pymatgen.analysis.structure_matcher — pymatgen 2020.7.3 documentation

# Source code for pymatgen.analysis.structure_matcher

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

    """
    This module provides classes to perform fitting of structures.
    """

    import numpy as np
    import itertools
    import abc

    from monty.json import MSONable
    from pymatgen.core import PeriodicSite
    from pymatgen.core.structure import Structure
    from pymatgen.core.lattice import Lattice
    from pymatgen.core.composition import Composition

    from pymatgen.core.periodic_table import get_el_sp
    from pymatgen.optimization.linear_assignment import LinearAssignment  # type: ignore
    from pymatgen.util.coord_cython import pbc_shortest_vectors, is_coord_subset_pbc  # type: ignore
    from pymatgen.util.coord import lattice_points_in_supercell
    from pymatgen.analysis.defects.core import Interstitial, \
        Defect, Vacancy, Substitution

    __author__ = "William Davidson Richards, Stephen Dacek, Shyue Ping Ong"
    __copyright__ = "Copyright 2011, The Materials Project"
    __version__ = "1.0"
    __maintainer__ = "William Davidson Richards"
    __email__ = "wrichard@mit.edu"
    __status__ = "Production"
    __date__ = "Dec 3, 2012"

    class AbstractComparator(MSONable, metaclass=abc.ABCMeta):
        """
        Abstract Comparator class. A Comparator defines how sites are compared in
        a structure.
        """

        @abc.abstractmethod
        def are_equal(self, sp1, sp2):
            """
            Defines how the species of two sites are considered equal. For
            example, one can consider sites to have the same species only when
            the species are exactly the same, i.e., Fe2+ matches Fe2+ but not
            Fe3+. Or one can define that only the element matters,
            and all oxidation state information are ignored.

            Args:
                sp1: First species. A dict of {specie/element: amt} as per the
                    definition in Site and PeriodicSite.
                sp2: Second species. A dict of {specie/element: amt} as per the
                    definition in Site and PeriodicSite.

            Returns:
                Boolean indicating whether species are considered equal.
            """
            return

        @abc.abstractmethod
        def get_hash(self, composition):
            """
            Defines a hash to group structures. This allows structures to be
            grouped efficiently for comparison. The hash must be invariant under
            supercell creation. (e.g. composition is not a good hash, but
            fractional_composition might be). Reduced formula is not a good formula,
            due to weird behavior with fractional occupancy.

            Composition is used here instead of structure because for anonymous
            matches it is much quicker to apply a substitution to a composition
            object than a structure object.

            Args:
                composition (Composition): composition of the structure

            Returns:
                A hashable object. Examples can be string formulas, integers etc.
            """
            return

        @classmethod
        def from_dict(cls, d):
            """
            :param d: Dict representation
            :return: Comparator.
            """
            for trans_modules in ['structure_matcher']:
                mod = __import__('pymatgen.analysis.' + trans_modules,
                                 globals(), locals(), [d['@class']], 0)
                if hasattr(mod, d['@class']):
                    trans = getattr(mod, d['@class'])
                    return trans()
            raise ValueError("Invalid Comparator dict")

        def as_dict(self):
            """
            :return: MSONable dict
            """
            return {"version": __version__, "@module": self.__class__.__module__,
                    "@class": self.__class__.__name__}

    class SpeciesComparator(AbstractComparator):
        """
        A Comparator that matches species exactly. The default used in
        StructureMatcher.
        """

        def are_equal(self, sp1, sp2):
            """
            True if species are exactly the same, i.e., Fe2+ == Fe2+ but not Fe3+.

            Args:
                sp1: First species. A dict of {specie/element: amt} as per the
                    definition in Site and PeriodicSite.
                sp2: Second species. A dict of {specie/element: amt} as per the
                    definition in Site and PeriodicSite.

            Returns:
                Boolean indicating whether species are equal.
            """
            return sp1 == sp2

        def get_hash(self, composition):
            """
            Returns: Fractional composition
            """
            return composition.fractional_composition

    class SpinComparator(AbstractComparator):
        """
        A Comparator that matches magnetic structures to their inverse spins.
        This comparator is primarily used to filter magnetically ordered
        structures with opposite spins, which are equivalent.
        """

        def are_equal(self, sp1, sp2):
            """
            True if species are exactly the same, i.e., Fe2+ == Fe2+ but not
            Fe3+. and the spins are reversed. i.e., spin up maps to spin down,
            and vice versa.

            Args:
                sp1: First species. A dict of {specie/element: amt} as per the
                    definition in Site and PeriodicSite.
                sp2: Second species. A dict of {specie/element: amt} as per the
                    definition in Site and PeriodicSite.

            Returns:
                Boolean indicating whether species are equal.
            """
            for s1 in sp1.keys():
                spin1 = getattr(s1, "spin", 0)
                oxi1 = getattr(s1, "oxi_state", 0)
                for s2 in sp2.keys():
                    spin2 = getattr(s2, "spin", 0)
                    oxi2 = getattr(s2, "oxi_state", 0)
                    if (s1.symbol == s2.symbol and oxi1 == oxi2 and
                            spin2 == -spin1):
                        break
                else:
                    return False
            return True

        def get_hash(self, composition):
            """
            Returns: Fractional composition
            """
            return composition.fractional_composition

    class ElementComparator(AbstractComparator):
        """
        A Comparator that matches elements. i.e. oxidation states are
        ignored.
        """

        def are_equal(self, sp1, sp2):
            """
            True if element:amounts are exactly the same, i.e.,
            oxidation state is not considered.

