# -*- coding: utf-8 -*-
import pickle
import sys
import threading
from math import sqrt
import numpy as np
import ase.parallel as mpi
from ase.build import minimize_rotation_and_translation
from ase.calculators.calculator import Calculator
from ase.calculators.singlepoint import SinglePointCalculator
from ase.io import read
from ase.optimize import MDMin
from ase.geometry import find_mic
from ase.utils import basestring
[docs]class NEB:
def __init__(self, images, k=0.1, fmax=0.05, climb=False, parallel=False,
remove_rotation_and_translation=False, world=None,
method='aseneb', dynamic_relaxation=False, scale_fmax=0.):
"""Nudged elastic band.
Paper I:
G. Henkelman and H. Jonsson, Chem. Phys, 113, 9978 (2000).
https://doi.org/10.1063/1.1323224
Paper II:
G. Henkelman, B. P. Uberuaga, and H. Jonsson, Chem. Phys,
113, 9901 (2000).
https://doi.org/10.1063/1.1329672
Paper III:
E. L. Kolsbjerg, M. N. Groves, and B. Hammer, J. Chem. Phys,
145, 094107 (2016)
https://doi.org/10.1063/1.4961868
images: list of Atoms objects
Images defining path from initial to final state.
k: float or list of floats
Spring constant(s) in eV/Ang. One number or one for each spring.
climb: bool
Use a climbing image (default is no climbing image).
parallel: bool
Distribute images over processors.
remove_rotation_and_translation: bool
TRUE actives NEB-TR for removing translation and
rotation during NEB. By default applied non-periodic
systems
dynamic_relaxation: bool
TRUE calculates the norm of the forces acting on each image
in the band. An image is optimized only if its norm is above
the convergence criterion. The list fmax_images is updated
every force call; if a previously converged image goes out
of tolerance (due to spring adjustments between the image
and its neighbors), it will be optimized again. This routine
can speed up calculations if convergence is non-uniform.
Convergence criterion should be the same as that given to
the optimizer. Not efficient when parallelizing over images.
scale_fmax: float
Scale convergence criteria along band based on the distance
between a state and the state with the highest potential energy.
method: string of method
Choice betweeen three method:
* aseneb: standard ase NEB implementation
* improvedtangent: Paper I NEB implementation
* eb: Paper III full spring force implementation
"""
self.images = images
self.climb = climb
self.parallel = parallel
self.natoms = len(images[0])
pbc = images[0].pbc
atomic_numbers = images[0].get_atomic_numbers()
for img in images:
if len(img) != self.natoms:
raise ValueError('Images have different numbers of atoms')
if (pbc != img.pbc).any():
raise ValueError('Images have different boundary conditions')
if (atomic_numbers != img.get_atomic_numbers()).any():
raise ValueError('Images have atoms in different orders')
self.nimages = len(images)
self.emax = np.nan
self.remove_rotation_and_translation = remove_rotation_and_translation
self.dynamic_relaxation = dynamic_relaxation
self.fmax = fmax
self.scale_fmax = scale_fmax
if not self.dynamic_relaxation and self.scale_fmax:
msg = ('Scaled convergence criteria only implemented in series '
'with dynamic_relaxation.')
raise ValueError(msg)
if method in ['aseneb', 'eb', 'improvedtangent']:
self.method = method
else:
raise NotImplementedError(method)
if isinstance(k, (float, int)):
k = [k] * (self.nimages - 1)
self.k = list(k)
if world is None:
world = mpi.world
self.world = world
if parallel:
assert world.size == 1 or world.size % (self.nimages - 2) == 0
self.real_forces = None # ndarray of shape (nimages, natom, 3)
self.energies = None # ndarray of shape (nimages,)
[docs] def interpolate(self, method='linear', mic=False):
if self.remove_rotation_and_translation:
minimize_rotation_and_translation(self.images[0], self.images[-1])
interpolate(self.images, mic)
if method == 'idpp':
self.idpp_interpolate(traj=None, log=None, mic=mic)
[docs] def idpp_interpolate(self, traj='idpp.traj', log='idpp.log', fmax=0.1,
optimizer=MDMin, mic=False, steps=100):
d1 = self.images[0].get_all_distances(mic=mic)
d2 = self.images[-1].get_all_distances(mic=mic)
d = (d2 - d1) / (self.nimages - 1)
old = []
for i, image in enumerate(self.images):
old.append(image.calc)
image.calc = IDPP(d1 + i * d, mic=mic)
opt = optimizer(self, trajectory=traj, logfile=log)
