Quantum Calculations to Estimate the Heat of Hydrogenation Theoretically

Standard enthalpies of hydrogenation of 29 unsaturated hydrocarbon compounds were calculated in the gas phase by CCSD(T) theory with complete basis set cc-pVXZ, where X = DZ, TZ, as well as by complete basis set limit extrapolation. Geometries of reactants and products were optimized at the M06-2X/6-31g(d) level. This M06-2X geometries were used in the CCSD(T)/cc-pVXZ//M06-2X/6-31g (d) and cc-pV(DT)Z extrapolation calculations. (MAD) the mean absolute deviations of the enthalpies of hydrogenation between the calculated and experimental results that range from 8.8 to 3.4 kJ mol (cid:1) 1 based on the Comparison between the calculation at CCSD(T) and experimental results. The MAD value has improved and decreased to 1.5 kJ mol (cid:1) 1 after using complete basis set limit extrapolation. The deviations of the experimental values are located inside the “ chemical accuracy ” ( (cid:3) 1 kcal mol (cid:1) 1 ≈ (cid:3) 4.2 kJ mol (cid:1) 1 ) as some results showed. A very good linear correlations between experimental and calculated enthalpies of hydro-genation have been obtained at CCSD(T)/cc-pVTZ//M06-2X/6-31g(d) level and CCSD (T)/cc-pV(DT)Z extrapolation levels (SD =2.11 and 2.12 kJ mol (cid:1) 1 , respectively).


Introduction
The calculation of enthalpies of formation for the large unsaturated molecules, some of which are not included in the practical range of combustion thermochemistry, based on quantum mechanical first principles which have been possible basing on the recent important advances in computational chemistry. Necessary, Quantum mechanical calculations of molecular thermochemical properties are approximate. Approximations may be employed by the Composite quantum mechanical procedures at each of several computational steps and in the same time it may have an empirical factor to correct the cumulative error. When the error of the various approximations is known within narrow limits, but the question about the accuracy of the "known" value is noticed immediately because the uncertainty of the comparison between the approximate quantum mechanical result and the standard to which it is compared.
The most correct quantum mechanical procedure is been established after its ability to reproduce various accurate experimental results to calculate unknown thermochemical values of explosive compounds or unstable, unsuited to classical thermochemical methods, or to calculate thermochemical properties of radicals, molecules, or ions of fleeting existence [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Here where a major advantage to create the accuracy of inherent hydrogen thermochemical results lies, and it works for encouraging and renewing interest in the diverse literature devoted to hydrogen thermochemistry.
The main part of the quantum chemistry is contended in the total electronic energy of a molecule. And this total electronic energy is a function of the nuclear geometric configuration after Born-Oppenheimer separation of electronic and nuclear motion, therefore generating hyper surfaces of potential energy-for electronically excited states as well as for the ground state.
At the end of this work a very important fact is clear now which clarify that wave function-based quantum-chemical methods can produce molecular electronic energies with an accuracy that surpasses that of experimental measurements of molecular energies (in terms of enthalpies of formation).
An important feature of the wavefunction-based quantum-chemical methodology is the ability to access the exact characterizing of the molecular electronic structure in a systematic manner. To achieve systematic approach two basic steps have been taken, the first step using advanced hierarchy of wavefunction models and the second step using systematic sequence of basis setsor a nearly complete basis setof atomic orbitals.

Basis-set convergence
It is noted that the type of wavefunction model used, or density functional, plays an important role in determining accuracy of computed molecular_electronic energies. In addition to the important role played by the flexibility of the one electron basis set of atomic orbitals (AOs) by which the molecular orbitals (MOs) are expanded. It is worth noting that Slater's determinants are constructed using MOs for use in Kohn -Sham theory or to expand the n-electron wavefunction.
Different ways can be chosen from basis sets of AOs (the literature [16][17][18]). As for the approach followed in study of molecular electronic-structure based on wavefunction methods, a well-defined procedure must have been used in order to generating sequences of the basis sets in order to increase flexibility. This way is very useful by generating hierarchies for basis sets, Each next higher level describes an improved systematically of the molecular electronic structure compared to the next lower level. Within the hierarchy the calculated results converge within a prescribed accuracy, and then the basis-set hierarchy ends up effectively complete basis [16]. Therein lies the problem where approaches of wavefunction that depend on the electron-correlation effects of the convergence to an effectively complete basis are very slow. When increasing the number N of AOs in the basis according to an optimal manner, it reduces the basis-set error as ∝N À1 , thus obtaining a good approximation. It is often noticed that the accuracy of the electronic correlation calculations is limited by computational technical constraints. This is due to computing times growing at least as N4 and the computational effort grows more quickly compared to gain in accuracy.

Correlation-consistent basis sets
The correlation-consistent basis sets from Dunning and his co-workers [19,20] represent a popular hierarchy of basis sets. When the valence orbitals are correlated in a calculations only then can the expansion mainly in the Limit extrapolation, until it ends to an effectively complete basis [21][22][23][24][25][26]. The term X-tuple zeta basis sets represents to correlation-consistent polarized valence, denoted by cc-pVXZ, and is used in the calculations of this work, where X = D, T, (double, triple zeta). It should be noted that when all the electrons are correlated (core as well as valence orbitals), the cc-pCVXZ basis sets must be used provided 2 ≤ X ≤ 5.

