pymatgen.analysis.defects.utils — pymatgen 2020.7.3 documentation

# Source code for pymatgen.analysis.defects.utils

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

    """
    Utilities for defects module.
    """

    import math

    from monty.json import MSONable

    import itertools
    import pandas as pd
    import numpy as np
    from numpy.linalg import norm
    import logging

    from collections import defaultdict
    from scipy.spatial import Voronoi
    from scipy.spatial.distance import squareform
    from scipy.cluster.hierarchy import linkage, fcluster
    from pymatgen.analysis.local_env import LocalStructOrderParams, \
        MinimumDistanceNN, cn_opt_params
    from pymatgen.core.periodic_table import Element, get_el_sp
    from pymatgen.core.sites import PeriodicSite
    from pymatgen.core.structure import Structure
    from pymatgen.io.vasp.outputs import Chgcar
    from pymatgen.symmetry.analyzer import SpacegroupAnalyzer
    from pymatgen.analysis.phase_diagram import get_facets
    from pymatgen.util.coord import pbc_diff
    from pymatgen.vis.structure_vtk import StructureVis
    from monty.dev import requires
    from copy import deepcopy

    try:
        from skimage.feature import peak_local_max

        peak_local_max_found = True
    except ImportError:
        peak_local_max_found = False

    __author__ = "Danny Broberg, Shyam Dwaraknath, Bharat Medasani, Nils Zimmermann, Geoffroy Hautier"
    __copyright__ = "Copyright 2014, The Materials Project"
    __version__ = "1.0"
    __maintainer__ = "Danny Broberg, Shyam Dwaraknath"
    __email__ = "dbroberg@berkeley.edu, shyamd@lbl.gov"
    __status__ = "Development"
    __date__ = "January 11, 2018"

    logger = logging.getLogger(__name__)
    hart_to_ev = 27.2114
    ang_to_bohr = 1.8897
    invang_to_ev = 3.80986
    kb = 8.6173324e-5  # eV / K
    kumagai_to_V = 1.809512739e2  # = Electron charge * 1e10 / VacuumPermittivity Constant

    motif_cn_op = {}
    for cn, di in cn_opt_params.items():  # type: ignore
        for mot, li in di.items():
            motif_cn_op[mot] = {'cn': int(cn), 'optype': li[0]}
            motif_cn_op[mot]['params'] = deepcopy(li[1]) if len(li) > 1 else None

    class QModel(MSONable):
        """
        Model for the defect charge distribution.
        A combination of exponential tail and gaussian distribution is used
        (see Freysoldt (2011), DOI: 10.1002/pssb.201046289 )
        q_model(r) = q [x exp(-r/gamma) + (1-x) exp(-r^2/beta^2)]
                without normalization constants
        By default, gaussian distribution with 1 Bohr width is assumed.
        If defect charge is more delocalized, exponential tail is suggested.
        """

        def __init__(self, beta=1.0, expnorm=0.0, gamma=1.0):
            """
            Args:
                beta: Gaussian decay constant. Default value is 1 Bohr.
                      When delocalized (eg. diamond), 2 Bohr is more appropriate.
                expnorm: Weight for the exponential tail in the range of [0-1].
                         Default is 0.0 indicating no tail .
                         For delocalized charges ideal value is around 0.54-0.6.
                gamma: Exponential decay constant
            """
            self.beta = beta
            self.expnorm = expnorm
            self.gamma = gamma

            self.beta2 = beta * beta
            self.gamma2 = gamma * gamma
            if expnorm and not gamma:
                raise ValueError("Please supply exponential decay constant.")

        def rho_rec(self, g2):
            """
            Reciprocal space model charge value
            for input squared reciprocal vector.
            Args:
                g2: Square of reciprocal vector

            Returns:
                Charge density at the reciprocal vector magnitude
            """
            return (self.expnorm / np.sqrt(1 + self.gamma2 * g2) + (
                    1 - self.expnorm) * np.exp(-0.25 * self.beta2 * g2))

        @property
        def rho_rec_limit0(self):
            """
            Reciprocal space model charge value
            close to reciprocal vector 0 .
            rho_rec(g->0) -> 1 + rho_rec_limit0 * g^2
            """
            return -2 * self.gamma2 * self.expnorm - 0.25 * self.beta2 * (
                    1 - self.expnorm)

    def eV_to_k(energy):
        """
        Convert energy to reciprocal vector magnitude k via hbar*k^2/2m
        Args:
            a: Energy in eV.

        Returns:
            (double) Reciprocal vector magnitude (units of 1/Bohr).
        """
        return math.sqrt(energy / invang_to_ev) * ang_to_bohr

    def genrecip(a1, a2, a3, encut):
        """
        Args:
            a1, a2, a3: lattice vectors in bohr
            encut: energy cut off in eV
        Returns:
            reciprocal lattice vectors with energy less than encut
        """
        vol = np.dot(a1, np.cross(a2, a3))  # 1/bohr^3
        b1 = (2 * np.pi / vol) * np.cross(a2, a3)  # units 1/bohr
        b2 = (2 * np.pi / vol) * np.cross(a3, a1)
        b3 = (2 * np.pi / vol) * np.cross(a1, a2)

        # create list of recip space vectors that satisfy |i*b1+j*b2+k*b3|<=encut
        G_cut = eV_to_k(encut)
        # Figure out max in all recipricol lattice directions
        i_max = int(math.ceil(G_cut / norm(b1)))
        j_max = int(math.ceil(G_cut / norm(b2)))
        k_max = int(math.ceil(G_cut / norm(b3)))

        # Build index list
        i = np.arange(-i_max, i_max)
        j = np.arange(-j_max, j_max)
        k = np.arange(-k_max, k_max)

        # Convert index to vectors using meshgrid
        indicies = np.array(np.meshgrid(i, j, k)).T.reshape(-1, 3)
        # Multiply integer vectors to get recipricol space vectors
        vecs = np.dot(indicies, [b1, b2, b3])
        # Calculate radii of all vectors
        radii = np.sqrt(np.einsum('ij,ij->i', vecs, vecs))

        # Yield based on radii
        for vec, r in zip(vecs, radii):
            if r < G_cut and r != 0:
                yield vec

    def generate_reciprocal_vectors_squared(a1, a2, a3, encut):
        """
        Generate reciprocal vector magnitudes within the cutoff along the specied
        lattice vectors.
        Args:
            a1: Lattice vector a (in Bohrs)
            a2: Lattice vector b (in Bohrs)
            a3: Lattice vector c (in Bohrs)
            encut: Reciprocal vector energy cutoff

