Original Metadynamics

In metadynamics, a time-dependent bias potential is “grown” during the course of the simulation that acts to enhance sampling of the order parameter [48]. Rather than confining to local regions of order parameter space as in umbrella sampling, the metadynamics potential pushes the system away from easily sampled regions of order parameter space. This bias can be expressed:

$$V_b(\theta,t) = w\sum_{t^\prime < t} \exp\left(-\frac{\left[\theta(t^\prime)-\theta(t)\right]^2}{2\sigma^2}\right)$$ (363)

Here, $w$ is a weight of each Gaussian kernel deposited, and $\sigma$ is its width. Apart from them, another key parameter in running metadynamics is how frequently a new Gaussian kernel is added, i.e., what is the list of values for $t^\prime$? Apart from an irrelevant constant, the free energy along the order parameter is the time-average of the bias potential

$$F(\theta) = -\frac{1}{t_f-t_i}\sum_{t=t_i}^{t_f}V_b\left[\theta(t)\right]$$ (364)

NAMD includes native support for metadynamics using the colvars module. By default, kernels are deposited every 1,000 steps. To illustrate metadynamics, we return to the system of a single molecule of butane, this time at 273 K. It requires a 100-million time-step MD simulation to generate a smooth histogram for the C1-C4 distance at this temperature. Fig. 46 shows the free energy vs C1-C4 distance computed using a 10-million time-step metadynamics simulation for which $w$ = 0.1 kcal/mol, and $\sigma$ = 0.1 Å. We see excellent reconstruction of the true free energy at a much lower computational cost with metadynamics.

Figure 46: Free energy in kcal/mol vs. C1-C4 distance in Å, for butane in vacuum at 273 K computed using MD (green dash) and metadynamics (black solid). The MD simulation was run for 10$^8$ timesteps, and the metadynamics for 10$^7$. Intermediate values of the metadynamics free energy are also shown color-coded from purple (early) to yellow (late). The final metadynamics free energy is the average over all free-energy snapshots (i.e., it is the time-average negative bias potential).

This 10$^7$-time-step metadynamics simulation deposited 10,000 Gaussian kernels in total. Generally, it is most efficient for the simulation to keep track of the bias potential on a grid rather than as an explicit sum of Gaussians. Here, the order-parameter line was divided into increments of 0.02 Å between 1.5 and 5.5 Å, or roughly 400 points. Fig. 47 shows the evolution of the bias potential $V_b(t)$ from this simulation.

Figure 47: Evolution of the bias potential from the 10$^7$-step metadynamics simulation of butane along the C1-C4 distance. The instantaneous bias potential is drawn every 10$^5$ time-steps and shifted to its current minimum value. Each curve is drawn with an $\alpha$ of 0.2, so where the curves appear opaque signifies a temporarily static portion of the bias as kernels are being deposited elsewhere.

The accuracy of metadynamics is fairly sensitive to $w$ and $\sigma$. Fig.  shows free energies for the butane system at 273 K computing using metadynamics (and the long MD for reference) using various combinations of $w$ and $\sigma$, all for 10$^7$ steps. Among those considered, it appears $w$ = 0.1 kcal/mol, and $\sigma$ = 0.1 Å are the best choices. Generally, smaller $w$ will yield more accurate free energies, but at larger computational cost.