# BFGS was originally used by the paper, but testing shows that
# MDMin results in nearly the same results in 3-4 orders of magnitude
# less time. Known working optimizers = BFGS, MDMin, FIRE, HessLBFGS
# Internal testing shows BFGS is only needed in situations where MDMIN
# cannot converge easily and tends to be obvious on inspection.
#
# askhl: 3-4 orders of magnitude difference cannot possibly be
# true unless something is actually broken. Should it not be
# "3-4 times"?
opt.run(fmax=fmax, steps=steps)
for image, calc in zip(self.images, old):
image.calc = calc
def get_positions(self):
positions = np.empty(((self.nimages - 2) * self.natoms, 3))
n1 = 0
for image in self.images[1:-1]:
n2 = n1 + self.natoms
positions[n1:n2] = image.get_positions()
n1 = n2
return positions
def set_positions(self, positions):
n1 = 0
for i, image in enumerate(self.images[1:-1]):
if self.dynamic_relaxation:
if self.parallel:
msg = ('Dynamic relaxation does not work efficiently '
'when parallelizing over images. Try AutoNEB '
'routine for freezing images in parallel.')
raise ValueError(msg)
else:
forces_dyn = self.get_fmax_all(self.images)
if forces_dyn[i] < self.fmax:
n1 += self.natoms
else:
n2 = n1 + self.natoms
image.set_positions(positions[n1:n2])
n1 = n2
else:
n2 = n1 + self.natoms
image.set_positions(positions[n1:n2])
n1 = n2
def get_fmax_all(self, images):
n = self.natoms
f_i = self.get_forces()
fmax_images = []
for i in range(self.nimages-2):
n1 = n * i
n2 = n + n * i
fmax_images.append(np.sqrt((f_i[n1:n2]**2).sum(axis=1)).max())
return fmax_images
def get_forces(self):
"""Evaluate and return the forces."""
images = self.images
calculators = [image.calc for image in images
if image.calc is not None]
if len(set(calculators)) != len(calculators):
msg = ('One or more NEB images share the same calculator. '
'Each image must have its own calculator. '
'You may wish to use the ase.neb.SingleCalculatorNEB '
'class instead, although using separate calculators '
'is recommended.')
raise ValueError(msg)
forces = np.empty(((self.nimages - 2), self.natoms, 3))
energies = np.empty(self.nimages)
if self.remove_rotation_and_translation:
# Remove translation and rotation between
# images before computing forces:
for i in range(1, self.nimages):
minimize_rotation_and_translation(images[i - 1], images[i])
if self.method != 'aseneb':
energies[0] = images[0].get_potential_energy()
energies[-1] = images[-1].get_potential_energy()
if not self.parallel:
# Do all images - one at a time:
for i in range(1, self.nimages - 1):
energies[i] = images[i].get_potential_energy()
forces[i - 1] = images[i].get_forces()
elif self.world.size == 1:
def run(image, energies, forces):
energies[:] = image.get_potential_energy()
forces[:] = image.get_forces()
threads = [threading.Thread(target=run,
args=(images[i],
energies[i:i + 1],
forces[i - 1:i]))
for i in range(1, self.nimages - 1)]
for thread in threads:
thread.start()
for thread in threads:
thread.join()
else:
# Parallelize over images:
i = self.world.rank * (self.nimages - 2) // self.world.size + 1
try:
energies[i] = images[i].get_potential_energy()
forces[i - 1] = images[i].get_forces()
except Exception:
# Make sure other images also fail:
error = self.world.sum(1.0)
raise
else:
error = self.world.sum(0.0)
if error:
raise RuntimeError('Parallel NEB failed!')
