Transition state

Table of Contents

- Nature of Hessian
- Anharmonicity
- Seideman and Miller, J. Chem. Phys. 95, 1768–1780 (1991): anharmonic corrections, contribution of imaginary mode to zero-point energy

Imaginary frequencies is not included in thermo contributions.

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

- 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)
- Visualize the vibrational mode associated with the imaginary frequency and make sure the motion corresponds to the desired reaction.
- The transition structure must connect desired reactants and products.

怎么验证过渡态找没找对？

如果优化出来的过渡态结构在期望的反应物和产物之间，同时有且只有一个虚频，一般能有6成把握认为过渡态找对了。

如果通过gview查看，发现仅有的那个虚频的振动方向对应反应坐标方向，那8成可以认为过渡态找对了。

如果走IRC还能连通自己期望的反应物和产物结构，那么说明100%找对了。

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.

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 $\vec{F}_2 = 2 \vec{F}_0 - \vec{F}_1$

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