Potential Energy Surface Characterization 

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Chapter 2 Potential Energy Surface Characterization

Reaction Energy

Reaction energies for F + CH₄ → HF + CH₃ and F + C₂H₆ → HF + C₂H₅ were calculated using several electronic structure methods and basis sets. The methods used include second-order Møller-Plesset perturbation theory (MP2)⁴⁵, the Becke Three Parameter Hybrid Functional with the correlation functional of Lee, Yang, and Parr (B3LYP)^{46, 47}, coupled-cluster theory⁴⁸ with single, double, and perturbative triple excitations (CCSD, CCSD(T)), the parameter model 3 (PM3)⁴⁹, Austin model 1 (AM1)⁵⁰, and the modified symmetrically orthogonalized intermediate neglect of differential overlap method (MSINDO)⁵¹. For ab initio calculations, Dunning’s double-, triple-, and quadruple-zeta correlation consistent basis sets augmented with diffuse functions (aug-cc-pVXZ, X=D,T,Q) and Dunning’s triple-zeta correlation consistent basis set (cc-pVTZ) were used.⁵² The calculated reaction energies were extrapolated to the complete basis set (CBS) limit using a two-point formula. ⁵³
To allow for the calculation of energies with higher levels of theory and larger basis sets, dual-level calculations have been performed. Dual-level calculations use geometries and harmonic frequencies calculated with a lower level of theory/smaller basis set. This permits the energy to be calculated at a higher level when the computational cost of the geometry optimization and/or harmonic frequency calculation is prohibitive. This technique has been employed for most of the CCSD(T) calculations that have been performed.

