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| In metadynamics,<ref name="Laio02"/><ref name="Iannuzzi03"/> the bias potential | | In metadynamics,{{cite|laio:pnas:02}}{{cite|iannuzzi:prl:03}} the bias potential |
| that acts on a selected number of geometric parameters (collective variables) ξ={ξ<sub>1</sub>, ξ<sub>2</sub>, ...,ξ<sub>''m''</sub>} is constructed on-the-fly during the simulation. The Hamiltonian for the metadynamics <math>\tilde{H}(q,p)</math> can be written as: | | that acts on a selected number of geometric parameters (collective variables) ξ={ξ<sub>1</sub>, ξ<sub>2</sub>, ...,ξ<sub>''m''</sub>} is constructed on-the-fly during the simulation. The Hamiltonian for the metadynamics <math>\tilde{H}(q,p)</math> can be written as: |
| :<math> | | :<math> |
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| A(\xi) = - \lim_{t \to \infty} \tilde{V}(t,\xi) + const. | | A(\xi) = - \lim_{t \to \infty} \tilde{V}(t,\xi) + const. |
| </math> | | </math> |
| Practical hints as how to adjust the parameters used in metadynamics (''h'', ''w'', ''t''<sub>G</sub>) are given in Refs.<ref name="Ensing05"/><ref name="Laio05"/>. | | Practical hints as how to adjust the parameters used in metadynamics (''h'', ''w'', ''t''<sub>G</sub>) are given in Refs.{{cite|ensing:jpc:05}} and {{cite|laio:jpc:05}}. |
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| The error estimation in free-energy calculations with metadynamics is discussed in Ref.<ref name="Laio05"/>. | | The error estimation in free-energy calculations with metadynamics is discussed in Ref.{{cite|laio:jpc:05}}. |
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| | == References == |
| | <references/> |
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| * For a metadynamics run with Andersen thermostat, one has to:
| | <noinclude> |
| #Set the standard MD-related tags: {{TAG|IBRION}}=0, {{TAG|TEBEG}}, {{TAG|POTIM}}, and {{TAG|NSW}}
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| #Set {{TAG|MDALGO}}=11, and choose an appropriate setting for {{TAG|ANDERSEN_PROB}}
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| #Set the parameters {{TAG|HILLS_H}}, {{TAG|HILLS_W}}, and {{TAG|HILLS_BIN}}
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| #Define collective variables in the {{FILE|ICONST}}-file, and set the {{TAG|STATUS}} parameter for the collective variables to 5
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| #If needed, define the bias potential in the {{FILE|PENALTYPOT}}-file
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| The actual time-dependent bias potential is written to the {{FILE|HILLSPOT}}-file, which is updated after adding a new Gaussian. At the beginning of the simulation, VASP attempts to read the initial bias potential from the {{FILE|PENALTYPOT}}-file. For the continuation of a metadynamics run, copy {{FILE|HILLSPOT}} to {{FILE|PENALTYPOT}}. The values of all collective variables for each MD step are listed in {{FILE|REPORT}}-file, check the lines after the string <tt>Metadynamics</tt>.