        Returns:
            [[g1^2], [g2^2], ...] Square of reciprocal vectors (1/Bohr)^2
            determined by a1, a2, a3 and whose magntidue is less than gcut^2.
        """
        for vec in genrecip(a1, a2, a3, encut):
            yield np.dot(vec, vec)

    def closestsites(struct_blk, struct_def, pos):
        """
        Returns closest site to the input position
        for both bulk and defect structures
        Args:
            struct_blk: Bulk structure
            struct_def: Defect structure
            pos: Position
        Return: (site object, dist, index)
        """
        blk_close_sites = struct_blk.get_sites_in_sphere(pos, 5, include_index=True)
        blk_close_sites.sort(key=lambda x: x[1])
        def_close_sites = struct_def.get_sites_in_sphere(pos, 5, include_index=True)
        def_close_sites.sort(key=lambda x: x[1])

        return blk_close_sites[0], def_close_sites[0]

    class StructureMotifInterstitial:
        """
        Generate interstitial sites at positions
        where the interstitialcy is coordinated by nearest neighbors
        in a way that resembles basic structure motifs
        (e.g., tetrahedra, octahedra).  The algorithm is called InFiT
        (Interstitialcy Finding Tool), it was introducted by
        Nils E. R. Zimmermann, Matthew K. Horton, Anubhav Jain,
        and Maciej Haranczyk (Front. Mater., 4, 34, 2017),
        and it is used by the Python Charged Defect Toolkit
        (PyCDT: D. Broberg et al., Comput. Phys. Commun., in press, 2018).
        """

        def __init__(self,
                     struct,
                     inter_elem,
                     motif_types=("tetrahedral", "octahedral"),
                     op_threshs=(0.3, 0.5),
                     dl=0.2,
                     doverlap=1,
                     facmaxdl=1.01,
                     verbose=False):
            """
            Generates symmetrically distinct interstitial sites at positions
            where the interstitial is coordinated by nearest neighbors
            in a pattern that resembles a supported structure motif
            (e.g., tetrahedra, octahedra).

            Args:
                struct (Structure): input structure for which symmetrically
                    distinct interstitial sites are to be found.
                inter_elem (string): element symbol of desired interstitial.
                motif_types ([string]): list of structure motif types that are
                    to be considered.  Permissible types are:
                    tet (tetrahedron), oct (octahedron).
                op_threshs ([float]): threshold values for the underlying order
                    parameters to still recognize a given structural motif
                    (i.e., for an OP value >= threshold the coordination pattern
                    match is positive, for OP < threshold the match is
                    negative.
                dl (float): grid fineness in Angstrom.  The input
                    structure is divided into a grid of dimension
                    a/dl x b/dl x c/dl along the three crystallographic
                    directions, with a, b, and c being the lengths of
                    the three lattice vectors of the input unit cell.
                doverlap (float): distance that is considered
                    to flag an overlap between any trial interstitial site
                    and a host atom.
                facmaxdl (float): factor to be multiplied with the maximum grid
                    width that is then used as a cutoff distance for the
                    clustering prune step.
                verbose (bool): flag indicating whether (True) or not (False;
                    default) to print additional information to screen.
            """
            # Initialize interstitial finding.
            self._structure = struct.copy()
            self._motif_types = motif_types[:]
            if len(self._motif_types) == 0:
                raise RuntimeError("no motif types provided.")
            self._op_threshs = op_threshs[:]
            self.cn_motif_lostop = {}
            self.target_cns = []
            for motif in self._motif_types:
                if motif not in list(motif_cn_op.keys()):
                    raise RuntimeError("unsupported motif type: {}.".format(motif))
                cn = int(motif_cn_op[motif]['cn'])
                if cn not in self.target_cns:
                    self.target_cns.append(cn)
                if cn not in list(self.cn_motif_lostop.keys()):
                    self.cn_motif_lostop[cn] = {}
                tmp_optype = motif_cn_op[motif]['optype']
                if tmp_optype == 'tet_max':
                    tmp_optype = 'tet'
                if tmp_optype == 'oct_max':
                    tmp_optype = 'oct'
                self.cn_motif_lostop[cn][motif] = LocalStructOrderParams(
                    [tmp_optype], parameters=[motif_cn_op[motif]['params']],
                    cutoff=-10.0)
            self._dl = dl
            self._defect_sites = []
            self._defect_types = []
            self._defect_site_multiplicity = []
            self._defect_cns = []
            self._defect_opvals = []

            rots, trans = SpacegroupAnalyzer(struct)._get_symmetry()
            nbins = [int(struct.lattice.a / dl), int(struct.lattice.b / dl),
                     int(struct.lattice.c / dl)]
            dls = [
                struct.lattice.a / float(nbins[0]),
                struct.lattice.b / float(nbins[1]),
                struct.lattice.c / float(nbins[2])
            ]
            maxdl = max(dls)
            if verbose:
                print("Grid size: {} {} {}".format(nbins[0], nbins[1], nbins[2]))
                print("dls: {} {} {}".format(dls[0], dls[1], dls[2]))
            struct_w_inter = struct.copy()
            struct_w_inter.append(inter_elem, [0, 0, 0])
            natoms = len(list(struct_w_inter.sites))
            trialsites = []

            # Build index list
            i = np.arange(0, nbins[0]) + 0.5
            j = np.arange(0, nbins[1]) + 0.5
            k = np.arange(0, nbins[2]) + 0.5

            # Convert index to vectors using meshgrid
            indicies = np.array(np.meshgrid(i, j, k)).T.reshape(-1, 3)
            # Multiply integer vectors to get recipricol space vectors
            vecs = np.multiply(indicies, np.divide(1, nbins))

            # Loop over trial positions that are based on a regular
            # grid in fractional coordinate space
            # within the unit cell.
            for vec in vecs:
                struct_w_inter.replace(natoms - 1, inter_elem, coords=vec,
                                       coords_are_cartesian=False)
                if len(struct_w_inter.get_sites_in_sphere(
                        struct_w_inter.sites[natoms - 1].coords, doverlap)) == 1:
                    neighs_images_weigths = MinimumDistanceNN(tol=0.8,
                                                              cutoff=6).get_nn_info(
                        struct_w_inter, natoms - 1)
                    neighs_images_weigths_sorted = sorted(neighs_images_weigths,
                                                          key=lambda x: x['weight'],
                                                          reverse=True)
                    for nsite in range(1, len(neighs_images_weigths_sorted) + 1):
                        if nsite not in self.target_cns:
                            continue

