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# Biased molecular dynamics

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

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

The term 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 acting only on a selected internal parameter of the system ξ=ξ(*q*), the Hamiltonian takes a form:

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

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

More generally, an observable :

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

Simulation methods such as umbrella sampling^{[1]} 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 above.

A more detailed description of the method can be found in Ref.^{[2]}.
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:

- Set the standard MD-related tags: IBRION=0, TEBEG, POTIM, and NSW
- Set MDALGO=1 (MDALGO=11 in VASP 5.x), and choose an appropriate setting for ANDERSEN_PROB
- In order to avoid updating of the bias potential, set HILLS_BIN=NSW
- Define collective variables in the ICONST-file, and set the
`STATUS`parameter for the collective variables to 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:

- Set the standard MD-related tags: IBRION=0, TEBEG, POTIM, and NSW
- Set MDALGO=2 (MDALGO=21 in VASP 5.x), and choose an appropriate setting for SMASS
- In order to avoid updating of the bias potential, set HILLS_BIN=NSW
- Define collective variables in the ICONST-file, and set the
`STATUS`parameter for the collective variables to 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`.

## References

- ↑ G. M. Torrie and J. P. Valleau, J. Comp. Phys. 23, 187 (1977).
- ↑ D. Frenkel and B. Smit,
*Understanding molecular simulations: from algorithms to applications*, Academic Press: San Diego, 2002.