            Args:
                sp1: First species. A dict of {specie/element: amt} as per the
                    definition in Site and PeriodicSite.
                sp2: Second species. A dict of {specie/element: amt} as per the
                    definition in Site and PeriodicSite.

            Returns:
                Boolean indicating whether species are the same based on element
                and amounts.
            """
            comp1 = Composition(sp1)
            comp2 = Composition(sp2)
            return comp1.get_el_amt_dict() == comp2.get_el_amt_dict()

        def get_hash(self, composition):
            """
            Returns: Fractional element composition
            """
            return composition.element_composition.fractional_composition

    class FrameworkComparator(AbstractComparator):
        """
        A Comparator that matches sites, regardless of species.
        """

        def are_equal(self, sp1, sp2):
            """
            True if there are atoms on both sites.

            Args:
                sp1: First species. A dict of {specie/element: amt} as per the
                    definition in Site and PeriodicSite.
                sp2: Second species. A dict of {specie/element: amt} as per the
                    definition in Site and PeriodicSite.

            Returns:
                True always
            """
            return True

        def get_hash(self, composition):
            """
            No hash possible
            """
            return 1

    class OrderDisorderElementComparator(AbstractComparator):
        """
        A Comparator that matches sites, given some overlap in the element
        composition
        """

        def are_equal(self, sp1, sp2):
            """
            True if there is some overlap in composition between the species

            Args:
                sp1: First species. A dict of {specie/element: amt} as per the
                    definition in Site and PeriodicSite.
                sp2: Second species. A dict of {specie/element: amt} as per the
                    definition in Site and PeriodicSite.

            Returns:
                True always
            """
            set1 = set(sp1.elements)
            set2 = set(sp2.elements)
            return set1.issubset(set2) or set2.issubset(set1)

        def get_hash(self, composition):
            """
            Returns: Fractional composition
            """
            return composition.fractional_composition

    class OccupancyComparator(AbstractComparator):
        """
        A Comparator that matches occupancies on sites,
        irrespective of the species of those sites.
        """

        def are_equal(self, sp1, sp2):
            """
            Args:
                sp1: First species. A dict of {specie/element: amt} as per the
                    definition in Site and PeriodicSite.
                sp2: Second species. A dict of {specie/element: amt} as per the
                    definition in Site and PeriodicSite.

            Returns:
                True if sets of occupancies (amt) are equal on both sites.
            """
            set1 = set(sp1.element_composition.values())
            set2 = set(sp2.element_composition.values())
            if set1 == set2:
                return True
            else:
                return False

        def get_hash(self, composition):
            """
            :param composition: Composition.
            :return: 1. Difficult to define sensible hash
            """
            return 1

    class StructureMatcher(MSONable):
        """
        Class to match structures by similarity.

        Algorithm:

        1. Given two structures: s1 and s2
        2. Optional: Reduce to primitive cells.
        3. If the number of sites do not match, return False
        4. Reduce to s1 and s2 to Niggli Cells
        5. Optional: Scale s1 and s2 to same volume.
        6. Optional: Remove oxidation states associated with sites
        7. Find all possible lattice vectors for s2 within shell of ltol.
        8. For s1, translate an atom in the smallest set to the origin
        9. For s2: find all valid lattices from permutations of the list
           of lattice vectors (invalid if: det(Lattice Matrix) < half
           volume of original s2 lattice)
        10. For each valid lattice:

            a. If the lattice angles of are within tolerance of s1,
               basis change s2 into new lattice.
            b. For each atom in the smallest set of s2:

                i. Translate to origin and compare fractional sites in
                structure within a fractional tolerance.
                ii. If true:

                    ia. Convert both lattices to cartesian and place
                    both structures on an average lattice
                    ib. Compute and return the average and max rms
                    displacement between the two structures normalized
                    by the average free length per atom

                    if fit function called:
                        if normalized max rms displacement is less than
                        stol. Return True

                    if get_rms_dist function called:
                        if normalized average rms displacement is less
                        than the stored rms displacement, store and
                        continue. (This function will search all possible
                        lattices for the smallest average rms displacement
                        between the two structures)
        """

        def __init__(self, ltol=0.2, stol=0.3, angle_tol=5, primitive_cell=True,
                     scale=True, attempt_supercell=False, allow_subset=False,
                     comparator=SpeciesComparator(), supercell_size='num_sites',
                     ignored_species=None):
            """
            Args:
                ltol (float): Fractional length tolerance. Default is 0.2.
                stol (float): Site tolerance. Defined as the fraction of the
                    average free length per atom := ( V / Nsites ) ** (1/3)
                    Default is 0.3.
                angle_tol (float): Angle tolerance in degrees. Default is 5 degrees.
                primitive_cell (bool): If true: input structures will be reduced to
                    primitive cells prior to matching. Default to True.
                scale (bool): Input structures are scaled to equivalent volume if
                   true; For exact matching, set to False.
                attempt_supercell (bool): If set to True and number of sites in
                    cells differ after a primitive cell reduction (divisible by an
                    integer) attempts to generate a supercell transformation of the
                    smaller cell which is equivalent to the larger structure.
                allow_subset (bool): Allow one structure to match to the subset of
                    another structure. Eg. Matching of an ordered structure onto a
                    disordered one, or matching a delithiated to a lithiated
                    structure. This option cannot be combined with
                    attempt_supercell, or with structure grouping.
                comparator (Comparator): A comparator object implementing an equals
                    method that declares declaring equivalency of sites. Default is
                    SpeciesComparator, which implies rigid species
                    mapping, i.e., Fe2+ only matches Fe2+ and not Fe3+.