for i in range(1, self.nimages - 1):
root = (i - 1) * self.world.size // (self.nimages - 2)
self.world.broadcast(energies[i:i + 1], root)
self.world.broadcast(forces[i - 1], root)
# Save for later use in iterimages:
self.energies = energies
self.real_forces = np.zeros((self.nimages, self.natoms, 3))
self.real_forces[1:-1] = forces
self.imax = 1 + np.argsort(energies[1:-1])[-1]
self.emax = energies[self.imax]
t1 = find_mic(images[1].get_positions() -
images[0].get_positions(),
images[0].get_cell(), images[0].pbc)[0]
if self.method == 'eb':
beeline = (images[self.nimages - 1].get_positions() -
images[0].get_positions())
beelinelength = np.linalg.norm(beeline)
eqlength = beelinelength / (self.nimages - 1)
nt1 = np.linalg.norm(t1)
for i in range(1, self.nimages - 1):
t2 = find_mic(images[i + 1].get_positions() -
images[i].get_positions(),
images[i].get_cell(), images[i].pbc)[0]
nt2 = np.linalg.norm(t2)
if self.method == 'eb':
# Tangents are bisections of spring-directions
# (formula C8 of paper III)
tangent = t1 / nt1 + t2 / nt2
# Normalize the tangent vector
tangent /= np.linalg.norm(tangent)
elif self.method == 'improvedtangent':
# Tangents are improved according to formulas 8, 9, 10,
# and 11 of paper I.
if energies[i + 1] > energies[i] > energies[i - 1]:
tangent = t2.copy()
elif energies[i + 1] < energies[i] < energies[i - 1]:
tangent = t1.copy()
else:
deltavmax = max(abs(energies[i + 1] - energies[i]),
abs(energies[i - 1] - energies[i]))
deltavmin = min(abs(energies[i + 1] - energies[i]),
abs(energies[i - 1] - energies[i]))
if energies[i + 1] > energies[i - 1]:
tangent = t2 * deltavmax + t1 * deltavmin
else:
tangent = t2 * deltavmin + t1 * deltavmax
# Normalize the tangent vector
tangent /= np.linalg.norm(tangent)
else:
if i < self.imax:
tangent = t2
elif i > self.imax:
tangent = t1
else:
tangent = t1 + t2
tt = np.vdot(tangent, tangent)
f = forces[i - 1]
ft = np.vdot(f, tangent)
if i == self.imax and self.climb:
# imax not affected by the spring forces. The full force
# with component along the elestic band converted
# (formula 5 of Paper II)
if self.method == 'aseneb':
f -= 2 * ft / tt * tangent
else:
f -= 2 * ft * tangent
elif self.method == 'eb':
f -= ft * tangent
# Spring forces
# (formula C1, C5, C6 and C7 of Paper III)
f1 = -(nt1 - eqlength) * t1 / nt1 * self.k[i - 1]
f2 = (nt2 - eqlength) * t2 / nt2 * self.k[i]
if self.climb and abs(i - self.imax) == 1:
deltavmax = max(abs(energies[i + 1] - energies[i]),
abs(energies[i - 1] - energies[i]))
deltavmin = min(abs(energies[i + 1] - energies[i]),
abs(energies[i - 1] - energies[i]))
f += (f1 + f2) * deltavmin / deltavmax
else:
f += f1 + f2
elif self.method == 'improvedtangent':
f -= ft * tangent
# Improved parallel spring force (formula 12 of paper I)
f += (nt2 * self.k[i] - nt1 * self.k[i - 1]) * tangent
else:
f -= ft / tt * tangent
f -= np.vdot(t1 * self.k[i - 1] -
t2 * self.k[i], tangent) / tt * tangent
t1 = t2
nt1 = nt2
if self.dynamic_relaxation:
n = self.natoms
k = i - 1
n1 = n * k
n2 = n1 + n
force_i = np.sqrt((forces.reshape((-1, 3))[n1:n2]**2.)
.sum(axis=1)).max()
n1_imax = (self.imax - 1) * n
positions = self.get_positions()
pos_imax = positions[n1_imax:n1_imax + n]
rel_pos = np.sqrt(((positions[n1:n2] - pos_imax)**2).sum())
if force_i < self.fmax * (1 + rel_pos * self.scale_fmax):
if k == self.imax - 1:
pass
else:
forces[k, :, :] = np.zeros((1, self.natoms, 3))
return forces.reshape((-1, 3))
def get_potential_energy(self, force_consistent=False):
"""Return the maximum potential energy along the band.
