Rare Events, Transition Pathways and Reaction Rates

Introduction Zero-Temperature
String Method 		Finite-Temperature
String Method 		Modified
String Method 		Reference

Minimal Energy Path

Dynamics of String

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Consider two local minima A and B on the multi-dimensional potential energy landscape. A can be taken as the initial state (in the configuration space) and B be the final state (the algorithm actually works fine as long as B is on the other side of the line dividing the two basins). Let $V$ be the potential which characterizes the PES and $\varphi$ be a path in configuration space that connects A and B. This path is parameterized by $\alpha$ which can be the position of the system in the multidimentional as is diffusion PES or size of loop for a nucleation process. The unit tangent vector is denote by $\hat \tau$. $\varphi$ is a minimal energy path (MEP) if at each point on the path the force perpendicular to the path is zero. This means
\begin{displaymath} \left( \nabla V \right) ^\perp \left( \varphi(\alpha)\right) =0 \end{displaymath} (1)

i.e.

\begin{displaymath} \left( \nabla V\right) \left( \varphi(\alpha) \right) \parallel \hat \tau (\alpha). \end{displaymath} (2)

Here $\perp$ denotes projection to the hyperplane normal to $\hat \tau$.