# Source code for pymatgen.analysis.ewald

# coding: utf-8
# Copyright (c) Pymatgen Development Team.

"""
This module provides classes for calculating the ewald sum of a structure.
"""

from math import pi, sqrt, log
from datetime import datetime
from copy import deepcopy, copy
from warnings import warn
import bisect

import numpy as np
from scipy.special import erfc, comb
import scipy.constants as constants

__author__ = "Shyue Ping Ong, William Davidson Richard"
__credits__ = "Christopher Fischer"
__version__ = "1.0"
__maintainer__ = "Shyue Ping Ong"
__email__ = "shyuep@gmail.com"
__status__ = "Production"
__date__ = "Aug 1 2012"

[docs]class EwaldSummation:
"""
Calculates the electrostatic energy of a periodic array of charges using
the Ewald technique.

Ref:
Ewald summation techniques in perspective: a survey
Abdulnour Y. Toukmaji and John A. Board Jr.
DOI: 10.1016/0010-4655(96)00016-1
URL: http://www.ee.duke.edu/~ayt/ewaldpaper/ewaldpaper.html

This matrix can be used to do fast calculations of ewald sums after species
removal.

E = E_recip + E_real + E_point

Atomic units used in the code, then converted to eV.
"""

# Converts unit of q*q/r into eV
CONV_FACT = 1e10 * constants.e / (4 * pi * constants.epsilon_0)

def __init__(self, structure, real_space_cut=None, recip_space_cut=None,
eta=None, acc_factor=12.0, w=1 / sqrt(2), compute_forces=False):
"""
Initializes and calculates the Ewald sum. Default convergence
parameters have been specified, but you can override them if you wish.

Args:
structure (Structure): Input structure that must have proper
Specie on all sites, i.e. Element with oxidation state. Use
real_space_cut (float): Real space cutoff radius dictating how
many terms are used in the real space sum. Defaults to None,
which means determine automagically using the formula given
in gulp 3.1 documentation.
recip_space_cut (float): Reciprocal space cutoff radius.
Defaults to None, which means determine automagically using
the formula given in gulp 3.1 documentation.
eta (float): The screening parameter. Defaults to None, which means
determine automatically.
acc_factor (float): No. of significant figures each sum is
converged to.
w (float): Weight parameter, w, has been included that represents
the relative computational expense of calculating a term in
real and reciprocal space. Default of 0.7 reproduces result
similar to GULP 4.2. This has little effect on the total
energy, but may influence speed of computation in large
systems. Note that this parameter is used only when the
cutoffs are set to None.
compute_forces (bool): Whether to compute forces. False by
default since it is usually not needed.
"""
self._s = structure
self._charged = abs(structure.charge) > 1e-8
self._vol = structure.volume
self._compute_forces = compute_forces

self._acc_factor = acc_factor
# set screening length
self._eta = eta if eta \
else (len(structure) * w / (self._vol ** 2)) ** (1 / 3) * pi
self._sqrt_eta = sqrt(self._eta)

# acc factor used to automatically determine the optimal real and
self._accf = sqrt(log(10 ** acc_factor))

self._rmax = real_space_cut if real_space_cut \
else self._accf / self._sqrt_eta
self._gmax = recip_space_cut if recip_space_cut \
else 2 * self._sqrt_eta * self._accf

# The next few lines pre-compute certain quantities and store them.
# Ewald summation is rather expensive, and these shortcuts are
# necessary to obtain several factors of improvement in speedup.
self._oxi_states = [compute_average_oxidation_state(site)
for site in structure]

self._coords = np.array(self._s.cart_coords)

# Now we call the relevant private methods to calculate the reciprocal
# and real space terms.
(self._recip, recip_forces) = self._calc_recip()
(self._real, self._point, real_point_forces) = \
self._calc_real_and_point()
if self._compute_forces:
self._forces = recip_forces + real_point_forces

# Compute the correction for a charged cell
self._charged_cell_energy = - EwaldSummation.CONV_FACT / 2 * np.pi / \
structure.volume / self._eta * structure.charge ** 2

[docs]    def compute_partial_energy(self, removed_indices):
"""
Gives total ewald energy for certain sites being removed, i.e. zeroed
out.
"""
total_energy_matrix = self.total_energy_matrix.copy()
for i in removed_indices:
total_energy_matrix[i, :] = 0
total_energy_matrix[:, i] = 0
return sum(sum(total_energy_matrix))

[docs]    def compute_sub_structure(self, sub_structure, tol=1e-3):
"""
Gives total ewald energy for an sub structure in the same
lattice. The sub_structure must be a subset of the original
structure, with possible different charges.

