"""Calculator for the Embedded Atom Method Potential"""
# eam.py
# Embedded Atom Method Potential
# These routines integrate with the ASE simulation environment
# Paul White (Oct 2012)
# UNCLASSIFIED
# License: See accompanying license files for details
import os
import numpy as np
from scipy.interpolate import InterpolatedUnivariateSpline as spline
from ase.calculators.calculator import Calculator, all_changes
from ase.data import chemical_symbols
from ase.neighborlist import NeighborList
from ase.stress import full_3x3_to_voigt_6_stress
from ase.units import Bohr, Hartree
[docs]
class EAM(Calculator):
r"""EAM Interface Documentation
Introduction
============
The Embedded Atom Method (EAM) [1]_ is a classical potential which is
good for modelling metals, particularly fcc materials. Because it is
an equiaxial potential the EAM does not model directional bonds
well. However, the Angular Dependent Potential (ADP) [2]_ which is an
extended version of EAM is able to model directional bonds and is also
included in the EAM calculator.
Generally all that is required to use this calculator is to supply a
potential file or as a set of functions that describe the potential.
The files containing the potentials for this calculator are not
included but many suitable potentials can be downloaded from The
Interatomic Potentials Repository Project at
https://www.ctcms.nist.gov/potentials/
Theory
======
A single element EAM potential is defined by three functions: the
embedded energy, electron density and the pair potential. A two
element alloy contains the individual three functions for each element
plus cross pair interactions. The ADP potential has two additional
sets of data to define the dipole and quadrupole directional terms for
each alloy and their cross interactions.
The total energy :math:`E_{\rm tot}` of an arbitrary arrangement of atoms is
given by the EAM potential as
.. math::
E_\text{tot} =
\sum_i F(\bar\rho_i) + \frac{1}{2}\sum_{i\ne j} \phi(r_{ij})
and
.. math::
\bar\rho_i = \sum_j \rho(r_{ij})
where :math:`F` is an embedding function, namely the energy to embed an
atom :math:`i` in the combined electron density :math:`\bar\rho_i` which is
contributed from each of its neighbouring atoms :math:`j` by an amount
:math:`\rho(r_{ij})`, :math:`\phi(r_{ij})` is the pair potential function
representing the energy in bond :math:`ij` which is due to the
short-range electro-static interaction between atoms, and :math:`r_{ij}`
is the distance between an atom and its neighbour for that bond.
The ADP potential is defined as
.. math::
E_\text{tot} = \sum_i F(\bar\rho_i)
+ \frac{1}{2}\sum_{i\ne j} \phi(r_{ij})
+ \frac{1}{2} \sum_{i,\alpha} (\mu_i^\alpha)^2
+ \frac{1}{2} \sum_{i,\alpha,\beta} (\lambda_i^{\alpha\beta})^2
- \frac{1}{6} \sum_i \nu_i^2
where :math:`\mu_i^\alpha` is the dipole vector,
:math:`\lambda_i^{\alpha\beta}`
is the quadrupole tensor and :math:`\nu_i` is the trace of
:math:`\lambda_i^{\alpha\beta}`.
The fs potential is defined as
.. math::
E_i = F_\alpha (\sum_{j\neq i} \rho_{\alpha \beta}(r_{ij}))
+ \frac{1}{2}\sum_{j\neq i}\phi_{\alpha \beta}(r_{ij})
where :math:`\alpha` and :math:`\beta` are element types of atoms.
This form
is similar to original EAM formula above, except that :math:`\rho` and
:math:`\phi` are determined by element types.
Running the Calculator
======================
EAM calculates the cohesive atom energy, forces and
stress. Internally the potential functions are defined by splines
which may be directly supplied or created by reading the spline
points from a data file from which a spline function is created.
The LAMMPS compatible ``.alloy``, ``.fs`` and ``.adp`` formats are
supported. The LAMMPS ``.eam`` format is slightly different from
the ``.alloy`` format and is currently not supported.
For example::
from ase.calculators.eam import EAM
mishin = EAM(potential='Al99.eam.alloy')
mishin.write_potential('new.eam.alloy')
mishin.plot()
slab.calc = mishin
slab.get_potential_energy()
slab.get_forces()
slab.get_stress()
The breakdown of energy contribution from the indvidual components are
stored in the calculator instance ``.results['energy_components']``
Arguments:
==========
========================= ================================================
Keyword Description
========================= ================================================
``potential`` file of potential in ``.eam``, ``.alloy``,
``.adp`` or ``.fs``
format or file object
(This is generally all you need to supply).
