# Source code for pymatgen.analysis.phase_diagram

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

    """
    This module defines tools to generate and analyze phase diagrams.
    """

    import re
    import collections
    import itertools
    import math
    import logging

    from monty.json import MSONable, MontyDecoder
    from functools import lru_cache

    import numpy as np
    from scipy.spatial import ConvexHull

    from pymatgen.core.composition import Composition
    from pymatgen.core.periodic_table import Element, DummySpecie, get_el_sp
    from pymatgen.util.coord import Simplex, in_coord_list
    from pymatgen.util.string import latexify
    from pymatgen.util.plotting import pretty_plot
    from pymatgen.analysis.reaction_calculator import Reaction, \
        ReactionError
    from pymatgen.entries import Entry

    __author__ = "Shyue Ping Ong"
    __copyright__ = "Copyright 2011, The Materials Project"
    __version__ = "1.0"
    __maintainer__ = "Shyue Ping Ong"
    __email__ = "shyuep@gmail.com"
    __status__ = "Production"
    __date__ = "May 16, 2011"

    logger = logging.getLogger(__name__)

    class PDEntry(Entry):
        """
        An object encompassing all relevant data for phase diagrams.

        .. attribute:: composition

            The composition associated with the PDEntry.

        .. attribute:: energy

            The energy associated with the entry.

        .. attribute:: name

            A name for the entry. This is the string shown in the phase diagrams.
            By default, this is the reduced formula for the composition, but can be
            set to some other string for display purposes.

        .. attribute:: attribute

            A arbitrary attribute. Can be used to specify that the entry is a newly
            found compound, or to specify a particular label for the entry, etc.
            An attribute can be anything but must be MSONable.
        """

        def __init__(self, composition: Composition, energy: float,
                     name: str = None, attribute: object = None):
            """
            Args:
                composition (Composition): Composition
                energy (float): Energy for composition.
                name (str): Optional parameter to name the entry. Defaults
                    to the reduced chemical formula.
                attribute: Optional attribute of the entry. Must be MSONable.
            """
            super().__init__(composition, energy)
            self.name = name if name else self.composition.reduced_formula
            self.attribute = attribute

        @property
        def energy(self) -> float:
            """
            :return: the energy of the entry.
            """
            return self._energy

        def __repr__(self):
            return "PDEntry : {} with energy = {:.4f}".format(self.composition,
                                                              self.energy)

        def as_dict(self):
            """
            :return: MSONable dict.
            """
            return_dict = super().as_dict()
            return_dict.update({"name": self.name,
                                "attribute": self.attribute})
            return return_dict

        def __eq__(self, other):
            if isinstance(other, self.__class__):
                return self.as_dict() == other.as_dict()
            else:
                return False

        def __hash__(self):
            return id(self)

        @classmethod
        def from_dict(cls, d):
            """
            :param d: Dict representation
            :return: PDEntry
            """
            return cls(Composition(d["composition"]), d["energy"],
                       d["name"] if "name" in d else None,
                       d["attribute"] if "attribute" in d else None)

    class GrandPotPDEntry(PDEntry):
        """
        A grand potential pd entry object encompassing all relevant data for phase
        diagrams.  Chemical potentials are given as a element-chemical potential
        dict.
        """

        def __init__(self, entry, chempots, name=None):
            """
            Args:
                entry: A PDEntry-like object.
                chempots: Chemical potential specification as {Element: float}.
                name: Optional parameter to name the entry. Defaults to the reduced
                    chemical formula of the original entry.
            """
            comp = entry.composition
            self.original_entry = entry
            self.original_comp = comp
            grandpot = entry.energy - sum([comp[el] * pot
                                           for el, pot in chempots.items()])
            self.chempots = chempots
            new_comp_map = {el: comp[el] for el in comp.elements
                            if el not in chempots}
            super().__init__(new_comp_map, grandpot, entry.name)
            self.name = name if name else entry.name