Args:
substructure (Structure): Substructure to compute Ewald sum for.
tol (float): Tolerance for site matching in fractional coordinates.

Returns:
Ewald sum of substructure.
"""
total_energy_matrix = self.total_energy_matrix.copy()

def find_match(site):
for test_site in sub_structure:
frac_diff = abs(np.array(site.frac_coords)
- np.array(test_site.frac_coords)) % 1
frac_diff = [abs(a) < tol or abs(a) > 1 - tol
for a in frac_diff]
if all(frac_diff):
return test_site
return None

matches = []
for i, site in enumerate(self._s):
matching_site = find_match(site)
if matching_site:
new_charge = compute_average_oxidation_state(matching_site)
old_charge = self._oxi_states[i]
scaling_factor = new_charge / old_charge
matches.append(matching_site)
else:
scaling_factor = 0
total_energy_matrix[i, :] *= scaling_factor
total_energy_matrix[:, i] *= scaling_factor

if len(matches) != len(sub_structure):
output = ["Missing sites."]
for site in sub_structure:
if site not in matches:
output.append("unmatched = {}".format(site))
raise ValueError("\n".join(output))

return sum(sum(total_energy_matrix))

@property
def reciprocal_space_energy(self):
"""
The reciprocal space energy.
"""
return sum(sum(self._recip))

@property
def reciprocal_space_energy_matrix(self):
"""
The reciprocal space energy matrix. Each matrix element (i, j)
corresponds to the interaction energy between site i and site j in
reciprocal space.
"""
return self._recip

@property
def real_space_energy(self):
"""
The real space space energy.
"""
return sum(sum(self._real))

@property
def real_space_energy_matrix(self):
"""
The real space energy matrix. Each matrix element (i, j) corresponds to
the interaction energy between site i and site j in real space.
"""
return self._real

@property
def point_energy(self):
"""
The point energy.
"""
return sum(self._point)

@property
def point_energy_matrix(self):
"""
The point space matrix. A diagonal matrix with the point terms for each
site in the diagonal elements.
"""
return self._point

@property
def total_energy(self):
"""
The total energy.
"""
return sum(sum(self._recip)) + sum(sum(self._real)) + sum(self._point) + self._charged_cell_energy

@property
def total_energy_matrix(self):
"""
The total energy matrix. Each matrix element (i, j) corresponds to the
total interaction energy between site i and site j.

Note that this does not include the charged-cell energy, which is only important
when the simulation cell is not charge balanced.
"""
totalenergy = self._recip + self._real
for i in range(len(self._point)):
totalenergy[i, i] += self._point[i]

@property
def forces(self):
"""
The forces on each site as a Nx3 matrix. Each row corresponds to a
site.
"""
if not self._compute_forces:
raise AttributeError(
"Forces are available only if compute_forces is True!")
return self._forces

[docs]    def get_site_energy(self, site_index):
"""Compute the energy for a single site in the structure

Args:
site_index (int): Index of site
ReturnS:
(float) - Energy of that site"""
if self._charged:
warn('Per atom energies for charged structures not supported in EwaldSummation')
return np.sum(self._recip[:, site_index]) + np.sum(self._real[:, site_index]) + self._point[site_index]

def _calc_recip(self):
"""
Perform the reciprocal space summation. Calculates the quantity
E_recip = 1/(2PiV) sum_{G < Gmax} exp(-(G.G/4/eta))/(G.G) S(G)S(-G)
where
S(G) = sum_{k=1,N} q_k exp(-i G.r_k)
S(G)S(-G) = |S(G)|**2