F + CH₄ → HF + CH₃

The experimental reaction energy at room temperature (298 K) is -31.4 kcal/mol^54-56. Here the 0 K estimate of -32.1 kcal/mol² will be used to compare with the electronic structure calculations that we have carried out in an attempt to calibrate the performance of the computational methods. The results of the calculations are shown in Table 2.1.
Zero-point corrected reaction energies calculated using MP2 theory show a trend of decreasing energy with increasing basis set size. For instance, MP2/aug-cc-pVDZ calculations yielded a reaction energy 3.15 kcal/mol below the experimental value, and MP2/aug-cc-pVTZ calculations predict a reaction energy 4.56 kcal/mol more exothermic than experiment. When the complete basis set estimate is calculated using these MP2 energies, the reaction energy found is over 6 kcal/mol below the experimental value. Although larger basis sets should improve the agreement with experiment, these results indicate that MP2 theory is not a very accurate method for this reaction because increasing the basis set does not produce a closer match with experiment. Therefore, the agreement of MP2/aug-cc-pVDZ calculations with experiments is a fortuitous counterbalancing of the errors of MP2, which tends to overestimate the reaction energy, and a low basis set (i.e. aug-cc-pVDZ), which tends to underestimate the reaction energy. When the reaction energy is calculated with diffuse functions removed, using the cc-pVTZ basis set, the reaction energy comes within 2.5 kcal/mol of the experimental values. The difference of the MP2/cc-pVTZ and MP2/aug-cc-pVTZ zero-point corrected reaction energies is about 3 kcal/mol, indicating that omission of diffuse functions in the basis set will result in inaccurate energies for the F + CH₄ → HF + CH₃ reaction.
B3LYP calculations yield reaction energies within 1 kcal/mol of the experimental value when aug-cc-pVDZ and aug-cc-pVTZ basis sets are used. The energies for these basis sets are actually quite close to each other with a difference of about 0.1 kcal/mol. When diffuse functions are removed from the aug-cc-pVTZ basis set the reaction energy increases by 2 kcal/mol. Seen here as well as with MP2 theory, the removal of diffuse functions yields more positive, less accurate reaction energies. Therefore, it can be concluded that diffuse functions in the basis set are required to obtain accurate predictions of the reaction energy. Roberto-Neto et al.²⁸ concluded that the importance of diffuse functions is mainly related to the ionic character of the fluorine atom for which the electron affinity is more accurately computed using diffuse functions.
Reaction energies calculated with the CCSD method and aug-cc-pVDZ and aug-cc-pVTZ basis sets are within 1 kcal/mol of each other. The energy calculated using the aug-cc-pVDZ basis set is about 2.4 kcal/mol higher than the experimental energy, while the aug-cc-pVTZ energy reduces the difference to 1.6 kcal/mol. Removing diffuse functions from the aug-cc-pVTZ basis set yields a reaction energy that is 3.7 kcal/mol above the experimental value.
The reaction energy at the CCSD(T) level was calculated with various basis sets using MP2/aug-cc-pVDZ geometries and harmonic frequencies. The CCSD(T) reaction energy was found to decrease as the size of the basis set was increased, which follows the trend mentioned above for the other ab initio calculations. Calculations performed with the aug-cc-pVQZ basis set yielded reaction energy within 0.3 kcal/mol of the experimental value. The complete basis set extrapolation of the CCSD(T) energies was found to be about 1.0 kcal/mol lower than the experimental value (0.5 kcal/mol considering the experimental error bar), which is the level of accuracy expected for this method.
CCSD(T) reaction energies were also calculated using geometries and harmonic frequencies calculated with MP2/aug-cc-pVTZ, CCSD/aug-cc-pVTZ, and CCSD(T)/aug-cc-pVDZ. Results similar to those of MP2/aug-cc-pVDZ were found for each of these methods and basis sets. Quantitatively, the nonzero-point corrected CCSD(T)/CBS reaction energies mentioned above are within 0.1 kcal/mol of each other. These results validate the use of dual-level calculations in this work, as the geometry of reactants and products seems to be well captured by a variety of methods. The results also suggest that it is more important to select an accurate method to calculate the energies than to carry out the geometry optimizations and harmonic frequencies calculation.
The semiempirical Hamiltonians PM3, AM1, and MSINDO were used to calculate the reaction energy. Both unrestricted (U) and restricted open-shell (RO) wave functions have been used. Reaction energies calculated with AM1 were found to be 25 kcal/mol below the experimental value. PM3 reaction energies were found to be higher than AM1, yet still 10.7 – 13.1 kcal/mol below the experimental value. MSINDO yielded the closest energies being 2.5 – 5.1 kcal/mol below the experimental value. For all of the semiempirical Hamiltonians, energies calculated with unrestricted wave functions were slightly lower (at most 2 kcal/mol) than those calculated with restricted open-shell wave functions.
In summary, for all first-principles methods used, the reaction energy decreases as the size of the basis set is increased. When diffuse functions are removed from the aug-cc-pVTZ basis set, the reaction energies become about 2 kcal/mol less negative than when diffuse functions were included. The most chemically-accurate results are produced with B3LYP/aug-cc-pVDZ, B3LYP/aug-cc-pVTZ, and dual-level CCSD(T)/aug-cc-pVTZ regardless of the method used to calculate geometries and harmonic frequencies. Our best calculations, CCSD(T)/CBS, are within 1 kcal/mol of the experimental reaction energy.

