Constrained molecular dynamics: Difference between revisions
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In general, constrained molecular dynamics generates biased statistical averages. | |||
It can be shown that the correct average for a quantity <math>a(\xi)</math> can be obtained using the formula: | |||
:<math> | |||
a(\xi)=\frac{\langle |\mathbf{Z}|^{-1/2} a(\xi^*) \rangle_{\xi^*}}{\langle |\mathbf{Z}|^{-1/2}\rangle_{\xi^*}}, | |||
</math> | |||
where <math>\langle ... \rangle_{\xi^*}</math> stands for the statistical average of the quantity enclosed in angular parentheses computed for a constrained ensemble and <math>Z</math> is a mass metric tensor defined as: | |||
:<math> | |||
Z_{\alpha,\beta}={\sum}_{i=1}^{3N} m_i^{-1} \nabla_i \xi_\alpha \cdot \nabla_i \xi_\beta, \, \alpha=1,...,r, \, \beta=1,...,r, | |||
</math> | |||
It can be shown that the free energy gradient can be computed using the equation:<ref name="Carter89"/><ref name="Otter00"/><ref name="Darve02"/><ref name="Fleurat05"/> | |||
:<math> | |||
\Bigl(\frac{\partial A}{\partial \xi_k}\Bigr)_{\xi^*}=\frac{1}{\langle|Z|^{-1/2}\rangle_{\xi^*}}\langle |Z|^{-1/2} [\lambda_k +\frac{k_B T}{2 |Z|} \sum_{j=1}^{r}(Z^{-1})_{kj} \sum_{i=1}^{3N} m_i^{-1}\nabla_i \xi_j \cdot \nabla_i |Z|]\rangle_{\xi^*}, | |||
</math> | |||
where <math>\lambda_{\xi_k}</math> is the Lagrange multiplier associated with the parameter <math>{\xi_k}</math> used in the [[#SHAKE|SHAKE algorithm]].<ref name="Ryckaert77"/> | |||
The free-energy difference between states (1) and (2) can be computed by integrating the free-energy gradients over a connecting path: | |||
:<math> | |||
{\Delta}A_{1 \rightarrow 2} = \int_{{\xi(1)}}^{{\xi(2)}}\Bigl( \frac{\partial {A}} {\partial \xi} \Bigr)_{\xi^*} \cdot d{\xi}. | |||
</math> | |||
Note that as the free-energy is a state quantity, the choice of path connecting (1) with (2) is irrelevant. | |||
<div id="SHAKE"></div> | <div id="SHAKE"></div> | ||
Constrained molecular dynamics is performed using the SHAKE algorithm.<ref name="Ryckaert77"/>. | Constrained molecular dynamics is performed using the SHAKE algorithm.<ref name="Ryckaert77"/>. |
Revision as of 15:02, 13 March 2019
In general, constrained molecular dynamics generates biased statistical averages. It can be shown that the correct average for a quantity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): a(\xi ) can be obtained using the formula:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): a(\xi )={\frac {\langle |{\mathbf {Z}}|^{{-1/2}}a(\xi ^{*})\rangle _{{\xi ^{*}}}}{\langle |{\mathbf {Z}}|^{{-1/2}}\rangle _{{\xi ^{*}}}}},
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \langle ...\rangle _{{\xi ^{*}}} stands for the statistical average of the quantity enclosed in angular parentheses computed for a constrained ensemble and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): Z is a mass metric tensor defined as:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): Z_{{\alpha ,\beta }}={\sum }_{{i=1}}^{{3N}}m_{i}^{{-1}}\nabla _{i}\xi _{\alpha }\cdot \nabla _{i}\xi _{\beta },\,\alpha =1,...,r,\,\beta =1,...,r,
It can be shown that the free energy gradient can be computed using the equation:[1][2][3][4]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\Bigl (}{\frac {\partial A}{\partial \xi _{k}}}{\Bigr )}_{{\xi ^{*}}}={\frac {1}{\langle |Z|^{{-1/2}}\rangle _{{\xi ^{*}}}}}\langle |Z|^{{-1/2}}[\lambda _{k}+{\frac {k_{B}T}{2|Z|}}\sum _{{j=1}}^{{r}}(Z^{{-1}})_{{kj}}\sum _{{i=1}}^{{3N}}m_{i}^{{-1}}\nabla _{i}\xi _{j}\cdot \nabla _{i}|Z|]\rangle _{{\xi ^{*}}},
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \lambda _{{\xi _{k}}} is the Lagrange multiplier associated with the parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\xi _{k}} used in the SHAKE algorithm.[5]
The free-energy difference between states (1) and (2) can be computed by integrating the free-energy gradients over a connecting path:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\Delta }A_{{1\rightarrow 2}}=\int _{{{\xi (1)}}}^{{{\xi (2)}}}{\Bigl (}{\frac {\partial {A}}{\partial \xi }}{\Bigr )}_{{\xi ^{*}}}\cdot d{\xi }.
