"Proton on a string"

In the preprint "Modified string method for finding minimum energy path" (arXiv:1009.5612), Amit Samanta and Weinan E describe a method for finding so-called minimum energy paths (MEP). These paths are the ones with most statistical weight wrt. a transition in configuration space. This could e.g. be a diffusional process, which is shown below for proton conduction in SrTiO₃ (Sr, Ti, O and H depicted as green, gray, red and white spheres, respectively; the potential energy surface has been calculated using the Quantum Espresso package), see the preprint for other examples.

I have compared their presented modification of the string method to the nudged elastic band method (NEB). Both methods sample the path using a finite number of images. An initial guess for the path is optimized by gradient following. The two methods differ in how the sliding down of intermediate images from barriers is prevented. For the NEB, this is achieved by introducing virtual spring forces, keeping the images apart. For the string method, the path is iteratively re-parametrized such that the path is evenly sampled.
Both methods perform similarly well. Below are shown the potential energies for MEPs obtained using the string and NEB methods for the above proton diffusion path in SrTiO₃, respectively:


The residual gradients in the initial and final configurations seem to be a little too large, therefore the NEB path shows the tendency to have minima away from these configurations. The string method, which is implemented here to fulfill Eq. 6 of the preprint at each optimization step, is more stable against this problem (in the case of the NEB, intermediate configurations at ~0.5 and ~4 Å should be relaxed separately as new boundary configurations).
What is interesting is that the new string method shows a slightly better convergence of the MEP (here optimized using Broyden's method with rank one Quasi-Newton updates):

Instead of plain re-parametrization at each step, the more sophisticated schemes outlined in the preprint might even yield better convergence. Despite the simplicity of what was implemented here, this seems to be basically as good as the NEB method.