This method is heavily vectorized to utilize numpy's C backend for
speed.
"""
numsites = self._s.num_sites
prefactor = 2 * pi / self._vol
erecip = np.zeros((numsites, numsites), dtype=np.float)
forces = np.zeros((numsites, 3), dtype=np.float)
coords = self._coords
rcp_latt = self._s.lattice.reciprocal_lattice
recip_nn = rcp_latt.get_points_in_sphere([[0, 0, 0]], [0, 0, 0],
self._gmax)

frac_coords = [fcoords for (fcoords, dist, i, img) in recip_nn if dist != 0]

gs = rcp_latt.get_cartesian_coords(frac_coords)
g2s = np.sum(gs ** 2, 1)
expvals = np.exp(-g2s / (4 * self._eta))
grs = np.sum(gs[:, None] * coords[None, :], 2)

oxistates = np.array(self._oxi_states)

# create array where q_2[i,j] is qi * qj
qiqj = oxistates[None, :] * oxistates[:, None]

# calculate the structure factor
sreals = np.sum(oxistates[None, :] * np.cos(grs), 1)
simags = np.sum(oxistates[None, :] * np.sin(grs), 1)

for g, g2, gr, expval, sreal, simag in zip(gs, g2s, grs, expvals,
sreals, simags):

# Uses the identity sin(x)+cos(x) = 2**0.5 sin(x + pi/4)
m = (gr[None, :] + pi / 4) - gr[:, None]
np.sin(m, m)
m *= expval / g2

erecip += m

if self._compute_forces:
pref = 2 * expval / g2 * oxistates
factor = prefactor * pref * (
sreal * np.sin(gr) - simag * np.cos(gr))

forces += factor[:, None] * g[None, :]

forces *= EwaldSummation.CONV_FACT
erecip *= prefactor * EwaldSummation.CONV_FACT * qiqj * 2 ** 0.5
return erecip, forces

def _calc_real_and_point(self):
"""
Determines the self energy -(eta/pi)**(1/2) * sum_{i=1}^{N} q_i**2
"""
fcoords = self._s.frac_coords
forcepf = 2.0 * self._sqrt_eta / sqrt(pi)
coords = self._coords
numsites = self._s.num_sites
ereal = np.empty((numsites, numsites), dtype=np.float)

forces = np.zeros((numsites, 3), dtype=np.float)

qs = np.array(self._oxi_states)

epoint = - qs ** 2 * sqrt(self._eta / pi)

for i in range(numsites):
nfcoords, rij, js, _ = self._s.lattice.get_points_in_sphere(fcoords,
coords[i], self._rmax, zip_results=False)

# remove the rii term
inds = rij > 1e-8
js = js[inds]
rij = rij[inds]
nfcoords = nfcoords[inds]

qi = qs[i]
qj = qs[js]

erfcval = erfc(self._sqrt_eta * rij)
new_ereals = erfcval * qi * qj / rij

# insert new_ereals
for k in range(numsites):
ereal[k, i] = np.sum(new_ereals[js == k])

if self._compute_forces:
nccoords = self._s.lattice.get_cartesian_coords(nfcoords)

fijpf = qj / rij ** 3 * (erfcval + forcepf * rij *
np.exp(-self._eta * rij ** 2))
forces[i] += np.sum(np.expand_dims(fijpf, 1) *
(np.array([coords[i]]) - nccoords) *
qi * EwaldSummation.CONV_FACT, axis=0)

ereal *= 0.5 * EwaldSummation.CONV_FACT
epoint *= EwaldSummation.CONV_FACT
return ereal, epoint, forces

@property
def eta(self):
"""
Returns: eta value used in Ewald summation.
"""
return self._eta

def __str__(self):
if self._compute_forces:
output = ["Real = " + str(self.real_space_energy),
"Reciprocal = " + str(self.reciprocal_space_energy),
"Point = " + str(self.point_energy),
"Total = " + str(self.total_energy),
"Forces:\n" + str(self.forces)
]
else:
output = ["Real = " + str(self.real_space_energy),
"Reciprocal = " + str(self.reciprocal_space_energy),
"Point = " + str(self.point_energy),
"Total = " + str(self.total_energy),
"Forces were not computed"]
return "\n".join(output)

[docs]class EwaldMinimizer:
"""
This class determines the manipulations that will minimize an ewald matrix,
given a list of possible manipulations. This class does not perform the
manipulations on a structure, but will return the list of manipulations
that should be done on one to produce the minimal structure. It returns the
manipulations for the n lowest energy orderings. This class should be used
to perform fractional species substitution or fractional species removal to
produce a new structure. These manipulations create large numbers of
candidate structures, and this class can be used to pick out those with the
lowest ewald sum.