For file object the ``form`` argument is
required
``elements[N]`` array of N element abbreviations
``embedded_energy[N]`` arrays of embedded energy functions
``electron_density[N]`` arrays of electron density functions
``phi[N,N]`` arrays of pair potential functions
``d_embedded_energy[N]`` arrays of derivative embedded energy functions
``d_electron_density[N]`` arrays of derivative electron density functions
``d_phi[N,N]`` arrays of derivative pair potentials functions
``d[N,N], q[N,N]`` ADP dipole and quadrupole function
``d_d[N,N], d_q[N,N]`` ADP dipole and quadrupole derivative functions
``skin`` skin distance passed to NeighborList().
If no atom
has moved more than the skin-distance since the
last
call to the ``update()`` method then the neighbor
list can be reused. Defaults to 1.0.
``form`` the form of the potential
``eam``, ``alloy``, ``adp`` or
``fs``. This will be determined from the file
suffix
or must be set if using equations or file object
========================= ================================================
Additional parameters for writing potential files
=================================================
The following parameters are only required for writing a potential in
``.alloy``, ``.adp`` or ``fs`` format file.
========================= ================================================
Keyword Description
========================= ================================================
``header`` Three line text header. Default is standard
message.
``Z[N]`` Array of atomic number of each element
``mass[N]`` Atomic mass of each element
``a[N]`` Array of lattice parameters for each element
``lattice[N]`` Lattice type
``nrho`` No. of rho samples along embedded energy curve
``drho`` Increment for sampling density
``nr`` No. of radial points along density and pair
potential curves
``dr`` Increment for sampling radius
========================= ================================================
Special features
================
``.plot()``
Plots the individual functions. This may be called from multiple EAM
potentials to compare the shape of the individual curves. This
function requires the installation of the Matplotlib libraries.
Notes/Issues
============
* Although currently not fast, this calculator can be good for trying
small calculations or for creating new potentials by matching baseline
data such as from DFT results. The format for these potentials is
compatible with LAMMPS_ and so can be used either directly by LAMMPS or
with the ASE LAMMPS calculator interface.
* Supported formats are the LAMMPS_ ``.alloy`` and ``.adp``. The
``.eam`` format is currently not supported. The form of the
potential will be determined from the file suffix.
* Any supplied values will override values read from the file.
* The derivative functions, if supplied, are only used to calculate
forces.
* There is a bug in early versions of scipy that will cause eam.py to
crash when trying to evaluate splines of a potential with one
neighbor such as caused by evaluating a dimer.
.. _LAMMPS: https://www.lammps.org/
.. [1] M.S. Daw and M.I. Baskes, Phys. Rev. Letters 50 (1983)
1285.
.. [2] Y. Mishin, M.J. Mehl, and D.A. Papaconstantopoulos,
Acta Materialia 53 2005 4029--4041.
