import numpy as np
from numpy import cos, sqrt, exp
from ase.data import atomic_numbers
from ase.calculators.calculator import Parameters
from ..utilities import Data, Logger, importer
from .cutoffs import Cosine, dict2cutoff
NeighborList = importer('NeighborList')
[docs]class Bispectrum(object):
"""Class that calculates spherical harmonic bispectrum fingerprints.
Parameters
----------
cutoff : object or float
Cutoff function, typically from amp.descriptor.cutoffs. Can be also
fed as a float representing the radius above which neighbor
interactions are ignored; in this case a cosine cutoff function will be
employed. Default is a 6.5-Angstrom cosine cutoff.
Gs : dict
Dictionary of symbols and dictionaries for making fingerprints. Either
auto-genetrated, or given in the following form, for example:
>>> Gs = {"Au": {"Au": 3., "O": 2.}, "O": {"Au": 5., "O": 10.}}
jmax : integer or half-integer or dict
Maximum degree of spherical harmonics that will be included in the
fingerprint vector. Can be also fed as a dictionary with chemical
species as keys.
dblabel : str
Optional separate prefix/location for database files, including
fingerprints, fingerprint derivatives, and neighborlists. This file
location can be shared between calculator instances to avoid
re-calculating redundant information. If not supplied, just uses the
value from label.
elements : list
List of allowed elements present in the system. If not provided, will
be found automatically.
version : str
Version of fingerprints.
Raises:
-------
RuntimeError, TypeError
"""
def __init__(self, cutoff=Cosine(6.5), Gs=None, jmax=5, dblabel=None,
elements=None, version='2016.02', mode='atom-centered'):
# Check of the version of descriptor, particularly if restarting.
compatibleversions = ['2016.02', ]
if (version is not None) and version not in compatibleversions:
raise RuntimeError('Error: Trying to use bispectrum fingerprints'
' version %s, but this module only supports'
' versions %s. You may need an older or '
' newer version of Amp.' %
(version, compatibleversions))
else:
version = compatibleversions[-1]
# Check that the mode is atom-centered.
if mode != 'atom-centered':
raise RuntimeError('Bispectrum scheme only works '
'in atom-centered mode. %s '
'specified.' % mode)
# If the cutoff is provided as a number, Cosine function will be used
# by default.
if isinstance(cutoff, int) or isinstance(cutoff, float):
cutoff = Cosine(cutoff)
# If the cutoff is provided as a dictionary, assume we need to load it
# with dict2cutoff.
if type(cutoff) is dict:
cutoff = dict2cutoff(cutoff)
# The parameters dictionary contains the minimum information
# to produce a compatible descriptor; that is, one that gives
# an identical fingerprint when fed an ASE image.
p = self.parameters = Parameters(
{'importname': '.descriptor.bispectrum.Bispectrum',
'mode': 'atom-centered'})
p.version = version
p.cutoff = cutoff.todict()
p.Gs = Gs
p.jmax = jmax
p.elements = elements
self.dblabel = dblabel
self.parent = None # Can hold a reference to main Amp instance.
[docs] def tostring(self):
"""Returns an evaluatable representation of the calculator that can
be used to restart the calculator."""
return self.parameters.tostring()
[docs] def calculate_fingerprints(self, images, parallel=None, log=None,
calculate_derivatives=False):
"""Calculates the fingerpints of the images, for the ones not already
done.
Parameters
----------
images : list or str
List of ASE atoms objects with positions, symbols, energies, and
forces in ASE format. This is the training set of data. This can
also be the path to an ASE trajectory (.traj) or database (.db)
file. Energies can be obtained from any reference, e.g. DFT
calculations.
parallel : dict
Configuration for parallelization. Should be in same form as in
amp.Amp.
log : Logger object
Write function at which to log data. Note this must be a callable
function.
calculate_derivatives : bool
Decides whether or not fingerprintprimes should also be calculated.
"""
if parallel is None:
parallel = {'cores': 1}
if calculate_derivatives is True:
import warnings
warnings.warn('Zernike descriptor cannot train forces yet. '
'Force training automatically turnned off. ')
calculate_derivatives = False
log = Logger(file=None) if log is None else log
if (self.dblabel is None) and hasattr(self.parent, 'dblabel'):
self.dblabel = self.parent.dblabel
self.dblabel = 'amp-data' if self.dblabel is None else self.dblabel
p = self.parameters
log('Cutoff function: %s' % repr(dict2cutoff(p.cutoff)))
if p.elements is None:
log('Finding unique set of elements in training data.')