An alternative (possibly more intuitive) interface to this class is the
order disordered structure transformation.

Author - Will Richards
"""

ALGO_FAST = 0
ALGO_COMPLETE = 1
ALGO_BEST_FIRST = 2

"""
ALGO_TIME_LIMIT: Slowly increases the speed (with the cost of decreasing
accuracy) as the minimizer runs. Attempts to limit the run time to
approximately 30 minutes.
"""
ALGO_TIME_LIMIT = 3

def __init__(self, matrix, m_list, num_to_return=1, algo=ALGO_FAST):
"""
Args:
matrix: A matrix of the ewald sum interaction energies. This is stored
in the class as a diagonally symmetric array and so
self._matrix will not be the same as the input matrix.
m_list: list of manipulations. each item is of the form
(multiplication fraction, number_of_indices, indices, species)
These are sorted such that the first manipulation contains the
most permutations. this is actually evaluated last in the
recursion since I'm using pop.
num_to_return: The minimizer will find the number_returned lowest
energy structures. This is likely to return a number of duplicate
structures so it may be necessary to overestimate and then
remove the duplicates later. (duplicate checking in this
process is extremely expensive)
"""
# Setup and checking of inputs
self._matrix = copy(matrix)
# Make the matrix diagonally symmetric (so matrix[i,:] == matrix[:,j])
for i in range(len(self._matrix)):
for j in range(i, len(self._matrix)):
value = (self._matrix[i, j] + self._matrix[j, i]) / 2
self._matrix[i, j] = value
self._matrix[j, i] = value

# sort the m_list based on number of permutations
self._m_list = sorted(m_list, key=lambda x: comb(len(x), x),
reverse=True)

for mlist in self._m_list:
if mlist > 1:
raise ValueError('multiplication fractions must be <= 1')
self._current_minimum = float('inf')
self._num_to_return = num_to_return
self._algo = algo
if algo == EwaldMinimizer.ALGO_COMPLETE:
raise NotImplementedError('Complete algo not yet implemented for '
'EwaldMinimizer')

self._output_lists = []
# Tag that the recurse function looks at at each level. If a method
# sets this to true it breaks the recursion and stops the search.
self._finished = False

self._start_time = datetime.utcnow()

self.minimize_matrix()

self._best_m_list = self._output_lists
self._minimized_sum = self._output_lists

[docs]    def minimize_matrix(self):
"""
This method finds and returns the permutations that produce the lowest
ewald sum calls recursive function to iterate through permutations
"""
if self._algo == EwaldMinimizer.ALGO_FAST or \
self._algo == EwaldMinimizer.ALGO_BEST_FIRST:
return self._recurse(self._matrix, self._m_list,
set(range(len(self._matrix))))

"""
minimum if the list is full.
"""
if self._output_lists is None:
self._output_lists = [[matrix_sum, m_list]]
else:
bisect.insort(self._output_lists, [matrix_sum, m_list])
if self._algo == EwaldMinimizer.ALGO_BEST_FIRST and \
len(self._output_lists) == self._num_to_return:
self._finished = True
if len(self._output_lists) > self._num_to_return:
self._output_lists.pop()
if len(self._output_lists) == self._num_to_return:
self._current_minimum = self._output_lists[-1]

[docs]    def best_case(self, matrix, m_list, indices_left):
"""
Computes a best case given a matrix and manipulation list.