                    Other comparators are provided, e.g., ElementComparator which
                    matches only the elements and not the species.

                    The reason why a comparator object is used instead of
                    supplying a comparison function is that it is not possible to
                    pickle a function, which makes it otherwise difficult to use
                    StructureMatcher with Python's multiprocessing.
                supercell_size (str or list): Method to use for determining the
                    size of a supercell (if applicable). Possible values are
                    num_sites, num_atoms, volume, or an element or list of elements
                    present in both structures.
                ignored_species (list): A list of ions to be ignored in matching.
                    Useful for matching structures that have similar frameworks
                    except for certain ions, e.g., Li-ion intercalation frameworks.
                    This is more useful than allow_subset because it allows better
                    control over what species are ignored in the matching.
            """

            self.ltol = ltol
            self.stol = stol
            self.angle_tol = angle_tol
            self._comparator = comparator
            self._primitive_cell = primitive_cell
            self._scale = scale
            self._supercell = attempt_supercell
            self._supercell_size = supercell_size
            self._subset = allow_subset
            self._ignored_species = [] if ignored_species is None else \
                ignored_species[:]

        def _get_supercell_size(self, s1, s2):
            """
            Returns the supercell size, and whether the supercell should
            be applied to s1. If fu == 1, s1_supercell is returned as
            true, to avoid ambiguity.
            """
            if self._supercell_size == 'num_sites':
                fu = s2.num_sites / s1.num_sites
            elif self._supercell_size == 'num_atoms':
                fu = s2.composition.num_atoms / s1.composition.num_atoms
            elif self._supercell_size == 'volume':
                fu = s2.volume / s1.volume
            elif not isinstance(self._supercell_size, str):
                s1comp, s2comp = 0, 0
                for el in self._supercell_size:
                    el = get_el_sp(el)
                    s1comp += s1.composition[el]
                    s2comp += s2.composition[el]
                fu = s2comp / s1comp
            else:
                el = get_el_sp(self._supercell_size)
                if (el in s2.composition) and (el in s1.composition):
                    fu = s2.composition[el] / s1.composition[el]
                else:
                    raise ValueError('Invalid argument for supercell_size.')

            if fu < 2 / 3:
                return int(round(1 / fu)), False
            else:
                return int(round(fu)), True

        def _get_lattices(self, target_lattice, s, supercell_size=1):
            """
            Yields lattices for s with lengths and angles close to the
            lattice of target_s. If supercell_size is specified, the
            returned lattice will have that number of primitive cells
            in it

            Args:
                s, target_s: Structure objects
            """
            lattices = s.lattice.find_all_mappings(
                target_lattice, ltol=self.ltol, atol=self.angle_tol,
                skip_rotation_matrix=True)
            for l, _, scale_m in lattices:
                if abs(abs(np.linalg.det(scale_m)) - supercell_size) < 0.5:
                    yield l, scale_m

        def _get_supercells(self, struct1, struct2, fu, s1_supercell):
            """
            Computes all supercells of one structure close to the lattice of the
            other
            if s1_supercell == True, it makes the supercells of struct1, otherwise
            it makes them of s2

            yields: s1, s2, supercell_matrix, average_lattice, supercell_matrix
            """

            def av_lat(l1, l2):
                params = (np.array(l1.parameters) +
                          np.array(l2.parameters)) / 2
                return Lattice.from_parameters(*params)

            def sc_generator(s1, s2):
                s2_fc = np.array(s2.frac_coords)
                if fu == 1:
                    cc = np.array(s1.cart_coords)
                    for l, sc_m in self._get_lattices(s2.lattice, s1, fu):
                        fc = l.get_fractional_coords(cc)
                        fc -= np.floor(fc)
                        yield fc, s2_fc, av_lat(l, s2.lattice), sc_m
                else:
                    fc_init = np.array(s1.frac_coords)
                    for l, sc_m in self._get_lattices(s2.lattice, s1, fu):
                        fc = np.dot(fc_init, np.linalg.inv(sc_m))
                        lp = lattice_points_in_supercell(sc_m)
                        fc = (fc[:, None, :] + lp[None, :, :]).reshape((-1, 3))
                        fc -= np.floor(fc)
                        yield fc, s2_fc, av_lat(l, s2.lattice), sc_m

            if s1_supercell:
                for x in sc_generator(struct1, struct2):
                    yield x
            else:
                for x in sc_generator(struct2, struct1):
                    # reorder generator output so s1 is still first
                    yield x[1], x[0], x[2], x[3]

        def _cmp_fstruct(self, s1, s2, frac_tol, mask):
            """
            Returns true if a matching exists between s2 and s2
            under frac_tol. s2 should be a subset of s1
            """
            if len(s2) > len(s1):
                raise ValueError("s1 must be larger than s2")
            if mask.shape != (len(s2), len(s1)):
                raise ValueError("mask has incorrect shape")

            return is_coord_subset_pbc(s2, s1, frac_tol, mask)

        def _cart_dists(self, s1, s2, avg_lattice, mask, normalization, lll_frac_tol=None):
            """
            Finds a matching in cartesian space. Finds an additional
            fractional translation vector to minimize RMS distance