"""
implemented_properties = ['energy', 'free_energy', 'forces', 'stress']
default_parameters = dict(
skin=1.0,
potential=None,
header=[
'EAM/ADP potential file\n',
'Generated from eam.py\n',
'blank\n',
],
)
def __init__(
self,
restart=None,
ignore_bad_restart_file=Calculator._deprecated,
label=os.curdir,
atoms=None,
form=None,
**kwargs,
):
self.form = form
if 'potential' in kwargs:
self.read_potential(kwargs['potential'])
Calculator.__init__(
self, restart, ignore_bad_restart_file, label, atoms, **kwargs
)
valid_args = (
'potential',
'elements',
'header',
'drho',
'dr',
'cutoff',
'atomic_number',
'mass',
'a',
'lattice',
'embedded_energy',
'electron_density',
'phi',
# derivatives
'd_embedded_energy',
'd_electron_density',
'd_phi',
'd',
'q',
'd_d',
'd_q', # adp terms
'skin',
'Z',
'nr',
'nrho',
'mass',
)
# set any additional keyword arguments
for arg, val in self.parameters.items():
if arg in valid_args:
if getattr(self, arg, None) is None:
setattr(self, arg, val)
else:
raise RuntimeError(
f'unknown keyword arg "{arg}" : not in {valid_args}'
)
def set_form(self, name):
"""set the form variable based on the file name suffix"""
extension = os.path.splitext(name)[1]
if extension == '.eam':
self.form = 'eam'
elif extension == '.alloy':
self.form = 'alloy'
elif extension == '.adp':
self.form = 'adp'
elif extension == '.fs':
self.form = 'fs'
else:
raise RuntimeError(f'unknown file extension type: {extension}')
def read_potential(self, filename):
"""Reads a LAMMPS EAM file in alloy or adp format
and creates the interpolation functions from the data
"""
if isinstance(filename, str):
with open(filename) as fd:
self._read_potential(fd)
else:
fd = filename
self._read_potential(fd)
def _read_potential(self, fd):
if self.form is None:
self.set_form(fd.name)
lines = fd.readlines()
def lines_to_list(lines):
"""Make data one long line so as not to care how its formatted"""
data = []
for line in lines:
data.extend(line.split())
return data
if self.form == 'eam': # single element eam file (aka funcfl)
self.header = lines[:1]
data = lines_to_list(lines[1:])
# eam form is just like an alloy form for one element
self.Nelements = 1
self.elements = [chemical_symbols[int(data[0])]]
self.Z = np.array([data[0]], dtype=int)
self.mass = np.array([data[1]], dtype=float)
self.a = np.array([data[2]], dtype=float)
self.lattice = [data[3]]
self.nrho = int(data[4])
self.drho = float(data[5])
self.nr = int(data[6])
self.dr = float(data[7])
self.cutoff = float(data[8])
n = 9 + self.nrho
self.embedded_data = np.array([np.float64(data[9:n])])
self.rphi_data = np.zeros([self.Nelements, self.Nelements, self.nr])
effective_charge = np.float64(data[n : n + self.nr])
# convert effective charges to rphi according to
# https://docs.lammps.org/pair_eam.html
self.rphi_data[0, 0] = Bohr * Hartree * (effective_charge**2)
self.density_data = np.array(
[np.float64(data[n + self.nr : n + 2 * self.nr])]
)
elif self.form in ['alloy', 'adp']:
self.header = lines[:3]
i = 3
data = lines_to_list(lines[i:])
self.Nelements = int(data[0])
d = 1
self.elements = data[d : d + self.Nelements]
d += self.Nelements
self.nrho = int(data[d])
self.drho = float(data[d + 1])
self.nr = int(data[d + 2])
self.dr = float(data[d + 3])
self.cutoff = float(data[d + 4])
self.embedded_data = np.zeros([self.Nelements, self.nrho])
self.density_data = np.zeros([self.Nelements, self.nr])
self.Z = np.zeros([self.Nelements], dtype=int)
self.mass = np.zeros([self.Nelements])
self.a = np.zeros([self.Nelements])
self.lattice = []
d += 5
# reads in the part of the eam file for each element
for elem in range(self.Nelements):
self.Z[elem] = int(data[d])
self.mass[elem] = float(data[d + 1])
self.a[elem] = float(data[d + 2])
self.lattice.append(data[d + 3])
d += 4
self.embedded_data[elem] = np.float64(data[d : (d + self.nrho)])
d += self.nrho
