import numpy as np
from ase.calculators.calculator import Calculator, all_changes
from ase.neighborlist import NeighborList
from ase.stress 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 |``
Therefore:
``d r_ij / d d_ij = + d_ij / r_ij``
``d r_ij / d d_i = - d_ij / r_ij``
The derivative of u_ij is:
::
d u_ij / d r_ij
= (-24 epsilon / r_ij) ( 2 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 combined into a "general derivative".
``du_ij = (d u_ij / d d_ij) / r_ij``
We do this for convenience: `du_ij` is purely scalar The pairwise force is:
``f_ij = du_ij * d_ij``
The total 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 (`bothways=True` in the neighbor list):
``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``
Differentiating `u` with respect to `r_i` yields the force,
independent of the choice of partitioning.
::
f_i = - d u / d r_i = - sum_ij d u_ij / d r_i
= - sum_ij d u_ij / d r_ij * d r_ij / d r_i
= sum_ij du_ij d_ij = sum_ij f_ij
This justifies calling `f_ij` pairwise forces.
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``
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`.
This implementation offers two ways to deal with this:
Either, we shift the pairwise 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! If this option is chosen, the forces discontinuously
jump to zero at the cutoff.
An alternative is to modify the pairwise potential by multiplying
it with a cutoff function that goes from 1 to 0 between an onset radius
ro and the cutoff rc. If the function is chosen suitably, it can also
smoothly push the forces down to zero, ensuring continuous forces as well.
In order for this to work well, the onset radius has to be set suitably,
typically around 2*sigma.
In this case, we introduce a modified pairwise potential:
``u'_ij = fc * u_ij``
The pairwise forces have to be modified accordingly:
``f'_ij = fc * f_ij + fc' * u_ij``
Where `fc' = d fc / d d_ij`.
This approach is taken from Jax-MD (https://github.com/google/jax-md),
which in turn is inspired by HOOMD Blue
(https://glotzerlab.engin.umich.edu/hoomd-blue/).
"""
implemented_properties = ['energy', 'energies', 'forces', 'free_energy']
implemented_properties += ['stress', 'stresses'] # bulk properties
default_parameters = {
'epsilon': 1.0,
'sigma': 1.0,
'rc': None,
'ro': None,
'smooth': False,
}
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
ro: float, None
Onset of cutoff function in 'smooth' mode. Defaults to 0.66 * rc.
smooth: bool, False
Cutoff mode. False means that the pairwise energy is simply shifted
to be 0 at r = rc, leading to the energy going to 0 continuously,
but the forces jumping to zero discontinuously at the cutoff.
True means that a smooth cutoff function is multiplied to the pairwise
energy that smoothly goes to 0 between ro and rc. Both energy and
forces are continuous in that case.
If smooth=True, make sure to check the tail of the
forces for kinks, ro might have to be adjusted to avoid distorting
the potential too much.
"""
Calculator.__init__(self, **kwargs)
if self.parameters.rc is None:
self.parameters.rc = 3 * self.parameters.sigma
if self.parameters.ro is None:
self.parameters.ro = 0.66 * self.parameters.rc
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
ro = self.parameters.ro
smooth = self.parameters.smooth
if self.nl is None or 'numbers' in system_changes:
self.nl = NeighborList(
[rc / 2] * natoms, self_interaction=False, bothways=True
)
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
if smooth:
cutoff_fn = cutoff_function(r2, rc**2, ro**2)
d_cutoff_fn = d_cutoff_function(r2, rc**2, ro**2)
pairwise_energies = 4 * epsilon * (c12 - c6)
pairwise_forces = -24 * epsilon * (2 * c12 - c6) / r2 # du_ij
if smooth:
# order matters, otherwise the pairwise energy is already
# modified
pairwise_forces = (
cutoff_fn * pairwise_forces + 2 * d_cutoff_fn
* pairwise_energies
)
pairwise_energies *= cutoff_fn
else:
pairwise_energies -= e0 * (c6 != 0.0)
pairwise_forces = pairwise_forces[:, np.newaxis] * distance_vectors
energies[ii] += 0.5 * pairwise_energies.sum() # atomic energies
forces[ii] += pairwise_forces.sum(axis=0)
stresses[ii] += 0.5 * np.dot(
pairwise_forces.T, distance_vectors
) # equivalent to outer product
# no lattice, no stress
if self.atoms.cell.rank == 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
def cutoff_function(r, rc, ro):
"""Smooth cutoff function.
Goes from 1 to 0 between ro and rc, ensuring
that u(r) = lj(r) * cutoff_function(r) is C^1.
Defined as 1 below ro, 0 above rc.
Note that r, rc, ro are all expected to be squared,
i.e. `r = r_ij^2`, etc.
Taken from https://github.com/google/jax-md.
"""
return np.where(
r < ro,
1.0,
np.where(r < rc, (rc - r) ** 2 * (rc + 2 *
r - 3 * ro) / (rc - ro) ** 3, 0.0),
)
def d_cutoff_function(r, rc, ro):
"""Derivative of smooth cutoff function wrt r.
Note that `r = r_ij^2`, so for the derivative wrt to `r_ij`,
we need to multiply `2*r_ij`. This gives rise to the factor 2
above, the `r_ij` is cancelled out by the remaining derivative
`d r_ij / d d_ij`, i.e. going from scalar distance to distance vector.
"""
return np.where(
r < ro,
0.0,
np.where(r < rc, 6 * (rc - r) * (ro - r) / (rc - ro) ** 3, 0.0),
)