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Biased molecular dynamics

The probability density for a geometric parameter $\xi$ of the system driven by a Hamiltonian:

$$H(q,p) = T(p) + V(q),$$ (6.10)

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

$$P(\xi_i)=\frac{\int \delta\Big(\xi(q)-\xi_i\Big) exp\left\{-H(q,p... ...ft\{-H(q,p)/k_B\,T\right\}dq\,dp} = \Big<\delta\Big(\xi(q)-\xi_i\Big)\Big>_{H}.$$ (6.11)

The term $\Big< X \Big>_H$ 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 $\xi=\xi(q)$, the Hamiltonian takes a form:

$$\tilde{H}(q,p) = H(q,p) + \tilde{V}(\xi),$$ (6.12)

and the probability density of $\xi$ in the biased ensemble is:

$$\tilde{P}(\xi_i)= \frac{\int \delta\Big(\xi(q)-\xi_i\Big) exp\lef... ...q,p)/k_B\,T\right\}dq\,dp} = \Big<\delta\Big(\xi(q)-\xi_i\Big)\Big>_{\tilde{H}}$$ (6.13)

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

$$<tex2html_comment_mark> P(\xi_i)=\tilde{P}(\xi_i) \frac{exp\left\... ...T\right\}}{\left< exp\left\{\tilde{V}(\xi)/k_B\,T\right\} \right>_{\tilde{H}}}.$$ (6.14)

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

$$\langle A \rangle_{H} = \frac{\int A(q) exp\left\{-H(q,p)/k_B\,T\right\} dq\,dp}{\int exp\left\{-H(q,p)/k_B\,T\right\}dq\,dp}$$ (6.15)

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

$$\langle A \rangle_{H} =\frac{\left< A(q) \,exp\left\{\tilde{V}(\x... ...tilde{H}}}{\left< exp\left\{\tilde{V}(\xi)/k_B\,T\right\} \right>_{\tilde{H}}}.$$ (6.16)

Simulation methods such as umbrella sampling [82] use a bias potential to enhance sampling of $\xi$ in regions with low $P(\xi_i)$ 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. [71]. 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.