p.elements = set([atom.symbol for atoms in images.values()
for atom in atoms])
p.elements = sorted(p.elements)
log('%i unique elements included: ' % len(p.elements) +
', '.join(p.elements))
log('Maximum degree of spherical harmonic bispectrum:')
if isinstance(p.jmax, dict):
for _ in p.jmax.keys():
log(' %2s: %d' % (_, p.jmax[_]))
else:
log('jmax: %d' % p.jmax)
if p.Gs is None:
log('No coefficient for atomic density function supplied; '
'creating defaults.')
p.Gs = generate_coefficients(p.elements)
log('Coefficients of atomic density function for each element:')
for _ in p.Gs.keys():
log(' %2s: %s' % (_, str(p.Gs[_])))
# Counts the number of descriptors for each element.
no_of_descriptors = {}
for element in p.elements:
count = 0
if isinstance(p.jmax, dict):
for _2j1 in xrange(int(2 * p.jmax[element]) + 1):
for j in xrange(int(min(_2j1, p.jmax[element])) + 1):
count += 1
else:
for _2j1 in xrange(int(2 * p.jmax) + 1):
for j in xrange(int(min(_2j1, p.jmax)) + 1):
count += 1
no_of_descriptors[element] = count
log('Number of descriptors for each element:')
for element in p.elements:
log(' %2s: %d' % (element, no_of_descriptors.pop(element)))
log('Calculating neighborlists...', tic='nl')
if not hasattr(self, 'neighborlist'):
calc = NeighborlistCalculator(cutoff=p.cutoff['kwargs']['Rc'])
self.neighborlist = Data(filename='%s-neighborlists'
% self.dblabel,
calculator=calc)
self.neighborlist.calculate_items(images, parallel=parallel, log=log)
log('...neighborlists calculated.', toc='nl')
log('Fingerprinting images...', tic='fp')
if not hasattr(self, 'fingerprints'):
calc = FingerprintCalculator(neighborlist=self.neighborlist,
Gs=p.Gs,
jmax=p.jmax,
cutoff=p.cutoff,)
self.fingerprints = Data(filename='%s-fingerprints'
% self.dblabel,
calculator=calc)
self.fingerprints.calculate_items(images, parallel=parallel, log=log)
log('...fingerprints calculated.', toc='fp')
# Calculators #################################################################
# Neighborlist Calculator
[docs]class NeighborlistCalculator:
"""For integration with .utilities.Data
For each image fed to calculate, a list of neighbors with offset
distances is returned.
"""
def __init__(self, cutoff):
self.globals = Parameters({'cutoff': cutoff})
self.keyed = Parameters()
self.parallel_command = 'calculate_neighborlists'
[docs] def calculate(self, image, key):
cutoff = self.globals.cutoff
n = NeighborList(cutoffs=[cutoff / 2.] * len(image),
self_interaction=False,
bothways=True,
skin=0.)
n.update(image)
return [n.get_neighbors(index) for index in xrange(len(image))]
[docs]class FingerprintCalculator:
"""For integration with .utilities.Data
"""
def __init__(self, neighborlist, Gs, jmax, cutoff,):
self.globals = Parameters({'cutoff': cutoff,
'Gs': Gs,
'jmax': jmax})
self.keyed = Parameters({'neighborlist': neighborlist})
self.parallel_command = 'calculate_fingerprints'
self.factorial = [1]
for _ in xrange(int(3. * jmax) + 2):
if _ > 0:
self.factorial += [_ * self.factorial[_ - 1]]
[docs] def calculate(self, image, key):
"""Makes a list of fingerprints, one per atom, for the fed image.
Parameters
----------
image : object
ASE atoms object.
key : str
key of the image after being hashed.
"""
nl = self.keyed.neighborlist[key]
fingerprints = []
for atom in image:
symbol = atom.symbol
index = atom.index
neighbors, offsets = nl[index]
neighborsymbols = [image[_].symbol for _ in neighbors]
Rs = [image.positions[neighbor] + np.dot(offset, image.cell)
for (neighbor, offset) in zip(neighbors, offsets)]
self.atoms = image
indexfp = self.get_fingerprint(index, symbol, neighborsymbols, Rs)
fingerprints.append(indexfp)
return fingerprints
[docs] def get_fingerprint(self, index, symbol, n_symbols, Rs):
"""Returns the fingerprint of symmetry function values for atom
specified by its index and symbol.
n_symbols and Rs are lists of
neighbors' symbols and Cartesian positions, respectively.
Parameters
----------
index : int
Index of the center atom.
symbol : str
Symbol of the center atom.
n_symbols : list of str
List of neighbors' symbols.