                        allsites = [neighs_images_weigths_sorted[i]['site'] for i in
                                    range(nsite)]
                        indices_neighs = [i for i in range(len(allsites))]
                        allsites.append(struct_w_inter.sites[natoms - 1])
                        for mot, ops in self.cn_motif_lostop[nsite].items():
                            opvals = ops.get_order_parameters(
                                allsites, len(allsites) - 1,
                                indices_neighs=indices_neighs)
                            if opvals[0] > op_threshs[motif_types.index(mot)]:
                                cns = {}
                                for isite in range(nsite):
                                    site = neighs_images_weigths_sorted[isite][
                                        'site']
                                    if isinstance(site.specie, Element):
                                        elem = site.specie.symbol
                                    else:
                                        elem = site.specie.element.symbol
                                    if elem in list(cns.keys()):
                                        cns[elem] = cns[elem] + 1
                                    else:
                                        cns[elem] = 1
                                trialsites.append({
                                    "mtype": mot,
                                    "opval": opvals[0],
                                    "coords": struct_w_inter.sites[
                                                  natoms - 1].coords[:],
                                    "fracs": vec,
                                    "cns": dict(cns)
                                })
                                break

            # Prune list of trial sites by clustering and find the site
            # with the largest order parameter value in each cluster.
            nintersites = len(trialsites)
            unique_motifs = []
            for ts in trialsites:
                if ts["mtype"] not in unique_motifs:
                    unique_motifs.append(ts["mtype"])
            labels = {}
            connected = []
            for i in range(nintersites):
                connected.append([])
                for j in range(nintersites):
                    dist, image = struct_w_inter.lattice.get_distance_and_image(
                        trialsites[i]["fracs"],
                        trialsites[j]["fracs"])
                    connected[i].append(
                        True if dist < (maxdl * facmaxdl) else False)
            include = []
            for motif in unique_motifs:
                labels[motif] = []
                for i, ts in enumerate(trialsites):
                    labels[motif].append(i if ts["mtype"] == motif else -1)
                change = True
                while change:
                    change = False
                    for i in range(nintersites - 1):
                        if change:
                            break
                        if labels[motif][i] == -1:
                            continue
                        for j in range(i + 1, nintersites):
                            if labels[motif][j] == -1:
                                continue
                            if connected[i][j] and labels[motif][i] != \
                                    labels[motif][j]:
                                if labels[motif][i] < labels[motif][j]:
                                    labels[motif][j] = labels[motif][i]
                                else:
                                    labels[motif][i] = labels[motif][j]
                                change = True
                                break
                unique_ids = []
                for l in labels[motif]:
                    if l != -1 and l not in unique_ids:
                        unique_ids.append(l)
                if verbose:
                    print("unique_ids {} {}".format(motif, unique_ids))
                for uid in unique_ids:
                    maxq = 0.0
                    imaxq = -1
                    for i in range(nintersites):
                        if labels[motif][i] == uid:
                            if imaxq < 0 or trialsites[i]["opval"] > maxq:
                                imaxq = i
                                maxq = trialsites[i]["opval"]
                    include.append(imaxq)

            # Prune by symmetry.
            multiplicity = {}
            discard = []
            for motif in unique_motifs:
                discard_motif = []
                for indi, i in enumerate(include):
                    if trialsites[i]["mtype"] != motif or \
                            i in discard_motif:
                        continue
                    multiplicity[i] = 1
                    symposlist = [
                        trialsites[i]["fracs"].dot(np.array(m, dtype=float)) for m
                        in rots]
                    for t in trans:
                        symposlist.append(trialsites[i]["fracs"] + np.array(t))
                    for indj in range(indi + 1, len(include)):
                        j = include[indj]
                        if trialsites[j]["mtype"] != motif or \
                                j in discard_motif:
                            continue
                        for sympos in symposlist:
                            dist, image = struct.lattice.get_distance_and_image(
                                sympos, trialsites[j]["fracs"])
                            if dist < maxdl * facmaxdl:
                                discard_motif.append(j)
                                multiplicity[i] += 1
                                break
                for i in discard_motif:
                    if i not in discard:
                        discard.append(i)

            if verbose:
                print("Initial trial sites: {}\nAfter clustering: {}\n"
                      "After symmetry pruning: {}".format(len(trialsites),
                                                          len(include),
                                                          len(include) - len(
                                                              discard)))
            for i in include:
                if i not in discard:
                    self._defect_sites.append(
                        PeriodicSite(
                            Element(inter_elem),
                            trialsites[i]["fracs"],
                            self._structure.lattice,
                            to_unit_cell=False,
                            coords_are_cartesian=False,
                            properties=None))
                    self._defect_types.append(trialsites[i]["mtype"])
                    self._defect_cns.append(trialsites[i]["cns"])
                    self._defect_site_multiplicity.append(multiplicity[i])
                    self._defect_opvals.append(trialsites[i]["opval"])

        def enumerate_defectsites(self):
            """
            Get all defect sites.

            Returns:
                defect_sites ([PeriodicSite]): list of periodic sites
                        representing the interstitials.
            """
            return self._defect_sites

        def get_motif_type(self, i):
            """
            Get the motif type of defect with index i (e.g., "tet").

            Returns:
                motif (string): motif type.
            """
            return self._defect_types[i]

        def get_defectsite_multiplicity(self, n):
            """
            Returns the symmtric multiplicity of the defect site at the index.
            """
            return self._defect_site_multiplicity[n]

        def get_coordinating_elements_cns(self, i):
            """
            Get element-specific coordination numbers of defect with index i.

            Returns:
                elem_cn (dict): dictionary storing the coordination numbers (int)
                        with string representation of elements as keys.
                        (i.e., {elem1 (string): cn1 (int), ...}).
            """
            return self._defect_cns[i]

        def get_op_value(self, i):
            """
            Get order-parameter value of defect with index i.

            Returns:
                opval (float): OP value.
            """
            return self._defect_opvals[i]

        def make_supercells_with_defects(self, scaling_matrix):
            """
            Generate a sequence of supercells
            in which each supercell contains a single interstitial,
            except for the first supercell in the sequence
            which is a copy of the defect-free input structure.

            Args:
                scaling_matrix (3x3 integer array): scaling matrix
                    to transform the lattice vectors.
            Returns:
                scs ([Structure]): sequence of supercells.