Basis-set extrapolation
The correlation-consistent basis sets form a basis-set hierarchy well suited for complete basis-set limit extrapolations of the correlation-energy. Then the correlation energy can have been determined depending on X when X ! ∞.
In 1977, 27 formulas for two-point linear extrapolation were introduced by Helgaker et al. [27] via the basis sets cc-pCVXZ and cc-pCVYZ, the electroncorrelation energies and are computed E X and E Y , and the C XY = X 3 /(X 3 Y 3 ) coefficient is determined in Eq. (1), The EX correlation energy can be recovered in correlation-consistent basis of cc-pVXZ via the Eq. (2), When performing calculations according to the basis sets, cc-pVXZ and cc-pVYZ, two equations with two unknowns are obtained, a and E, whose Solving leads to Eq. (2), with that in mind E = E XY Note that through experience [28][29][30][31][32] that the best estimate of complete basis set limit extrapolation is by using two consecutive basis set when X = Y -1, for example the pair cc-pCVDZ/cc-pCVTZ, and so on).
The level at which the results were obtained by extrapolation is denoted by cc-pCV(XY)Z, where XY = DT, TQ , and so forth. It is only at this level that the correlation energy is extrapolated.
The CCSD(T)/cc-pV(XY)Z extrapolation will be applied to obtain estimates of the basis-set limit of corrections for M06-2X/6-31g(d) level of theory by calculations of hydrogenation enthalpies of some unsaturated hydrocarbon compounds ( Table 1).
In Section 3, we shall describe the methods used, the molecular equilibrium geometries, and the basis sets. Results will be shown in Section 3.2, including molecular electronic energies, enthalpies of hydrogenation, and statistical analysis of the computational results.

Computational methods
Electronic energies were computed by the density-functional (M06-2X) [34] approach with 6-31g(d) basis set. The M06-2X/6-31g(d) equilibrium geometries of the reactants and products was optimized with the Gaussian 09 program [35]. All CCSD(T)/cc-pV(DT)Z calculations were performed at fixed molecular equilibrium geometries that were optimized at the M06-2X/6-31g(d) level can be found in supporting information.

Nonrelativistic electronic energies
The total electronic M06-2X/6-31g(d) energies, H corr, G corr , and the zero-point vibrational energies (ZPE) are reported in Table 2 (T)/cc-pVXZ energies computed at the M06-2X/6-31g(d) geometries, where X = D, T. Table 3 also shows extrapolated values by the Eq. (2) (denoted E ∞ ). It is noted from Table 3 that the cc-pV(DT)Z extrapolated level yield electronic energies for all reactants and products less than CCSD(T)/cc-pVDZ and CCSD (T)/cc-pVTZ levels, and it is expected that the calculations of the hydrogenation enthalpies by cc-pV(DT)Z extrapolated level are compatible with the experimental values.

Enthalpies of hydrogenation at 298.15 K in gas phase
The usual way to calculate enthalpies of reaction is to calculate heats of formation, and take the appropriate sums and difference (Eq. (3)).
However, since Gaussian program provides the sum of electronic and thermal enthalpies, there is a short cut: namely, to simply take the difference of the sums of these values for the reactants and the products. This works since the number of atoms of each element is the same on both sides of the reaction, therefore all the atomic information cancels out, and you need only the molecular data. For example,  using the information in Table 2 (or Table 3 for energies), the enthalpy of reaction can be calculated simply by Eq. (4): E 0 can represent either E DZ , E TZ , E QZ or E ∞ keeping the calculated H corr value at M06-2X/6-31g(d) level is fixed. The calculated enthalpies of hydrogenation are CCSD(T)/cc-pVXZ//M06-2X/6-13(d)  reported in Table 4, along with the experimental values. Table 5 shows statistical parameters for all used computational methods. Figure 1 shows a linear analysis of the best calculated results in terms of experimental results. Our best theoretical estimates of the enthalpies of hydrogenation are based on basis set limit extrapolation calculations, when the pair cc-pVDZ and cc-pVTZ are used, and the mean absolute deviation (MAD) between experimental and calculated values is 1.5 kJ mol À1 ( Table 5). The enthalpies of hydrogenation of some alkenes (12 compounds) have been calculated at the HF, B3LYP, M06, MP2, G3, G4, CBS-QB3, CBS-APNO, and W1BD levels and, in the case of the first four methods, using a variety of basis sets up to aug-ccpVTZ [37], and it is found that the MAD decreases gradually from the first to the last method (18.4-4.2 kJ mol À1 ). Moreover, Rogers et al. [38][39][40] calculated the hydrogenation enthalpy at 298.15 K for reactions involving cyclic and acyclic C4 (20 reactions), cyclic C5 (23 reactions) and C6 (24 reactions) hydrocarbons using the G2 and G2(MP2) ab initio methods, and it is found that the MAD is about 3.3, 3.7 and 5.0 kJ À1 , respectively.

Conclusion
Enthalpies of hydrogenation are relatively easy to calculate with CCSD(T)/cc-pVXZ/M06-31g(d)/6-31g(d) level, where X = D, T, giving fairly good agreement  with experiment, especially when cc-pVTZ basis set are used, and basis set extrapolation techniques by Eq. (2) seem to represent an easy-to-use alternative, especially when the pair cc-pVDZ and cc-pVTZ are used.