            Args:
                s1, s2: numpy arrays of fractional coordinates. len(s1) >= len(s2)
                avg_lattice: Lattice on which to calculate distances
                mask: numpy array of booleans. mask[i, j] = True indicates
                    that s2[i] cannot be matched to s1[j]
                normalization (float): inverse normalization length

            Returns:
                Distances from s2 to s1, normalized by (V/Natom) ^ 1/3
                Fractional translation vector to apply to s2.
                Mapping from s1 to s2, i.e. with numpy slicing, s1[mapping] => s2
            """
            if len(s2) > len(s1):
                raise ValueError("s1 must be larger than s2")
            if mask.shape != (len(s2), len(s1)):
                raise ValueError("mask has incorrect shape")

            # vectors are from s2 to s1
            vecs, d_2 = pbc_shortest_vectors(avg_lattice, s2, s1, mask,
                                             return_d2=True,
                                             lll_frac_tol=lll_frac_tol)
            lin = LinearAssignment(d_2)
            s = lin.solution
            short_vecs = vecs[np.arange(len(s)), s]
            translation = np.average(short_vecs, axis=0)
            f_translation = avg_lattice.get_fractional_coords(translation)
            new_d2 = np.sum((short_vecs - translation) ** 2, axis=-1)

            return new_d2 ** 0.5 * normalization, f_translation, s

        def _get_mask(self, struct1, struct2, fu, s1_supercell):
            """
            Returns mask for matching struct2 to struct1. If struct1 has sites
            a b c, and fu = 2, assumes supercells of struct2 will be ordered
            aabbcc (rather than abcabc)

            Returns:
            mask, struct1 translation indices, struct2 translation index
            """
            mask = np.zeros((len(struct2), len(struct1), fu), dtype=np.bool)

            inner = []
            for sp2, i in itertools.groupby(enumerate(struct2.species_and_occu),
                                            key=lambda x: x[1]):
                i = list(i)
                inner.append((sp2, slice(i[0][0], i[-1][0] + 1)))

            for sp1, j in itertools.groupby(enumerate(struct1.species_and_occu),
                                            key=lambda x: x[1]):
                j = list(j)
                j = slice(j[0][0], j[-1][0] + 1)
                for sp2, i in inner:
                    mask[i, j, :] = not self._comparator.are_equal(sp1, sp2)

            if s1_supercell:
                mask = mask.reshape((len(struct2), -1))
            else:
                # supercell is of struct2, roll fu axis back to preserve
                # correct ordering
                mask = np.rollaxis(mask, 2, 1)
                mask = mask.reshape((-1, len(struct1)))

            # find the best translation indices
            i = np.argmax(np.sum(mask, axis=-1))
            inds = np.where(np.invert(mask[i]))[0]
            if s1_supercell:
                # remove the symmetrically equivalent s1 indices
                inds = inds[::fu]
            return np.array(mask, dtype=np.int_), inds, i

        def fit(self, struct1, struct2, symmetric=False):
            """
            Fit two structures.

            Args:
                struct1 (Structure): 1st structure
                struct2 (Structure): 2nd structure
                symmetric (Bool): Defaults to False
                    If True, check the equality both ways.
                    This only impacts a small percentage of structures

            Returns:
                True or False.
            """
            struct1, struct2 = self._process_species([struct1, struct2])

            if not self._subset and self._comparator.get_hash(struct1.composition) \
                    != self._comparator.get_hash(struct2.composition):
                return None

            if not symmetric:
                struct1, struct2, fu, s1_supercell = self._preprocess(struct1, struct2)
                match = self._match(struct1, struct2, fu, s1_supercell,
                                    break_on_match=True)
                if match is None:
                    return False
                else:
                    return match[0] <= self.stol
            else:
                struct1, struct2, fu, s1_supercell = self._preprocess(struct1, struct2)
                match1 = self._match(struct1, struct2, fu, s1_supercell,
                                     break_on_match=True)
                struct1, struct2 = struct2, struct1
                struct1, struct2, fu, s1_supercell = self._preprocess(struct1, struct2)
                match2 = self._match(struct1, struct2, fu, s1_supercell,
                                     break_on_match=True)

                if match1 is None or match2 is None:
                    return False
                else:
                    return max(match1[0], match2[0]) <= self.stol

        def get_rms_dist(self, struct1, struct2):
            """
            Calculate RMS displacement between two structures

            Args:
                struct1 (Structure): 1st structure
                struct2 (Structure): 2nd structure

