with , and being kinetic, and potential energies, respectively, can be written as:

(6.11) |

The term stands for a thermal average of quantity evaluated for the system driven by the Hamiltonian . If the system is modified by adding a bias potential acting only on a selected internal parameter of the system , the Hamiltonian takes a form:

(6.12) |

and the probability density of in the biased ensemble is:

(6.13) |

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

(6.14) |

More generally, an observable :

(6.15) |

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

Simulation methods such as umbrella sampling [82] use a bias potential to enhance sampling of in regions with low such as transition regions of chemical reactions. The correct distributions are recovered afterwards using the equation for 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.