Blue-moon ensemble

From VASP Wiki
Revision as of 06:46, 19 July 2022 by Karsai (talk | contribs)

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 (blue moon ensemble average):

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

The information needed to determine the blue moon ensemble averages within a Constrained molecular dynamics can be obtained by setting LBLUEOUT=.TRUE. The following output is written for each MD step in the file REPORT: 

>Blue_moon
       lambda        |z|^(-1/2)    GkT           |z|^(-1/2)*(lambda+GkT)
  b_m>  0.585916E+01  0.215200E+02 -0.117679E+00  0.123556E+03


with the four numerical terms indicating 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}}} , 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/":): {\displaystyle |Z|^{-1/2}} , 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/":): {\displaystyle \left ( \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| \right ) } , 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/":): {\displaystyle \left ( |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|] \right ) } , respectively. Note that one line introduced by the string 'b_m>' is written for each constrained coordinate. With this output, the free energy gradient with respected to the fixed coordinate 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}} can conveniently be determined (by equation given above) as a ratio between averages of the last and the second numerical terms. In the simplest case when only one constraint is used, the free energy gradient can be obtained as follows: 

grep b_m REPORT |awk 'BEGIN {a=0.;b=0.} {a+=$5;b+=$3} END {print a/b}'

As example of a blue moon ensemble average, let us consider calculation of unbiased potential energy average from constrained MD. For simplicity, only a single constraint is assumed. Here we extract 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/":): {\displaystyle |Z|^{-1/2}} for each step and store the data in an auxiliary file zet.dat: 

grep b_m REPORT |awk '{print $3}' > zet.dat

Here we extract potential energy for each step and store the data in an auxiliary file energy.dat: 

grep e_b REPORT |awk '{print $3}' > energy.dat

Finally, the weighted average is determined according to the first formula shown above: 

paste energy.dat zet.dat |awk 'BEGIN {a=0.;b=0.} {a+=$1*$2;b+=$2} END {print a/b}'

References

  1. ↑ E. A. Carter, G. Ciccotti, J. T. Hynes, and R. Kapral, Chem. Phys. Lett. 156, 472 (1989).
  2. ↑ W. K. Den Otter and W. J. Briels, Mol. Phys. 98, 773 (2000).
  3. ↑ E. Darve, M. A. Wilson, and A. Pohorille, Mol. Simul. 28, 113 (2002).
  4. ↑ P. Fleurat-Lessard and T. Ziegler, J. Chem. Phys. 123, 084101 (2005).
  5. ↑ J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J. Comp. Phys. 23, 327 (1977).
Retrieved from "https://www.vasp.at/wiki/index.php?title=Blue-moon_ensemble&oldid=17890"