### Using Energies to calculate energies of reaction and equilibrium constants

• Force field, semi-empirical, or low-level ab initio methods can all give geometries of lesser or greater accuracy, but getting accurate enough energies is more difficult
• Errors in energy differences, as shown by using different methods for the calculation, translate into very different equilibrium constants
• Consider the Boltzman distribution for two species in equilibrium:

• K = exp (-DE / RT )
• We can use this to write a table of K against energy difference
 Compound ratio at 294K Energy difference (kJ mol-1) 1 : 1 0 1 : 2 1.69 1 : 5 3.93 1 : 10 5.63 1 : 100 11.26
• These data are also given as graphs on p 187 of Goodman's 'Chemical Applications of Molecular Modelling'
• The range of compound distribution which can be measured easily by NMR integration is between 1 : 1 and 1 : 10, which is only 5.63 kJ mol-1
• Neither force field nor semi-empirical methods achieve that kind of accuracy, even for small organic molecules:  ab initio methods must be used
• In synthesis, an equilibrium with a  products : reactants ratio of 100 : 1 would be very useful
• The energy difference between going forwards or backwards, to that extent, is only about 22 kJ mol-1
• The best discussion of what methods can be used for calculating reaction energies, is given in chapter 4 of Hehre's  'Practical Strategies for Electronic Structure Calculations'
• The energies in the tables in Hehre's book are all in kcal mol-1 so to obtain these in kJ mol-1, multiply by 4.184

#### Reaction categories

Hehre divides reactions into four categories, which present different levels of difficulty in modelling reaction energies
•  The number of each kind of bond stays the same
• These are called isodesmic reactions
• Examples are:
• proton transfer, e.g.

• Me3N  +  [NH4]+[Me3NH]+   +  NH3
Here, there are four N-H bonds, three C-N bonds, and one lone pair, on each side of the equation
• redistribution reactions, e.g. transamination or transesterification
• all conformer interconversions
• comparisons between diastereomers
• Isodesmic reaction energies are the easiest to calculate
• The HF method with a medium-sized basis set is sufficient, but a density functional calculation should be better
• DE for the protonation of trimethylamine, above, was calculated as -92 kJ mol-1, at the RHF/6-31G* level, compared with an experimental value of -79 kJ mol-1.  A B3LYP/6-31G* calculation reproduced the experimental value.  (Hehre, p 145)
• The number of bonds and the number of lone pairs stays the same, but the bonds are between different elements
• Example:
• hydrogenation, e.g.

• CH2=CH2  +  2 H22 CH4
Here, C-H bonds are formed instead of H-H or C-C bonds, but there are eight bonds altogether on each side of the equation
• The HF method with a fairly large basis set - at least 6-31G*  - is needed
• DF methods with the same size basis set often produce poorer results for these
• The total number of electron pairs stays the same, but lone pairs become bonding pairs or vice versa
• Example:
• heterolytic cleavage

• Here, the HF method and a large basis set may work
• Anions have more diffuse electrons, so diffuse functions have to be added to the basis set to allow them to be modelled well
• Beware that free ions are imaginary:  the calculation should really be done for complexes of the ions to Lewis acids or bases respectively, e.g. solvent molecules
• This may be too difficult
• The total number of electron pairs is not conserved, i.e. odd electrons are involved
• Example:
• homolytic cleavage

• ROOR RO.  +  RO.
• Here, it is essential to use a method which calculates correlation energy.  The Hartree Fock method is not sufficient

#### Heats of formation by isodesmic reactions

• Heats of reaction can be calculated by simple addition and subtraction, if  heats of formation of all the components in the reaction are known
• DHf can be looked up for many simple molecules, so the problem reduces to calculating DHf for one or two reactants or products, rather than doing ab initio calculations on all species involved in the reaction
• While the reaction of interest may not be isodesmic, it is usually possible to invent isodesmic reactions involving only the compound whose DHf is required, along with simple molecules whose DHf are already known
• It is then possible to do ab initio calculations on the molecule of interest and on the simple molecules in the equation, and because the reaction is isodesmic, the calculated reaction energy should be of usable accuracy:  this will give the unknown DHf with the same accuracy
• It does not matter if the isodesmic reaction is completely unrealistic:  DHf is itself the energy of a completely unrealistic reaction (in most cases), but is useful in calculating the energy of a realistic reaction, because of the additivity of heats of reaction
• For example, suppose we are considering a reaction of cyclopentadiene, and hence wish to know its DHf
• Consider the hypothetical reaction

• This is called a bond separation reaction, because we have separated CpH into three C-C single bonds and two C=C double bonds
• We have 26 C-H bonds on both sides of the equation, so this is an isodesmic reaction
• We can calculate ab initio energies for the CpH and very easily for the simple organic molecules, at whatever same level is necessary, so we can obtain the energy of the isodesmic reaction
• DHf is well known for methane, ethane and ethylene
• We thus have DHf for CpH
• DHf for CpH was calculated as 151 kJ mol-1, at the B3LYP/6-31G* level, compared with an experimental value of 131 kJ mol-1  (Hehre, p 140)

#### Thermal energies and entropies

• These can be obtained by vibrational analysis, which is available in the major modelling packages
• If you have to do vibrational analysis to demonstrate that you have not found a saddle point (see  Energy minimisation used to optimise geometry  ) then thermal energies and entropies are calculated automatically (by Gaussian) as part of this
• Differences in thermal energies are often not a very significant part of an overall reaction energy, compared with differences in electronic energy, but they may be sometimes, so they are worth considering
• Ab initio calculations are usually done on isolated gas-phase molecules
• Their vibrational frequencies and associated vibrational energies bear some similarity to those for the molecule in solution, so are probably relevant
• The entropy of gas phase molecules is dominated by translational and rotational entropy, which will be quite different in solution
• Gas phase calculations may be sufficient to obtain a useful approximation to DH for a reaction in solution, but cannot give DG
• Often, people hope that entropy does not change significantly in a reaction, and assume DG  =  DH.   For comparison of conformers, diastereomers, etc., this may be a valid assumption
• If you have your unix id set up, you can view the results of a vibrational analysis for twist-boat cyclohexane, and see an animation by molden of the vibrations.  The instructions are given in a separate web document  Using more and molden to view a vibrational analysis