# Source code for ase.calculators.lj

import numpy as np

from ase.neighborlist import NeighborList
from ase.calculators.calculator import Calculator, all_changes
from ase.constraints import full_3x3_to_voigt_6_stress

[docs]class LennardJones(Calculator):
"""Lennard Jones potential calculator

see https://en.wikipedia.org/wiki/Lennard-Jones_potential

The fundamental definition of this potential is a pairwise energy:

u_ij = 4 epsilon ( sigma^12/r_ij^12 - sigma^6/r_ij^6 )

For convenience, we'll use d_ij to refer to "distance vector" and
r_ij to refer to "scalar distance". So, with position vectors r_i:

r_ij = | r_j - r_i | = | d_ij |

The derivative of u_ij is:

::

d u_ij / d r_ij
= (-24 epsilon / r_ij) ( sigma^12/r_ij^12 - sigma^6/r_ij^6 )

We can define a pairwise force

f_ij = d u_ij / d d_ij = d u_ij / d r_ij * d_ij / r_ij

The terms in front of d_ij are often combined into a "general derivative"

du_ij = (d u_ij / d d_ij) / r_ij

The force on an atom is:

f_i = sum_(j != i) f_ij

There is some freedom of choice in assigning atomic energies, i.e.
choosing a way to partition the total energy into atomic contributions.

We choose a symmetric approach:

u_i = 1/2 sum_(j != i) u_ij

The total energy of a system of atoms is then:

u = sum_i u_i = 1/2 sum_(i, j != i) u_ij

The stress can be written as ( (x) denoting outer product):

sigma = 1/2 sum_(i, j != i) f_ij (x) d_ij = sum_i sigma_i ,
with atomic contributions

sigma_i  = 1/2 sum_(j != i) f_ij (x) d_ij

Implementation note:

For computational efficiency, we minimise the number of
pairwise evaluations, so we iterate once over all the atoms,
and use NeighbourList with bothways=False. In terms of the
equations, we therefore effectively restrict the sum over i != j
to j > i, and need to manually re-add the "missing" j < i contributions.

Another consideration is the cutoff. We have to ensure that the potential
goes to zero smoothly as an atom moves across the cutoff threshold,
otherwise the potential is not continuous. In cases where the cutoff is
so large that u_ij is very small at the cutoff this is automatically
ensured, but in general, u_ij(rc) != 0.

In order to catch this case, this implementation shifts the total energy

u'_ij = u_ij - u_ij(rc)

which ensures that it is precisely zero at the cutoff.
However, this means that the energy effectively depends
on the cutoff, which might lead to unexpected results!

"""

implemented_properties = ['energy', 'energies', 'forces', 'free_energy']
implemented_properties += ['stress', 'stresses']  # bulk properties
default_parameters = {'epsilon': 1.0, 'sigma': 1.0, 'rc': None}
nolabel = True

def __init__(self, **kwargs):
"""
Parameters
----------
sigma: float
The potential minimum is at  2**(1/6) * sigma, default 1.0
epsilon: float
The potential depth, default 1.0
rc: float, None
Cut-off for the NeighborList is set to 3 * sigma if None.
The energy is upshifted to be continuous at rc.
Default None
"""

Calculator.__init__(self, **kwargs)

if self.parameters.rc is None:
self.parameters.rc = 3 * self.parameters.sigma

self.nl = None

def calculate(
self, atoms=None, properties=None, system_changes=all_changes,
):
if properties is None:
properties = self.implemented_properties

Calculator.calculate(self, atoms, properties, system_changes)

natoms = len(self.atoms)

sigma = self.parameters.sigma
epsilon = self.parameters.epsilon
rc = self.parameters.rc

if self.nl is None or 'numbers' in system_changes:
self.nl = NeighborList([rc / 2] * natoms, self_interaction=False)

self.nl.update(self.atoms)

positions = self.atoms.positions
cell = self.atoms.cell

# potential value at rc
e0 = 4 * epsilon * ((sigma / rc) ** 12 - (sigma / rc) ** 6)

energies = np.zeros(natoms)
forces = np.zeros((natoms, 3))
stresses = np.zeros((natoms, 3, 3))

for ii in range(natoms):
neighbors, offsets = self.nl.get_neighbors(ii)
cells = np.dot(offsets, cell)

# pointing *towards* neighbours
distance_vectors = positions[neighbors] + cells - positions[ii]

r2 = (distance_vectors ** 2).sum(1)
c6 = (sigma ** 2 / r2) ** 3
c6[r2 > rc ** 2] = 0.0
c12 = c6 ** 2

pairwise_energies = 4 * epsilon * (c12 - c6) - e0 * (c6 != 0.0)
energies[ii] += 0.5 * pairwise_energies.sum()  # atomic energies

pairwise_forces = (-24 * epsilon * (2 * c12 - c6) / r2)[
:, np.newaxis
] * distance_vectors

forces[ii] += pairwise_forces.sum(axis=0)
stresses[ii] += 0.5 * np.dot(
pairwise_forces.T, distance_vectors
)  # equivalent to outer product

# add j < i contributions
for jj, atom_j in enumerate(neighbors):
energies[atom_j] += 0.5 * pairwise_energies[jj]
forces[atom_j] += -pairwise_forces[jj]  # f_ji = - f_ij
stresses[atom_j] += 0.5 * np.outer(
pairwise_forces[jj], distance_vectors[jj]
)

# no lattice, no stress
if self.atoms.number_of_lattice_vectors == 3:
stresses = full_3x3_to_voigt_6_stress(stresses)
self.results['stress'] = (
stresses.sum(axis=0) / self.atoms.get_volume()
)
self.results['stresses'] = stresses / self.atoms.get_volume()

energy = energies.sum()
self.results['energy'] = energy
self.results['energies'] = energies

self.results['free_energy'] = energy

self.results['forces'] = forces