   This is part of the bias module

Used to performed MetaDynamics on one or more collective variables.

In a metadynamics simulations a history dependent bias composed of intermittently added Gaussian functions is added to the potential .

$V(\vec{s},t) = \sum_{ k \tau < t} W(k \tau) \exp\left( -\sum_{i=1}^{d} \frac{(s_i-s_i^{(0)}(k \tau))^2}{2\sigma_i^2} \right).$

This potential forces the system away from the kinetic traps in the potential energy surface and out into the unexplored parts of the energy landscape. Information on the Gaussian functions from which this potential is composed is output to a file called HILLS, which is used both the restart the calculation and to reconstruct the free energy as a function of the CVs. The free energy can be reconstructed from a metadynamics calculation because the final bias is given by:

$V(\vec{s}) = -F(\vec(s))$

During post processing the free energy can be calculated in this way using the sum_hills utility.

In the simplest possible implementation of a metadynamics calculation the expense of a metadynamics calculation increases with the length of the simulation as one has to, at every step, evaluate the values of a larger and larger number of Gaussians. To avoid this issue you can store the bias on a grid. This approach is similar to that proposed in  but has the advantage that the grid spacing is independent on the Gaussian width. Notice that 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 in plumed is well-tempered metadynamics . In this varient of metadynamics the heights of the Gaussian hills are rescaled at each step so the bias is now given by:

$V({s},t)= \sum_{t'=0,\tau_G,2\tau_G,\dots}^{t'<t} W e^{-V({s}({q}(t'),t')/\Delta T} \exp\left( -\sum_{i=1}^{d} \frac{(s_i({q})-s_i({q}(t'))^2}{2\sigma_i^2} \right),$

This method ensures that the 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 file does 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  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 gaussians 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 . If, for example, metadynamics is performed on a CV s and one is interested only to the free energy for s > sw, the history dependent potential is still updated according to the above equations but the metadynamics force is set to zero for s < sw. Notice that Gaussians are added also if s < sw, as the tails of these Gaussians influence VG in the relevant region s > sw. In this way, the force on the system in the region s > sw comes from both metadynamics and the force field, in the region s < sw 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 sw in a region where the free energy derivative is not large;
• If in the region outside the limit sw the system has a free energy minimum, the INTERVAL keyword should be used together with a UPPER_WALLS or LOWER_WALLS at sw.

As a final note, since version 2.0.2 when the system is outside of the selected interval the force is set to zero and the bias value to the value at the corresponding boundary. This allows acceptances for replica exchange methods to be computed correctly.

Multiple walkers  can also be used. See below the examples.

The c(t) reweighting factor can also be calculated on the fly using the equations presented in . The expression used to calculate c(t) follows directly from using Eq. 12 in Eq. 3 in  and gives smoother results than equivalent Eqs. 13 and Eqs. 14 in that paper. The c(t) is given by the rct component while the bias normalized by c(t) is given by the rbias component (rbias=bias-ct) which can be used to obtain a reweighted histogram. The calculation of c(t) is enabled by using the keyword REWEIGHTING_NGRID where the grid used for the calculation is specified. This grid should have a size that is equal or larger than the grid given in GRID_BIN./ By default c(t) is updated every 50 Gaussian hills but this can be changed by using the REWEIGHTING_NHILLS keyword. This option can only be employed together with Well-Tempered Metadynamics and requires that a grid is used.

Additional material and examples can be also found in the tutorials:

Notice that at variance with PLUMED 1.3 it is now straightforward to apply concurrent metadynamics as done e.g. in Ref.  . This indeed can be obtained by using the METAD action multiple times in the same input file.

Description of components

By default this Action calculates the following quantities. These quanties can be referenced elsewhere in the input by using this Action's label followed by a dot and the name of the quantity required from the list below.

 Quantity Description bias the instantaneous value of the bias potential work accumulator for work

In addition the following quantities can be calculated by employing the keywords listed below