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| <div id="BiasedMD"></div>
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| == References ==
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| <references>
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| <ref name="Andersen80">[http://dx.doi.org/10.1063/1.439486 H. C. Andersen, J. Chem. Phys. 72, 2384 (1980).]</ref>
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| <ref name="Ryckaert77">[http://dx.doi.org/10.1016/0021-9991(77)90098-5 J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J. Comp. Phys. 23, 327 (1977).]</ref>
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| <ref name="Carter89">[http://dx.doi.org/10.1016/S0009-2614(89)87314-2 E. A. Carter, G. Ciccotti, J. T. Hynes, and R. Kapral, Chem. Phys. Lett. 156, 472 (1989).]</ref>
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| <ref name="Otter00">[http://dx.doi.org/10.1080/00268970009483348 W. K. Den Otter and W. J. Briels, Mol. Phys. 98, 773 (2000).]</ref>
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| <ref name="Darve02">[http://dx.doi.org/10.1080/08927020211975 E. Darve, M. A. Wilson, and A. Pohorille, Mol. Simul. 28, 113 (2002).]</ref>
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| <ref name="Fleurat05">[http://dx.doi.org/10.1063/1.1948367 P. Fleurat-Lessard and T. Ziegler, J. Chem. Phys. 123, 084101 (2005).]</ref>
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| <ref name="Allen91">M. P. Allen and D. J. Tildesley, ''Computer simulation of liquids'', Oxford university press: New York, 1991.</ref>
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| <ref name="Parrinello80">[http://dx.doi.org/10.1103/PhysRevLett.45.1196 M. Parrinello and A. Rahman, Phys. Rev. Lett. 45, 1196 (1980).]</ref>
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| <ref name="Parrinello81">[http://dx.doi.org/10.1063/1.328693 M. Parrinello and A. Rahman, J. Appl. Phys. 52, 7182 (1981).]</ref>
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| <ref name="Toton10">[http://dx.doi.org/10.1088/0953-8984/22/7/074205 D. Toton, C. D. Lorenz, N. Rompotis, N. Martsinovich, and L. Kantorovich, J. Phys.: Condens. Matter 22, 074205 (2010).]</ref>
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| <ref name="Kantorovich08">[http://dx.doi.org/10.1103/PhysRevB.78.094305 L. Kantorovich and N. Rompotis, Phys. Rev. B 78, 094305 (2008).]</ref>
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| <ref name="Laio02">[http://dx.doi.org/10.1073/pnas.202427399 A. Laio and M. Parrinello, Proc. Natl. Acad, Sci. USA 99, 12562 (2002).]</ref>
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| <ref name="Iannuzzi03">[http://dx.doi.org/10.1103/PhysRevLett.90.238302 M. Iannuzzi, A. Laio, and M. Parrinello, Phys. Rev. Lett. 90, 238302 (2003).]</ref>
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| <ref name="Ensing05">[http://dx.doi.org/10.1021/jp045571i B. Ensing, A. Laio, M. Parrinello, and M. L. Klein, J. Phys. Chem. B 109, 6676 (2005).]</ref>
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| <ref name="Laio05">[http://dx.doi.org/10.1021/jp045424k A. Laio, A. Rodriguez-Fortea, F. L. Gervasio, M. Ceccarelli, and M. Parrinello, J. Phys. Chem. B 109, 6714 (2005).]</ref>
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| <ref name="Torrie77">[http://dx.doi.org/10.1016/0021-9991(77)90121-8 G. M. Torrie and J. P. Valleau, J. Comp. Phys. 23, 187 (1977).]</ref>
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| <ref name="FrenkelSmit">D. Frenkel and B. Smit, ''Understanding molecular simulations: from algorithms to applications'', Academic Press: San Diego, 2002.</ref>
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| </references>
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| ---- | | ---- |
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| [[Category:Molecular Dynamics]][[Category:Howto]] | | [[Category:Advanced molecular-dynamics sampling]][[Category:Theory]] |
In metadynamics,[1][2] the bias potential
that acts on a selected number of geometric parameters (collective variables) ξ={ξ1, ξ2, ...,ξm} is constructed on-the-fly during the simulation. The Hamiltonian for the metadynamics 
can be written as:
where 
is the Hamiltonian for the original (unbiased) system, and 
is the time-dependent bias potential. The latter term is usually defined as a sum of Gaussian hills with height h and width w:
In practice, is updated by adding a new Gaussian with a time increment tG, which is typically one or two orders of magnitude greater than the time step used in the MD simulation.
In the limit of infinite simulation time, the bias potential is related to the free energy via:
Practical hints as how to adjust the parameters used in metadynamics (h, w, tG) are given in Refs.[3] and [4].
The error estimation in free-energy calculations with metadynamics is discussed in Ref.[4].
References
- ↑ R. A. Laio and M. Parrinello, Proc. Natl. Acad, Sci. USA 99, 12562 (2002).
- ↑ M. Iannuzzi, A. Laio, and M. Parrinello, Phys. Rev. Lett. 90, 238302 (2003).
- ↑ B. Ensing, A. Laio, M. Parrinello, and M. L. Klein, J. Phys. Chem. B 109, 6676 (2005).
- ↑ a b A. Laio, A. Rodriguez-Fortea, F. L. Gervasio, M. Ceccarelli, and M. Parrinello, J. Phys. Chem. B 109, 6714 (2005).
