From Metadynamics to Dynamics

Metadynamics is a commonly used and successful enhanced sampling method. By the introduction of a history dependent bias which depends on a restricted number of collective variables it can explore complex free energy surfaces characterized by several metastable states separated by large free energy barriers. Here we extend its scope by introducing a simple yet powerful method for calculating the rates of transition between different metastable states. The method does not rely on a previous knowledge of the transition states or reaction coordinates, as long as collective variables are known that can distinguish between the various stable minima in free energy space. We demonstrate that our method recovers the correct escape rates out of these stable states and also preserves the correct sequence of state-to-state transitions, with minimal extra computational effort needed over ordinary metadynamics. We apply the formalism to three different problems and in each case find excellent agreement with the results of long unbiased molecular dynamics runs.

Figure 1

(a) Model potential energy surface from Ref. 19 showing various stable states. The highest contour is at $V=2$ energy units and contours are separated by 0.25. (b) Logarithm of the one-dimensional diffusion constant versus inverse temperature (bottom) and temperature (top). The stars (blue), circles (red), and asterisks (green) correspond to calculations for three values of the well-tempered metadynamics effective temperature [4, 25]: 0.75, 0.625, and 0.5, respectively, while the solid line is for kinetic Monte Carlo calculations reported in Ref. 19. 95% confidence intervals are provided.

Figure 2

(a) Logarithm of one over the average escape time in ns (log $\nu$) from $C_{7eq}$ to $C_{7ax}$ versus inverse temperature (bottom) and temperature (top). Stars (red) are from metadynamics, circles (blue) are from long unbiased MD. Below 300 K, we did not observe transitions in unbiased MD within constraints of computer power. 95% confidence intervals are also provided. (b) The $\theta -\Phi$ anticorrelation in trajectories as they cross over from $C_{7eq}$ to $C_{7ax}$, obtained from metadynamics without using $\theta$ as a CV. (c) Acceleration [Eq. (3)] versus metadynamics time at $T=300\mathrm{K}$, overlaid on the system’s trajectory in the $\Phi$ dimension showing distinct kinks each time a TS is crossed.