            Returns:
                rms displacement normalized by (Vol / nsites) ** (1/3)
                and maximum distance between paired sites. If no matching
                lattice is found None is returned.
            """
            struct1, struct2 = self._process_species([struct1, struct2])
            struct1, struct2, fu, s1_supercell = self._preprocess(struct1, struct2)
            match = self._match(struct1, struct2, fu, s1_supercell, use_rms=True,
                                break_on_match=False)

            if match is None:
                return None
            else:
                return match[0], max(match[1])

        def _process_species(self, structures):
            copied_structures = []
            for s in structures:
                # We need the copies to be actual Structure to work properly, not
                # subclasses. So do type(s) == Structure.
                ss = s.copy() if type(s) == Structure else \
                    Structure.from_sites(s)
                if self._ignored_species:
                    ss.remove_species(self._ignored_species)
                copied_structures.append(ss)
            return copied_structures

        def _preprocess(self, struct1, struct2, niggli=True):
            """
            Rescales, finds the reduced structures (primitive and niggli),
            and finds fu, the supercell size to make struct1 comparable to
            s2
            """
            struct1 = struct1.copy()
            struct2 = struct2.copy()

            if niggli:
                struct1 = struct1.get_reduced_structure(reduction_algo="niggli")
                struct2 = struct2.get_reduced_structure(reduction_algo="niggli")

            # primitive cell transformation
            if self._primitive_cell:
                struct1 = struct1.get_primitive_structure()
                struct2 = struct2.get_primitive_structure()

            if self._supercell:
                fu, s1_supercell = self._get_supercell_size(struct1, struct2)
            else:
                fu, s1_supercell = 1, True
            mult = fu if s1_supercell else 1 / fu

            # rescale lattice to same volume
            if self._scale:
                ratio = (struct2.volume / (struct1.volume * mult)) ** (1 / 6)
                nl1 = Lattice(struct1.lattice.matrix * ratio)
                struct1.lattice = nl1
                nl2 = Lattice(struct2.lattice.matrix / ratio)
                struct2.lattice = nl2

            return struct1, struct2, fu, s1_supercell

        def _match(self, struct1, struct2, fu, s1_supercell=True, use_rms=False,
                   break_on_match=False):
            """
            Matches one struct onto the other
            """
            ratio = fu if s1_supercell else 1 / fu
            if len(struct1) * ratio >= len(struct2):
                return self._strict_match(
                    struct1, struct2, fu, s1_supercell=s1_supercell,
                    break_on_match=break_on_match, use_rms=use_rms)
            else:
                return self._strict_match(
                    struct2, struct1, fu, s1_supercell=(not s1_supercell),
                    break_on_match=break_on_match, use_rms=use_rms)

        def _strict_match(self, struct1, struct2, fu, s1_supercell=True,
                          use_rms=False, break_on_match=False):
            """
            Matches struct2 onto struct1 (which should contain all sites in
            struct2).

            Args:
                struct1, struct2 (Structure): structures to be matched
                fu (int): size of supercell to create
                s1_supercell (bool): whether to create the supercell of
                    struct1 (vs struct2)
                use_rms (bool): whether to minimize the rms of the matching
                break_on_match (bool): whether to stop search at first
                    valid match
            """
            if fu < 1:
                raise ValueError("fu cannot be less than 1")

            mask, s1_t_inds, s2_t_ind = self._get_mask(struct1, struct2,
                                                       fu, s1_supercell)

            if mask.shape[0] > mask.shape[1]:
                raise ValueError('after supercell creation, struct1 must '
                                 'have more sites than struct2')

            # check that a valid mapping exists
            if (not self._subset) and mask.shape[1] != mask.shape[0]:
                return None

            if LinearAssignment(mask).min_cost > 0:
                return None

            best_match = None
            # loop over all lattices
            for s1fc, s2fc, avg_l, sc_m in \
                    self._get_supercells(struct1, struct2, fu, s1_supercell):
                # compute fractional tolerance
                normalization = (len(s1fc) / avg_l.volume) ** (1 / 3)
                inv_abc = np.array(avg_l.reciprocal_lattice.abc)
                frac_tol = inv_abc * self.stol / (np.pi * normalization)
                # loop over all translations
                for s1i in s1_t_inds:
                    t = s1fc[s1i] - s2fc[s2_t_ind]
                    t_s2fc = s2fc + t
                    if self._cmp_fstruct(s1fc, t_s2fc, frac_tol, mask):
                        inv_lll_abc = np.array(avg_l.get_lll_reduced_lattice().reciprocal_lattice.abc)
                        lll_frac_tol = inv_lll_abc * self.stol / (np.pi * normalization)
                        dist, t_adj, mapping = self._cart_dists(
                            s1fc, t_s2fc, avg_l, mask, normalization, lll_frac_tol)
                        if use_rms:
                            val = np.linalg.norm(dist) / len(dist) ** 0.5
                        else:
                            val = max(dist)
                        if best_match is None or val < best_match[0]:
                            total_t = t + t_adj
                            total_t -= np.round(total_t)
                            best_match = val, dist, sc_m, total_t, mapping
                            if (break_on_match or val < 1e-5) and val < self.stol:
                                return best_match

            if best_match and best_match[0] < self.stol:
                return best_match

        def group_structures(self, s_list, anonymous=False):
            """
            Given a list of structures, use fit to group
            them by structural equality.

            Args:
                s_list ([Structure]): List of structures to be grouped
                anonymous (bool): Whether to use anonymous mode.

