Blue-moon ensemble

In general, constrained molecular dynamics generates biased statistical averages. It can be shown that the correct average for a quantity $a(\xi )$ can be obtained using the formula (blue moon ensemble average):

$a(\xi )={\frac {\langle |\mathbf {Z} |^{-1/2}a(\xi ^{*})\rangle _{\xi ^{*}}}{\langle |\mathbf {Z} |^{-1/2}\rangle _{\xi ^{*}}}},$

where $\langle ...\rangle _{\xi ^{*}}$ stands for the statistical average of the quantity enclosed in angular parentheses computed for a constrained ensemble and $Z$ is a mass metric tensor defined as:

$Z_{\alpha ,\beta }={\sum }_{i=1}^{3N}m_{i}^{-1}\nabla _{i}\xi _{\alpha }\cdot \nabla _{i}\xi _{\beta },\,\alpha =1,...,r,\,\beta =1,...,r,$

It can be shown that the free energy gradient can be computed using the equation:

${\Bigl (}{\frac {\partial A}{\partial \xi _{k}}}{\Bigr )}_{\xi ^{*}}={\frac {1}{\langle |Z|^{-1/2}\rangle _{\xi ^{*}}}}\langle |Z|^{-1/2}[\lambda _{k}+{\frac {k_{B}T}{2|Z|}}\sum _{j=1}^{r}(Z^{-1})_{kj}\sum _{i=1}^{3N}m_{i}^{-1}\nabla _{i}\xi _{j}\cdot \nabla _{i}|Z|]\rangle _{\xi ^{*}},$

where $\lambda _{\xi _{k}}$ is the Lagrange multiplier associated with the parameter ${\xi _{k}}$ used in the SHAKE algorithm.

The free-energy difference between states (1) and (2) can be computed by integrating the free-energy gradients over a connecting path:

${\Delta }A_{1\rightarrow 2}=\int _{\xi (1)}^{\xi (2)}{\Bigl (}{\frac {\partial {A}}{\partial \xi }}{\Bigr )}_{\xi ^{*}}\cdot d{\xi }.$

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 $\lambda _{\xi _{k}}$, $|Z|^{-1/2}$, $\left({\frac {k_{B}T}{2|Z|}}\sum _{j=1}^{r}(Z^{-1})_{kj}\sum _{i=1}^{3N}m_{i}^{-1}\nabla _{i}\xi _{j}\cdot \nabla _{i}|Z|\right)$, and $\left(|Z|^{-1/2}[\lambda _{k}+{\frac {k_{B}T}{2|Z|}}\sum _{j=1}^{r}(Z^{-1})_{kj}\sum _{i=1}^{3N}m_{i}^{-1}\nabla _{i}\xi _{j}\cdot \nabla _{i}|Z|]\right)$, 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 ${\xi _{k}}$ 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 $|Z|^{-1/2}$ 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}'

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