Note that as the free-energy is a state quantity, the choice of path connecting (1) with (2) is irrelevant.
Constrained molecular dynamics is performed using the SHAKE algorithm.[5]. In this algorithm, the Lagrangian for the system Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\mathcal {L}} is extended as follows:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\mathcal {L}}^{*}({\mathbf {q,{\dot {q}}}})={\mathcal {L}}({\mathbf {q,{\dot {q}}}})+\sum _{{i=1}}^{{r}}\lambda _{i}\sigma _{i}(q),
where the summation is over r geometric constraints, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\mathcal {L}}^{*} is the Lagrangian for the extended system, and λi is a Lagrange multiplier associated with a geometric constraint σi:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \sigma _{i}(q)=\xi _{i}({q})-\xi _{i}\;
with ξi(q) being a geometric parameter and ξi is the value of ξi(q) fixed during the simulation.
In the SHAKE algorithm, the Lagrange multipliers λi are determined in the iterative procedure:
- Perform a standard MD step (leap-frog algorithm):
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): v_{i}^{{t+{\Delta }t/2}}=v_{i}^{{t-{\Delta }t/2}}+{\frac {a_{i}^{{t}}}{m_{i}}}{\Delta }t
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): q_{i}^{{t+{\Delta }t}}=q_{i}^{{t}}+v_{i}^{{t+{\Delta }t/2}}{\Delta }t
- Use the new positions q(t+Δt) to compute Lagrange multipliers for all constraints:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\lambda }_{k}={\frac {1}{{\Delta }t^{2}}}{\frac {\sigma _{k}(q^{{t+{\Delta }t}})}{\sum _{{i=1}}^{N}m_{i}^{{-1}}\bigtriangledown _{i}{\sigma }_{k}(q^{{t}})\bigtriangledown _{i}{\sigma }_{k}(q^{{t+{\Delta }t}})}}
- Update the velocities and positions by adding a contribution due to restoring forces (proportional to λk):
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): v_{i}^{{t+{\Delta }t/2}}=v_{i}^{{t-{\Delta }t/2}}+\left(a_{i}^{{t}}-\sum _{k}{\frac {{\lambda }_{k}}{m_{i}}}\bigtriangledown _{i}{\sigma }_{k}(q^{{t}})\right){\Delta }t
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): q_{i}^{{t+{\Delta }t}}=q_{i}^{{t}}+v_{i}^{{t+{\Delta }t/2}}{\Delta }t
- repeat steps 2-4 until either |σi(q)| are smaller than a predefined tolerance (determined by SHAKETOL), or the number of iterations exceeds SHAKEMAXITER.
Anderson thermostat
- For a constrained molecular dynamics run with Andersen thermostat, one has to:
- Set the standard MD-related tags: IBRION=0, TEBEG, POTIM, and NSW
- Set MDALGO=1, and choose an appropriate setting for ANDERSEN_PROB
- Define geometric constraints in the ICONST-file, and set the STATUS parameter for the constrained coordinates to 0
- When the free-energy gradient is to be computed, set LBLUEOUT=.TRUE.
References
- ↑ Cite error: Invalid
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- ↑ Cite error: Invalid
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- ↑ Cite error: Invalid
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- ↑ Cite error: Invalid
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- ↑ a b J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J. Comp. Phys. 23, 327 (1977).