The Shape of V(t)

Several shapes have been tested (and more might be proposed in the future). The default choice is the construction of $V(t)$ by the accumulation of Gaussian-like hills, i.e. (within the Lagrangian formulation, but the expressions are the same for the direct MTD approach, providing to exchange $\bf s$ with $\mathbf{S(\cdots)}$ )

$$V(t,\mathbf{s}) = \sum_{t_{i} < t } \Bigg[W_{i}\exp \left \{- \frac{(\mathbf{s}-\mathbf{s}^{i})^{2}}{2 (\Delta s^{\perp})^{2}} \right \}$$

$$\exp \left \{-\frac{\left( (\mathbf{s}^{i+1}-\mathbf{s}^{i}) \cdo... ...}-\mathbf{s}^{i})\right)^{2}} {2 (\Delta s^{\vert\vert}_{i})^4} \right\}\Bigg],$$ (263)

Here, $t$ indicates the actual simulation time, $i$ counts the metadynamics steps, the first exponential gives the hill's shape in the direction perpendicular to the trajectory, whereas the second exponential tunes the shape along the trajectory. In this form, the width of the hill along the trajectory is determined by the displacement in the CV-space, walked between two consecutive metadynamics steps, $\Delta s^{\vert\vert}_{i} = f_{b}\sqrt{\Big[\sum_{\alpha}(s_{\alpha}^{i+1}-(s_{\alpha}^{i})^2\Big]}$ . $f_b$ is a factor, which is read in input and can be used to change the size of the hills along the trajectory, by default it is 1. The height $W$ and the width $\Delta s^{\perp}$ are input parameters that can also be tuned during the MTD, in order to better fit the hill shape to the curvature of the underlying energy surface (in the CV-space). As a rule of thumb, $\Delta s^{\perp}$ should have roughly the size of the fluctuations of CV at equilibrium (half the amplitude of the well) and $W$ should not exceed few percents of the barrier's height. These information can be obtained by some short MD runs at equilibrium (without MTD) and from some insight in the chemical/physical problem at hand. Since, in general, different CV fluctuate in wells of different size, it is important to define one scaling factor $scf_{\alpha}$ for each component $s_{\alpha}$ , so that $\langle \delta s_{\alpha}\rangle /scf_{\alpha} = \Delta s^{\perp} \forall \alpha$ .

Other implemented shapes of $V(t)$ are:
Shift: the tails of the Gaussians are cut off, by zeroing the Gaussian at a distance $R_{cutoff}\Delta s^{\perp}$ from its center. In this way the problem of the overlap of the tails in regions far from the actual trajectory is reduced.
Rational: instead of Gaussian-like hills, some kind of rational functions are used,

$$V(t,\mathbf{s}) = \sum_{t_{i} < t } \Bigg[W_{i} \frac{1- \left(\frac{\sqrt{(\mathbf... ...eft(\frac{\sqrt{(\mathbf{s}-\mathbf{s}^{i})^{2}}}{\Delta s^{\perp}}\right)^{m}}$$

$$\exp \left \{-\frac{\left( (\mathbf{s}^{i+1}-\mathbf{s}^{i}) \cdo... ...}-\mathbf{s}^{i})\right)^{2}} {2 (\Delta s^{\vert\vert}_{i})^4} \right\}\Bigg],$$ (264)

where the exponents $n$ and $m$ determine the decay.
Lorentzian: Lorentzian functions are used in place of Gaussians.

In all the cases, a new hill is added at each step of MTD, where $\Delta t _{meta}=t_{i+1}-t_{i}$ is usually chosen equal to $10\div500$ steps of CP-MD (it depends on the relaxation time of the system and the size of the hills). The center of the new hill at time $t_{i+1}$ is positioned along the vector $\mathbf{s} - \mathbf{s^{i}}$ .