pymatgen.analysis.ewald — pymatgen 2020.7.3 documentation

# Source code for pymatgen.analysis.ewald

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

    """
    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"
    __copyright__ = "Copyright 2011, The Materials Project"
    __credits__ = "Christopher Fischer"
    __version__ = "1.0"
    __maintainer__ = "Shyue Ping Ong"
    __email__ = "shyuep@gmail.com"
    __status__ = "Production"
    __date__ = "Aug 1 2012"

    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
                    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

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

        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))

        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]
            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 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)

    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[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]

        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))))

        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]

        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

        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):
            """
            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

    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.")