self.density_data[elem] = np.float64(data[d : (d + self.nr)])
d += self.nr
# reads in the r*phi data for each interaction between elements
self.rphi_data = np.zeros([self.Nelements, self.Nelements, self.nr])
for i in range(self.Nelements):
for j in range(i + 1):
self.rphi_data[j, i] = np.float64(data[d : (d + self.nr)])
d += self.nr
elif self.form == 'fs':
self.header = lines[:3]
i = 3
data = lines_to_list(lines[i:])
self.Nelements = int(data[0])
d = 1
self.elements = data[d : d + self.Nelements]
d += self.Nelements
self.nrho = int(data[d])
self.drho = float(data[d + 1])
self.nr = int(data[d + 2])
self.dr = float(data[d + 3])
self.cutoff = float(data[d + 4])
self.embedded_data = np.zeros([self.Nelements, self.nrho])
self.density_data = np.zeros(
[self.Nelements, self.Nelements, self.nr]
)
self.Z = np.zeros([self.Nelements], dtype=int)
self.mass = np.zeros([self.Nelements])
self.a = np.zeros([self.Nelements])
self.lattice = []
d += 5
# reads in the part of the eam file for each element
for elem in range(self.Nelements):
self.Z[elem] = int(data[d])
self.mass[elem] = float(data[d + 1])
self.a[elem] = float(data[d + 2])
self.lattice.append(data[d + 3])
d += 4
self.embedded_data[elem] = np.float64(data[d : (d + self.nrho)])
d += self.nrho
self.density_data[elem, :, :] = np.float64(
data[d : (d + self.nr * self.Nelements)]
).reshape([self.Nelements, self.nr])
d += self.nr * self.Nelements
# reads in the r*phi data for each interaction between elements
self.rphi_data = np.zeros([self.Nelements, self.Nelements, self.nr])
for i in range(self.Nelements):
for j in range(i + 1):
self.rphi_data[j, i] = np.float64(data[d : (d + self.nr)])
d += self.nr
self.r = np.arange(0, self.nr) * self.dr
self.rho = np.arange(0, self.nrho) * self.drho
# choose the set_splines method according to the type
if self.form == 'fs':
self.set_fs_splines()
else:
self.set_splines()
if self.form == 'adp':
self.read_adp_data(data, d)
self.set_adp_splines()
def set_splines(self):
# this section turns the file data into three functions (and
# derivative functions) that define the potential
self.embedded_energy = np.empty(self.Nelements, object)
self.electron_density = np.empty(self.Nelements, object)
self.d_embedded_energy = np.empty(self.Nelements, object)
self.d_electron_density = np.empty(self.Nelements, object)
for i in range(self.Nelements):
self.embedded_energy[i] = spline(
self.rho, self.embedded_data[i], k=3
)
self.electron_density[i] = spline(self.r, self.density_data[i], k=3)
self.d_embedded_energy[i] = self.deriv(self.embedded_energy[i])
self.d_electron_density[i] = self.deriv(self.electron_density[i])
self.phi = np.empty([self.Nelements, self.Nelements], object)
self.d_phi = np.empty([self.Nelements, self.Nelements], object)
# ignore the first point of the phi data because it is forced
# to go through zero due to the r*phi format in alloy and adp
for i in range(self.Nelements):
for j in range(i, self.Nelements):
self.phi[i, j] = spline(
self.r[1:], self.rphi_data[i, j][1:] / self.r[1:], k=3
)
self.d_phi[i, j] = self.deriv(self.phi[i, j])
if j != i:
self.phi[j, i] = self.phi[i, j]
self.d_phi[j, i] = self.d_phi[i, j]
def set_fs_splines(self):
self.embedded_energy = np.empty(self.Nelements, object)
self.electron_density = np.empty(
[self.Nelements, self.Nelements], object
)
self.d_embedded_energy = np.empty(self.Nelements, object)
self.d_electron_density = np.empty(
[self.Nelements, self.Nelements], object
)
for i in range(self.Nelements):
self.embedded_energy[i] = spline(
self.rho, self.embedded_data[i], k=3
)
self.d_embedded_energy[i] = self.deriv(self.embedded_energy[i])
for j in range(self.Nelements):
self.electron_density[i, j] = spline(
self.r, self.density_data[i, j], k=3
)
self.d_electron_density[i, j] = self.deriv(
self.electron_density[i, j]
)
self.phi = np.empty([self.Nelements, self.Nelements], object)
