Techniques

Consider a simple reaction (R $\longleftrightarrow$ P) in solution and the differential expressions for the reaction rate

$$\frac{dc_R}{dt} = -k_R c_R(t) + k_P c_P(t)$$

$$\frac{dc_P}{dt} = k_R c_R(t) - k_P c_P(t)$$ (2)

where $c_R(t)$ and $c_P(t)$ are the concentrations of the reactant R and the product P at time $t$ and $k_R$ and $k_P$ are the rate constants for the forward and backward reactions, respectively. We assume $k$ to be time independent (cf. ref zwanzig90 for a discussion of interesting cases for which $k$ is a function of time). Furthermore, we assume that the temperature dependence of the reaction is Arrhenius-like, which means it can be written in the form:

$$k = A(T) \exp{[-\Delta G^\ddagger/RT]} \quad ,$$ (3)

$$\frac{\mathrm{d}\ln(k)}{\mathrm{d}(1/T)} \approx -\frac{\Delta H^\ddagger}{R}$$ (4)

where $\Delta G^\ddagger$ and $\Delta H^\ddagger$, the activation free energy and activation enthalpy, respectively, are approximately independent of temperature and $A(T)$ is the pre-exponential factor with only a weak temperature dependence. The gas constant is $R=8.31451$ J K$^{-1}$mol$^{-1}$. In the condensed phase, we can distinguish between the relatively strong intramolecular bonds, with bonding energies which are much higher than the energy associated with the thermal motions ( $E_\mathrm{intra}\gg k_BT$) and the much weaker intermolecular solvent-solvent, solute-solute, and solute-solvent interactions, with binding energies in the order of that of the thermal motions ( $E_\mathrm{inter}\approx k_BT$). Traditionally, the intramolecular bonds belong to the territory of the quantum chemists as the making and breaking of these bonds (i.e. chemistry) is governed by the electronic structure. The weak intermolecular interactions on the other hand, are in the dominion of statistical thermodynamics as these bonds are broken and formed continuously on the time scale of the thermal motions, so that measurements (computations) require averaging over the different configurations of the interacting particles. To appreciate the title study of Chemistry in water, in which the estimation of the rate constant $k$ and its dependence of solvent effects belong to the important parameters, some background information is essential on both statistical thermodynamics and on electronic structure calculations. We will therefore start with elementary statistical thermodynamics in this section and the section hereafter, and then introduce the basics of the electronic structure calculations and the happy marriage of these two fields in the method of Car-Parrinello molecular dynamics in the sections 2.3 and 2.4.