Biased molecular dynamics

The probability density for a geometric parameter ξ of the system driven by a Hamiltonian:

$H(q,p) = T(p) + V(q), \;$

with T(p), and V(q) being kinetic, and potential energies, respectively, can be written as:

$P(\xi_i)=\frac{\int\delta\Big(\xi(q)-\xi_i\Big) \exp\left\{-H(q,p)/k_B\,T\right\} dq\,dp}{\int \exp\left\{-H(q,p)/k_B\,T\right\}dq\,dp} = \langle\delta\Big(\xi(q)-\xi_i\Big)\rangle_{H}.$

The term $\langle X \rangle_H$ stands for a thermal average of quantity X evaluated for the system driven by the Hamiltonian H.

If the system is modified by adding a bias potential $\tilde{V}(\xi)$ acting only on a selected internal parameter of the system ξ=ξ(q), the Hamiltonian takes a form:

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

and the probability density of ξ in the biased ensemble is:

$\tilde{P}(\xi_i)= \frac{\int \delta\Big(\xi(q)-\xi_i\Big) \exp\left\{-\tilde{H}(q,p)/k_B\,T\right\} dq\,dp}{\int \exp\left\{-\tilde{H}(q,p)/k_B\,T\right\}dq\,dp} = \langle\delta\Big(\xi(q)-\xi_i\Big)\rangle_{\tilde{H}}$

It can be shown that the biased and unbiased averages are related via a simple formula:

$P(\xi_i)=\tilde{P}(\xi_i) \frac{\exp\left\{\tilde{V}(\xi)/k_B\,T\right\}}{\langle \exp\left\{\tilde{V}(\xi)/k_B\,T\right\} \rangle_{\tilde{H}}}.$

More generally, an observable $\langle A \rangle_{H}$:

$\langle A \rangle_{H} = \frac{\int A(q) \exp\left\{-H(q,p)/k_B\,T\right\} dq\,dp}{\int \exp\left\{-H(q,p)/k_B\,T\right\}dq\,dp}$

can be expressed in terms of thermal averages within the biased ensemble:

$\langle A \rangle_{H} =\frac{\langle A(q) \,\exp\left\{\tilde{V}(\xi)/k_B\,T\right\} \rangle_{\tilde{H}}}{\langle \exp\left\{\tilde{V}(\xi)/k_B\,T\right\} \rangle_{\tilde{H}}}.$

Simulation methods such as umbrella sampling use a bias potential to enhance sampling of ξ in regions with low P(ξ_i) such as transition regions of chemical reactions. The correct distributions are recovered afterwards using the equation for $\langle A \rangle_{H}$ above.

A more detailed description of the method can be found in Ref.. Biased molecular dynamics can be used also to introduce soft geometric constraints in which the controlled geometric parameter is not strictly constant, instead it oscillates in a narrow interval of values.

Andersen thermostat

• For a biased molecular dynamics 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. In order to avoid updating of the bias potential, set HILLS_BIN=NSW
4. Define collective variables in the ICONST-file, and set the STATUS parameter for the collective variables to 5
5. Define the bias potential in the PENALTYPOT-file

Nose-Hoover thermostat

• For a biased molecular dynamics 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. In order to avoid updating of the bias potential, set HILLS_BIN=NSW
4. Define collective variables in the ICONST-file, and set the STATUS parameter for the collective variables to 5
5. Define the bias potential in the PENALTYPOT-file

The values of all collective variables for each MD step are listed in the REPORT-file, check the lines after the string Metadynamics.

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