self.d_phi = np.empty([self.Nelements, self.Nelements], object)
for i in range(self.Nelements):
for j in range(i, self.Nelements):
self.phi[i, j] = spline(
self.r[1:], self.rphi_data[i, j][1:] / self.r[1:], k=3
)
self.d_phi[i, j] = self.deriv(self.phi[i, j])
if j != i:
self.phi[j, i] = self.phi[i, j]
self.d_phi[j, i] = self.d_phi[i, j]
def set_adp_splines(self):
self.d = np.empty([self.Nelements, self.Nelements], object)
self.d_d = np.empty([self.Nelements, self.Nelements], object)
self.q = np.empty([self.Nelements, self.Nelements], object)
self.d_q = np.empty([self.Nelements, self.Nelements], object)
for i in range(self.Nelements):
for j in range(i, self.Nelements):
self.d[i, j] = spline(self.r[1:], self.d_data[i, j][1:], k=3)
self.d_d[i, j] = self.deriv(self.d[i, j])
self.q[i, j] = spline(self.r[1:], self.q_data[i, j][1:], k=3)
self.d_q[i, j] = self.deriv(self.q[i, j])
# make symmetrical
if j != i:
self.d[j, i] = self.d[i, j]
self.d_d[j, i] = self.d_d[i, j]
self.q[j, i] = self.q[i, j]
self.d_q[j, i] = self.d_q[i, j]
def read_adp_data(self, data, d):
"""read in the extra adp data from the potential file"""
self.d_data = np.zeros([self.Nelements, self.Nelements, self.nr])
# should be non symmetrical combinations of 2
for i in range(self.Nelements):
for j in range(i + 1):
self.d_data[i, j] = self.d_data[j, i] = data[d : d + self.nr]
d += self.nr
self.q_data = np.zeros([self.Nelements, self.Nelements, self.nr])
# should be non symmetrical combinations of 2
for i in range(self.Nelements):
for j in range(i + 1):
self.q_data[i, j] = self.q_data[j, i] = data[d : d + self.nr]
d += self.nr
def write_potential(self, filename, nc=1, numformat='%.17e'):
"""Writes out the potential in the format given by the form
variable to 'filename' with a data format that is nc columns
wide. Note: array lengths need to be an exact multiple of nc
"""
with open(filename, 'wb') as fd:
self._write_potential(fd, nc=nc, numformat=numformat)
def _write_potential(self, fd, nc, numformat):
assert self.nr % nc == 0
assert self.nrho % nc == 0
if self.form == 'eam':
return self._write_potential_eam(fd, nc, numformat)
for i in range(3):
fd.write(self.header[i].encode())
fd.write(f'{self.Nelements} '.encode())
fd.write(' '.join(self.elements).encode() + b'\n')
fmt = f'%d {numformat} %d {numformat} {numformat}\n'
fd.write(
(
fmt % (self.nrho, self.drho, self.nr, self.dr, self.cutoff)
).encode()
)
# start of each section for each element
# rs = np.linspace(0, self.nr * self.dr, self.nr)
# rhos = np.linspace(0, self.nrho * self.drho, self.nrho)
rs = np.arange(0, self.nr) * self.dr
rhos = np.arange(0, self.nrho) * self.drho
for i in range(self.Nelements):
fd.write(
(
'%d %f %f %s\n'
% (self.Z[i], self.mass[i], self.a[i], str(self.lattice[i]))
).encode()
)
np.savetxt(
fd,
self.embedded_energy[i](rhos).reshape(self.nrho // nc, nc),
fmt=nc * [numformat],
)
if self.form == 'fs':
for j in range(self.Nelements):
np.savetxt(
fd,
self.electron_density[i, j](rs).reshape(
self.nr // nc, nc
),
fmt=nc * [numformat],
)
else:
np.savetxt(
fd,
self.electron_density[i](rs).reshape(self.nr // nc, nc),
fmt=nc * [numformat],
)
# write out the pair potentials in Lammps DYNAMO setfl format
# as r*phi for alloy format
for i in range(self.Nelements):
for j in range(i + 1):
np.savetxt(
fd,
(rs * self.phi[i, j](rs)).reshape(self.nr // nc, nc),
fmt=nc * [numformat],
)
if self.form == 'adp':
# these are the u(r) or dipole values
for i in range(self.Nelements):
for j in range(i + 1):
np.savetxt(fd, self.d_data[i, j])
# these are the w(r) or quadrupole values
for i in range(self.Nelements):
for j in range(i + 1):
np.savetxt(fd, self.q_data[i, j])
def _write_potential_eam(self, fd, nc, numformat):
"""Write the potential in the DYNAMO single-element funcfl format."""
fd.write(self.header[0].encode())
i = 0 # The DYNAMO funcfl format can store only one element.