 Quantity Keyword Description rbias REWEIGHTING_NGRID the instantaneous value of the bias normalized using the $$c(t)$$ reweighting factor [rbias=bias-c(t)]. This component can be used to obtain a reweighted histogram. rct REWEIGHTING_NGRID the reweighting factor $$c(t)$$. acc ACCELERATION the metadynamics acceleration factor
Compulsory keywords
 ARG the input for this action is the scalar output from one or more other actions. The particular scalars that you will use are referenced using the label of the action. If the label appears on its own then it is assumed that the Action calculates a single scalar value. The value of this scalar is thus used as the input to this new action. If * or *.* appears the scalars calculated by all the proceding actions in the input file are taken. Some actions have multi-component outputs and each component of the output has a specific label. For example a DISTANCE action labelled dist may have three componets x, y and z. To take just the x component you should use dist.x, if you wish to take all three components then use dist.*.More information on the referencing of Actions can be found in the section of the manual on the PLUMED Getting started. Scalar values can also be referenced using POSIX regular expressions as detailed in the section on Regular Expressions. To use this feature you you must compile PLUMED with the appropriate flag. SIGMA the widths of the Gaussian hills PACE the frequency for hill addition FILE ( default=HILLS ) a file in which the list of added hills is stored
Options
 NUMERICAL_DERIVATIVES ( default=off ) calculate the derivatives for these quantities numerically GRID_SPARSE ( default=off ) use a sparse grid to store hills GRID_NOSPLINE ( default=off ) don't use spline interpolation with grids STORE_GRIDS ( default=off ) store all the grid files the calculation generates. They will be deleted if this keyword is not present WALKERS_MPI ( default=off ) Switch on MPI version of multiple walkers - not compatible with other WALKERS_* options ACCELERATION ( default=off ) Set to TRUE if you want to compute the metadynamics acceleration factor.
 HEIGHT the heights of the Gaussian hills. Compulsory unless TAU, TEMP and BIASFACTOR are given FMT specify format for HILLS files (useful for decrease the number of digits in regtests) BIASFACTOR use well tempered metadynamics and use this biasfactor. Please note you must also specify temp TEMP the system temperature - this is only needed if you are doing well-tempered metadynamics TAU in well tempered metadynamics, sets height to (kb*DeltaT*pace*timestep)/tau GRID_MIN the lower bounds for the grid GRID_MAX the upper bounds for the grid GRID_BIN the number of bins for the grid GRID_SPACING the approximate grid spacing (to be used as an alternative or together with GRID_BIN) REWEIGHTING_NGRID calculate the c(t) reweighting factor and use that to obtain the normalized bias [rbias=bias-c(t)]. Here you should specify the number of grid points required in each dimension. The number of grid points should be equal or larger to the number of grid points given in GRID_BIN.This method is not compatible with metadynamics not on a grid. REWEIGHTING_NHILLS how many Gaussian hills should be deposited between calculating the c(t) reweighting factor. The default is to do this every 50 hills. GRID_WSTRIDE write the grid to a file every N steps GRID_WFILE the file on which to write the grid ADAPTIVE use a geometric (=GEOM) or diffusion (=DIFF) based hills width scheme. Sigma is one number that has distance units or timestep dimensions WALKERS_ID walker id WALKERS_N number of walkers WALKERS_DIR shared directory with the hills files from all the walkers WALKERS_RSTRIDE stride for reading hills files INTERVAL monodimensional lower and upper limits, outside the limits the system will not feel the biasing force. GRID_RFILE a grid file from which the bias should be read at the initial step of the simulation SIGMA_MAX the upper bounds for the sigmas (in CV units) when using adaptive hills. Negative number means no bounds SIGMA_MIN the lower bounds for the sigmas (in CV units) when using adaptive hills. Negative number means no bounds RESTART allows per-action setting of restart (YES/NO/AUTO) UPDATE_FROM Only update this action from this time UPDATE_UNTIL Only update this action until this time
Examples
The following input is for a standard metadynamics 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 metadynamics bias potential are written to the COLVAR file every 100 steps.
DISTANCE ATOMS=3,5 LABEL=d1
DISTANCE ATOMS=2,4 LABEL=d2
METAD ARG=d1,d2 SIGMA=0.2,0.2 HEIGHT=0.3 PACE=500 LABEL=restraint
PRINT ARG=d1,d2,restraint.bias STRIDE=100  FILE=COLVAR

If you use adaptive Gaussians, with diffusion scheme where you use a Gaussian that should cover the space of 20 timesteps in collective variables. Note that in this case the histogram correction is needed when summing up hills.
DISTANCE ATOMS=3,5 LABEL=d1
DISTANCE ATOMS=2,4 LABEL=d2
PRINT ARG=d1,d2,restraint.bias STRIDE=100  FILE=COLVAR

If you use adaptive Gaussians, with geometrical scheme where you use a Gaussian that should cover the space of 0.05 nm in Cartesian space. Note that in this case the histogram correction is needed when summing up hills.
DISTANCE ATOMS=3,5 LABEL=d1
DISTANCE ATOMS=2,4 LABEL=d2
PRINT ARG=d1,d2,restraint.bias STRIDE=100  FILE=COLVAR

When using adaptive Gaussians you might want to limit how the hills width can change. You can use SIGMA_MIN and SIGMA_MAX keywords. The sigmas should specified in terms of CV so you should use the CV units. Note that if you use a negative number, this means that the limit is not set. Note also that in this case the histogram correction is needed when summing up hills.
DISTANCE ATOMS=3,5 LABEL=d1
DISTANCE ATOMS=2,4 LABEL=d2
ARG=d1,d2 SIGMA=0.05 HEIGHT=0.3 PACE=500 LABEL=restraint ADAPTIVE=GEOM
SIGMA_MIN=0.2,0.1 SIGMA_MAX=0.5,1.0
PRINT ARG=d1,d2,restraint.bias STRIDE=100  FILE=COLVAR

Multiple walkers can be also use as in  These are enabled by setting the number of walker used, the id of the current walker which interprets the input file, the directory where the hills containing files resides, and the frequency to read the other walkers. Here is an example
DISTANCE ATOMS=3,5 LABEL=d1
ARG=d1 SIGMA=0.05 HEIGHT=0.3 PACE=500 LABEL=restraint
WALKERS_N=10
WALKERS_ID=3
WALKERS_DIR=../
WALKERS_RSTRIDE=100

METAD ...