Metadynamics

Link to PDF containing derivation for both single and multiple CVs

According to well-tempered metadynamics [1], bias potential (Vbias)(V_{\text{bias}}) deposited at any time (t)(t) along a single collective variable (CV) with respect to CV visited at an iteration (niter)(niter),

Vbias(CV,tniter)=Vbias(CV,tniter−1)+w×exp⁡(−(CV−CVniter)22σ2)V_{\text{bias}}(\text{CV},t_{\text{niter}}) = V_{\text{bias}}(\text{CV},t_{niter - 1})+w\times\exp\Big(-\frac{(\text{CV}-\text{CV}_{\text{niter}})^{2}}{2\sigma^{2}}\Big)

  • In standard metadynamics [2], w=w0w = w_{0}
  • In well-tempered metadynamics [1], w=w0×exp⁡(−Vbias(CV, tniter )kBΔT)w = w_{0} \times \exp\left( - \frac{V_{\text{bias}}\left( \text{CV},\ t_{\text{niter}}\ \right)}{k_{B}\Delta T} \right)

Force at every timestep (t) due to deposited bias potential:

Fbias(r,t)F_{\text{bias}}(\text{r},t)

=−(dVbias(CV,tniter)dCV)×dCVdr=-\Big(\frac{dV_{\text{bias}}(\text{CV},t_{\text{niter}})}{\text{dCV}}\Big)\times\frac{\text{dCV}}{\text{dr}}

Substituting Vbias(CV,tniter)V_{\text{bias}}(\text{CV},t_{\text{niter}}) in the first term of the RHS in the above equation results in

dVbias(CV,tniter)dCV=d[Vbias(CV,tniter−1)+w×exp⁡(−(CV−CVniter)22σ2)]dCV\frac{dV_{\text{bias}}(\text{CV},t_{\text{niter}})}{\text{dCV}}=\frac{d\lbrack V_{\text{bias}(\text{CV},t_{niter-1})}+w\times\exp\Big(-\frac{(\text{CV}-\text{CV}_{\text{niter}})^{2}}{2\sigma^{2}}\Big)\rbrack}{\text{dCV}}

  • Helper function: Derivative of Gaussian at any iteration

d[w×exp⁡(−(CV−CVniter)22σ2)]dCV= \frac{d\lbrack w\times \exp\Big(-\frac{(\text{CV}-\text{CV}_{\text{niter}})^{2}}{2\sigma^{2}}\Big)\rbrack}{\text{dCV}}=

−w×(CV−CVniter)σ2 -w\times\frac{(\text{CV}-\text{CV}_{\text{niter}})}{\sigma^{2}}

×exp⁡(−(CV−CVniter)22σ2) \times \exp\Big(-\frac{(\text{CV}-\text{CV}_{\text{niter}})^{2}}{2\sigma^{2}}\Big)

Substituting helper function in dVbias(CV,tniter)dCV\frac{dV_{\text{bias}}(\text{CV},t_{\text{niter}})}{\text{dCV}},

dVbias(CV,tniter)dCV=∑iter≤niter−1[−w×(CV−CVniter)σ2\frac{dV_{\text{bias}}(\text{CV},t_{\text{niter}})}{\text{dCV}}=\sum_{iter\leq niter-1}\Big\lbrack -w\times\frac{(\text{CV}-\text{CV}_{\text{niter}})}{\sigma^{2}}

×exp⁡(−(CV−CVniter)22σ2)] \times \exp\Big(-\frac{(\text{CV}-\text{CV}_{\text{niter}})^{2}}{2\sigma^{2}}\Big)\Big\rbrack

−w×(CV−CVniter)σ2 -w\times\frac{(\text{CV}-\text{CV}_{\text{niter}})}{\sigma^{2}}

×exp⁡(−(CV−CVniter)22σ2) \times \exp\Big(-\frac{(\text{CV}-\text{CV}_{\text{niter}})^{2}}{2\sigma^{2}}\Big)

[1] A. Barducci, G. Bussi, and M. Parrinello, Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free-Energy Method, Physical Review Letters 100, 020603 (2008).

[2] A. Laio and M. Parrinello, Escaping Free-Energy Minima, Proceedings of the National Academy of Sciences 99, 12562 (2002).