text: # 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 tesselation. 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
  
  
  def parse_defect_states(structure, defect_site, wavecar, procar,
                          sampling_radius = None):
      """
      This module parses Procar and Wavecar for defect localization.
       Algorithm:
          i) Find band indices of interest from wavecar data,
              defined as a window of +/- 10 bands away from the fermi level
          ii) Store eigenvalue data for each band
          iii) If no sampling_radius is defined, then determined the ws-radius and use as sampling_radius
          iii) Calculate percentage of single particle level contained within
  
          procar percentage contained within sampling_radius
          iv) For all bands which have caluclate wavecar percentage contained within radius
              (with charge center as starting point)
  
      Args:
          structure (Structure):
          defect_site (PeriodicSite):
          wavecar (Wavecar):
          procar (Procar):
          sampling_radius (float):
  
      Returns:
           dictionary of defect metadata for localization
      """
      # find interesting bands
      fermi_band_set = []
      if wavecar.spin == 1:
          for kpt_ind in range(wavecar.nk):
              for band_ind in range(wavecar.nb):
                  if not wavecar.band_energy[kpt_ind][band_ind][2]:
                      fermi_band_set.append(band_ind)
                      break
      elif wavecar.spin == 2:
          for spin_ind in range(len(wavecar.band_energy)):
              for kpt_ind in range(wavecar.nk):
                  for band_ind in range(wavecar.nb):
                      if not wavecar.band_energy[spin_ind][kpt_ind][band_ind][2]:
                          fermi_band_set.append(band_ind)
                          break
      fermi_band_set = list(set(fermi_band_set))
      sample_bands = np.arange(min(fermi_band_set) - 10, max(fermi_band_set) + 11)
      logger.info('sampling {} bands'.format(len(sample_bands)))
  
       # store global energy data about these bands
      store_eigen_dat = {spin_ind: {band_ind: [] for band_ind in sample_bands}
                         for spin_ind in range(wavecar.spin)}
      if wavecar.spin == 1:
          for kpt_ind in range(wavecar.nk):
              for band_ind in sample_bands:
                  occu = [wavecar.band_energy[kpt_ind][band_ind][0],
                          wavecar.band_energy[kpt_ind][band_ind][2]]
                  store_eigen_dat[0][band_ind].append(occu)
      elif wavecar.spin == 2:
          for spin_ind in range(wavecar.spin):
              for kpt_ind in range(wavecar.nk):
                  for band_ind in sample_bands:
                      occu = [wavecar.band_energy[spin_ind][kpt_ind][band_ind][0],
                              wavecar.band_energy[spin_ind][kpt_ind][band_ind][2]]
                      store_eigen_dat[spin_ind][band_ind].append(occu)
  
      if not sampling_radius:
          # get ws radius
          wz = structure.lattice.get_wigner_seitz_cell()
          dist = []
          for facet in wz:
              midpt = np.mean(np.array(facet), axis=0)
              dist.append(np.linalg.norm(midpt))
          sampling_radius = min(dist)
          logger.info('Sampling radius determined to be {}'.format(sampling_radius))
  
       # check procar to see bands which may be of interest
      followup_bands = []
      stored_procar_dat = {}
      for spinind, spinkey in enumerate(procar.data.keys()):
          stored_procar_dat[spinind] = []
          for kptindex in range(procar.nkpoints):
              stored_procar_dat[spinind].append({})
              for band_ind in sample_bands:
                  perc, rad_dat = procar.perc_contained_in_radius_from_site(structure, defect_site, sampling_radius,
                                                                   spinkey, kptindex, band_ind)
                  stored_procar_dat[spinind][kptindex].update(
                      {band_ind: {'perc': perc, 'rad_dat': rad_dat, 'eigen': store_eigen_dat[spinind][band_ind]}})
  
       # now use weighting to find bands with 70% localization overall
      for band_ind in sample_bands:
          for spinind in range(wavecar.spin):
              perc = 0.
              for kwt, kptindex in zip(procar.weights, range(procar.nkpoints)):
                  perc += stored_procar_dat[spinind][kptindex][band_ind]['perc'] * kwt
  
              if perc >= 0.7:
                  followup_bands.append(band_ind)
  
      followup_bands = list(set(followup_bands))
      followup_bands.sort()
  
      # run wavecar analysis on follow up bands
      followup_wf_parse_md = {}  # bandindex as key, spin as second key,
      perc90_local = {spin: [] for spin in range(wavecar.spin)}
      if len(followup_bands):
          logger.info('found {} bands for follow up\n\t{}'.format(len(followup_bands), followup_bands))
          for band_ind in followup_bands:
              followup_wf_parse_md[band_ind] = {}
              for spin in range(wavecar.spin):
                  print(band_ind, spin)
                  dat = wavecar.get_total_radial_distrib_from_coords(procar.weights, band_ind, spin, defect_site.coords,
                                                               coords_are_cartesian=True, reduce_size=True)
  
                  # quick check if this is 90% localized within the sampling radius
                  x = dat['tot'][0]  # [ reduced_x, reduced_y, percentage_y]
                  y = dat['tot'][2]
                  cont90rad = None
                  for yind, yval in enumerate(y):
                      if yval >= 0.9 and cont90rad is None:
                          cont90rad = x[yind]
  
                  followup_wf_parse_md[band_ind][spin] = {'rad_dist_data': dat,
                                                          'eigen': store_eigen_dat[spin][band_ind],
                                                          'cont90rad': cont90rad}
                  if cont90rad <= sampling_radius:
                      perc90_local[spin].append(band_ind)
  
      else:
          logger.info('found NO bands for follow up...')
  
      defect_out = {'localized_band_indices': perc90_local,
                    'stored_procar_dat': stored_procar_dat,
                    'sampling_radius': sampling_radius,
                    'followup_wf_parse': followup_wf_parse_md}
  
      return defect_out