Blue moon ensemble calculations

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 [math]\displaystyle{ \lambda_{\xi_k} }[/math], [math]\displaystyle{ |Z|^{-1/2} }[/math], [math]\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 ) }[/math], and [math]\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 ) }[/math], respectively.

Mind: [math]\displaystyle{ \left ( \frac{1}{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 ) }[/math] is defined as [math]\displaystyle{ G }[/math] in the REPORT file. This is an arbitrary character and has no relation to Green's functions, reciprocal lattice vectors, etc.

Note that one line introduced by the string 'b_m>' is written for each constrained coordinate. With this output, the free energy gradient with respect to the fixed coordinate [math]\displaystyle{ {\xi_k} }[/math] can conveniently be determined (by the 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 an example of a blue moon ensemble average, let us consider the calculation of an unbiased potential energy average from constrained MD. For simplicity, only a single constraint is assumed. Here we extract [math]\displaystyle{ |Z|^{-1/2} }[/math] 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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