            """
            scs = []
            sc = self._structure.copy()
            sc.make_supercell(scaling_matrix)
            scs.append(sc)
            for ids, defect_site in enumerate(self._defect_sites):
                sc_with_inter = sc.copy()
                sc_with_inter.append(
                    defect_site.species_string,
                    defect_site.frac_coords,
                    coords_are_cartesian=False,
                    validate_proximity=False,
                    properties=None)
                if not sc_with_inter:
                    raise RuntimeError(
                        "could not generate supercell with" " interstitial {}".format(
                            ids + 1))
                scs.append(sc_with_inter.copy())
            return scs

    class TopographyAnalyzer:
        """
        This is a generalized module to perform topological analyses of a crystal
        structure using Voronoi tessellations. It can be used for finding potential
        interstitial sites. Applications including using these sites for
        inserting additional atoms or for analyzing diffusion pathways.

        Note that you typically want to do some preliminary postprocessing after
        the initial construction. The initial construction will create a lot of
        points, especially for determining potential insertion sites. Some helper
        methods are available to perform aggregation and elimination of nodes. A
        typical use is something like::

            a = TopographyAnalyzer(structure, ["O"], ["P"])
            a.cluster_nodes()
            a.remove_collisions()
        """

        def __init__(self,
                     structure,
                     framework_ions,
                     cations,
                     tol=0.0001,
                     max_cell_range=1,
                     check_volume=True,
                     constrained_c_frac=0.5,
                     thickness=0.5):
            """
            Init.

            Args:
                structure (Structure): An initial structure.
                framework_ions ([str]): A list of ions to be considered as a
                    framework. Typically, this would be all anion species. E.g.,
                    ["O", "S"].
                cations ([str]): A list of ions to be considered as non-migrating
                    cations. E.g., if you are looking at Li3PS4 as a Li
                    conductor, Li is a mobile species. Your cations should be [
                    "P"]. The cations are used to exclude polyhedra from
                    diffusion analysis since those polyhedra are already occupied.
                tol (float): A tolerance distance for the analysis, used to
                    determine if something are actually periodic boundary images of
                    each other. Default is usually fine.
                max_cell_range (int): This is the range of periodic images to
                    construct the Voronoi tessellation. A value of 1 means that we
                    include all points from (x +- 1, y +- 1, z+- 1) in the
                    voronoi construction. This is because the Voronoi poly
                    extends beyond the standard unit cell because of PBC.
                    Typically, the default value of 1 works fine for most
                    structures and is fast. But for really small unit
                    cells with high symmetry, you may need to increase this to 2
                    or higher.
                check_volume (bool): Set False when ValueError always happen after
                    tuning tolerance.
                constrained_c_frac (float): Constraint the region where users want
                    to do Topology analysis the default value is 0.5, which is the
                    fractional coordinate of the cell
                thickness (float): Along with constrained_c_frac, limit the
                    thickness of the regions where we want to explore. Default is
                    0.5, which is mapping all the site of the unit cell.

            """
            self.structure = structure
            self.framework_ions = set([get_el_sp(sp) for sp in framework_ions])
            self.cations = set([get_el_sp(sp) for sp in cations])

            # Let us first map all sites to the standard unit cell, i.e.,
            # 0 ≤ coordinates < 1.
            # structure = Structure.from_sites(structure, to_unit_cell=True)
            # lattice = structure.lattice

            # We could constrain the region where we want to dope/explore by setting
            # the value of constrained_c_frac and thickness. The default mode is
            # mapping all sites to the standard unit cell
            s = structure.copy()
            constrained_sites = []
            for i, site in enumerate(s):
                if site.frac_coords[2] >= constrained_c_frac - thickness and \
                        site.frac_coords[
                            2] <= constrained_c_frac + thickness:
                    constrained_sites.append(site)
            structure = Structure.from_sites(sites=constrained_sites)
            lattice = structure.lattice

            # Divide the sites into framework and non-framework sites.
            framework = []
            non_framework = []
            for site in structure:
                if self.framework_ions.intersection(site.species.keys()):
                    framework.append(site)
                else:
                    non_framework.append(site)

            # We construct a supercell series of coords. This is because the
            # Voronoi polyhedra can extend beyond the standard unit cell. Using a
            # range of -2, -1, 0, 1 should be fine.
            coords = []
            cell_range = list(range(-max_cell_range, max_cell_range + 1))
            for shift in itertools.product(cell_range, cell_range, cell_range):
                for site in framework:
                    shifted = site.frac_coords + shift
                    coords.append(lattice.get_cartesian_coords(shifted))

            # Perform the voronoi tessellation.
            voro = Voronoi(coords)

            # Store a mapping of each voronoi node to a set of points.
            node_points_map = defaultdict(set)
            for pts, vs in voro.ridge_dict.items():
                for v in vs:
                    node_points_map[v].update(pts)

            logger.debug("%d total Voronoi vertices" % len(voro.vertices))

            # Vnodes store all the valid voronoi polyhedra. Cation vnodes store
            # the voronoi polyhedra that are already occupied by existing cations.
            vnodes = []
            cation_vnodes = []

            def get_mapping(poly):
                """
                Helper function to check if a vornoi poly is a periodic image
                of one of the existing voronoi polys.
                """
                for v in vnodes:
                    if v.is_image(poly, tol):
                        return v
                return None

            # Filter all the voronoi polyhedra so that we only consider those
            # which are within the unit cell.
            for i, vertex in enumerate(voro.vertices):
                if i == 0:
                    continue
                fcoord = lattice.get_fractional_coords(vertex)
                poly = VoronoiPolyhedron(lattice, fcoord, node_points_map[i],
                                         coords, i)
                if np.all([-tol <= c < 1 + tol for c in fcoord]):
                    if len(vnodes) == 0:
                        vnodes.append(poly)
                    else:
                        ref = get_mapping(poly)
                        if ref is None:
                            vnodes.append(poly)

            logger.debug("%d voronoi vertices in cell." % len(vnodes))

            # Eliminate all voronoi nodes which are closest to existing cations.
            if len(cations) > 0:
                cation_coords = [
                    site.frac_coords for site in non_framework if
                    self.cations.intersection(site.species.keys())
                ]

                vertex_fcoords = [v.frac_coords for v in vnodes]
                dist_matrix = lattice.get_all_distances(cation_coords,
                                                        vertex_fcoords)
                indices = \
                    np.where(dist_matrix == np.min(dist_matrix, axis=1)[:, None])[1]
                cation_vnodes = [v for i, v in enumerate(vnodes) if i in indices]
                vnodes = [v for i, v in enumerate(vnodes) if i not in indices]

            logger.debug("%d vertices in cell not with cation." % len(vnodes))
            self.coords = coords
            self.vnodes = vnodes
            self.cation_vnodes = cation_vnodes
            self.framework = framework
            self.non_framework = non_framework
            if check_volume:
                self.check_volume()

        def check_volume(self):
            """
            Basic check for volume of all voronoi poly sum to unit cell volume
            Note that this does not apply after poly combination.
            """
            vol = sum((v.volume for v in self.vnodes)) + sum(
                (v.volume for v in self.cation_vnodes))
            if abs(vol - self.structure.volume) > 1e-8:
                raise ValueError(
                    "Sum of voronoi volumes is not equal to original volume of "
                    "structure! This may lead to inaccurate results. You need to "
                    "tweak the tolerance and max_cell_range until you get a "
                    "correct mapping.")

        def cluster_nodes(self, tol=0.2):
            """
            Cluster nodes that are too close together using a tol.

