Blue-moon ensemble: Difference between revisions
No edit summary |
No edit summary |
||
| Line 20: | Line 20: | ||
Note that as the free-energy is a state quantity, the choice of path connecting (1) with (2) is irrelevant. | Note that as the free-energy is a state quantity, the choice of path connecting (1) with (2) is irrelevant. | ||
== | == How to == | ||
#When the free-energy gradient is to be computed, set {{TAG|LBLUEOUT}}=.TRUE. | #For the constrained MD simulation [:Category:Constrained molecular dynamics|Constrained molecular dynamics] The blue moon ensemble calculations When the free-energy gradient is to be computed, set {{TAG|LBLUEOUT}}=.TRUE. | ||
== References == | == References == | ||
Revision as of 15:11, 18 April 2022
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.
How to
- For the constrained MD simulation [:Category:Constrained molecular dynamics|Constrained molecular dynamics] The blue moon ensemble calculations When the free-energy gradient is to be computed, set LBLUEOUT=.TRUE.
References
- ↑ E. A. Carter, G. Ciccotti, J. T. Hynes, and R. Kapral, Chem. Phys. Lett. 156, 472 (1989).
- ↑ W. K. Den Otter and W. J. Briels, Mol. Phys. 98, 773 (2000).
- ↑ E. Darve, M. A. Wilson, and A. Pohorille, Mol. Simul. 28, 113 (2002).
- ↑ P. Fleurat-Lessard and T. Ziegler, J. Chem. Phys. 123, 084101 (2005).
- ↑ J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J. Comp. Phys. 23, 327 (1977).