# Part 1 – Reaction rate constant with transition state theory (TST): partition functions

As part of the work carried out to understand origins of life, it is crucial to determine which processes/reactions were most likely to occur. The thermal rate constant of a reaction defines the efficiency of the process at a given temperature. To get an exact rate, one needs an accurate potential energy surface and quantum time-dependent dynamics of reactants on the potential energy surface. This is computationally very demanding.

One simple theoretical approach which avoids having to propagate in time is classical Transition State Theory (TST). For more details on TST see the original papers by Wigner, Eyring, Evans and Polanyi:

• Pelzer and Wigner, Z. Phys. Chem. B 15, 445 (1932)
• Eyring, J. Chem. Phys. 3, 105 (1935)
• Evans and Polyani, Trans. Faraday Soc. 31, 857 (1935)

The main idea of TST is that the reaction dynamics can be described by a one dimensional minimum energy path on the potential energy surface which connects reactants to products through a transition state. The transition state is found at the maximum energy along the one dimensional path and it is a first order saddle point in the multidimensional potential energy surface. The imaginary frequency of the transition state is not included in the computation of the transition state partition function as it corresponds to the reaction coordinate degree of freedom.

The approximations required to derive TST are

• the Born-Oppenheimer approximation is valid
• reactants are at equilibrium and in the Boltzmann distribution
• no recrossing: molecules which have crossed the transition state may not return no recrossing of transition state and classical dynamics of the nuclei
• at the transition state, the motion along the one dimensional reaction coordinate can be separated from other motion and treated classically as translational motion

The expression for the TST rate constant is then

$\sf k(T)=$ $\sf \frac{k_BT}{h}\frac{Q^{\ddagger}(T)}{Q_R(T)}e^{-\beta E_a}$

Here we see the ratio of the canonical partition function for the transition state, $\sf Q^{\ddagger}$, and the canonical partition function for the reactants, $\sf Q_R$, as well as the exponential of the reaction energy barrier $\sf E_a$. As long as the ratio of partition functions stays constant as a function of temperature, the rate constant will decay linearly with temperature in log scale.

### Example

Let’s see how it works for a simple reaction

CH4 + H → [CH5+]→H2+CH3

Here for simplicity the transition state is indicated in the square brackets. We’ll begin by some general expressions and then implement these for the above reaction.

• Compute partition functions of reactants and transition state

The most accurate way to compute the partition function of the molecules in their ground state would be to diagonalize the reactant and transition state Hamiltonians and sum over all eigenstates, i.e

$\sf Q(T)=\sum_i g_i e^{-\beta E_i}$

With $\sf \beta=1/(k_BT)$ and $\sf g_i$ is the degeneracy of the i-th level. This expression is difficult to evaluate as one needs a full potential energy surface. Diagonalizing the Hamiltonian is expensive and gets worse as the number of degrees of freedom increases. Therefore, further approximations are made.

Assuming we have one of each of these molecules, the overall canonical partition function can be expressed as the product of the partition function for electronic, vibrational, translational and rotational degrees of freedom:

$\sf Q(T)=Q_{\textsf{el}}(T)\cdot Q_{\textsf{vib}}(T)\cdot Q_{\textsf{tras}}(T)\cdot Q_{\textsf{rot}}(T)$

For N identical molecules this expression needs to be elevated to the power of N and rescaled by N!.

Each partition function can then expressed as

$\sf Q_\textsf{tras}(T)=$ $\sf \left(\frac{2\pi \textit{\textsf {m}} \sf k_B T}{h^2}\right)^{3/2}\cdot V%s=2$

$\sf Q_\textsf{vib}(T)=$ $\sf \prod_i^{n_{\textsf{vib}}}\frac{e^{\sf -\beta\hslash\omega_i/2}}{1-e^{\sf -\beta\hslash\omega_i}}%s=2$

$\sf Q_\textsf{rot}(T)=$ $\sf \frac{\pi^{1/2}}{\sigma_r}\left(\frac{T^3}{\Theta_A\cdot\Theta_B\cdot\Theta_C}\right)^{1/2} %s=2$

with m the mass and V the volume, $\sf n_{\textsf{vib}}$=3N-6 for non linear molecules and 3N-5 for linear molecules. The rotational constants are obtained from the three moments of inertia $\sf I_j$  as $\sf \Theta_j=\frac{h^2}{8\pi^2 I_j k_b}$. The paramerter $\sf \sigma_r$ is the rotational constant which can be obtained from the point group of the molecule. In general if your molecule is not at equilibrium it may be complicated to estimate this value, for more details see e.g. M. K. Gilson and K. K. Irikura, J. Phys. Chem. B, 114:16304 (2010). For a linear molecule the rotational partition function is simply $\sf Q_{\textsf{rot}}=\frac{T}{\sigma_r\Theta_r}$.

The electronic partition function is approximated to be $\sf Q_{\textsf{el}}=g_{el}^{\textsf{ground}}$ i.e the ground state electronic degeneracy.

Note that in the above expressions for the partition functions we have assumed that

• the harmonic approximation is valid
• there is no roto-vibrational coupling
• the molecules can be seen as linear or non linear rigid rotor
• the main contribution to the electronic partition function comes from the ground electronic state

For this reaction at T=300K, we find

$\sf{ Q_{CH_4}=7.93e+12\text{; }Q_{H}=9.77e+29 \textsf{ and } Q_{CH_5^+}=1.27e+15}$

and we have used

$\sf{\sigma_r(CH_4)=12}$$\sf{\sigma_r(CH_5^+)=2}$,

$\sf{\omega(CH_4)=}$ $\sf{\left(\textsf{1341 1341 1341 1569 1569 3031 3151 3151 3151}\right) cm^{-1}}$

and

$\sf{\omega(CH_5^+)=}$ $\sf{\left(\textsf{534 534 1074 1125 1125 1442 1442 1795 3076 3223 3223}\right) cm^{-1}}$

excluding the imaginary frequency at the saddle point and

$\sf{I(CH_4)}=$ $\sf{ \left(\textsf{3.173 3.173 3.173}\right) au\cdot\AA^2}$

and

$\sf{I(CH_5^+)}=$ $\sf{ \left(\textsf{4.401 4.436 4.460}\right) au\cdot\AA^2}$

Some of these values (e.g. frequencies) were taken from J. Chem. Phys. 124:164307 (2006). For the electronic partition function we used the spin multiplicity of 2 for hydrogen, 1 for methane and 2 for the transition state.

Next post we’ll put this all in and get the rate constant as a function of temperature.