Mechanism and Kinetics in Homogeneous Catalysis …
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potential energies. Yet at the time, including these corrections had not yet become
mandatory in computational chemistry. The reasons for this are manifold. First,
calculating the required vibrational frequencies was and remains computationally
demanding for larger systems (the model treated in our study, while of very modest
size by today’s standards, was not completely trivial at the time). Next, the accuracy
of vibrational frequencies computed for systems whose structure was optimized
with PCM solvation was not always very high, due to various numerical issues.
Finally, there was some doubt about the need to include ‘ideal-gas-like’ free energy
corrections for systems in solution. In retrospect, these doubts were incorrect, but they
were quite widespread, and can perhaps be attributed to the fact that in many cases,
free energy corrections to the potential energy roughly match in magnitude—but
with opposite sign—the dispersion corrections needed to improve DFT energies.
The fact that these two apparently very different effects roughly match in magnitude
is associated with the fact that both dispersion effects and entropy effects tend to be
large when dealing with processes in which two fragments come together to form
a larger species (or the reverse of course). Hence for unimolecular processes, both
effects tend to be small, whereas they become large for addition steps (like the first
step in Figs. 1 and 2), and fortuitously roughly match in magnitude.
Given these many differences in the theoretical protocol used in [9] and the current
state of the art, it is natural to wonder to what extent the conclusions reached are still
applicable. In Table 1, the electronic energies from reference [9] are compared to
new results generated for this review, including a ‘best estimate’ of the relative free
energies at a level of theory similar to that used by us in a recent study of another
reaction [16].
As can be seen in the table, including the corrections for entropy and for dispersion does indeed have a very large impact on relative energies. For example, the
betaine complexes are stabilized by 3–5 kcal mol
−1 compared to the earlier study.
The CCSD(T) calculations match the B3LYP-D3 results quite well, though the cisoid
betaines are somewhat more stable relative to reactants with the CCSD(T) approach.
Another difference concerns the effect of the free energy corrections. As expected,
these destabilize all the species relative to reactants, which again makes sense since
one is bringing two separate molecules together to form one complex. As mentioned
above, then, the dispersion and free energy corrections are in opposite directions, but
the free energy correction is in this case somewhat larger, so the relative free energy
of the TSs compared to reactants is somewhat higher than the published [9] relative
potential energies with B3LYP.
In terms of the qualitative picture emerging from the calculations, the comforting
outcome is that the new calculations remain broadly in favor of the previous conclusion: The deciding factor making the trans epoxide product dominate is the high
energy and free energy of the syn torsional TS, which is confirmed to be rate-limiting.
As a final note on this reaction, it is worth noting that carrying out the new calculations was greatly facilitated by the fact that the original paper [9] contained optimized
structures for all species in the Supporting Information, so that this data was available
long after the original work had been done. In the past, quantum chemical studies
often included full structural data for all optimized species in Figures, and various
295
potential energies. Yet at the time, including these corrections had not yet become
mandatory in computational chemistry. The reasons for this are manifold. First,
calculating the required vibrational frequencies was and remains computationally
demanding for larger systems (the model treated in our study, while of very modest
size by today’s standards, was not completely trivial at the time). Next, the accuracy
of vibrational frequencies computed for systems whose structure was optimized
with PCM solvation was not always very high, due to various numerical issues.
Finally, there was some doubt about the need to include ‘ideal-gas-like’ free energy
corrections for systems in solution. In retrospect, these doubts were incorrect, but they
were quite widespread, and can perhaps be attributed to the fact that in many cases,
free energy corrections to the potential energy roughly match in magnitude—but
with opposite sign—the dispersion corrections needed to improve DFT energies.
The fact that these two apparently very different effects roughly match in magnitude
is associated with the fact that both dispersion effects and entropy effects tend to be
large when dealing with processes in which two fragments come together to form
a larger species (or the reverse of course). Hence for unimolecular processes, both
effects tend to be small, whereas they become large for addition steps (like the first
step in Figs. 1 and 2), and fortuitously roughly match in magnitude.
Given these many differences in the theoretical protocol used in [9] and the current
state of the art, it is natural to wonder to what extent the conclusions reached are still
applicable. In Table 1, the electronic energies from reference [9] are compared to
new results generated for this review, including a ‘best estimate’ of the relative free
energies at a level of theory similar to that used by us in a recent study of another
reaction [16].
As can be seen in the table, including the corrections for entropy and for dispersion does indeed have a very large impact on relative energies. For example, the
betaine complexes are stabilized by 3–5 kcal mol
−1 compared to the earlier study.
The CCSD(T) calculations match the B3LYP-D3 results quite well, though the cisoid
betaines are somewhat more stable relative to reactants with the CCSD(T) approach.
Another difference concerns the effect of the free energy corrections. As expected,
these destabilize all the species relative to reactants, which again makes sense since
one is bringing two separate molecules together to form one complex. As mentioned
above, then, the dispersion and free energy corrections are in opposite directions, but
the free energy correction is in this case somewhat larger, so the relative free energy
of the TSs compared to reactants is somewhat higher than the published [9] relative
potential energies with B3LYP.
In terms of the qualitative picture emerging from the calculations, the comforting
outcome is that the new calculations remain broadly in favor of the previous conclusion: The deciding factor making the trans epoxide product dominate is the high
energy and free energy of the syn torsional TS, which is confirmed to be rate-limiting.
As a final note on this reaction, it is worth noting that carrying out the new calculations was greatly facilitated by the fact that the original paper [9] contained optimized
structures for all species in the Supporting Information, so that this data was available
long after the original work had been done. In the past, quantum chemical studies
often included full structural data for all optimized species in Figures, and various