Rs : list of list of float
List of Cartesian atomic positions of neighbors.
Returns
-------
symbols, fingerprints : list of float
fingerprints for atom specified by its index and symbol.
"""
home = self.atoms[index].position
cutoff = self.globals.cutoff
Rc = cutoff['kwargs']['Rc']
jmax = self.globals.jmax
if cutoff['name'] == 'Cosine':
cutoff_fxn = Cosine(Rc)
elif cutoff['name'] == 'Polynomial':
# cutoff_fxn = Polynomial(cutoff)
raise NotImplementedError()
rs = []
psis = []
thetas = []
phis = []
for neighbor in Rs:
x = neighbor[0] - home[0]
y = neighbor[1] - home[1]
z = neighbor[2] - home[2]
r = np.linalg.norm(neighbor - home)
if r > 10.**(-10.):
psi = np.arcsin(r / Rc)
theta = np.arccos(z / r)
if abs((z / r) - 1.0) < 10.**(-8.):
theta = 0.0
elif abs((z / r) + 1.0) < 10.**(-8.):
theta = np.pi
if x < 0.:
phi = np.pi + np.arctan(y / x)
elif 0. < x and y < 0.:
phi = 2 * np.pi + np.arctan(y / x)
elif 0. < x and 0. <= y:
phi = np.arctan(y / x)
elif x == 0. and 0. < y:
phi = 0.5 * np.pi
elif x == 0. and y < 0.:
phi = 1.5 * np.pi
else:
phi = 0.
rs += [r]
psis += [psi]
thetas += [theta]
phis += [phi]
fingerprint = []
for _2j1 in xrange(int(2 * jmax) + 1):
j1 = 0.5 * _2j1
j2 = 0.5 * _2j1
for j in xrange(int(min(_2j1, jmax)) + 1):
value = calculate_B(j1, j2, 1.0 * j, self.globals.Gs[symbol],
Rc, cutoff['name'],
self.factorial, n_symbols,
rs, psis, thetas, phis)
value = value.real
fingerprint.append(value)
return symbol, fingerprint
# Auxiliary functions #########################################################
[docs]def calculate_B(j1, j2, j, G_element, cutoff, cutofffn, factorial, n_symbols,
rs, psis, thetas, phis):
"""Calculates bi-spectrum B_{j1, j2, j} according to Eq. (5) of "Gaussian
Approximation Potentials: The Accuracy of Quantum Mechanics, without the
Electrons", Phys. Rev. Lett. 104, 136403.
"""
mvals = m_values(j)
B = 0.
for m in mvals:
for mp in mvals:
c = calculate_c(j, mp, m, G_element, cutoff, cutofffn, factorial,
n_symbols, rs, psis, thetas, phis)
m1bound = min(j1, m + j2)
mp1bound = min(j1, mp + j2)
m1 = max(-j1, m - j2)
while m1 < (m1bound + 0.5):
mp1 = max(-j1, mp - j2)
while mp1 < (mp1bound + 0.5):
c1 = calculate_c(j1, mp1, m1, G_element, cutoff, cutofffn,
factorial, n_symbols, rs, psis, thetas,
phis)
c2 = calculate_c(j2, mp - mp1, m - m1, G_element, cutoff,
cutofffn, factorial, n_symbols, rs, psis,
thetas, phis)
B += CG(j1, m1, j2, m - m1, j, m, factorial) * \
CG(j1, mp1, j2, mp - mp1, j, mp, factorial) * \
np.conjugate(c) * c1 * c2
mp1 += 1.
m1 += 1.
return B
###############################################################################
[docs]def calculate_c(j, mp, m, G_element, cutoff, cutofffn, factorial, n_symbols,
rs, psis, thetas, phis):
"""Calculates c^{j}_{m'm} according to Eq. (4) of "Gaussian Approximation
Potentials: The Accuracy of Quantum Mechanics, without the Electrons",
Phys. Rev. Lett. 104, 136403
"""
if cutofffn is 'Cosine':
cutoff_fxn = Cosine(cutoff)
elif cutofffn is 'Polynomial':
# cutoff_fxn = Polynomial(cutoff)
raise NotImplementedError
value = 0.
for n_symbol, r, psi, theta, phi in zip(n_symbols, rs, psis, thetas, phis):
value += G_element[n_symbol] * \
np.conjugate(U(j, m, mp, psi, theta, phi, factorial)) * \
cutoff_fxn(r)
return value
###############################################################################
[docs]def m_values(j):
"""Returns a list of m values for a given j."""
assert j >= 0, '2*j should be a non-negative integer.'
