The reaction energy for a generalized reaction in which reactants A and B react in ratios given by the stochiometric constants a and b to products M and N such as



is defined as:



This equation can be evaluated using all types of energies discussed in the previous chapter (E_tot, E₀, E₂₉₈, H₂₉₈, G₂₉₈). The conceptually most simple reactions are those in which only one bond undergoes reaction, a typical case being the homolytic bond dissociation reaction yielding two radicals. Taking the C-H bond dissociation in methane as an example, the following results are obtained at the HF/6-31G(d) level of theory (the UHF(6-31G(d) level being used for the open shell products):

type of energy E(CH4) E(CH3) E(H) Erxn [kJ/mol) Etot -40.195172 -39.558992 -0.498233 +362.2 E0 -40.147395 -39.528021 -0.498233 +318.1 E298 -40.144543 -39.524770 -0.496817 +322.8 H298 -40.143599 -39.523825 -0.495872 +325.3 G298 -40.164692 -39.546258 -0.508887 +287.6

This bond dissociation reaction is strongly endothermic in all cases considered here. The most positive reaction energy (+362.2 kJ/mol) is obtained when using total energies at 0K describing a vibrationless system sitting on the potential energy surface. Additional consideration of zero point vibrational energies leads to a significant change, in the current case lowering the reaction energy to +318.1 kJ/mol. Correction to internal energies at 298 K appears to be rather modest in comparison. The difference between E₂₉₈ and H₂₉₈ equates to RT (=+2.479 kJ/mol at 298.15 K) since we have two products, but only one reactant. Finally, accounting for differences in reaction entropies we obtain a free energy of reaction of +287.6 kJ/mol.

Comparison to experiment can be made at the stage of H₂₉₈ as the standard heats of formation of all three species are known. From the NIST Chemistry WebBook we take the following heats of formation: CH₄ (-74.87 kJ/mol), CH₃ (+145.69 kJ/mol), and H (+218.0 kJ/mol). This yields a reaction enthalpy for C-H bond homolysis in methane of +438.56 kJ/mol. The recommended value in a recent compilation of bond dissociation energies (Yun-Ran Luo, "Handbook of Bond Dissociation Energies in Organic Compounds", CRC Press, 2003) is +439.3 +/-0.4 kJ/mol.

Comparison of this value with our theoretical results suggests that the HF/6-31G(d) level is quite inaccurate when it comes to predicting bond dissociation energies. The deviation of 113 kJ/mol is mainly due to the neglect of electron correlation in the Hartree-Fock (HF) treatment and can only be remedied by using a better theoretical method. That the deficiencies of Hartree-Fock theory have serious consequences here is, of course, due to the fact that bond dissociation processes are accompanied by a large change in correlation energy.

A much better performance of the HF-level can be observed in reactions, in which the number of electron pairs remains constant during the reaction. These types of reactions are known as isogyric reactions and are often used to reliably calculate thermochemical data at lower levels of theory. Taking the reaction of methane (CH₄) with the hydrogen atom to yield the methyl radical (CH₃) and molecular hydrogen (H₂) as an example, the following results are obtained at the HF/6-31G(d) level of theory (again using the UHF/6-31G(d) level for open shell systems):

type of energy E(CH4) E(H) E(CH3) E(H2) Erxn [kJ/mol) Etot -40.195172 -0.498233 -39.558992 -1.126828 +19.9 E0 -40.147395 -0.498233 -39.528021 -1.116243 +3.6 E298 -40.144543 -0.496817 -39.524770 -1.113883 +7.1 H298 -40.143599 -0.495872 -39.523825 -1.112939 +7.1 G298 -40.164692 -0.508887 -39.546258 -1.127698 -9.9

The reaction enegies for this hydrogen transfer reaction are much smaller in absolute terms than for the bond dissociation reaction considered before. This indicates that the C-H bond strength in methane and the H-H bond strength in molecular hydrogen is rather similar.

In order to calculate the reaction enthalpy at 298.15K we only need the three heats of formation for CH₄, CH₃, and H as before, and have to recall that molecular hydrogen H₂ (in the gas phase) represents the reference against which heats of formation are defined. The experimental reaction enthalpy for our isogyric model reaction therefore amounts to +2.56 kJ/mol.

The comparison of the experimentally measured and the theoretically predicted reaction enthalpy shows a much better agreement now, the theoretical value being too high by only 4.5 kJ/mol. Improving on this result does not only need significantly better theoretical methods but also a better thermochemical model than the rigid-rotor/harmonic oscillator model.

On a more general note the predictive value of HF theory will be larger for those reactions, in which the products are as similar as possible to the reactants (e.g. isomerization reaction). A subclass of these types of reactions are isodesmic reactions in which the numbers of bonds of each formal type are the same for reactants and products. Using the reaction of dimethyl ether with water to yield two methanol molecules as an example, we can see that there are 6 C-H bonds, 2 C-O bonds, and 2 H-O bonds on both sides of the reaction equation:




The experimentally measured reaction enthalpy for this reaction amounts to +23.9 kJ/mol at 298.15 K. Calculations at the HF/6-31G(d) level of theory predict a value of +14.9 kJ/mol. For further examples see "Ab Initio Molecular Orbital Theory", W. J. Hehre, L. Radom, P. v. R. Schleyer, J. A. Pople, Wiley&Sons, 1986.

last changes: 23.11.2006, HZ
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