vals = self.Z[i], self.mass[i], self.a[i], str(self.lattice[i])
fd.write(('%d %f %f %s\n' % vals).encode())
fmt = f'%d {numformat} %d {numformat} {numformat}\n'
vals = self.nrho, self.drho, self.nr, self.dr, self.cutoff
fd.write((fmt % vals).encode())
rs = np.arange(0, self.nr) * self.dr
rhos = np.arange(0, self.nrho) * self.drho
arr = self.embedded_energy[i](rhos).reshape(self.nrho // nc, nc)
np.savetxt(fd, arr, fmt=nc * [numformat])
z2s = rs * self.phi[i, i](rs) / (Hartree * Bohr)
z2s = np.where(np.isclose(z2s, 0.0), 0.0, z2s)
arr = np.sqrt(z2s).reshape(self.nr // nc, nc)
np.savetxt(fd, arr, fmt=nc * [numformat])
arr = self.electron_density[i](rs).reshape(self.nr // nc, nc)
np.savetxt(fd, arr, fmt=nc * [numformat])
def update(self, atoms):
# check all the elements are available in the potential
self.Nelements = len(self.elements)
elements = np.unique(atoms.get_chemical_symbols())
unavailable = np.logical_not(
np.array([item in self.elements for item in elements])
)
if np.any(unavailable):
raise RuntimeError(
'These elements are not in the potential: '
f'{elements[unavailable]}'
)
# cutoffs need to be a vector for NeighborList
cutoffs = self.cutoff * np.ones(len(atoms))
# convert the elements to an index of the position
# in the eam format
self.index = np.array(
[self.elements.index(el) for el in atoms.get_chemical_symbols()]
)
self.pbc = atoms.get_pbc()
# since we need the contribution of all neighbors to the
# local electron density we cannot just calculate and use
# one way neighbors
self.neighbors = NeighborList(
cutoffs,
skin=self.parameters.skin,
self_interaction=False,
bothways=True,
)
self.neighbors.update(atoms)
def calculate(
self, atoms=None, properties=['energy'], system_changes=all_changes
):
"""EAM Calculator
atoms: Atoms object
Contains positions, unit-cell, ...
properties: list of str
List of what needs to be calculated. Can be any combination
of 'energy', 'forces'
system_changes: list of str
List of what has changed since last calculation. Can be
any combination of these five: 'positions', 'numbers', 'cell',
'pbc', 'initial_charges' and 'initial_magmoms'.
"""
Calculator.calculate(self, atoms, properties, system_changes)
# we shouldn't really recalc if charges or magmos change
if len(system_changes) > 0: # something wrong with this way
self.update(self.atoms)
self.calculate_energy(self.atoms)
if 'forces' in properties or 'stress' in properties:
self.calculate_forces(self.atoms)
# check we have all the properties requested
for property in properties:
if property not in self.results:
if property == 'energy':
self.calculate_energy(self.atoms)
if property in ['forces', 'stress']:
self.calculate_forces(self.atoms)
# we need to remember the previous state of parameters
# if 'potential' in parameter_changes and potential != None:
# self.read_potential(potential)
def calculate_energy(self, atoms):
"""Calculate the energy
the energy is made up of the ionic or pair interaction and
the embedding energy of each atom into the electron cloud
generated by its neighbors
"""
pair_energy = 0.0
embedding_energy = 0.0
mu_energy = 0.0
lam_energy = 0.0
trace_energy = 0.0
self.total_density = np.zeros(len(atoms))
if self.form == 'adp':
self.mu = np.zeros([len(atoms), 3])
self.lam = np.zeros([len(atoms), 3, 3])
for i in range(len(atoms)): # this is the atom to be embedded
neighbors, offsets = self.neighbors.get_neighbors(i)
offset = np.dot(offsets, atoms.get_cell())
rvec = atoms.positions[neighbors] + offset - atoms.positions[i]
# calculate the distance to the nearest neighbors
r = np.sqrt(np.sum(np.square(rvec), axis=1))
