$$Cl^{-} + H_3C-NC \rightarrow Cl-H_3C + NC^{-}$$
The activation free energy is given by the difference between the free energy of the transition state and the separated species
$$\Delta G = G^{\ddagger} - G^{Reac}.$$
The free energy in solvent is obtained by correcting the energy of the solute, \(E_{solute}\), with the solvation free energy, i.e. the energy required to place the solute in a cavity of solvent
$$G_{solution} = E_{solute} + 1/2 \int_S \sigma(\vec{r}) V(\vec{r}) d\vec{a}.$$
\(G_{solution}\) is output from a PCM calculation as Free Energy in Solvent, meaning the program computes first \(E_{solute}\) and then the solvation free energy (obtained from integrating the electrostatic potential times surface charge density over the surface of the solute).
This value is corrected for free energy contributions from translation, rotation and vibration and ZPE. The translational contribution is adjusted to report the correction for a 1mol/L solution instead of the ideal gas (which is 1mol/24.5L).
The reaction is bimolecular, meaning two particles collide to form one particle (the transition state) which then decays to products. For both reacting particles, the translational free energy correction has to be evaluated (since in solution, other than in gas phase, it is not expected that the particles form a coordinated complex [Vayner et al.]).
From HF/3-21G//HF/3-21G (the only method from which genuine transition state structures could be located), the following values are obtained.
| React 1 | React 2 | Sum | TS | Products | Activation Energy | |||
| PCM | Cl- | H3C-NC | Cl- + H3C-NC | TS | Cl-H3C + NC- | TS - (Cl- + H3C-NC) | ||
| HF/3-21G//HF/3-21G | E_elec | Ha | -457.5 | -131.2 | -588.6 | -588.6 | -588.6 | |
| ZPE | ZPE | Ha | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | |
| E(0K) | E_elec + ZPE | Ha | -457.5 | -131.1 | -588.6 | -588.5 | -588.5 | |
| E(0K) | kcal/mol | -287068.1 | -82277.2 | -369345.2 | -369298.3 | -369298.9 | 46.9 | |
| G_elec | kcal/mol | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||
| G_trans | 24.5l | kcal/mol | -9.4 | -9.6 | -19.0 | -10.1 | -10.1 | 8.9 |
| G_t(1l) = G_t(24.5l) - RTln(24.5) | RT*ln(24.5) | kcal/mol | 1.9 | 1.9 | 1.9 | 1.9 | 1.9 | |
| G_trans | 1l | kcal/mol | -11.3 | -11.5 | -22.8 | -12.0 | -12.0 | 10.8 |
| G_rot | kcal/mol | 0.0 | -5.2 | -5.2 | -6.5 | -6.5 | -1.2 | |
| G_vib | kcal/mol | 0.0 | 30.5 | 30.5 | 28.2 | 28.5 | -2.2 | |
| G_trv(298) | 1mol/24.5l | kcal/mol | -9.4 | 15.7 | 6.3 | 11.7 | 11.9 | 5.4 |
| G_trv(298) | 1mol/l | kcal/mol | -11.3 | 13.8 | 2.5 | 9.8 | 10.0 | 7.3 |
| S_trans | 24.5l | cal/(mol*K) | 36.6 | 37.1 | 73.6 | 38.9 | -34.7 | |
| S_trans | 1l | cal/(mol*K) | 34.7 | 35.2 | 69.8 | 37.0 | -32.8 | |
| T*S_trans | 298K, 24.5l | kcal/(mol) | 10.9 | 11.0 | 21.9 | 11.6 | -10.4 | |
| T*S_trans | 298K, 1l | kcal/(mol) | 10.3 | 10.5 | 20.8 | 11.0 | -9.8 | |
| E(0K) + G_trv(298) | 24.5l | kcal/mol | -287077.5 | -82261.5 | -369339.0 | -369286.6 | -369287.0 | 52.346 |
| E(0K) + G_trv(298) | 1l | kcal/mol | -287079.4 | -82263.4 | -369342.8 | -369288.5 | -369288.9 | 54.250 |
| Free energy in solvent (PCM) | kcal/mol | -287067.9 | -82307.8 | -369375.7 | -369327.1 | -369328.7 | 48.530 | |
| Free energy in solvent + ZPE + G_trv | 24.5l | kcal/mol | -287077.3 | -82261.4 | -369338.7 | -369286.4 | -369286.7 | 52.348 |
| Free energy in solvent + ZPE + G_trv | 1l | kcal/mol | -287079.2 | -82263.3 | -369342.5 | -369288.3 | -369288.6 | 54.252 |
The increase of the barrier due to the loss in translational entropy is found in the third and fourth column, where '+' indicates the summation of the values from the first and second column (i.e. S\(_{trans, 1M}\) = 69.8 cal/(mol*K) = (34.69 + 35.16) cal/(mol*K), which are obtained from separate calculations of the two species (Cl\(^-\) and H\(_3\)C-NC). This is not the same as calculating S\(_{trans, 1M}\) from a coordinated complex, which consists only from one particle.
The translational entropy (for a 1M state) of the TS however is 37.0 cal/(mol*K). At \(T=298K\), the total entropic change evaluates to (11.59 - 21.95) kcal/mol = -10.35 kcal/mol (for 1mol/24.5L) and from
$$G = H - TS$$
it is seen that this loss of entropy will lead to an increase of the barrier. The correction in translational free energy when going from the gas phase standard state to the solution standard state is moderate (around 2 kcal/mol).
It is seen that given such an approach, the barrier would be estimated to be around 54kcal/mol.
The following points are unclear to me:
- Should the free energy in solution be corrected for ZPE and free energy contributions from translation, rotation and vibration?
- Are the free energy corrections of the reactants correct when calculated from a coordinated structure (as opposed to two individual structures in separate calculations)?
- Which effects not included (apart from electronic structure properties) would be most relevant when extending the model and how could they be obtained? Conformational free energy correction from MD snap shots?
Surprisingly, the activation energy in E(0K) + G\(_{trv, 298, 1L}\) without considering solvation (54.25 kcal/mol) is the same as the activation free energy in solvent. Is this a coincidence or does this make sense or not?