return [j - i for i in xrange(int(2 * j + 1))]
###############################################################################
[docs]def binomial(n, k, factorial):
"""Returns C(n,k) = n!/(k!(n-k)!)."""
assert n >= 0 and k >= 0 and n >= k, \
'n and k should be non-negative integers with n >= k.'
c = factorial[int(n)] / (factorial[int(k)] * factorial[int(n - k)])
return c
###############################################################################
[docs]def WignerD(j, m, mp, alpha, beta, gamma, factorial):
"""Returns the Wigner-D matrix. alpha, beta, and gamma are the Euler
angles."""
result = 0
if abs(beta - np.pi / 2.) < 10.**(-10.):
# Varshalovich Eq. (5), Section 4.16, Page 113.
# j, m, and mp here are J, M, and M', respectively, in Eq. (5).
for k in xrange(int(2 * j + 1)):
if k > j + mp or k > j - m:
break
elif k < mp - m:
continue
result += (-1)**k * binomial(j + mp, k, factorial) * \
binomial(j - mp, k + m - mp, factorial)
result *= (-1)**(m - mp) * \
sqrt(float(factorial[int(j + m)] * factorial[int(j - m)]) /
float((factorial[int(j + mp)] * factorial[int(j - mp)]))) / \
2.**j
result *= exp(-1j * m * alpha) * exp(-1j * mp * gamma)
else:
# Varshalovich Eq. (10), Section 4.16, Page 113.
# m, mpp, and mp here are M, m, and M', respectively, in Eq. (10).
mvals = m_values(j)
for mpp in mvals:
# temp1 = WignerD(j, m, mpp, 0, np.pi/2, 0) = d(j, m, mpp, np.pi/2)
temp1 = 0.
for k in xrange(int(2 * j + 1)):
if k > j + mpp or k > j - m:
break
elif k < mpp - m:
continue
temp1 += (-1)**k * binomial(j + mpp, k, factorial) * \
binomial(j - mpp, k + m - mpp, factorial)
temp1 *= (-1)**(m - mpp) * \
sqrt(float(factorial[int(j + m)] * factorial[int(j - m)]) /
float((factorial[int(j + mpp)] *
factorial[int(j - mpp)]))) / 2.**j
# temp2 = WignerD(j, mpp, mp, 0, np.pi/2, 0) = d(j, mpp, mp,
# np.pi/2)
temp2 = 0.
for k in xrange(int(2 * j + 1)):
if k > j - mp or k > j - mpp:
break
elif k < - mp - mpp:
continue
temp2 += (-1)**k * binomial(j - mp, k, factorial) * \
binomial(j + mp, k + mpp + mp, factorial)
temp2 *= (-1)**(mpp + mp) * \
sqrt(float(factorial[int(j + mpp)] * factorial[int(j - mpp)]) /
float((factorial[int(j - mp)] *
factorial[int(j + mp)]))) / 2.**j
result += temp1 * exp(-1j * mpp * beta) * temp2
# Empirical normalization factor so results match Varshalovich
# Tables 4.3-4.12
# Note that this exact normalization does not follow from the
# above equations
result *= (1j**(2 * j - m - mp)) * ((-1)**(2 * m))
result *= exp(-1j * m * alpha) * exp(-1j * mp * gamma)
return result
###############################################################################
[docs]def U(j, m, mp, omega, theta, phi, factorial):
"""Calculates rotation matrix U_{MM'}^{J} in terms of rotation angle omega as
well as rotation axis angles theta and phi, according to Varshalovich,
Eq. (3), Section 4.5, Page 81. j, m, mp, and mpp here are J, M, M', and M''
in Eq. (3).
"""
result = 0.
mvals = m_values(j)
for mpp in mvals:
result += WignerD(j, m, mpp, phi, theta, -phi, factorial) * \
exp(- 1j * mpp * omega) * \
WignerD(j, mpp, mp, phi, -theta, -phi, factorial)
return result
###############################################################################
[docs]def CG(a, alpha, b, beta, c, gamma, factorial):
"""Clebsch-Gordan coefficient C_{a alpha b beta}^{c gamma} is calculated
acoording to the expression given in Varshalovich Eq. (3), Section 8.2,
Page 238."""
if int(2. * a) != 2. * a or int(2. * b) != 2. * b or int(2. * c) != 2. * c:
raise ValueError("j values must be integer or half integer")
if int(2. * alpha) != 2. * alpha or int(2. * beta) != 2. * beta or \
int(2. * gamma) != 2. * gamma:
raise ValueError("m values must be integer or half integer")
if alpha + beta - gamma != 0.:
return 0.