            Returns:
                A list of lists of matched structures
                Assumption: if s1 == s2 but s1 != s3, than s2 and s3 will be put
                in different groups without comparison.
            """
            if self._subset:
                raise ValueError("allow_subset cannot be used with"
                                 " group_structures")

            original_s_list = list(s_list)
            s_list = self._process_species(s_list)

            # Use structure hash to pre-group structures
            if anonymous:
                def c_hash(c):
                    return c.anonymized_formula
            else:
                c_hash = self._comparator.get_hash

            def s_hash(s):
                return c_hash(s[1].composition)

            sorted_s_list = sorted(enumerate(s_list), key=s_hash)
            all_groups = []

            # For each pre-grouped list of structures, perform actual matching.
            for k, g in itertools.groupby(sorted_s_list, key=s_hash):
                unmatched = list(g)
                while len(unmatched) > 0:
                    i, refs = unmatched.pop(0)
                    matches = [i]
                    if anonymous:
                        inds = filter(lambda i: self.fit_anonymous(refs,
                                                                   unmatched[i][1]), list(range(len(unmatched))))
                    else:
                        inds = filter(lambda i: self.fit(refs, unmatched[i][1]),
                                      list(range(len(unmatched))))
                    inds = list(inds)
                    matches.extend([unmatched[i][0] for i in inds])
                    unmatched = [unmatched[i] for i in range(len(unmatched))
                                 if i not in inds]
                    all_groups.append([original_s_list[i] for i in matches])

            return all_groups

        def as_dict(self):
            """
            :return: MSONable dict
            """
            return {"version": __version__, "@module": self.__class__.__module__,
                    "@class": self.__class__.__name__,
                    "comparator": self._comparator.as_dict(),
                    "stol": self.stol,
                    "ltol": self.ltol,
                    "angle_tol": self.angle_tol,
                    "primitive_cell": self._primitive_cell,
                    "scale": self._scale,
                    "attempt_supercell": self._supercell,
                    "allow_subset": self._subset,
                    "supercell_size": self._supercell_size,
                    "ignored_species": self._ignored_species}

        @classmethod
        def from_dict(cls, d):
            """
            :param d: Dict representation
            :return: StructureMatcher
            """
            return StructureMatcher(
                ltol=d["ltol"], stol=d["stol"], angle_tol=d["angle_tol"],
                primitive_cell=d["primitive_cell"], scale=d["scale"],
                attempt_supercell=d["attempt_supercell"],
                allow_subset=d["allow_subset"],
                comparator=AbstractComparator.from_dict(d["comparator"]),
                supercell_size=d["supercell_size"],
                ignored_species=d["ignored_species"])

        def _anonymous_match(self, struct1, struct2, fu, s1_supercell=True,
                             use_rms=False, break_on_match=False, single_match=False):
            """
            Tries all permutations of matching struct1 to struct2.
            Args:
                struct1, struct2 (Structure): Preprocessed input structures
            Returns:
                List of (mapping, match)
            """
            if not isinstance(self._comparator, SpeciesComparator):
                raise ValueError('Anonymous fitting currently requires SpeciesComparator')

            # check that species lists are comparable
            sp1 = struct1.composition.elements
            sp2 = struct2.composition.elements
            if len(sp1) != len(sp2):
                return None

            ratio = fu if s1_supercell else 1 / fu
            swapped = len(struct1) * ratio < len(struct2)

            s1_comp = struct1.composition
            s2_comp = struct2.composition
            matches = []
            for perm in itertools.permutations(sp2):
                sp_mapping = dict(zip(sp1, perm))

                # do quick check that compositions are compatible
                mapped_comp = Composition({sp_mapping[k]: v
                                           for k, v in s1_comp.items()})
                if (not self._subset) and (
                        self._comparator.get_hash(mapped_comp) !=
                        self._comparator.get_hash(s2_comp)):
                    continue

                mapped_struct = struct1.copy()
                mapped_struct.replace_species(sp_mapping)
                if swapped:
                    m = self._strict_match(struct2, mapped_struct, fu,
                                           (not s1_supercell), use_rms,
                                           break_on_match)
                else:
                    m = self._strict_match(mapped_struct, struct2, fu, s1_supercell,
                                           use_rms, break_on_match)
                if m:
                    matches.append((sp_mapping, m))
                    if single_match:
                        break
            return matches

        def get_rms_anonymous(self, struct1, struct2):
            """
            Performs an anonymous fitting, which allows distinct species in one
            structure to map to another. E.g., to compare if the Li2O and Na2O
            structures are similar.

            Args:
                struct1 (Structure): 1st structure
                struct2 (Structure): 2nd structure

            Returns:
                (min_rms, min_mapping)
                min_rms is the minimum rms distance, and min_mapping is the
                corresponding minimal species mapping that would map
                struct1 to struct2. (None, None) is returned if the minimax_rms
                exceeds the threshold.
            """
            struct1, struct2 = self._process_species([struct1, struct2])
            struct1, struct2, fu, s1_supercell = self._preprocess(struct1, struct2)

            matches = self._anonymous_match(struct1, struct2, fu, s1_supercell,
                                            use_rms=True, break_on_match=False)
            if matches:
                best = sorted(matches, key=lambda x: x[1][0])[0]
                return best[1][0], best[0]
            else:
                return None, None

        def get_best_electronegativity_anonymous_mapping(self, struct1, struct2):
            """
            Performs an anonymous fitting, which allows distinct species in one
            structure to map to another. E.g., to compare if the Li2O and Na2O
            structures are similar. If multiple substitutions are within tolerance
            this will return the one which minimizes the difference in
            electronegativity between the matches species.

