2nd Force Constant From Lammps

In one of my project, I need to calculate the force constant between differnet paires of atoms, and there is how I do it.

Methods

• The main reference is the paper from Esfarjani in 2008 [PRB, 77, 144112 (2018)]
• Basically, if we only consider the 2nd terms, the force on atom $i$ is the summation of the force between different atoms pairs: \(F_i = - \sum_{j} \Phi_{ij} u_j\) we can then moving one atom at each time, and the from the force of other atoms, we can calculate the force constant $\Phi_{ij}$
• In detials, I build a equilibrium lattice in LAMMPS first, then I moved the atoms in center cell at one direction at each time, and dumped out the force on each atoms. The LAMMPS scirpts read as below:

group cAtom id 249
displace_atoms cAtom move 0.0 0.001 0.0 units box
dump 1 all custom 1 force_12.txt id fx fy fz
dump_modify 1 sort id format  "%d %10.15g %10.15g %10.15g"

• Then I used a python script to extrac the force constant between atom pairs.
• Make sure the 2nd force constants are correct, I check:
  1. the symmetry of of force constant tensor
  2. the acoustic summation rule:
  3. the dispersion from this force constant