            Args:
                tol (float): A distance tolerance. PBC is taken into account.
            """
            lattice = self.structure.lattice

            vfcoords = [v.frac_coords for v in self.vnodes]

            # Manually generate the distance matrix (which needs to take into
            # account PBC.
            dist_matrix = np.array(lattice.get_all_distances(vfcoords, vfcoords))
            dist_matrix = (dist_matrix + dist_matrix.T) / 2
            for i in range(len(dist_matrix)):
                dist_matrix[i, i] = 0
            condensed_m = squareform(dist_matrix)
            z = linkage(condensed_m)
            cn = fcluster(z, tol, criterion="distance")
            merged_vnodes = []
            for n in set(cn):
                poly_indices = set()
                frac_coords = []
                for i, j in enumerate(np.where(cn == n)[0]):
                    poly_indices.update(self.vnodes[j].polyhedron_indices)
                    if i == 0:
                        frac_coords.append(self.vnodes[j].frac_coords)
                    else:
                        fcoords = self.vnodes[j].frac_coords
                        # We need the image to combine the frac_coords properly.
                        d, image = lattice.get_distance_and_image(frac_coords[0],
                                                                  fcoords)
                        frac_coords.append(fcoords + image)
                merged_vnodes.append(
                    VoronoiPolyhedron(lattice, np.average(frac_coords, axis=0),
                                      poly_indices, self.coords))
            self.vnodes = merged_vnodes
            logger.debug("%d vertices after combination." % len(self.vnodes))

        def remove_collisions(self, min_dist=0.5):
            """
            Remove vnodes that are too close to existing atoms in the structure

            Args:
                min_dist(float): The minimum distance that a vertex needs to be
                    from existing atoms.
            """
            vfcoords = [v.frac_coords for v in self.vnodes]
            sfcoords = self.structure.frac_coords
            dist_matrix = self.structure.lattice.get_all_distances(vfcoords,
                                                                   sfcoords)
            all_dist = np.min(dist_matrix, axis=1)
            new_vnodes = []
            for i, v in enumerate(self.vnodes):
                if all_dist[i] > min_dist:
                    new_vnodes.append(v)
            self.vnodes = new_vnodes

        def get_structure_with_nodes(self):
            """
            Get the modified structure with the voronoi nodes inserted. The
            species is set as a DummySpecie X.
            """
            new_s = Structure.from_sites(self.structure)
            for v in self.vnodes:
                new_s.append("X", v.frac_coords)
            return new_s

        def print_stats(self):
            """
            Print stats such as the MSE dist.
            """
            latt = self.structure.lattice

            def get_min_dist(fcoords):
                n = len(fcoords)
                dist = latt.get_all_distances(fcoords, fcoords)
                all_dist = [dist[i, j] for i in range(n) for j in range(i + 1, n)]
                return min(all_dist)

            voro = [s[1] for s in self.vertices]
            print("Min dist between voronoi vertices centers = %.4f" % get_min_dist(
                voro))

            def get_non_framework_dist(fcoords):
                cations = [site.frac_coords for site in self.non_framework]
                dist_matrix = latt.get_all_distances(cations, fcoords)
                min_dist = np.min(dist_matrix, axis=1)
                if len(cations) != len(min_dist):
                    raise Exception("Could not calculate distance to all cations")
                return np.linalg.norm(min_dist), min(min_dist), max(min_dist)

            print(len(self.non_framework))
            print("MSE dist voro = %s" % str(get_non_framework_dist(voro)))

        def write_topology(self, fname="Topo.cif"):
            """
            Write topology to a file.

            :param fname: Filename
            """
            new_s = Structure.from_sites(self.structure)
            for v in self.vnodes:
                new_s.append("Mg", v.frac_coords)
            new_s.to(filename=fname)

        def analyze_symmetry(self, tol):
            """
            :param tol: Tolerance for SpaceGroupAnalyzer
            :return: List
            """
            s = Structure.from_sites(self.framework)
            site_to_vindex = {}
            for i, v in enumerate(self.vnodes):
                s.append("Li", v.frac_coords)
                site_to_vindex[s[-1]] = i

            print(len(s))
            finder = SpacegroupAnalyzer(s, tol)
            print(finder.get_space_group_operations())
            symm_structure = finder.get_symmetrized_structure()
            print(len(symm_structure.equivalent_sites))
            return [[site_to_vindex[site]
                     for site in sites]
                    for sites in symm_structure.equivalent_sites
                    if sites[0].specie.symbol == "Li"]

        def vtk(self):
            """
            Show VTK visualization.
            """
            if StructureVis is None:
                raise NotImplementedError("vtk must be present to view.")
            lattice = self.structure.lattice
            vis = StructureVis()
            vis.set_structure(Structure.from_sites(self.structure))
            for v in self.vnodes:
                vis.add_site(PeriodicSite("K", v.frac_coords, lattice))
                vis.add_polyhedron(
                    [PeriodicSite("S", c, lattice, coords_are_cartesian=True) for c
                     in v.polyhedron_coords],
                    PeriodicSite("Na", v.frac_coords, lattice),
                    color="element",
                    draw_edges=True,
                    edges_color=(0, 0, 0))
            vis.show()

    class VoronoiPolyhedron:
        """
        Convenience container for a voronoi point in PBC and its associated polyhedron.
        """

        def __init__(self, lattice, frac_coords, polyhedron_indices, all_coords,
                     name=None):
            """
            :param lattice:
            :param frac_coords:
            :param polyhedron_indices:
            :param all_coords:
            :param name:
            """
            self.lattice = lattice
            self.frac_coords = frac_coords
            self.polyhedron_indices = polyhedron_indices
            self.polyhedron_coords = np.array(all_coords)[list(polyhedron_indices), :]
            self.name = name

        def is_image(self, poly, tol):
            """
            :param poly: VoronoiPolyhedron
            :param tol: Coordinate tolerance.
            :return: Whether a poly is an image of the current one.
            """
            frac_diff = pbc_diff(poly.frac_coords, self.frac_coords)
            if not np.allclose(frac_diff, [0, 0, 0], atol=tol):
                return False
            to_frac = self.lattice.get_fractional_coords
            for c1 in self.polyhedron_coords:
                found = False
                for c2 in poly.polyhedron_coords:
                    d = pbc_diff(to_frac(c1), to_frac(c2))
                    if not np.allclose(d, [0, 0, 0], atol=tol):
                        found = True
                        break
                if not found:
                    return False
            return True