nearest = np.arange(len(r))[r <= self.cutoff]
for j_index in range(self.Nelements):
use = self.index[neighbors[nearest]] == j_index
if not use.any():
continue
pair_energy += (
np.sum(self.phi[self.index[i], j_index](r[nearest][use]))
/ 2.0
)
if self.form == 'fs':
density = np.sum(
self.electron_density[j_index, self.index[i]](
r[nearest][use]
)
)
else:
density = np.sum(
self.electron_density[j_index](r[nearest][use])
)
self.total_density[i] += density
if self.form == 'adp':
self.mu[i] += self.adp_dipole(
r[nearest][use],
rvec[nearest][use],
self.d[self.index[i], j_index],
)
self.lam[i] += self.adp_quadrupole(
r[nearest][use],
rvec[nearest][use],
self.q[self.index[i], j_index],
)
# add in the electron embedding energy
embedding_energy += self.embedded_energy[self.index[i]](
self.total_density[i]
)
components = dict(pair=pair_energy, embedding=embedding_energy)
if self.form == 'adp':
mu_energy += np.sum(self.mu**2) / 2.0
lam_energy += np.sum(self.lam**2) / 2.0
for i in range(len(atoms)): # this is the atom to be embedded
trace_energy -= np.sum(self.lam[i].trace() ** 2) / 6.0
adp_result = dict(
adp_mu=mu_energy, adp_lam=lam_energy, adp_trace=trace_energy
)
components.update(adp_result)
self.positions = atoms.positions.copy()
self.cell = atoms.get_cell().copy()
energy = 0.0
for i in components:
energy += components[i]
self.energy_free = energy
self.energy_zero = energy
self.results['energy_components'] = components
self.results['energy'] = energy
self.results['free_energy'] = energy
def calculate_forces(self, atoms):
# calculate the forces based on derivatives of the three EAM functions
self.update(atoms)
self.results['forces'] = np.zeros((len(atoms), 3))
stresses = np.zeros((len(atoms), 3, 3))
for i in range(len(atoms)): # this is the atom to be embedded
neighbors, offsets = self.neighbors.get_neighbors(i)
offset = np.dot(offsets, atoms.get_cell())
# create a vector of relative positions of neighbors
rvec = atoms.positions[neighbors] + offset - atoms.positions[i]
r = np.sqrt(np.sum(np.square(rvec), axis=1))
nearest = np.arange(len(r))[r < self.cutoff]
d_embedded_energy_i = self.d_embedded_energy[self.index[i]](
self.total_density[i]
)
urvec = rvec.copy() # unit directional vector
for j in np.arange(len(neighbors)):
urvec[j] = urvec[j] / r[j]
for j_index in range(self.Nelements):
use = self.index[neighbors[nearest]] == j_index
if not use.any():
continue
rnuse = r[nearest][use]
density_j = self.total_density[neighbors[nearest][use]]
if self.form == 'fs':
scale = (
self.d_phi[self.index[i], j_index](rnuse)
+ (
d_embedded_energy_i
* self.d_electron_density[j_index, self.index[i]](
rnuse
)
)
+ (
self.d_embedded_energy[j_index](density_j)
* self.d_electron_density[self.index[i], j_index](
rnuse
)
)
)
else:
scale = (
self.d_phi[self.index[i], j_index](rnuse)
+ (
d_embedded_energy_i
* self.d_electron_density[j_index](rnuse)
)
+ (
self.d_embedded_energy[j_index](density_j)
* self.d_electron_density[self.index[i]](rnuse)
)
)
self.results['forces'][i] += np.dot(scale, urvec[nearest][use])
stresses[i] += np.dot(
(scale[:, np.newaxis] * urvec[nearest][use]).T,
rvec[nearest][use],
)
if self.form == 'adp':
adp_forces, adp_stresses = self.angular_forces(
self.mu[i],
self.mu[neighbors[nearest][use]],
self.lam[i],
self.lam[neighbors[nearest][use]],
rnuse,
rvec[nearest][use],
self.index[i],
j_index,
)
self.results['forces'][i] += adp_forces
stresses[i] += adp_stresses
if self.atoms.cell.rank == 3:
stress = 0.5 * np.sum(stresses, axis=0) / self.atoms.get_volume()
self.results['stress'] = full_3x3_to_voigt_6_stress(stress)