Args:
matrix: the current matrix (with some permutations already
performed)
m_list: [(multiplication fraction, number_of_indices, indices,
species)] describing the manipulation
indices: Set of indices which haven't had a permutation
performed on them.
"""
m_indices = []
fraction_list = []
for m in m_list:
m_indices.extend(m)
fraction_list.extend([m] * m)

indices = list(indices_left.intersection(m_indices))

interaction_matrix = matrix[indices, :][:, indices]

fractions = np.zeros(len(interaction_matrix)) + 1
fractions[:len(fraction_list)] = fraction_list
fractions = np.sort(fractions)

# Sum associated with each index (disregarding interactions between
# indices)
sums = 2 * np.sum(matrix[indices], axis=1)
sums = np.sort(sums)

# Interaction corrections. Can be reduced to (1-x)(1-y) for x,y in
# fractions each element in a column gets multiplied by (1-x), and then
# the sum of the columns gets multiplied by (1-y) since fractions are
# less than 1, there is no effect of one choice on the other
step1 = np.sort(interaction_matrix) * (1 - fractions)
step2 = np.sort(np.sum(step1, axis=1))
step3 = step2 * (1 - fractions)
interaction_correction = np.sum(step3)

if self._algo == self.ALGO_TIME_LIMIT:
elapsed_time = datetime.utcnow() - self._start_time
speedup_parameter = elapsed_time.total_seconds() / 1800
avg_int = np.sum(interaction_matrix, axis=None)
avg_frac = np.average(np.outer(1 - fractions, 1 - fractions))
average_correction = avg_int * avg_frac

interaction_correction = average_correction * speedup_parameter \
+ interaction_correction * (1 - speedup_parameter)

best_case = np.sum(matrix) + np.inner(sums[::-1], fractions - 1) + interaction_correction

return best_case

[docs]    def get_next_index(self, matrix, manipulation, indices_left):
"""
Returns an index that should have the most negative effect on the
matrix sum
"""
f = manipulation
indices = list(indices_left.intersection(manipulation))
sums = np.sum(matrix[indices], axis=1)
if f < 1:
next_index = indices[sums.argmax(axis=0)]
else:
next_index = indices[sums.argmin(axis=0)]

return next_index

def _recurse(self, matrix, m_list, indices, output_m_list=[]):
"""
This method recursively finds the minimal permutations using a binary
tree search strategy.

Args:
matrix: The current matrix (with some permutations already
performed).
m_list: The list of permutations still to be performed
indices: Set of indices which haven't had a permutation
performed on them.
"""
# check to see if we've found all the solutions that we need
if self._finished:
return

# if we're done with the current manipulation, pop it off.
while m_list[-1] == 0:
m_list = copy(m_list)
m_list.pop()
# if there are no more manipulations left to do check the value
if not m_list:
matrix_sum = np.sum(matrix)
if matrix_sum < self._current_minimum:
return

# if we wont have enough indices left, return
if m_list[-1] > len(indices.intersection(m_list[-1])):
return

if len(m_list) == 1 or m_list[-1] > 1:
if self.best_case(matrix, m_list, indices) > self._current_minimum:
return

index = self.get_next_index(matrix, m_list[-1], indices)

m_list[-1].remove(index)

# Make the matrix and new m_list where we do the manipulation to the
# index that we just got
matrix2 = np.copy(matrix)
m_list2 = deepcopy(m_list)
output_m_list2 = copy(output_m_list)

matrix2[index, :] *= m_list[-1]
matrix2[:, index] *= m_list[-1]
output_m_list2.append([index, m_list[-1]])
indices2 = copy(indices)
indices2.remove(index)
m_list2[-1] -= 1

# recurse through both the modified and unmodified matrices

self._recurse(matrix2, m_list2, indices2, output_m_list2)
self._recurse(matrix, m_list, indices, output_m_list)

@property
def best_m_list(self):
"""
Returns: Best m_list found.
"""
return self._best_m_list

@property
def minimized_sum(self):
"""
Returns: Minimized sum
"""
return self._minimized_sum

@property
def output_lists(self):
"""
Returns: output lists.
"""
return self._output_lists

[docs]def compute_average_oxidation_state(site):
"""
Calculates the average oxidation state of a site

Args:
site: Site to compute average oxidation state

Returns:
Average oxidation state of site.
"""
try:
avg_oxi = sum([sp.oxi_state * occu
for sp, occu in site.species.items()
if sp is not None])
return avg_oxi
except AttributeError:
pass
try:
return site.charge
except AttributeError:
raise ValueError("Ewald summation can only be performed on structures "
"that are either oxidation state decorated or have "
"site charges.")