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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For a given initial guess path, the force perpendicular to the path will be non-zero. Thus the basic idea of string method is to solve equation (1) by evolving curves in configuration space according to the dynamics
\begin{displaymath} v_n=-\left( V\right)^\perp (\varphi) \end{displaymath} (3)

where $v_n$ is the normal velocity of the curve, subjected to the condition that $\varphi$ connects A and B. It is easier to express (3) in terms of a particular parameterization, $\varphi(\alpha,t)$. A simple choice is the equal-arclength parameterization. Equation (3) then becomes

\begin{displaymath} \partial_t \varphi = -\left( \nabla V\right)^\perp \left( \varphi\right)+\gamma \hat\tau \end{displaymath} (4)

where $\gamma$ is a Lagrange multiplier for the particular parameterization that is chosen. For the equal-arclength parameterization, one has

\begin{displaymath}\partial_\alpha \vert\varphi_\alpha\vert^2=0 \end{displaymath}

i.e.

\begin{displaymath}\gamma(\alpha,t)= \alpha \int_0^1 \nabla V\cdot \hat \tau_{\a... ...pha' - \int_0^\alpha \nabla V\cdot \hat \tau_{\alpha'}d\alpha'.\end{displaymath}

Other parameterizations can also be used.