def angular_forces(self, mu_i, mu, lam_i, lam, r, rvec, form1, form2):
# calculate the extra components for the adp forces
# rvec are the relative positions to atom i
psi = np.zeros(mu.shape)
for gamma in range(3):
term1 = (mu_i[gamma] - mu[:, gamma]) * self.d[form1][form2](r)
term2 = np.sum(
(mu_i - mu)
* self.d_d[form1][form2](r)[:, np.newaxis]
* (rvec * rvec[:, gamma][:, np.newaxis] / r[:, np.newaxis]),
axis=1,
)
term3 = 2 * np.sum(
(lam_i[:, gamma] + lam[:, :, gamma])
* rvec
* self.q[form1][form2](r)[:, np.newaxis],
axis=1,
)
term4 = 0.0
for alpha in range(3):
for beta in range(3):
rs = rvec[:, alpha] * rvec[:, beta] * rvec[:, gamma]
term4 += (
(lam_i[alpha, beta] + lam[:, alpha, beta])
* self.d_q[form1][form2](r)
* rs
) / r
term5 = (
(lam_i.trace() + lam.trace(axis1=1, axis2=2))
* (self.d_q[form1][form2](r) * r + 2 * self.q[form1][form2](r))
* rvec[:, gamma]
) / 3.0
# the minus for term5 is a correction on the adp
# formulation given in the 2005 Mishin Paper and is posted
# on the NIST website with the AlH potential
psi[:, gamma] = term1 + term2 + term3 + term4 - term5
return np.sum(psi, axis=0), np.dot(psi.T, rvec)
def adp_dipole(self, r, rvec, d):
# calculate the dipole contribution
mu = np.sum((rvec * d(r)[:, np.newaxis]), axis=0)
return mu # sign to agree with lammps
def adp_quadrupole(self, r, rvec, q):
# slow way of calculating the quadrupole contribution
r = np.sqrt(np.sum(rvec**2, axis=1))
lam = np.zeros([rvec.shape[0], 3, 3])
qr = q(r)
for alpha in range(3):
for beta in range(3):
lam[:, alpha, beta] += qr * rvec[:, alpha] * rvec[:, beta]
return np.sum(lam, axis=0)
def deriv(self, spline):
"""Wrapper for extracting the derivative from a spline"""
def d_spline(aspline):
return spline(aspline, 1)
return d_spline
def plot(self, name=''):
"""Plot the individual curves"""
import matplotlib.pyplot as plt
if self.form == 'eam' or self.form == 'alloy' or self.form == 'fs':
nrow = 2
elif self.form == 'adp':
nrow = 3
else:
raise RuntimeError(f'Unknown form of potential: {self.form}')
if hasattr(self, 'r'):
r = self.r
else:
r = np.linspace(0, self.cutoff, 50)
if hasattr(self, 'rho'):
rho = self.rho
else:
rho = np.linspace(0, 10.0, 50)
plt.subplot(nrow, 2, 1)
self.elem_subplot(
rho,
self.embedded_energy,
r'$\rho$',
r'Embedding Energy $F(\bar\rho)$',
name,
plt,
)
plt.subplot(nrow, 2, 2)
if self.form == 'fs':
self.multielem_subplot(
r,
self.electron_density,
r'$r$',
r'Electron Density $\rho(r)$',
name,
plt,
half=False,
)
else:
self.elem_subplot(
r,
self.electron_density,
r'$r$',
r'Electron Density $\rho(r)$',
name,
plt,
)
plt.subplot(nrow, 2, 3)
self.multielem_subplot(
r, self.phi, r'$r$', r'Pair Potential $\phi(r)$', name, plt
)
plt.ylim(-1.0, 1.0) # need reasonable values
if self.form == 'adp':
plt.subplot(nrow, 2, 5)
self.multielem_subplot(
r, self.d, r'$r$', r'Dipole Energy', name, plt
)
plt.subplot(nrow, 2, 6)
self.multielem_subplot(
r, self.q, r'$r$', r'Quadrupole Energy', name, plt
)
plt.plot()
def elem_subplot(self, curvex, curvey, xlabel, ylabel, name, plt):
plt.xlabel(xlabel)
plt.ylabel(ylabel)
for i in np.arange(self.Nelements):
label = name + ' ' + self.elements[i]
plt.plot(curvex, curvey[i](curvex), label=label)
plt.legend()
def multielem_subplot(
self, curvex, curvey, xlabel, ylabel, name, plt, half=True
):
plt.xlabel(xlabel)
plt.ylabel(ylabel)
for i in np.arange(self.Nelements):
for j in np.arange((i + 1) if half else self.Nelements):
label = name + ' ' + self.elements[i] + '-' + self.elements[j]
plt.plot(curvex, curvey[i, j](curvex), label=label)
plt.legend()
