Appendix B: Reaction rate

To calculate the reaction rate of the S$_\mathrm{N}$2 reaction in aqueous solution, it is not correct to take the computed free energy difference ( $\Delta A^\ddagger=27$ kcal/mol) and plug it into the textbook relations

$$\frac{d[{\rm X}^-]}{dt}=\frac{d[{\rm CH_3Y}]}{dt} = - k_f [{\rm X^\prime}^-][{\rm CH_3Y}] + k_b [{\rm Y}^-][{\rm CH_3X}]$$ (54)

$$k_f = \frac{k_B T}{h} e^{-\beta \Delta A^\ddagger}$$ (55)

for two reasons.

In the first place, is the exponential term, $e^{-\beta \Delta A^\ddagger}$ related to the Boltzmann probability $P(\xi^\ddagger)$ to find the reacting system in the transition state independent of the exact starting configuration in the reactant well. Thus, instead of taking $\Delta A^\ddagger=A(\xi=0.5)-A(\xi=0)$, the Boltzmann factor is obtained by integration over the reactant well of the free energy profile, using

$$P(\xi^\ddagger) = \frac{e^{-\beta A(0.5)}}{ \int\limits^{0.5}_{-\infty} d \xi e^{-\beta A(\xi)}}$$ (56)

As the energy will increase rapidly as $\xi$ decreases below zero (since it is associated with configurations in which none of the Cl's are chemically connected to the CH$_3$ part), it is not necessary to calculate the profile $A(\xi)$ up to $\xi\rightarrow\infty$. Also, we note that the profile is dependent on the size of the periodic supercell, simply because the available space, and therefore the entropy, of the system at a certain reactant state constraint value (say $\xi =0$) increases with the box size. (Obviously, the probability to find the two reactants in the transition state is much smaller in an ocean of liquid than in a small box with a few tens of water molecules). Normalizing the Boltzmann factor with the concentration gives the correct result for the transition state theory rate for the second order rate coefficient.

In the second place, does the free energy in the exponential term of equation 3.9 include a correction for the fact that not in all occasions that the system reaches the transition state, it actually crosses the barrier to end up in the product well. Instead, the computed Boltzmann factor (equation 3.10) is an ensemble average that only gives the (transition state theory) upper limit of the rate constant $k^{\rm {TST}}$, assuming that every time the barrier top is reached, the products are subsequently formed. Multiplying $k^{\rm {TST}}$ by the so-called transmission coefficient, $\kappa$, gives the true rate constant. The transmission coefficient is a time dependent function, which correlates the initial velocity of the system along the reaction coordinate, $\dot{\xi}$, on top of the barrier at time $t$ with $\xi$ at time $\tau=t+\Delta t$:

$$\kappa(\tau)= \frac{ \left< \dot{\xi}(0) \delta(\xi(0)-\xi^\... ...\right> } {\left< \frac{1}{2} \vert\dot{\xi}(0)\vert \right> }$$ (57)

For $\tau=0$, the transmission coefficient $\kappa=1$, but $\kappa$ rapidly drops with increasing $\tau$ to a plateau value for $\tau$ much larger than the time scale of the barrier recrossings. This plateau value gives the desired correction to $k^{\rm {TST}}$.

Figure 3.7: Transmission coefficient $\kappa$ as a function of the reactant-solvent coupling $\lambda$.

For a reaction in solution $\kappa$ is related to the coupling $\lambda$ between the reactants and the solvent, as schematically drawn in figure 3.7. At low reactant-solvent coupling values (aka Lindemann-Hinshelwood regime or energy diffusion regime), the energy obtained by the reactants from the solvent to reach the barrier top cannot be dumped rapidly back to the solvent environment after crossing the barrier. As a result, the reactant complex retains enough energy to cross the transition barrier back and forth until it finally loses the energy and the correlation between $\dot{\xi}(0)$ and the final value of $\xi$ is low, hence a small transmission coefficient $\kappa$ which is proportional to $\lambda$. At the other side (aka Kramers regime or spatial diffusion regime), the reactant complex is pushed back and forward by the fluctuation in the solvent environment, which also results in a small correlation and thus a small $\kappa$ which is inversely proportional to $\lambda$. In between these two reactant-solvent coupling extremes $\kappa$ has values closer to unity.

For the S$_\mathrm{N}$2 reaction in water, we see that a large force pulls the [Cl$\cdots$CH$_3 \cdots$Cl]$^-$ complex along the reaction coordinate in the either direction if the solvation of the two chloride groups becomes unbalanced (compare table 3.7 and figure 3.5). In order for the system to proceed to either side, one Cl has to gain solvent molecules in the solvation shell, whereas the other Cl has to expel H$_2$O molecules. Due to this very strong coupling between the solvent environment and the reaction complex, a low transmission coefficient is expected for the S$_\mathrm{N}$2 reaction in water.