        @property
        def coordination(self):
            """
            :return: Coordination number
            """
            return len(self.polyhedron_indices)

        @property
        def volume(self):
            """
            :return: Volume
            """
            return calculate_vol(self.polyhedron_coords)

        def __str__(self):
            return "Voronoi polyhedron %s" % self.name

    class ChargeDensityAnalyzer:
        """
        Analyzer to find potential interstitial sites based on charge density. The
        `total` charge density is used.
        """

        def __init__(self, chgcar):
            """
            Initialization.

            Args:
                chgcar (pmg.Chgcar): input Chgcar object.
            """
            self.chgcar = chgcar
            self.structure = chgcar.structure
            self.extrema_coords = []  # list of frac_coords of local extrema
            self.extrema_type = None  # "local maxima" or "local minima"
            self._extrema_df = None  # extrema frac_coords - chg density table
            self._charge_distribution_df = None  # frac_coords - chg density table

        @classmethod
        def from_file(cls, chgcar_filename):
            """
            Init from a CHGCAR.

            :param chgcar_filename:
            :return:
            """
            chgcar = Chgcar.from_file(chgcar_filename)
            return cls(chgcar=chgcar)

        @property
        def charge_distribution_df(self):
            """
            :return: Charge distribution.
            """
            if self._charge_distribution_df is None:
                return self._get_charge_distribution_df()
            else:
                return self._charge_distribution_df

        @property
        def extrema_df(self):
            """
            :return: The extrema in charge density.
            """
            if self.extrema_type is None:
                logger.warning(
                    "Please run ChargeDensityAnalyzer.get_local_extrema first!")
            return self._extrema_df

        def _get_charge_distribution_df(self):
            """
            Return a complete table of fractional coordinates - charge density.
            """
            # Fraction coordinates and corresponding indices
            axis_grid = np.array([np.array(self.chgcar.get_axis_grid(i)) /
                                  self.structure.lattice.abc[i] for i in range(3)])
            axis_index = np.array([range(len(axis_grid[i])) for i in range(3)])

            data = {}

            for index in itertools.product(*axis_index):
                a, b, c = index
                f_coords = (axis_grid[0][a], axis_grid[1][b], axis_grid[2][c])
                data[f_coords] = self.chgcar.data["total"][a][b][c]

            # Fraction coordinates - charge density table
            df = pd.Series(data).reset_index()
            df.columns = ['a', 'b', 'c', 'Charge Density']
            self._charge_distribution_df = df

            return df

        def _update_extrema(self, f_coords, extrema_type, threshold_frac=None,
                            threshold_abs=None):
            """Update _extrema_df, extrema_type and extrema_coords"""

            if threshold_frac is not None:
                if threshold_abs is not None:
                    logger.warning(  # Exit if both filter are set
                        "Filter can be either threshold_frac or threshold_abs!")
                    return
                if threshold_frac > 1 or threshold_frac < 0:
                    raise Exception("threshold_frac range is [0, 1]!")

            # Return empty result if coords list is empty
            if len(f_coords) == 0:
                df = pd.DataFrame({}, columns=['A', 'B', 'C', "Chgcar"])
                self._extrema_df = df
                self.extrema_coords = []
                logger.info("Find {} {}.".format(len(df), extrema_type))
                return

            data = {}
            unit = 1 / np.array(self.chgcar.dim)  # pixel along a, b, c

            for fc in f_coords:
                a, b, c = tuple(map(int, fc / unit))
                data[tuple(fc)] = self.chgcar.data["total"][a][b][c]

            df = pd.Series(data).reset_index()
            df.columns = ['a', 'b', 'c', 'Charge Density']
            ascending = (extrema_type == "local minima")

            if threshold_abs is None:
                threshold_frac = threshold_frac \
                    if threshold_frac is not None else 1.0
                num_extrema = int(threshold_frac * len(f_coords))
                df = df.sort_values(by="Charge Density", ascending=ascending)[
                     0:num_extrema]
                df.reset_index(drop=True, inplace=True)  # reset major index
            else:  # threshold_abs is set
                df = df.sort_values(by="Charge Density", ascending=ascending)
                df = df[df["Charge Density"] <= threshold_abs] if ascending \
                    else df[df["Charge Density"] >= threshold_abs]

            extrema_coords = []
            for row in df.iterrows():
                fc = np.array(row[1]["a":"c"])
                extrema_coords.append(fc)

            self._extrema_df = df
            self.extrema_type = extrema_type
            self.extrema_coords = extrema_coords
            logger.info("Find {} {}.".format(len(df), extrema_type))

        @requires(peak_local_max_found,
                  "get_local_extrema requires skimage.feature.peak_local_max module"
                  " to be installed. Please confirm your skimage installation.")
        def get_local_extrema(self, find_min=True, threshold_frac=None,
                              threshold_abs=None):
            """
            Get all local extrema fractional coordinates in charge density,
            searching for local minimum by default. Note that sites are NOT grouped
            symmetrically.

            Args:
                find_min (bool): True to find local minimum else maximum, otherwise
                    find local maximum.

                threshold_frac (float): optional fraction of extrema shown, which
                    returns `threshold_frac * tot_num_extrema` extrema fractional
                    coordinates based on highest/lowest intensity.

                    E.g. set 0.2 to show the extrema with 20% highest or lowest
                    intensity. Value range: 0 <= threshold_frac <= 1

                    Note that threshold_abs and threshold_frac should not set in the
                    same time.

                threshold_abs (float): optional filter. When searching for local
                    minima, intensity <= threshold_abs returns; when searching for
                    local maxima, intensity >= threshold_abs returns.

                    Note that threshold_abs and threshold_frac should not set in the
                    same time.