F + C₂H₆ → HF + C₂H₅

The F + C₂H₆ → HF + C₂H₅ reaction is more exothermic than the homologue reaction described above. The lower reaction energy for this reaction can be attributed to the C₂H₅ radical being more stable than the CH₃ radical. More energy is required to break the C-H bond in CH₄ and form the CH₃ radical than is needed to break the C-H bond in C₂H₆. Here we note a discrepancy among the various experimental values available. The reaction energy obtained from the enthalpies of formation of the reactant and product species reported in the Computational Chemistry Comparison and Benchmark Database² (CCCBDB) is over 3 kcal/mol more negative than that provided by the rest of the sources consulted here. This is attributed to an incorrect enthalpy of formation for the ethyl radical. The enthalpy of formation is about 120 kJ/mol for three of the sources. However, for the CCCBDB the enthalpy of formation is 107±6 kJ/mol; even considering the uncertainty, the enthalpies of formation do not overlap. It is concluded that the enthalpy of formation of the ethyl radical at 0K should be revised in the CCCBDB database. Therefore, in the following comparison with experiment we will leave aside the CCCBDB value. The results of the calculations are shown in Table 2.2.
Calculations performed using the MP2 level of theory yield reaction energies below the experimental value. MP2/aug-cc-pVDZ reaction energies are 2.1 – 2.7 kcal/mol below the experimental values. As the size of the basis set is increased the reaction energy decreases. The MP2/CBS reaction energy is found to be 5.6 – 6.2 kcal/mol below the experimental values. Increasing the size of the basis set should yield a more accurate reaction energy, bringing the calculated value closer to those found experimentally. However, for MP2 this is not the case, and the calculated reaction energy is found to be farther away from the experimental values as the size of the basis set is increased; therefore MP2 theory is not best for use with this reaction. These results are analogous to those described previously for F + CH₄ → HF + CH₃.
B3LYP/aug-cc-pVDZ and B3LYP/aug-cc-pVTZ calculations yield similar reaction energies with about a 0.3 kcal/mol difference. These energies are 0.4 – 1.3 kcal/mol lower than the experimental values. When diffuse functions are excluded and calculations performed with the cc-pVTZ basis set, the reaction energy is 1.6 kcal/mol higher than the reaction energy calculated with the aug-cc-pVTZ basis set.
Reaction energies calculated using the CCSD method exhibit a decrease of about 1 kcal/mol when the basis set is increased from aug-cc-pVDZ to aug-cc-pVTZ. These reaction energies are 1.5 – 3.2 kcal/mol higher than the experimental values. When the cc-pVTZ basis set is used the reaction energy is about 2 kcal/mol less negative than the corresponding result with the aug-cc-pVTZ basis set.
Using geometries and harmonic frequencies calculated with MP2/aug-cc-pVDZ, reaction energies were calculated using the CCSD(T) method and various basis sets. As the size of the basis set used was increased, the reaction energy decreased. The energies were found to be 0.1 – 2.5 kcal/mol away from the experimental values. Our best calculations, CCSD(T)/CBS, provide reaction energies within 1 kcal/mol of the experiment. As with F + CH₄, CCSD(T) yields the most accurate results among the ab initio methods employed in this work.
Reaction energies were also calculated using CCSD(T)/aug-cc-pVDZ, CCSD/aug-cc-pVTZ and MP2/aug-cc-pVTZ geometries and frequencies. These values are similar to the reaction energies determined using CCSD(T) with MP2/aug-cc-pVDZ geometries. In fact, for all of the geometries used for CCSD(T) calculations, the reaction energy was found to differ by less 0.2 kcal/mol. This suggests that the level of theory used to calculate the geometries is not as important as the level of theory used to find the energy.

List of Figures 
1. Introduction
1.1. Literature Review of the F + CH4 and F + C2H6 Reactions
1.2. Potential Energy Surfaces
1.2.1. Born-Oppenheimer Approximation
1.3. Quasiclassical Trajectories
2. Potential Energy Surface Characterization 
2.1. Reaction Energy.
2.2. Reaction Barrier
2.3. Minimum Energy Reaction Path
2.4. Potential Energy Surface Scans
3. Semiempirical Hamiltonian Parameter Fitting 
4. Trajectory Calculations
4.1. F + CH4 → HF + CH3.
4.2. F + C2H6 → HF + C2H5
5. Conclusions
5.1. Summary.
5.2. Future Work
References
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Ab initio and Direct Quasiclassical Trajectory Study of the F + CH4 → HF + CH3 and F + C2H6 → HF + C2H5 Reactions