            Args:
                struct1 (Structure): 1st structure
                struct2 (Structure): 2nd structure

            Returns:
                min_mapping (Dict): Mapping of struct1 species to struct2 species
            """
            struct1, struct2 = self._process_species([struct1, struct2])
            struct1, struct2, fu, s1_supercell = self._preprocess(struct1, struct2)

            matches = self._anonymous_match(struct1, struct2, fu, s1_supercell,
                                            use_rms=True, break_on_match=True)

            if matches:
                min_X_diff = np.inf
                for m in matches:
                    X_diff = 0
                    for k, v in m[0].items():
                        X_diff += struct1.composition[k] * (k.X - v.X) ** 2
                    if X_diff < min_X_diff:
                        min_X_diff = X_diff
                        best = m[0]
                return best

        def get_all_anonymous_mappings(self, struct1, struct2, niggli=True,
                                       include_dist=False):
            """
            Performs an anonymous fitting, which allows distinct species in one
            structure to map to another. Returns a dictionary of species
            substitutions that are within tolerance

            Args:
                struct1 (Structure): 1st structure
                struct2 (Structure): 2nd structure
                niggli (bool): Find niggli cell in preprocessing
                include_dist (bool): Return the maximin distance with each mapping

            Returns:
                list of species mappings that map struct1 to struct2.
            """
            struct1, struct2 = self._process_species([struct1, struct2])
            struct1, struct2, fu, s1_supercell = self._preprocess(struct1, struct2,
                                                                  niggli)

            matches = self._anonymous_match(struct1, struct2, fu, s1_supercell,
                                            break_on_match=not include_dist)
            if matches:
                if include_dist:
                    return [(m[0], m[1][0]) for m in matches]
                else:
                    return [m[0] for m in matches]

        def fit_anonymous(self, struct1, struct2, niggli=True):
            """
            Performs an anonymous fitting, which allows distinct species in one
            structure to map to another. E.g., to compare if the Li2O and Na2O
            structures are similar.

            Args:
                struct1 (Structure): 1st structure
                struct2 (Structure): 2nd structure

            Returns:
                True/False: Whether a species mapping can map struct1 to stuct2
            """
            struct1, struct2 = self._process_species([struct1, struct2])
            struct1, struct2, fu, s1_supercell = self._preprocess(struct1, struct2,
                                                                  niggli)

            matches = self._anonymous_match(struct1, struct2, fu, s1_supercell,
                                            break_on_match=True, single_match=True)

            if matches:
                return True
            else:
                return False

        def get_supercell_matrix(self, supercell, struct):
            """
            Returns the matrix for transforming struct to supercell. This
            can be used for very distorted 'supercells' where the primitive cell
            is impossible to find
            """
            if self._primitive_cell:
                raise ValueError("get_supercell_matrix cannot be used with the "
                                 "primitive cell option")
            struct, supercell, fu, s1_supercell = self._preprocess(struct,
                                                                   supercell, False)

            if not s1_supercell:
                raise ValueError("The non-supercell must be put onto the basis"
                                 " of the supercell, not the other way around")

            match = self._match(struct, supercell, fu, s1_supercell, use_rms=True,
                                break_on_match=False)

            if match is None:
                return None

            return match[2]

        def get_transformation(self, struct1, struct2):
            """
            Returns the supercell transformation, fractional translation vector,
            and a mapping to transform struct2 to be similar to struct1.

            Args:
                struct1 (Structure): Reference structure
                struct2 (Structure): Structure to transform.

            Returns:
                supercell (numpy.ndarray(3, 3)): supercell matrix
                vector (numpy.ndarray(3)): fractional translation vector
                mapping (list(int or None)):
                    The first len(struct1) items of the mapping vector are the
                    indices of struct1's corresponding sites in struct2 (or None
                    if there is no corresponding site), and the other items are
                    the remaining site indices of struct2.
            """
            if self._primitive_cell:
                raise ValueError("get_transformation cannot be used with the "
                                 "primitive cell option")

            struct1, struct2 = self._process_species((struct1, struct2))

            s1, s2, fu, s1_supercell = self._preprocess(struct1, struct2, False)
            ratio = fu if s1_supercell else 1 / fu
            if s1_supercell and fu > 1:
                raise ValueError("Struct1 must be the supercell, "
                                 "not the other way around")

            if len(s1) * ratio >= len(s2):
                # s1 is superset
                match = self._strict_match(s1, s2, fu=fu, s1_supercell=False,
                                           use_rms=True, break_on_match=False)
                if match is None:
                    return None
                # invert the mapping, since it needs to be from s1 to s2
                mapping = [list(match[4]).index(i) if i in match[4] else None
                           for i in range(len(s1))]
                return match[2], match[3], mapping
            else:
                # s2 is superset
                match = self._strict_match(s2, s1, fu=fu, s1_supercell=True,
                                           use_rms=True, break_on_match=False)
                if match is None:
                    return None
                # add sites not included in the mapping
                not_included = list(range(len(s2) * fu))
                for i in match[4]:
                    not_included.remove(i)
                mapping = list(match[4]) + not_included
                return match[2], -match[3], mapping

        def get_s2_like_s1(self, struct1, struct2, include_ignored_species=True):
            """
            Performs transformations on struct2 to put it in a basis similar to
            struct1 (without changing any of the inter-site distances)

            Args:
                struct1 (Structure): Reference structure
                struct2 (Structure): Structure to transform.
                include_ignored_species (bool): Defaults to True,
                    the ignored_species is also transformed to the struct1
                    lattice orientation, though obviously there is no direct
                    matching to existing sites.

