Metadynamics

Metadynamics is an enhanced sampling technique used in molecular dynamics to explore free energy landscapes along chosen collective variables. It was introduced by Laio and Parrinello in 2002. The core idea is to discourage the system from revisiting previously sampled regions by adding a time-dependent bias potential constructed as a sum of repulsive Gaussians centered at the current values of the selected collective variables. As the simulation proceeds, the bias fills free energy wells, enabling the system to escape metastable states and sample high-energy regions, allowing estimation of the underlying free energy surface along the CVs. The free energy surface F(s) is related to the negative of the bias potential V(s,t); in practice, after sufficient exploration, F(s) ≈ -V(s,t).

Variants include well-tempered metadynamics, which temper deposition of Gaussians over time to improve convergence and provide a probabilistic guarantee toward the true free energy; other methods include multiple-walker, bias-exchange, and parallel bias metadynamics. Choice of collective variables is crucial; CVs should capture slow degrees of freedom relevant to the process of interest, such as the distance between ligands, dihedral angles, or coordination numbers. Metadynamics is widely used to study protein folding, ligand binding, phase transitions, diffusion in materials, and crystal nucleation. Reweighting techniques allow estimation of unbiased ensemble averages from biased simulations. The method can be implemented in classical or ab initio MD, and is often used in conjunction with umbrella sampling or replica methods for enhanced reliability.

Limitations include dependence on CV choice, potential for poor convergence if sampling is insufficient, and sensitivity to Gaussian height, width, and deposition frequency. Overall, metadynamics provides a practical framework for mapping free energy landscapes and sampling rare events.