This is part of the bias module

Used to performed Parallel Bias metadynamics.

This action activate Parallel Bias Metadynamics (PBMetaD) [105], a version of metadynamics [80] in which multiple low-dimensional bias potentials are applied in parallel. In the current implementation, these have the form of mono-dimensional metadynamics bias potentials:

\[ {V(s_1,t), ..., V(s_N,t)} \]

where:

\[ V(s_i,t) = \sum_{ k \tau < t} W_i(k \tau) \exp\left( - \frac{(s_i-s_i^{(0)}(k \tau))^2}{2\sigma_i^2} \right). \]

To ensure the convergence of each mono-dimensional bias potential to the corresponding free energy, at each deposition step the Gaussian heights are multiplied by the so-called conditional term:

\[ W_i(k \tau)=W_0 \frac{\exp\left( - \frac{V(s_i,k \tau)}{k_B T} \right)}{\sum_{i=1}^N \exp\left( - \frac{V(s_i,k \tau)}{k_B T} \right)} \]

where \(W_0\) is the initial Gaussian height.

The PBMetaD bias potential is defined by:

\[ V_{PB}(\vec{s},t) = -k_B T \log{\sum_{i=1}^N \exp\left( - \frac{V(s_i,t)}{k_B T} \right)}. \]

Information on the Gaussian functions that build each bias potential are printed to multiple HILLS files, which are used both to restart the calculation and to reconstruct the mono-dimensional free energies as a function of the corresponding CVs. These can be reconstructed using the sum_hills utility because the final bias is given by:

\[ V(s_i) = -F(s_i) \]

Currently, only a subset of the METAD options are available in PBMetaD.

The bias potentials can be stored on a grid to increase performances of long PBMetaD simulations. You should provide either the number of bins for every collective variable (GRID_BIN) or the desired grid spacing (GRID_SPACING). In case you provide both PLUMED will use the most conservative choice (highest number of bins) for each dimension. In case you do not provide any information about bin size (neither GRID_BIN nor GRID_SPACING) and if Gaussian width is fixed PLUMED will use 1/5 of the Gaussian width as grid spacing. This default choice should be reasonable for most applications.

Another option that is available is well-tempered metadynamics [11]. In this variant of PBMetaD the heights of the Gaussian hills are scaled at each step by the additional well-tempered metadynamics term. This ensures that each bias converges more smoothly. It should be noted that, in the case of well-tempered metadynamics, in the output printed the Gaussian height is re-scaled using the bias factor. Also notice that with well-tempered metadynamics the HILLS files do not contain the bias, but the negative of the free-energy estimate. This choice has the advantage that one can restart a simulation using a different value for the \(\Delta T\). The applied bias will be scaled accordingly.

Note that you can use here also the flexible Gaussian approach [29] in which you can adapt the Gaussian to the extent of Cartesian space covered by a variable or to the space in collective variable covered in a given time. In this case the width of the deposited Gaussian potential is denoted by one value only that is a Cartesian space (ADAPTIVE=GEOM) or a time (ADAPTIVE=DIFF). Note that a specific integration technique for the deposited Gaussian kernels should be used in this case. Check the documentation for utility sum_hills.

With the keyword INTERVAL one changes the metadynamics algorithm setting the bias force equal to zero outside boundary [10]. If, for example, metadynamics is performed on a CV s and one is interested only to the free energy for s > boundary, the history dependent potential is still updated according to the above equations but the metadynamics force is set to zero for s < boundary. Notice that Gaussian kernels are added also if s < boundary, as the tails of these Gaussian kernels influence VG in the relevant region s > boundary. In this way, the force on the system in the region s > boundary comes from both metadynamics and the force field, in the region s < boundary only from the latter. This approach allows obtaining a history-dependent bias potential VG that fluctuates around a stable estimator, equal to the negative of the free energy far enough from the boundaries. Note that:

It works only for one-dimensional biases;
It works both with and without GRID;
The interval limit boundary in a region where the free energy derivative is not large;
If in the region outside the limit boundary the system has a free energy minimum, the INTERVAL keyword should be used together with a UPPER_WALLS or LOWER_WALLS at boundary.

For systems with multiple CVs that share identical properties, PBMetaD with partitioned families can be used to group them under one bias potential that each contributes to [114]. This is done with a list of PF keywords, where each PF* argument contains the list of CVs from ARG to be placed in that family. Once invoked, each CV in ARG must be placed in exactly one PF, even if it results in families containing only one CV. Additionally, in cases where each of SIGMA or GRID entry would correspond to each ARG entry, they now correspond to each PF and must be adjusted accordingly.

Multiple walkers [117] can also be used. See below the examples.

Examples

The following input is for PBMetaD calculation using as collective variables the distance between atoms 3 and 5 and the distance between atoms 2 and 4. The value of the CVs and the PBMetaD bias potential are written to the COLVAR file every 100 steps.