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| | | #REDIRECT [[:Category:Biased molecular dynamics]] |
| The probability density for a geometric parameter ξ of the system driven by a Hamiltonian:
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| :<math>
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| H(q,p) = T(p) + V(q), \;
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| </math>
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| with ''T''(''p''), and ''V''(''q'') being kinetic, and potential energies, respectively, can be written as:
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| :<math>
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| P(\xi_i)=\frac{\int\delta\Big(\xi(q)-\xi_i\Big) \exp\left\{-H(q,p)/k_B\,T\right\} dq\,dp}{\int \exp\left\{-H(q,p)/k_B\,T\right\}dq\,dp} =
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| \langle\delta\Big(\xi(q)-\xi_i\Big)\rangle_{H}.
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| </math>
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| The term <math>\langle X \rangle_H</math> stands for a thermal average of quantity ''X'' evaluated for the system driven by the Hamiltonian ''H''.
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| If the system is modified by adding a bias potential <math>\tilde{V}(\xi)</math> acting on one or multiple selected internal coordinates of the system ξ=ξ(''q''), the Hamiltonian takes a form:
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| :<math>
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| \tilde{H}(q,p) = H(q,p) + \tilde{V}(\xi),
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| </math>
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| and the probability density of ξ in the biased ensemble is:
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| :<math>
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| \tilde{P}(\xi_i)= \frac{\int \delta\Big(\xi(q)-\xi_i\Big) \exp\left\{-\tilde{H}(q,p)/k_B\,T\right\} dq\,dp}{\int \exp\left\{-\tilde{H}(q,p)/k_B\,T\right\}dq\,dp} = \langle\delta\Big(\xi(q)-\xi_i\Big)\rangle_{\tilde{H}}
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| </math>
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| It can be shown that the biased and unbiased averages are related via a simple formula:
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| :<math>
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| P(\xi_i)=\tilde{P}(\xi_i) \frac{\exp\left\{\tilde{V}(\xi)/k_B\,T\right\}}{\langle \exp\left\{\tilde{V}(\xi)/k_B\,T\right\} \rangle_{\tilde{H}}}.
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| </math>
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| More generally, an observable <math>\langle A \rangle_{H}</math>:
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| :<math>
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| \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}
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| </math>
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| can be expressed in terms of thermal averages within the biased ensemble:
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| :<math>
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| \langle A \rangle_{H} =\frac{\langle A(q) \,\exp\left\{\tilde{V}(\xi)/k_B\,T\right\} \rangle_{\tilde{H}}}{\langle \exp\left\{\tilde{V}(\xi)/k_B\,T\right\} \rangle_{\tilde{H}}}.
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| </math>
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| Simulation methods such as umbrella sampling<ref name="Torrie77"/> use a bias potential to enhance sampling of ξ in regions with low ''P''(ξ<sub>''i''</sub>) such as transition regions of chemical
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| reactions.
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| The correct distributions are recovered afterwards using the equation for <math>\langle A \rangle_{H}</math> above.
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| A more detailed description of the method can be found in Ref.<ref name="FrenkelSmit"/>.
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| 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
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| of values.
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| The bias potentials are supported both the NVT and NpT MD simulations regardless of the particular thermostat and/or barostat setting.
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| Different types of potentials can be freely combined.
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| [[File:Bias potentials.png|thumb|Graphical representation of (a) harmonic, (b) Fermi function-shaped, and (c) and Gauss function-shaped bias potentials. ]]
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| == Supported types of bias potentials ==
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| Presently, the following types of bias potential are supported:
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| *Harmonic potential (see curve (a) on the plot above)
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| :<math>
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| \tilde{V}(\xi_1,\dots,\xi_{M}) = \sum_{\mu=1}^{M}\frac{1}{2}\kappa_{\mu} (\xi_{\mu}(q)-\xi_{0,\mu})^2 \;
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| </math>
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| where the sum runs all (<math>M</math>) coordinates the potential acts upon, which are defined in the {{FILE|ICONST}}-file by setting the status to 8.
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| The parameters of the potential, <math>\kappa</math> (force constant) and <math>\xi_0</math> (minimum of potential), are defined, respectively, via the keywords {{TAG|SPRING_K}} and {{TAG|SPRING_R0}} in {{FILE|INCAR}}.
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| Optionally, it is also possible to change the value of <math>\xi_{0,\mu}</math> every MD step at a constant rate defined via parameter {{TAG|SPRING_V}}.
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| Note that the number of items defined via {{TAG|SPRING_K}}, {{TAG|SPRING_R0}}, and {{TAG|SPRING_V}} must be equal to <math>M</math>, otherwise the calculation terminates with an error message.
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| This form of bias potential is employed in several simulation protocols, such as the umbrella sampling<ref name="Torrie77"/>, umbrella integration, or steered MD, and is useful also in cases where the <math>\xi_{\mu}</math> values need restrained.
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| *Fermi function (see curve (b) on the plot above)
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| :<math>
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| \tilde{V}(\xi_1,\dots,\xi_{M}) = \sum_{\mu=1}^{M}\frac{A_{\mu}}{1+\text{exp}\left [-D_{\mu}\frac{\xi(q)}{\xi_{0,\mu}} -1 \right ]} \;
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| </math>
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| where the sum runs all (<math>M</math>) coordinates the potential acts upon, which are defined in the {{FILE|ICONST}}-file by setting the status to 4.
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| {{TAG|FBIAS_A}} {{TAG|FBIAS_D}} {{TAG|FBIAS_R0}}
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| *Gauss function (see curve (b) on the plot above)
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| :<math>
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| \tilde{V}(\xi_1,\dots,\xi_{M}) = h\,\sum_{\mu=1}^{M}\text{exp}\left [-\frac{(\xi(q)-\xi_0)^2}{2w^2} \right ], \;
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| </math>
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| == Andersen thermostat ==
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| * For a biased molecular dynamics 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}}
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| #Set {{TAG|MDALGO}}=1 ({{TAG|MDALGO}}=11 in VASP 5.x), and choose an appropriate setting for {{TAG|ANDERSEN_PROB}}
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| #In order to avoid updating of the bias potential, set {{TAG|HILLS_BIN}}={{TAG|NSW}}
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| #Define collective variables in the {{FILE|ICONST}}-file, and set the <tt>STATUS</tt> parameter for the collective variables to 5
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| #Define the bias potential in the {{FILE|PENALTYPOT}}-file
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| == Nose-Hoover thermostat ==
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| * For a biased molecular dynamics run with Nose-Hoover 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}}=2 ({{TAG|MDALGO}}=21 in VASP 5.x), and choose an appropriate setting for {{TAG|SMASS}}
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| #In order to avoid updating of the bias potential, set {{TAG|HILLS_BIN}}={{TAG|NSW}}
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| #Define collective variables in the {{FILE|ICONST}}-file, and set the <tt>STATUS</tt> parameter for the collective variables to 5
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| #Define the bias potential in the {{FILE|PENALTYPOT}}-file
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| The values of all collective variables for each MD step are listed in the {{FILE|REPORT}}-file, check the lines after the string <tt>Metadynamics</tt>.
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| == References ==
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| <references>
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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:Biased molecular dynamics]][[Category:Theory]][[Category:Howto]] | |
