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In metadynamics,[1][2] the bias potential that acts on a selected number of geometric parameters (collective variables) ξ={ξ1, ξ2, ...,ξm} is constructed on-the-fly during the simulation. The Hamiltonian for the metadynamics ${\displaystyle {\tilde {H}}(q,p)}$ can be written as:

${\displaystyle {\tilde {H}}(q,p,t)=H(q,p)+{\tilde {V}}(t,\xi ),}$

where ${\displaystyle H(q,p)}$ is the Hamiltonian for the original (unbiased) system, and ${\displaystyle {\tilde {V}}(t,\xi )}$ is the time-dependent bias potential. The latter term is usually defined as a sum of Gaussian hills with height h and width w:

${\displaystyle {\tilde {V}}(t,\xi )=h\sum _{i=1}^{\lfloor t/t_{G}\rfloor }\exp {\left\{-{\frac {|\xi ^{(t)}-\xi ^{(i\cdot t_{G})}|^{2}}{2w^{2}}}\right\}}.}$

In practice, ${\displaystyle {\tilde {V}}(t,\xi )}$ is updated by adding a new Gaussian with a time increment tG, which is typically one or two orders of magnitude greater than the time step used in the MD simulation.

In the limit of infinite simulation time, the bias potential is related to the free energy via:

${\displaystyle A(\xi )=-\lim _{t\to \infty }{\tilde {V}}(t,\xi )+const.}$

Practical hints as how to adjust the parameters used in metadynamics (h, w, tG) are given in Refs.[3][4].

The error estimation in free-energy calculations with metadynamics is discussed in Ref.[4].

## Anderson thermostat

• For a metadynamics run with Andersen thermostat, one has to:
1. Set the standard MD-related tags: IBRION=0, TEBEG, POTIM, and NSW
2. Set MDALGO=1 (MDALGO=11 in VASP 5.x), and choose an appropriate setting for ANDERSEN_PROB
3. Set the parameters HILLS_H, HILLS_W, and HILLS_BIN
4. Define collective variables in the ICONST-file, and set the STATUS parameter for the collective variables to 5
5. If needed, define the bias potential in the PENALTYPOT-file

The actual time-dependent bias potential is written to the HILLSPOT-file, which is updated after adding a new Gaussian. At the beginning of the simulation, VASP attempts to read the initial bias potential from the PENALTYPOT-file. For the continuation of a metadynamics run, copy HILLSPOT to PENALTYPOT. The values of all collective variables for each MD step are listed in REPORT-file, check the lines after the string Metadynamics.

## Nose-Hoover thermostat

• For a metadynamics run with Nose-Hoover thermostat, one has to:
1. Set the standard MD-related tags: IBRION=0, TEBEG, POTIM, and NSW
2. Set MDALGO=2 (MDALGO=21 in VASP 5.x), and choose an appropriate setting for SMASS
3. Set the parameters HILLS_H, HILLS_W, and HILLS_BIN
4. Define collective variables in the ICONST-file, and set the STATUS parameter for the collective variables to 5
5. If needed, define the bias potential in the PENALTYPOT-file

The actual time-dependent bias potential is written to the HILLSPOT-file, which is updated after adding a new Gaussian. At the beginning of the simulation, VASP attempts to read the initial bias potential from the PENALTYPOT-file. For the continuation of a metadynamics run, copy HILLSPOT to PENALTYPOT. The values of all collective variables for each MD step are listed in REPORT-file, check the lines after the string Metadynamics.