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.
Me_{3}N + [NH_{4}]^{+}[Me_{3}NH]^{+}
+ NH_{3} 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.
CH_{2}=CH_{2} + 2 H_{2}2
CH_{4} 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
DH_{f} can be looked up for many
simple molecules, so the problem reduces to calculating DH_{f}
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 DH_{f}
is required, along with simple molecules whose DH_{f}
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 DH_{f}
with the same accuracy
It does not matter if the isodesmic reaction is completely unrealistic:
DH_{f} 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 DH_{f}
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
DH_{f} is well known for methane,
ethane and ethylene
We thus have DH_{f} for CpH
DH_{f} 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