Metadynamics

Navigation: Documentation / Methods / Metadynamics

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Contents

• MTD:Description
• MTD:Controls
• MTD:Collective variables
• MTD:Post-processing
• MTD:Multiple walkers approach
• MTD:Examples
• MTD:Utilities

Description

Free Energy from Metadynamics Simulations

Metadynamics introduces a history-dependent potential, which is composed of Gaussian functions added in regular intervals. This potential is employed as a bias in molecular dynamics simulation to suppress visiting already sampled regions.

$V_{MTD}(\boldsymbol \xi) = \sum_{t=1}^{T_{MTD}} h_t exp \left( - \frac{1}{2} \frac{(\boldsymbol \xi - \boldsymbol \xi_t)^2}{\boldsymbol \sigma} \right)$ ... (8)

As a result, the system evolves by free diffusions alongside biased collective variables. Moreover, the history-dependent potential converges to the negative value of the free energy in a limit of infinitesimally long metadynamics simulation.

The application of the history dependent potential thus leads to the modified equations of motions:

$m_{i} \frac { d^2 \bold r_i }{dt^2} = \bold F_{pot,i}(\bold r_i) - \bold F_{bias,i}(\bold r_i) = - \frac{\partial V(\bold R)} {\partial \bold r_i} - \frac{\partial G(\boldsymbol \xi)}{\partial \boldsymbol \xi} \frac{\partial \boldsymbol \xi}{\partial \bold r_i}$ ... (8)

where $m_i$ is mass of atom i, $\bold r_i$ is atom position, and $t$ is time.

Since the biasing force is not known prior to the simulation, it is calculated during the simulation and adaptively applied. To accelerate sampling, the biasing force is applied even if an inadequate number of samples is collected in a bin. In this case, the biasing force is scaled in the early stages to avoid artifacts from applications of overestimated biasing forces. The biasing force can also be smoothed to decrease noise in collected data. For further details, see feimode in ABF:Controls.

Sampling Space Discretization

Due to numerical reasons, mean forces are collected on a regular grid. The averaging of instantaneous collective force is then done in small intervals centered at discrete CV values:

$\frac{\partial G(\boldsymbol \xi^{*})}{\partial \boldsymbol \xi} = - {\langle \bold F_{ICF} \rangle}_{\boldsymbol \xi=\boldsymbol \xi^{*} \pm \bold h/2}$ ... (5)

with the standard error:

$\sigma_e \left( \frac{\partial G(\boldsymbol \xi^{*})}{\partial \boldsymbol \xi} \right) = \sqrt{ \frac {N_{corr}} {N_b} } \sigma(\bold F_{ICF})_ {\boldsymbol \xi=\boldsymbol \xi^{*} \pm \bold h/2}$ ... (6)

where the standard deviation is given by:

$\sigma(\bold F_{ICF})_ {\boldsymbol \xi=\boldsymbol \xi^{*} \pm \bold h/2} = \sqrt { \frac{1}{N_b} \sum_{i=1}^{N_b} \left( F_{ICF}(\boldsymbol \xi_i) - \overline {F_{ICF}} \right)^2 \vert _{\boldsymbol \xi=\boldsymbol \xi^{*} \pm \bold h/2} }$ ... (7)

where $\bold h$ is the interval size also called a bin, $N_b$ is the number of samples collected in a bin centered at $\boldsymbol \xi^{*}$, and $N_{corr}$ is a statistical inefficiency due to correlation in time series.

Therefore, each CV involved in ABF simulations must be discretized by specifying an interval in which the sampling is performed and the number of intervals (bins) for discretization, for further details, see ABF:Collective variables. The increasing number of bins improves the accuracy of Equation 5 and subsequently the quality of the integrated free energy (Equation 1) but it also increases the noise because of a smaller number of samples collected in a bin. A reasonable compromise is the number of bins, which leads to 0.1 Å or 1-2° bin widths.