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| In metadynamics,<ref name="Laio02"/><ref name="Iannuzzi03"/> the bias potential
| | #REDIRECT [[Category:Metadynamics]] |
| 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:
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| :<math>
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| \tilde{H}(q,p,t) = H(q,p) + \tilde{V}(t,\xi),
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| </math>
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| where <math>H(q,p)</math> is the Hamiltonian for the original (unbiased) system, and <math>\tilde{V}(t,\xi)</math> is the time-dependent bias potential. The latter term is usually defined as a sum of Gaussian hills with height ''h'' and width ''w'':
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| :<math>
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| \tilde{V}(t,\xi) = h \sum_{i=1}^{\lfloor t/t_G \rfloor} \exp{\left\{ -\frac{|\xi^{(t)}-\xi^{(i \cdot t_G)}|^2}{2
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| w^2} \right\}}.
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| </math>
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| In practice, <math>\tilde{V}(t,\xi)</math> is updated by adding a new Gaussian with a time increment ''t''<sub>G</sub>, which is typically one or two orders of magnitude greater than the time step used in the MD simulation.
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| In the limit of infinite simulation time, the bias potential is related to the free energy via:
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| :<math>
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| A(\xi) = - \lim_{t \to \infty} \tilde{V}(t,\xi) + const.
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| </math>
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| 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"/>.
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| The error estimation in free-energy calculations with metadynamics is discussed in Ref.<ref name="Laio05"/>.
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| * For a metadynamics run with Andersen thermostat, one has to:
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| #Set the standard MD-related tags: {{TAG|IBRION}}=0, {{TAG|TEBEG}}, {{TAG|POTIM}}, and {{TAG|NSW}} | |
| #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]] | |