else:
minimum = min(a + b - c, a - b + c, -a + b + c, a + b + c + 1.,
a - abs(alpha), b - abs(beta), c - abs(gamma))
if minimum < 0.:
return 0.
else:
sqrtarg = \
factorial[int(a + alpha)] * \
factorial[int(a - alpha)] * \
factorial[int(b + beta)] * \
factorial[int(b - beta)] * \
factorial[int(c + gamma)] * \
factorial[int(c - gamma)] * \
(2. * c + 1.) * \
factorial[int(a + b - c)] * \
factorial[int(a - b + c)] * \
factorial[int(-a + b + c)] / \
factorial[int(a + b + c + 1.)]
sqrtres = sqrt(sqrtarg)
zmin = max(a + beta - c, b - alpha - c, 0.)
zmax = min(b + beta, a - alpha, a + b - c)
sumres = 0.
for z in xrange(int(zmin), int(zmax) + 1):
value = \
factorial[int(z)] * \
factorial[int(a + b - c - z)] * \
factorial[int(a - alpha - z)] * \
factorial[int(b + beta - z)] * \
factorial[int(c - b + alpha + z)] * \
factorial[int(c - a - beta + z)]
sumres += (-1.)**z / value
result = sqrtres * sumres
return result
###############################################################################
[docs]def generate_coefficients(elements):
"""Automatically generates coefficients if not given by the user.
Parameters
---------
elements : list of str
List of symbols of all atoms.
Returns
-------
G : dict of dicts
"""
_G = {}
for element in elements:
_G[element] = atomic_numbers[element]
G = {}
for element in elements:
G[element] = _G
return G
###############################################################################
if __name__ == "__main__":
"""Directly calling this module; apparently from another node.
Calls should come as
python -m amp.descriptor.example id hostname:port
This session will then start a zmq session with that socket, labeling
itself with id. Instructions on what to do will come from the socket.
"""
import sys
import tempfile
import zmq
from ..utilities import MessageDictionary
hostsocket = sys.argv[-1]
proc_id = sys.argv[-2]
msg = MessageDictionary(proc_id)
# Send standard lines to stdout signaling process started and where
# error is directed. This should be caught by pxssh. (This could
# alternatively be done by zmq, but this works.)
print('<amp-connect>') # Signal that program started.
sys.stderr = tempfile.NamedTemporaryFile(mode='w', delete=False,
suffix='.stderr')
print('Log and error written to %s<stderr>' % sys.stderr.name)
# Establish client session via zmq; find purpose.
context = zmq.Context()
socket = context.socket(zmq.REQ)
socket.connect('tcp://%s' % hostsocket)
socket.send_pyobj(msg('<purpose>'))
purpose = socket.recv_pyobj()
if purpose == 'calculate_neighborlists':
# Request variables.
socket.send_pyobj(msg('<request>', 'cutoff'))
cutoff = socket.recv_pyobj()
socket.send_pyobj(msg('<request>', 'images'))
images = socket.recv_pyobj()
# sys.stderr.write(str(images)) # Just to see if they are there.
# Perform the calculations.
calc = NeighborlistCalculator(cutoff=cutoff)
neighborlist = {}
# for key in images.iterkeys():
while len(images) > 0:
key, image = images.popitem() # Reduce memory.
neighborlist[key] = calc.calculate(image, key)
# Send the results.
socket.send_pyobj(msg('<result>', neighborlist))
socket.recv_string() # Needed to complete REQ/REP.
elif purpose == 'calculate_fingerprints':
# Request variables.
socket.send_pyobj(msg('<request>', 'cutoff'))
cutoff = socket.recv_pyobj()
socket.send_pyobj(msg('<request>', 'Gs'))
Gs = socket.recv_pyobj()
socket.send_pyobj(msg('<request>', 'jmax'))
jmax = socket.recv_pyobj()
socket.send_pyobj(msg('<request>', 'neighborlist'))
neighborlist = socket.recv_pyobj()
socket.send_pyobj(msg('<request>', 'images'))
images = socket.recv_pyobj()
calc = FingerprintCalculator(neighborlist, Gs, jmax, cutoff,)
result = {}
while len(images) > 0:
key, image = images.popitem() # Reduce memory.
result[key] = calc.calculate(image, key)
if len(images) % 100 == 0:
socket.send_pyobj(msg('<info>', len(images)))
socket.recv_string() # Needed to complete REQ/REP.
# Send the results.
socket.send_pyobj(msg('<result>', result))
socket.recv_string() # Needed to complete REQ/REP.
else:
raise NotImplementedError('purpose %s unknown.' % purpose)