        @property
        def is_element(self):
            """
            True if the entry is an element.
            """
            return self.original_comp.is_element

        def __repr__(self):
            chempot_str = " ".join(["mu_%s = %.4f" % (el, mu)
                                    for el, mu in self.chempots.items()])
            return "GrandPotPDEntry with original composition " + \
                   "{}, energy = {:.4f}, {}".format(self.original_entry.composition,
                                                    self.original_entry.energy,
                                                    chempot_str)

        def __str__(self):
            return self.__repr__()

        def as_dict(self):
            """
            :return: MSONAble dict
            """
            return {"@module": self.__class__.__module__,
                    "@class": self.__class__.__name__,
                    "entry": self.original_entry.as_dict(),
                    "chempots": {el.symbol: u for el, u in self.chempots.items()},
                    "name": self.name}

        @classmethod
        def from_dict(cls, d):
            """
            :param d: Dict representation
            :return: PDStructureEntry
            """
            chempots = {Element(symbol): u for symbol, u in d["chempots"].items()}
            entry = MontyDecoder().process_decoded(d["entry"])
            return cls(entry, chempots, d["name"])

        def __getattr__(self, a):
            """
            Delegate attribute to original entry if available.
            """
            if hasattr(self.original_entry, a):
                return getattr(self.original_entry, a)
            raise AttributeError(a)

    class TransformedPDEntry(PDEntry):
        """
        This class repesents a TransformedPDEntry, which allows for a PDEntry to be
        transformed to a different composition coordinate space. It is used in the
        construction of phase diagrams that do not have elements as the terminal
        compositions.
        """

        def __init__(self, comp, original_entry):
            """
            Args:
                comp (Composition): Transformed composition as a Composition.
                original_entry (PDEntry): Original entry that this entry arose from.
            """
            super().__init__(comp, original_entry.energy)
            self.original_entry = original_entry
            self.name = original_entry.name

        def __getattr__(self, a):
            """
            Delegate attribute to original entry if available.
            """
            if hasattr(self.original_entry, a):
                return getattr(self.original_entry, a)
            raise AttributeError(a)

        def __repr__(self):
            output = ["TransformedPDEntry {}".format(self.composition),
                      " with original composition {}".format(self.original_entry.composition),
                      ", E = {:.4f}".format(self.original_entry.energy)]
            return "".join(output)

        def __str__(self):
            return self.__repr__()

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

        @classmethod
        def from_dict(cls, d):
            """
            :param d: Dict representation
            :return: TransformedPDEntry
            """
            entry = MontyDecoder().process_decoded(d["entry"])
            return cls(d["composition"], entry)

    class PhaseDiagram(MSONable):
        """
        Simple phase diagram class taking in elements and entries as inputs.
        The algorithm is based on the work in the following papers:

        1. S. P. Ong, L. Wang, B. Kang, and G. Ceder, Li-Fe-P-O2 Phase Diagram from
           First Principles Calculations. Chem. Mater., 2008, 20(5), 1798-1807.
           doi:10.1021/cm702327g

        2. S. P. Ong, A. Jain, G. Hautier, B. Kang, G. Ceder, Thermal stabilities
           of delithiated olivine MPO4 (M=Fe, Mn) cathodes investigated using first
           principles calculations. Electrochem. Comm., 2010, 12(3), 427-430.
           doi:10.1016/j.elecom.2010.01.010

        .. attribute: elements:

            Elements in the phase diagram.

        ..attribute: all_entries

            All entries provided for Phase Diagram construction. Note that this
            does not mean that all these entries are actually used in the phase
            diagram. For example, this includes the positive formation energy
            entries that are filtered out before Phase Diagram construction.

        .. attribute: qhull_data

            Data used in the convex hull operation. This is essentially a matrix of
            composition data and energy per atom values created from qhull_entries.

        .. attribute: qhull_entries:

            Actual entries used in convex hull. Excludes all positive formation
            energy entries.

        .. attribute: dim

            The dimensionality of the phase diagram.

        .. attribute: facets

            Facets of the phase diagram in the form of  [[1,2,3],[4,5,6]...].
            For a ternary, it is the indices (references to qhull_entries and
            qhull_data) for the vertices of the phase triangles. Similarly
            extended to higher D simplices for higher dimensions.

        .. attribute: el_refs:

            List of elemental references for the phase diagrams. These are
            entries corresponding to the lowest energy element entries for simple
            compositional phase diagrams.

