Source code for pymatgen.analysis.ewald

# coding: utf-8
# Copyright (c) Pymatgen Development Team.
# Distributed under the terms of the MIT License.

from __future__ import division, unicode_literals

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
from scipy.misc import comb

import scipy.constants as constants

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

__author__ = "Shyue Ping Ong, William Davidson Richard"
__copyright__ = "Copyright 2011, The Materials Project"
__credits__ = "Christopher Fischer"
__version__ = "1.0"
__maintainer__ = "Shyue Ping Ong"
__email__ = ""
__status__ = "Production"
__date__ = "Aug 1 2012"

[docs]class EwaldSummation(object): """ Calculates the electrostatic energy of a periodic array of charges using the Ewald technique. Ref: 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 Structure.add_oxidation_state... for example. 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 # reciprocal space cutoff radii 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
[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. """ if self._charged: warn('Charged structures not supported in EwaldSummation, but ' 'charged input structures can be used for ' 'EwaldSummation.compute_sub_structure') return sum(sum(self._recip)) + sum(sum(self._real)) + sum(self._point) @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. """ totalenergy = self._recip + self._real for i in range(len(self._point)): totalenergy[i, i] += self._point[i] return totalenergy @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 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) 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 If cell is charged a compensating background is added (i.e. a G=0 term) """ 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): 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 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) """ 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): # 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[2]), x[1]), reverse=True) for mlist in self._m_list: if mlist[0] > 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[0][1] self._minimized_sum = self._output_lists[0][0]
[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))))
[docs] def add_m_list(self, matrix_sum, m_list): """ This adds an m_list to the output_lists and updates the current 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][0]
[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[2]) fraction_list.extend([m[0]] * m[1]) 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[0] indices = list(indices_left.intersection(manipulation[2])) 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][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: self.add_m_list(matrix_sum, output_m_list) return # if we wont have enough indices left, return if m_list[-1][1] > len(indices.intersection(m_list[-1][2])): return if len(m_list) == 1 or m_list[-1][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][2].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][0] matrix2[:, index] *= m_list[-1][0] output_m_list2.append([index, m_list[-1][3]]) indices2 = copy(indices) indices2.remove(index) m_list2[-1][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): return self._best_m_list @property def minimized_sum(self): return self._minimized_sum @property def output_lists(self): 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_and_occu.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.")