Transition state

# Coordination

• Nature of Hessian
• Anharmonicity

# General

## Tricks

• use a cheap method like xTB to generate an initial pathway, then switch to a higher level of theory like B3LYP

## How to check if the TS is found

1. Perform an analytical or numerical frequency calculation on the found structure. There must be one and only one imaginary frequency (corresponding to one and only one negative value in Hessian)
2. Visualize the vibrational mode associated with the imaginary frequency and make sure the motion corresponds to the desired reaction.
3. The transition structure must connect desired reactants and products.

## Two end methods

Two end: reactant and product - how the points between reactant and product are generated - what function of these points is minimized - what constrains are imposed to control the optimization

The advantage NEB is that multiple transition states if existed can be located in one calculation. Another advantage is that NEB is easy for parallelization.

## One end method

### Dimer

The dimer method only needs the structure of initial state and first derivative information to locate TS. - Transition state theory (TST): finding the free energy barrier for the transition - Harmonic transition state theory (hTST): finding the saddle point on the potential energy surface corresponding to a maximum along a minimum energy path

Two replicas $\vec{R} _ 1$ and $\vec{R} _ 2$ are separated from their midpoint $\vec{R} _ 0$. The distance between $\vec{R} _ 0$ and $\vec{R} _ 1$ is $\Delta R$ which is often very small (~0.001Å). The direction of dimer $\vec{N}$ is from $\vec{R} _ 2$ to $\vec{R} _ 1$. The curvature $C$ at $\vec{R} _ 0$ is calculated with the finite difference method. \begin{aligned} C &= \dfrac{(\vec{g} _ 1 - \vec{g} _ 2) \cdot \vec{N}}{2\Delta R} = \dfrac{[- \vec{F} _ 1 - (- \vec{F} _ 2)] \cdot \vec{N}}{2\Delta R} = \dfrac{( \vec{F} _ 2 - \vec{F} _ 1) \cdot \vec{N}}{2\Delta R} \\ &= \dfrac{[ \frac{E _ 1 - E _ 0}{\Delta R} - \frac{E _ 0 - E _ 2}{\Delta R}] }{\Delta R} = \dfrac{ E _ 1 + E _ 2 - 2 E _ 0 }{(\Delta R)^2} = \dfrac{ E - 2 E _ 0 }{(\Delta R)^2} \end{aligned}

• The default setting in ASE implementation is to calculate forces on $\vec{R} _ 0$ and $\vec{R} _ 1$ and evaluating forces at $\vec{R} _ 2$ with $\vec{F}_2 = 2 \vec{F}_0 - \vec{F}_1$

#### Rotation

The dimer will orient itself along the lowest curvature mode when the energy of the dimer is minimized by rotation.