Note that the force_consistent keyword is ignored and is only
present for compatibility with ase.Atoms.get_potential_energy."""
return self.emax
def __len__(self):
# Corresponds to number of optimizable degrees of freedom, i.e.
# virtual atom count for the optimization algorithm.
return (self.nimages - 2) * self.natoms
def iterimages(self):
# Allows trajectory to convert NEB into several images
if not self.parallel or self.world.size == 1:
for atoms in self.images:
yield atoms
return
for i, atoms in enumerate(self.images):
if i == 0 or i == self.nimages - 1:
yield atoms
else:
atoms = atoms.copy()
atoms.calc = SinglePointCalculator(energy=self.energies[i],
forces=self.real_forces[i],
atoms=atoms)
yield atoms
class IDPP(Calculator):
"""Image dependent pair potential.
See:
Improved initial guess for minimum energy path calculations.
Søren Smidstrup, Andreas Pedersen, Kurt Stokbro and Hannes Jónsson
Chem. Phys. 140, 214106 (2014)
"""
implemented_properties = ['energy', 'forces']
def __init__(self, target, mic):
Calculator.__init__(self)
self.target = target
self.mic = mic
def calculate(self, atoms, properties, system_changes):
Calculator.calculate(self, atoms, properties, system_changes)
P = atoms.get_positions()
d = []
D = []
for p in P:
Di = P - p
if self.mic:
Di, di = find_mic(Di, atoms.get_cell(), atoms.get_pbc())
else:
di = np.sqrt((Di**2).sum(1))
d.append(di)
D.append(Di)
d = np.array(d)
D = np.array(D)
dd = d - self.target
d.ravel()[::len(d) + 1] = 1 # avoid dividing by zero
d4 = d**4
e = 0.5 * (dd**2 / d4).sum()
f = -2 * ((dd * (1 - 2 * dd / d) / d**5)[..., np.newaxis] * D).sum(0)
self.results = {'energy': e, 'forces': f}
class SingleCalculatorNEB(NEB):
def __init__(self, images, k=0.1, climb=False):
if isinstance(images, basestring):
# this is a filename
images = read(images)
NEB.__init__(self, images, k, climb, False)
self.calculators = [None] * self.nimages
self.energies_ok = False
self.first = True
def interpolate(self, initial=0, final=-1, mic=False):
"""Interpolate linearly between initial and final images."""
if final < 0:
final = self.nimages + final
n = final - initial
pos1 = self.images[initial].get_positions()
pos2 = self.images[final].get_positions()
dist = (pos2 - pos1)
if mic:
cell = self.images[initial].get_cell()
assert((cell == self.images[final].get_cell()).all())
pbc = self.images[initial].get_pbc()
assert((pbc == self.images[final].get_pbc()).all())
dist, D_len = find_mic(dist, cell, pbc)
dist /= n
for i in range(1, n):
self.images[initial + i].set_positions(pos1 + i * dist)
def refine(self, steps=1, begin=0, end=-1, mic=False):
"""Refine the NEB trajectory."""
if end < 0:
end = self.nimages + end
j = begin
n = end - begin
for i in range(n):
for k in range(steps):
self.images.insert(j + 1, self.images[j].copy())
self.calculators.insert(j + 1, None)
self.k[j:j + 1] = [self.k[j] * (steps + 1)] * (steps + 1)
self.nimages = len(self.images)
self.interpolate(j, j + steps + 1, mic=mic)
j += steps + 1
def set_positions(self, positions):
# new positions -> new forces
if self.energies_ok:
# restore calculators
self.set_calculators(self.calculators[1:-1])
NEB.set_positions(self, positions)
def get_calculators(self):
"""Return the original calculators."""
calculators = []
for i, image in enumerate(self.images):
if self.calculators[i] is None:
calculators.append(image.get_calculator())
else:
calculators.append(self.calculators[i])
return calculators
def set_calculators(self, calculators):
"""Set new calculators to the images."""