            Returns:
                A structure object similar to struct1, obtained by making a
                supercell, sorting, and translating struct2.
            """
            s1, s2 = self._process_species([struct1, struct2])
            trans = self.get_transformation(s1, s2)
            if trans is None:
                return None
            sc, t, mapping = trans
            sites = [site for site in s2]
            # Append the ignored sites at the end.
            sites.extend([site for site in struct2 if site not in s2])
            temp = Structure.from_sites(sites)

            temp.make_supercell(sc)
            temp.translate_sites(list(range(len(temp))), t)
            # translate sites to correct unit cell
            for i, j in enumerate(mapping[:len(s1)]):
                if j is not None:
                    vec = np.round(struct1[i].frac_coords - temp[j].frac_coords)
                    temp.translate_sites(j, vec, to_unit_cell=False)

            sites = [temp.sites[i] for i in mapping if i is not None]

            if include_ignored_species:
                start = int(round(len(temp) / len(struct2) * len(s2)))
                sites.extend(temp.sites[start:])

            return Structure.from_sites(sites)

        def get_mapping(self, superset, subset):
            """
            Calculate the mapping from superset to subset.

            Args:
                superset (Structure): Structure containing at least the sites in
                    subset (within the structure matching tolerance)
                subset (Structure): Structure containing some of the sites in
                    superset (within the structure matching tolerance)

            Returns:
                numpy array such that superset.sites[mapping] is within matching
                tolerance of subset.sites or None if no such mapping is possible
            """
            if self._supercell:
                raise ValueError("cannot compute mapping to supercell")
            if self._primitive_cell:
                raise ValueError("cannot compute mapping with primitive cell "
                                 "option")
            if len(subset) > len(superset):
                raise ValueError("subset is larger than superset")

            superset, subset, _, _ = self._preprocess(superset, subset, True)
            match = self._strict_match(superset, subset, 1, break_on_match=False)

            if match is None or match[0] > self.stol:
                return None

            return match[4]

    class PointDefectComparator(MSONable):
        """
        A class that matches pymatgen Point Defect objects even if their
        cartesian co-ordinates are different (compares sublattices for the defect)

        NOTE: for defect complexes (more than a single defect),
        this comparator will break.
        """

        def __init__(self, check_charge=False, check_primitive_cell=False,
                     check_lattice_scale=False):
            """
            Args:
                check_charge (bool): Gives option to check
                    if charges are identical.
                    Default is False (different charged defects can be same)
                check_primitive_cell (bool): Gives option to
                    compare different supercells of bulk_structure,
                    rather than directly compare supercell sizes
                    Default is False (requires bulk_structure in each defect to be same size)
                check_lattice_scale (bool): Gives option to scale volumes of
                    structures to each other identical lattice constants.
                    Default is False (enforces same
                    lattice constants in both structures)
            """
            self.check_charge = check_charge
            self.check_primitive_cell = check_primitive_cell
            self.check_lattice_scale = check_lattice_scale

        def are_equal(self, d1, d2):
            """
            Args:
                d1: First defect. A pymatgen Defect object.
                d2: Second defect. A pymatgen Defect object.

            Returns:
                True if defects are identical in type and sublattice.
            """
            possible_defect_types = (Defect, Vacancy, Substitution, Interstitial)

            if not isinstance(d1, possible_defect_types) or not isinstance(d2, possible_defect_types):
                raise ValueError("Cannot use PointDefectComparator to"
                                 " compare non-defect objects...")

            if not isinstance(d1, d2.__class__):
                return False
            elif d1.site.specie != d2.site.specie:
                return False
            elif self.check_charge and (d1.charge != d2.charge):
                return False

            sm = StructureMatcher(ltol=0.01,
                                  primitive_cell=self.check_primitive_cell,
                                  scale=self.check_lattice_scale)

            if not sm.fit(d1.bulk_structure, d2.bulk_structure):
                return False

            d1 = d1.copy()
            d2 = d2.copy()
            if self.check_primitive_cell or self.check_lattice_scale:
                # if allowing for base structure volume or supercell modifications,
                # then need to preprocess defect objects to allow for matching
                d1_mod_bulk_structure, d2_mod_bulk_structure, _, _ = sm._preprocess(
                    d1.bulk_structure, d2.bulk_structure)
                d1_defect_site = PeriodicSite(d1.site.specie, d1.site.coords,
                                              d1_mod_bulk_structure.lattice,
                                              to_unit_cell=True,
                                              coords_are_cartesian=True)
                d2_defect_site = PeriodicSite(d2.site.specie, d2.site.coords,
                                              d2_mod_bulk_structure.lattice,
                                              to_unit_cell=True,
                                              coords_are_cartesian=True)

                d1._structure = d1_mod_bulk_structure
                d2._structure = d2_mod_bulk_structure
                d1._defect_site = d1_defect_site
                d2._defect_site = d2_defect_site

            return sm.fit(d1.generate_defect_structure(),
                          d2.generate_defect_structure())