Kramers Reaction Rate Theory
===============================

:Date: 28 July 2019

Dependency
-------------

- `Statistical Mechanics`_
- `Boltzmann Factor`_

.. _Statistical Mechanics: 
.. _Boltzmann Factor:


What is the problem ?
------------------------

.. figure:: Kramers-Reaction-Rate-Theory-1.webp
   
   *TO BE MODIFIED*

.. math::
	\require{mhchem}
	\ce{A <=> B ->[k^+] C}

Let the transition state :math:`B` correspond to **reaction coordinate** :math:`x=x^{\ne}`

Derivation
-------------

**Step 1** **Transition State Theory (TST)**

.. math::
	\begin{align*}
	& k^+ = {\text{flux} \over N_A} \\
	& \text{and}\\
	& \text{flux} = \underbrace{{N^{\ne} \over \Delta x}}_{\text{density}}
		\underbrace{\langle v^{\ne} \rangle}_{velocity} \\
	& {N^{\ne} \over N_A} = {z^{\ne} \over z_A} \; (\text{Boltzmann Factor})\\
	\Rightarrow & k^+ = {z^{\ne} \over z_A} {\langle v^{\ne} \rangle \over \Delta x}
	\end{align*}

**Step 2** Find :math:`z^{\ne}`

.. math::
	\begin{align*}
	& z^{\ne} = {1\over h} \int dp 
		\int_{x^{\ne}-{\Delta x \over 2}}^{x^{\ne}+{\Delta x \over 2}}
		dx \; \exp[{p^2\over 2m}+V(x)] \\
	&\text{Assume } V=const \text{ around } x^{\ne}\\
	\Rightarrow z^{\ne} & = {1\over h} \Delta x\; e^{-\beta V(x^{\ne})} 
		\sqrt{2\pi m \beta}
	\end{align*}

**Step 3** Find :math:`z^A`

*TO BE ADDED*

**Step 4** Find :math:`\langle v^{\ne} \rangle`

*TO BE ADDED*

**Step 5** Combining above

.. math::
	\boxed{
		k^+ = {\omega_A \over 2\pi}e^{-\beta V(x^{\ne})} \;\;\; \text{TST}
	}

**Step 6** Modify **TST** to get **Kramers Theory**

*TO BE FURTHER EXPLAINED*

.. math::
	\boxed{
		k^+ = {1\over \omega_B}\left(-{\gamma \over 2}+\sqrt{{\gamma^2 \over 4}+\omega_B^2}\right) 
		\left\{
			{\omega_A \over 2\pi}e^{-\beta V(x^{\ne})}
		\right\} \;\;\; \text{Kramers}
	}