self.energies_ok = False
self.first = True
if not isinstance(calculators, list):
calculators = [calculators] * self.nimages
n = len(calculators)
if n == self.nimages:
for i in range(self.nimages):
self.images[i].set_calculator(calculators[i])
elif n == self.nimages - 2:
for i in range(1, self.nimages - 1):
self.images[i].set_calculator(calculators[i - 1])
else:
raise RuntimeError(
'len(calculators)=%d does not fit to len(images)=%d'
% (n, self.nimages))
def get_energies_and_forces(self):
"""Evaluate energies and forces and hide the calculators"""
if self.energies_ok:
return
self.emax = -1.e32
def calculate_and_hide(i):
image = self.images[i]
calc = image.get_calculator()
if self.calculators[i] is None:
self.calculators[i] = calc
if calc is not None:
if not isinstance(calc, SinglePointCalculator):
self.images[i].set_calculator(
SinglePointCalculator(
image,
energy=image.get_potential_energy(
apply_constraint=False),
forces=image.get_forces(apply_constraint=False)))
self.emax = min(self.emax, image.get_potential_energy())
if self.first:
calculate_and_hide(0)
# Do all images - one at a time:
for i in range(1, self.nimages - 1):
calculate_and_hide(i)
if self.first:
calculate_and_hide(-1)
self.first = False
self.energies_ok = True
def get_forces(self):
self.get_energies_and_forces()
return NEB.get_forces(self)
def n(self):
return self.nimages
def write(self, filename):
from ase.io.trajectory import Trajectory
traj = Trajectory(filename, 'w', self)
traj.write()
traj.close()
def __add__(self, other):
for image in other:
self.images.append(image)
return self
def fit0(E, F, R, cell=None, pbc=None):
"""Constructs curve parameters from the NEB images."""
E = np.array(E) - E[0]
n = len(E)
Efit = np.empty((n - 1) * 20 + 1)
Sfit = np.empty((n - 1) * 20 + 1)
s = [0]
dR = np.zeros_like(R)
for i in range(n):
if i < n - 1:
dR[i] = R[i + 1] - R[i]
if cell is not None and pbc is not None:
dR[i], _ = find_mic(dR[i], cell, pbc)
s.append(s[i] + sqrt((dR[i]**2).sum()))
else:
dR[i] = R[i] - R[i - 1]
if cell is not None and pbc is not None:
dR[i], _ = find_mic(dR[i], cell, pbc)
lines = []
dEds0 = None
for i in range(n):
d = dR[i]
if i == 0:
ds = 0.5 * s[1]
elif i == n - 1:
ds = 0.5 * (s[-1] - s[-2])
else:
ds = 0.25 * (s[i + 1] - s[i - 1])
d = d / sqrt((d**2).sum())
dEds = -(F[i] * d).sum()
x = np.linspace(s[i] - ds, s[i] + ds, 3)
y = E[i] + dEds * (x - s[i])
lines.append((x, y))
if i > 0:
s0 = s[i - 1]
s1 = s[i]
x = np.linspace(s0, s1, 20, endpoint=False)
c = np.linalg.solve(np.array([(1, s0, s0**2, s0**3),
(1, s1, s1**2, s1**3),
(0, 1, 2 * s0, 3 * s0**2),
(0, 1, 2 * s1, 3 * s1**2)]),
np.array([E[i - 1], E[i], dEds0, dEds]))
y = c[0] + x * (c[1] + x * (c[2] + x * c[3]))
Sfit[(i - 1) * 20:i * 20] = x
Efit[(i - 1) * 20:i * 20] = y
dEds0 = dEds
Sfit[-1] = s[-1]
Efit[-1] = E[-1]
return s, E, Sfit, Efit, lines
def plot_band_from_fit(s, E, Sfit, Efit, lines, ax=None):
if ax is None:
import matplotlib.pyplot as plt
ax = plt.gca()
ax.plot(s, E, 'o')
for x, y in lines:
ax.plot(x, y, '-g')
ax.plot(Sfit, Efit, 'k-')
ax.set_xlabel(r'path [$\AA$]')
ax.set_ylabel('energy [eV]')
Ef = max(Efit) - E[0]
Er = max(Efit) - E[-1]
dE = E[-1] - E[0]
ax.set_title('$E_\\mathrm{f} \\approx$ %.3f eV; '
'$E_\\mathrm{r} \\approx$ %.3f eV; '
'$\\Delta E$ = %.3f eV'
% (Ef, Er, dE))
return ax
NEBtools = NEBTools # backwards compatibility
def interpolate(images, mic=False):
"""Given a list of images, linearly interpolate the positions of the
interior images."""
pos1 = images[0].get_positions()
pos2 = images[-1].get_positions()
d = pos2 - pos1
if mic:
d = find_mic(d, images[0].get_cell(), images[0].pbc)[0]
d /= (len(images) - 1.0)
for i in range(1, len(images) - 1):
images[i].set_positions(pos1 + i * d)
if __name__ == '__main__':
# This stuff is used by ASE's GUI
import matplotlib.pyplot as plt
fit = pickle.load(sys.stdin)
plot_band_from_fit(*fit)
plt.show()