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

1. For the constrained MD simulation Constrained molecular dynamics The blue moon ensemble calculations When the free-energy gradient is to be computed, set LBLUEOUT=.TRUE.

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