Anharmonic Free Energy Update

[ejmeitz]

This PR takes code from Alois's anharmonic_free_energy branch which already fixed some bugs in the existing anharmonic_free_energy and fixes all the remaining bugs to the best of my knowledge.

The code calculates the first and second cumulant contributions (for F, S, U and Cv) and assumes that you are using self-consistent phonons. The constant correction terms (i.e., $$V_0$$) from the paper cannot be calculated in TDEP as it requires interfacing with LAMMPS or some other energy calculator.

This should match exactly with my own implementation that I used to collect data for a paper which was compared to ground truth values from TI. I will verify and then mark the PR as ready for review.

Example Output:

# Temperature:          0.800000000000E+02
 # Harmonic contribution
    # Free energy [eV/at]  Internal energy [eV/at]             Entropy [kB]       Heat capacity [kB]
          -0.002161720125           0.020681592720           3.313571612390           3.000000000000
 # First Cumulant Contribution
    # Free energy [eV/at]  Internal energy [eV/at]             Entropy [kB]       Heat capacity [kB]
           0.000055701287          -0.000055701287          -0.016159670355          -0.016159670355
 # Second Cumulant Contribution
    # Free energy [eV/at]  Internal energy [eV/at]             Entropy [kB]       Heat capacity [kB]
          -0.000787906127           0.000787906127           0.228581851748           0.228581851748

[ejmeitz]

@mjv500 @OrbitalC this is good to go. Just had some issues getting reference data for the tests. It seems the floating point error from using MPI or not when doing the sums matters.

[mjv500]

@ejmeitz looks quite good. I merged latest trunk. Nice addition of tests as well.

Namespace issue: lo_thermodynamic_helpers.f90 should be in libolle. Either it stays in the current position and you chop off the lo, or you move it to libolle/ in case other executables might use it someday.

once this is fixed we can merge

[ejmeitz]

@mjv500 All good. I removed the lo_ prefixes as the classes are specific to this algorithm. I also updated the citation with a link to the pre-print. I will try to remember to update that once its actually published.

[mjv500]

all tests pass and overview of changes looks good