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PMFT

Textbook guide to the potential of mean force and torque — free energy from structure, including orientation-resolved 2-D maps.

Conventions

  • Free energy in \(k_B T\) units when \(w = -\ln g\).
  • Optional orientations topology block rotates bonds into a local frame.

1. From \(g(r)\) to free energy

A pair distribution is a Boltzmann factor in disguise:

\[ w(r) = -k_B T \ln g(r). \]

Minima are coordination shells; barriers are desolvation penalties. PMFTXY generalizes this to a 2-D map of neighbour positions in the local \((x,y)\) frame of each reference particle — face-to-face vs edge-to-edge contacts that isotropic \(g(r)\) averages away.

With an orientations block (head, tail) per particle, every bond is rotated into that particle's body frame (atan2 of head - tail). Without it, the analyzer works in the lab frame.

2. Usage

import numpy as np
import molpy as mp

rng = np.random.default_rng(0)
xyz = rng.uniform(0.0, 20.0, size=(200, 3))
frame = mp.Frame()
frame["atoms"] = {"x": xyz[:, 0], "y": xyz[:, 1], "z": xyz[:, 2]}
frame.box = mp.Box.cubic(20.0)

from molpy.compute import NeighborList, PMFTXY

nlist = NeighborList(cutoff=6.0)(frame)
result = PMFTXY(x_max=6.0, y_max=6.0, n_x=120, n_y=120)([frame], [nlist])

3. Pitfalls

  1. Neighbor cutoff too short for the free-energy region of interest.
  2. Misaligned orientation endpoints → garbage body frame.
  3. Sparse sampling of 2-D histograms — average many frames.

See also