            Returns:
                extrema_coords (list): list of fractional coordinates corresponding
                    to local extrema.
            """
            sign, extrema_type = 1, "local maxima"

            if find_min:
                sign, extrema_type = -1, "local minima"

            # Make 3x3x3 supercell
            # This is a trick to resolve the periodical boundary issue.
            total_chg = sign * self.chgcar.data["total"]
            total_chg = np.tile(total_chg, reps=(3, 3, 3))
            coordinates = peak_local_max(total_chg, min_distance=1)

            # Remove duplicated sites introduced by supercell.
            f_coords = [coord / total_chg.shape * 3 for coord in coordinates]
            f_coords = [f - 1 for f in f_coords if
                        all(np.array(f) < 2) and all(np.array(f) >= 1)]

            # Update information
            self._update_extrema(f_coords, extrema_type,
                                 threshold_frac=threshold_frac,
                                 threshold_abs=threshold_abs)

            return self.extrema_coords

        def cluster_nodes(self, tol=0.2):
            """
            Cluster nodes that are too close together using a tol.

            Args:
                tol (float): A distance tolerance. PBC is taken into account.
            """
            lattice = self.structure.lattice
            vf_coords = self.extrema_coords

            if len(vf_coords) == 0:
                if self.extrema_type is None:
                    logger.warning(
                        "Please run ChargeDensityAnalyzer.get_local_extrema first!")
                    return
                new_f_coords = []
                self._update_extrema(new_f_coords, self.extrema_type)
                return new_f_coords

            # Manually generate the distance matrix (which needs to take into
            # account PBC.
            dist_matrix = np.array(lattice.get_all_distances(vf_coords, vf_coords))
            dist_matrix = (dist_matrix + dist_matrix.T) / 2

            for i in range(len(dist_matrix)):
                dist_matrix[i, i] = 0
            condensed_m = squareform(dist_matrix)
            z = linkage(condensed_m)
            cn = fcluster(z, tol, criterion="distance")
            merged_fcoords = []

            for n in set(cn):
                frac_coords = []
                for i, j in enumerate(np.where(cn == n)[0]):
                    if i == 0:
                        frac_coords.append(self.extrema_coords[j])
                    else:
                        f_coords = self.extrema_coords[j]
                        # We need the image to combine the frac_coords properly.
                        d, image = lattice.get_distance_and_image(frac_coords[0],
                                                                  f_coords)
                        frac_coords.append(f_coords + image)
                merged_fcoords.append(np.average(frac_coords, axis=0))

            merged_fcoords = [f - np.floor(f) for f in merged_fcoords]
            merged_fcoords = [f * (np.abs(f - 1) > 1E-15) for f in merged_fcoords]
            # the second line for fringe cases like
            # np.array([ 5.0000000e-01 -4.4408921e-17  5.0000000e-01])
            # where the shift to [0,1) does not work due to float precision
            self._update_extrema(merged_fcoords, extrema_type=self.extrema_type)
            logger.debug(
                "{} vertices after combination.".format(len(self.extrema_coords)))

        def remove_collisions(self, min_dist=0.5):
            """
            Remove predicted sites that are too close to existing atoms in the
            structure.

            Args:
                min_dist (float): The minimum distance (in Angstrom) that
                    a predicted site needs to be from existing atoms. A min_dist
                    with value <= 0 returns all sites without distance checking.
            """
            s_f_coords = self.structure.frac_coords
            f_coords = self.extrema_coords
            if len(f_coords) == 0:
                if self.extrema_type is None:
                    logger.warning(
                        "Please run ChargeDensityAnalyzer.get_local_extrema first!")
                    return
                new_f_coords = []
                self._update_extrema(new_f_coords, self.extrema_type)
                return new_f_coords

            dist_matrix = self.structure.lattice.get_all_distances(f_coords,
                                                                   s_f_coords)
            all_dist = np.min(dist_matrix, axis=1)
            new_f_coords = []

            for i, f in enumerate(f_coords):
                if all_dist[i] > min_dist:
                    new_f_coords.append(f)
            self._update_extrema(new_f_coords, self.extrema_type)

            return new_f_coords

        def get_structure_with_nodes(self, find_min=True, min_dist=0.5, tol=0.2,
                                     threshold_frac=None, threshold_abs=None):
            """
            Get the modified structure with the possible interstitial sites added.
            The species is set as a DummySpecie X.

            Args:
                find_min (bool): True to find local minimum else maximum, otherwise
                    find local maximum.

                min_dist (float): The minimum distance (in Angstrom) that
                    a predicted site needs to be from existing atoms. A min_dist
                    with value <= 0 returns all sites without distance checking.

                tol (float): A distance tolerance of nodes clustering that sites too
                    closed to other predicted sites will be merged. PBC is taken
                    into account.

                threshold_frac (float): optional fraction of extrema, which returns
                    `threshold_frac * tot_num_extrema` extrema fractional
                    coordinates based on highest/lowest intensity.

                    E.g. set 0.2 to insert DummySpecie atom at the extrema with 20%
                    highest or lowest intensity.
                    Value range: 0 <= threshold_frac <= 1

                    Note that threshold_abs and threshold_frac should not set in the
                    same time.

                threshold_abs (float): optional filter. When searching for local
                    minima, intensity <= threshold_abs returns; when searching for
                    local maxima, intensity >= threshold_abs returns.

                    Note that threshold_abs and threshold_frac should not set in the
                    same time.

            Returns:
                structure (Structure)
            """

            structure = self.structure.copy()
            self.get_local_extrema(find_min=find_min, threshold_frac=threshold_frac,
                                   threshold_abs=threshold_abs)

            self.remove_collisions(min_dist)
            self.cluster_nodes(tol=tol)
            for fc in self.extrema_coords:
                structure.append("X", fc)

            return structure

        def sort_sites_by_integrated_chg(self, r=0.4):
            """
            Get the average charge density around each local minima in the charge density
            and store the result in _extrema_df
            Args:
                r (float): radius of sphere around each site to evaluate the average
            """

            if self.extrema_type is None:
                self.get_local_extrema()
            int_den = []
            for isite in self.extrema_coords:
                mask = self._dist_mat(isite) < r
                vol_sphere = self.chgcar.structure.volume * (mask.sum() / self.chgcar.ngridpts)
                chg_in_sphere = np.sum(self.chgcar.data['total'] * mask) / mask.size / vol_sphere
                int_den.append(chg_in_sphere)
            self._extrema_df['avg_charge_den'] = int_den
            self._extrema_df.sort_values(by=['avg_charge_den'], inplace=True)
            self._extrema_df.reset_index(drop=True, inplace=True)

        def _dist_mat(self, pos_frac):
            # return a matrix that contains the distances
            aa = np.linspace(0, 1, len(self.chgcar.get_axis_grid(0)),
                             endpoint=False)
            bb = np.linspace(0, 1, len(self.chgcar.get_axis_grid(1)),
                             endpoint=False)
            cc = np.linspace(0, 1, len(self.chgcar.get_axis_grid(2)),
                             endpoint=False)
            AA, BB, CC = np.meshgrid(aa, bb, cc, indexing='ij')
            dist_from_pos = self.chgcar.structure.lattice.get_all_distances(
                fcoords1=np.vstack([AA.flatten(), BB.flatten(), CC.flatten()]).T,
                fcoords2=pos_frac)
            return dist_from_pos.reshape(AA.shape)

    def calculate_vol(coords):
        """
        Calculate volume given a set of coords.

