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:

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

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 $\lambda _{\xi _{k}}$, $|Z|^{-1/2}$, ${\frac {k_{B}T}{2|Z|}}\sum _{j=1}^{r}(Z^{-1})_{kj}\sum _{i=1}^{3N}m_{i}^{-1}\nabla _{i}\xi _{j}\cdot \nabla _{i}|Z|$, and , respectively.

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