values were observed from the many papers. Aluminium (or halogenated) compounds, or molecules containing hypervalent atoms, are usually a source of large deviations. These deviations have constantly been found, and so far have not been solved by functionals and basis sets. Enthalpies of formation for compounds containing these atoms presented a variety of large positive and negative deviations with respect to experimental data. Presently, composite methods are shown to be one of the best options for the calculation of thermochemical properties, with low computational cost and applicability to chemical systems. These methods incorporate additive corrections to the order of electronic correlations, and some considered extrapolation techniques.
4 Conclusion
We have employed the EnAt1 and EnAt2 methods to study the enthalpies of formation for 29 compounds of aluminium. EnAt1 and EnAt2 presented the best results compared to G3X-CEP, G3X(CCSD)-CEP, G4, B3LYP/6-311G(d,p), B3LYP/cc-pVDZ, B3LYP/cc-pVTZ, B3LYP/extrapolation, B2PLYP/cc-pVDZ, B2PLYP/cc-pVTZ, B2PLYP/extrapolation, MP2/cc-pVDZ, MP2/cc-pVTZ, MP2/extrapolation and LC-wPBE, PBE1PBE, BMK, M06 and M06-2X with the aug-cc-pVTZ basis set.
The MAE between the enthalpy of formation (298.15 K) computed at the EnAt1 and EnAt2 levels was quite small, namely 0.14 kcal mol−1. In the meantime, the MAE between the enthalpy of formation calculated at the EnAt2 and the M06-2X (best functional calculated) levels was quite large at 13.05 kcal mol−1.
In general, average unspecified errors in the enthalpy of formation (298.15 K) have steadily decreased as composite methods with compact effective pseudopotentials (CEP) have evolved. Earlier Gn-CEP composite methods such as G3CEP and G3CEP(MP2) had significantly larger MAE than the EnAt1 and EnAt2 composite methods.
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The author would like to acknowledge The National Center of High Performance Computing in Ceará (CENAPAD-UFC) for access to their computational facilities.
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de Souza Silva, C. Heats of formation for aluminium compounds with EnAt1 and EnAt2. Theor Chem Acc 139, 135 (2020). https://doi.org/10.1007/s00214-020-02642-8
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DOI: https://doi.org/10.1007/s00214-020-02642-8