        :param coords: List of coords.
        :return: Volume
        """
        if len(coords) == 4:
            coords_affine = np.ones((4, 4))
            coords_affine[:, 0:3] = np.array(coords)
            return abs(np.linalg.det(coords_affine)) / 6
        else:
            simplices = get_facets(coords, joggle=True)
            center = np.average(coords, axis=0)
            vol = 0
            for s in simplices:
                c = list(coords[i] for i in s)
                c.append(center)
                vol += calculate_vol(c)
            return vol

    def converge(f, step, tol, max_h):
        """
        simple newton iteration based convergence function
        """
        g = f(0)
        dx = 10000
        h = step
        while (dx > tol):
            g2 = f(h)
            dx = abs(g - g2)
            g = g2
            h += step

            if h > max_h:
                raise Exception("Did not converge before {}".format(h))
        return g

    def tune_for_gamma(lattice, epsilon):
        """
        This tunes the gamma parameter for Kumagai anisotropic
        Ewald calculation. Method is to find a gamma parameter which generates a similar
        number of reciprocal and real lattice vectors,
        given the suggested cut off radii by Kumagai and Oba
        """
        logger.debug("Converging for ewald parameter...")
        prec = 25  # a reasonable precision to tune gamma for

        gamma = (2 * np.average(lattice.abc)) ** (-1 / 2.)
        recip_set, _, real_set, _ = generate_R_and_G_vecs(gamma, prec, lattice, epsilon)
        recip_set = recip_set[0]
        real_set = real_set[0]

        logger.debug("First approach with gamma ={}\nProduced {} real vecs and {} recip "
                     "vecs.".format(gamma, len(real_set), len(recip_set)))

        while float(len(real_set)) / len(recip_set) > 1.05 or \
                float(len(recip_set)) / len(real_set) > 1.05:
            gamma *= (float(len(real_set)) / float(len(recip_set))) ** 0.17
            logger.debug("\tNot converged...Try modifying gamma to {}.".format(gamma))
            recip_set, _, real_set, _ = generate_R_and_G_vecs(gamma, prec, lattice, epsilon)
            recip_set = recip_set[0]
            real_set = real_set[0]
            logger.debug("Now have {} real vecs and {} recip vecs.".format(len(real_set), len(recip_set)))

        logger.debug("Converged with gamma = {}".format(gamma))

        return gamma

    def generate_R_and_G_vecs(gamma, prec_set, lattice, epsilon):
        """
        This returns a set of real and reciprocal lattice vectors
        (and real/recip summation values)
        based on a list of precision values (prec_set)

        gamma (float): Ewald parameter
        prec_set (list or number): for prec values to consider (20, 25, 30 are sensible numbers)
        lattice: Lattice object of supercell in question

        """
        if type(prec_set) != list:
            prec_set = [prec_set]

        [a1, a2, a3] = lattice.matrix  # Angstrom
        volume = lattice.volume
        [b1, b2, b3] = lattice.reciprocal_lattice.matrix  # 1/ Angstrom
        invepsilon = np.linalg.inv(epsilon)
        rd_epsilon = np.sqrt(np.linalg.det(epsilon))

        # generate reciprocal vector set (for each prec_set)
        recip_set = [[] for prec in prec_set]
        recip_summation_values = [0. for prec in prec_set]
        recip_cut_set = [(2 * gamma * prec) for prec in prec_set]

        i_max = int(math.ceil(max(recip_cut_set) / np.linalg.norm(b1)))
        j_max = int(math.ceil(max(recip_cut_set) / np.linalg.norm(b2)))
        k_max = int(math.ceil(max(recip_cut_set) / np.linalg.norm(b3)))
        for i in np.arange(-i_max, i_max + 1):
            for j in np.arange(-j_max, j_max + 1):
                for k in np.arange(-k_max, k_max + 1):
                    if not i and not j and not k:
                        continue
                    gvec = i * b1 + j * b2 + k * b3
                    normgvec = np.linalg.norm(gvec)
                    for recip_cut_ind, recip_cut in enumerate(recip_cut_set):
                        if normgvec <= recip_cut:
                            recip_set[recip_cut_ind].append(gvec)

                            Gdotdiel = np.dot(gvec, np.dot(epsilon, gvec))
                            summand = math.exp(-Gdotdiel / (4 * (gamma ** 2))) / Gdotdiel
                            recip_summation_values[recip_cut_ind] += summand

        recip_summation_values = np.array(recip_summation_values)
        recip_summation_values /= volume

        # generate real vector set (for each prec_set)
        real_set = [[] for prec in prec_set]
        real_summation_values = [0. for prec in prec_set]
        real_cut_set = [(prec / gamma) for prec in prec_set]

        i_max = int(math.ceil(max(real_cut_set) / np.linalg.norm(a1)))
        j_max = int(math.ceil(max(real_cut_set) / np.linalg.norm(a2)))
        k_max = int(math.ceil(max(real_cut_set) / np.linalg.norm(a3)))
        for i in np.arange(-i_max, i_max + 1):
            for j in np.arange(-j_max, j_max + 1):
                for k in np.arange(-k_max, k_max + 1):
                    rvec = i * a1 + j * a2 + k * a3
                    normrvec = np.linalg.norm(rvec)
                    for real_cut_ind, real_cut in enumerate(real_cut_set):
                        if normrvec <= real_cut:
                            real_set[real_cut_ind].append(rvec)
                            if normrvec > 1e-8:
                                sqrt_loc_res = np.sqrt(np.dot(rvec, np.dot(invepsilon, rvec)))
                                nmr = math.erfc(gamma * sqrt_loc_res)
                                real_summation_values[real_cut_ind] += nmr / sqrt_loc_res

        real_summation_values = np.array(real_summation_values)
        real_summation_values /= (4 * np.pi * rd_epsilon)

        return recip_set, recip_summation_values, real_set, real_summation_values