Rare Events, Transition Pathways and Reaction Rates

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

Algorithm

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The string is discretized into a number of points, each being a configuration in space. Equation (4) is solved by a time-splitting scheme:
  • Evolve the string by solving
    \begin{displaymath}\partial_t \varphi = -\left( \nabla V\right)^\perp\left(\varphi\right) \end{displaymath}

  • Reparameterize the string by interpolation to enforce, e.g. equal-arclength.

Reparameterization can be carried out once for every few steps.


Here are some simple examples. The codes are some what crude, may require further optimization!
  1. c code:  main.c PotentialVal.c

    The 2D potential described under section 'D. Illustrative Examples' in
    Simplified and improved string method for computing the minimum energy paths in barrier-crossing events, J. Chem. Phys., 126 (16), 164103 (2007)

    The red dashed curve shows the MEP and red solid curve is the initial guess path. Point D is minimized using CG along the normal EF. Line AB is the minimization direction according to the algorithm proposed by Elber and co-workers.

    This code requires GNU Science Library - GSL for cubic spline interpolation for reparameterization and conjugated gradient minimization


  2. The 2D potential : Free energy profile along a discretized reaction path via the hyperplane constraint force and torque, J. Chem. Phys., 122, 114108 (2005). In this code the initial guess path is pretty crude - a straight line joining the two local minimum points (red dash line). The converged MEP is the solid red line. The local minimum points are (-2.7127, 0.1509) and (2.7127, -0.1509) and the saddle is at (0,0).

    This code requires GNU Science Library - GSL for cubic spline interpolation for reparameterization. The constrained minimization is performed with steepest descent minimization scheme.

Send comments to Amit Samanta: asamanta AT math.princeton.edu or Weinan E: weinan AT math.princeton.edu

-Last updated on May 25, 2010.