RunningIntegral

Running integral over the energy coordinate, and the estimation of the excluded-volume contribution

Effective as of ver 0.3.alpha5, and the upgrade to treat the last case in this page is incorporated in ver 0.3.7.

In an slvfe calculation, the solvation free energy is calculated through integration over the energy coordinate ¥epsilon from -¥infty to ¥infty.
When cumuint = 'yes', slvfe outputs the running integral from -¥infty to ¥epsilon as a function of ¥epsilon to the cumuintfl file; cumuintfl is the name of the output file storing the running integral when the option cumuint is 'yes'.
The default for cumuint is 'not', and the default for cumuintfl is 'cumsfe'.
When the trajectory is divided into blocks as specified by numdiv and cumuintfl = 'cumsfe', for example, the outputs from the 1st, 2nd, 3rd, ... blocks are cumsfe01, cumsfe02, cumsfe03, respectively, and the average over the blocks is output to cumsfe.

Either of the self-energy correction or the Lennard-Jone long-range correction is not applied to the running-integral values. In this sense, the values in the cumuintfl file reflects only the intermolecular effects within the cutoff for the Lennard-Jones interaction.
The average sum of the solute-solvent interaction is calculated from the distribution function for cumuintfl, furthermore, while it is computed directly from the trajectory when the slvfe output is obtained at default setting. The mesh error is then absent for the solute-solvent energy in the latter case, while it is present in the former.
In fact, the solute-solvent energy at default setting is computed directly from the trajectory, instead of the distribution function, because much of the mesh error is carried by the solute-solvent energy when computed from the distribution function obtained over discretized energy coordinates.
Thus, although the last line in the cumuintfl file corresponds to the integral from -¥infty to ¥infty, it deviates from the total solvation free energy given as the slvfe output, due possibly to the self-energy correction, Lennard-Jone long-range correction, and mesh error for the solute-solvent energy.
When slfslt = 'not', ljlrc = 'not', and uvread = 'not' are set in parameters_fe, the last line in cumuintfl will agree with the slvfe output; it is not necessary to declare ljlrc = 'not' explicitly since this is the default in an slvfe run.
See Parameter files for slvfe for the meanings of slfslt, ljlrc, and uvread.

When the excluded-volume contribution is evaluated with the procedures described below, it is not necessary to take into account the notes in the preceding paragraph. The self-energy correction, Lennard-Jones long-range correction, and mesh error in the solute-solvent energy are canceled there.
The self-energy and Lennard-Jones corrections appear as offsets to the total solvation free energy, and when the excluded-volume contribution is calculated with an appropriate threshold for the energy coordinate,
the solute-solvent energy is canceled anyway.

When the number of solvent species is 1, the cumuintfl file reads, for example,

    -19.945         0.00000
    -19.795         0.00000
...
      9.8450       -10.87787
      9.9950       -10.86492
      10.145       -10.85187
      10.295       -10.83914
...
     0.60618E+11    -4.39875
     0.10000E+12    -4.32528

The first column is the energy coordinate ¥epsilon, and the second column is the value of the running integral up to ¥epsilon.

The running integral is useful to estimate the excluded-volume contribution to the solvation free energy.
The excluded-volume contribution can be estimated as the integral value from a certain threshold to ¥infty.
This threshold is set by user, and it should be larger than the energy coordinate above which the distribution function in the solution is (numerically) zero.
The maximum value of the solute-solvent pair energy in the solution run is provided in uvrange.tt calculated from ermod for the solution system. The threshold for the excluded-volume contribution is taken to be larger than the maximum value thus determined.
When the threshold is taken to be 10 kcal/mol, for example, the running integral up to 10 kcal/mol is obtained from interpolation in the cumuintfl output.
The excluded-volume contribution is then determined as the difference of the running integral at the largest energy coordinate (the last line in cumuintfl) from the running integral up to 10 kcal/mol.

When the number of solvent species is larger than 1, the cumuintfl output has two formats.
If the energy coordinates are common among all the solvent species, the output is provided, for example, as

    -19.945         0.00000     0.00000     0.00000
    -19.795         0.00000     0.00000     0.00000
...
      9.8450       -12.12173    -3.70731    -8.41441
      9.9950       -12.10846    -3.70619    -8.40227
      10.145       -12.09489    -3.70500    -8.38989
      10.295       -12.08190    -3.70379    -8.37811
...
     0.60618E+11    -4.93938    -3.33859    -1.60079
     0.10000E+12    -4.85527    -3.33766    -1.51761

The first column is the energy coordinate.
The 3rd, 4th, ... columns are the running integrals for the 1st, 2nd, ... solvent species, respectively, and the 2nd column is the sum of the running-integral values for all the solvent species.
The value of the 2nd column is equal to the sum of the values in the following columns.
The excluded-volume contribution can be estimated as in the above case for the number of solvent species being 1.
The contributions from the respective solvent species are obtained from the running integrals in the 3rd and following columns, and the total contribution is calculated from the values in the 2nd column.

If the energy coordinates are not common among all the solvent species, the output is given, for example, as

    -19.945        1     0.00000
    -19.795        1     0.00000
...
     9.8450        1    -3.63640
     9.9950        1    -3.63533
     10.145        1    -3.63403
     10.295        1    -3.63265
...
    0.60618E+11    1    -3.25820
    0.10000E+12    1    -3.25726
    -25.395        2     0.00000
    -25.245        2     0.00000
...
     9.8500        2    -8.39195
     9.9500        2    -8.37981
     10.050        2    -8.36721
     10.150        2    -8.35538
...
    0.25312E+12    2    -3.25820
    0.10000E+13    2    -3.25726

The second column refers to the solvent species, and the first and third columns are the energy coordinate and the running integral for that species, respectively.
The sum of the running integrals over the solvent species is not in the cumuintfl output since the energy coordinate is not common for all the species.
The excluded-volume contribution from each solvent species can be obtained with the procedure described above.
The total contribuion is to be estimated by manually adding the contributions for all the solvent species.