        .. attribute: simplices:

            The simplices of the phase diagram as a list of np.ndarray, i.e.,
            the list of stable compositional coordinates in the phase diagram.
        """

        # Tolerance for determining if formation energy is positive.
        formation_energy_tol = 1e-11
        numerical_tol = 1e-8

        def __init__(self, entries, elements=None):
            """
            Standard constructor for phase diagram.

            Args:
                entries ([PDEntry]): A list of PDEntry-like objects having an
                    energy, energy_per_atom and composition.
                elements ([Element]): Optional list of elements in the phase
                    diagram. If set to None, the elements are determined from
                    the the entries themselves and are sorted alphabetically.
                    If specified, element ordering (e.g. for pd coordinates)
                    is preserved.
            """
            if elements is None:
                elements = set()
                for entry in entries:
                    elements.update(entry.composition.elements)
                elements = sorted(list(elements))

            elements = list(elements)
            dim = len(elements)

            entries = sorted(entries, key=lambda e: e.composition.reduced_composition)

            el_refs = {}
            min_entries = []
            all_entries = []
            for c, g in itertools.groupby(entries, key=lambda e: e.composition.reduced_composition):
                g = list(g)
                min_entry = min(g, key=lambda e: e.energy_per_atom)
                if c.is_element:
                    el_refs[c.elements[0]] = min_entry
                min_entries.append(min_entry)
                all_entries.extend(g)

            if len(el_refs) != dim:
                raise PhaseDiagramError(
                    "There are no entries associated with a terminal element!.")

            data = np.array([
                [e.composition.get_atomic_fraction(el) for el in elements] + [
                    e.energy_per_atom]
                for e in min_entries
            ])

            # Use only entries with negative formation energy
            vec = [el_refs[el].energy_per_atom for el in elements] + [-1]
            form_e = -np.dot(data, vec)
            inds = np.where(form_e < -self.formation_energy_tol)[0].tolist()

            # Add the elemental references
            inds.extend([min_entries.index(el) for el in el_refs.values()])

            qhull_entries = [min_entries[i] for i in inds]
            qhull_data = data[inds][:, 1:]

            # Add an extra point to enforce full dimensionality.
            # This point will be present in all upper hull facets.
            extra_point = np.zeros(dim) + 1 / dim
            extra_point[-1] = np.max(qhull_data) + 1
            qhull_data = np.concatenate([qhull_data, [extra_point]], axis=0)

            if dim == 1:
                self.facets = [qhull_data.argmin(axis=0)]
            else:
                facets = get_facets(qhull_data)
                finalfacets = []
                for facet in facets:
                    # Skip facets that include the extra point
                    if max(facet) == len(qhull_data) - 1:
                        continue
                    m = qhull_data[facet]
                    m[:, -1] = 1
                    if abs(np.linalg.det(m)) > 1e-14:
                        finalfacets.append(facet)
                self.facets = finalfacets

            self.simplexes = [Simplex(qhull_data[f, :-1]) for f in self.facets]
            self.all_entries = all_entries
            self.qhull_data = qhull_data
            self.dim = dim
            self.el_refs = el_refs
            self.elements = elements
            self.qhull_entries = qhull_entries
            self._stable_entries = set(self.qhull_entries[i] for i in
                                       set(itertools.chain(*self.facets)))

        def pd_coords(self, comp):
            """
            The phase diagram is generated in a reduced dimensional space
            (n_elements - 1). This function returns the coordinates in that space.
            These coordinates are compatible with the stored simplex objects.
            """
            if set(comp.elements).difference(self.elements):
                raise ValueError('{} has elements not in the phase diagram {}'
                                 ''.format(comp, self.elements))
            return np.array(
                [comp.get_atomic_fraction(el) for el in self.elements[1:]])

        @property
        def all_entries_hulldata(self):
            """
            :return: The actual ndarray used to construct the convex hull.
            """
            data = []
            for entry in self.all_entries:
                comp = entry.composition
                row = [comp.get_atomic_fraction(el) for el in self.elements]
                row.append(entry.energy_per_atom)
                data.append(row)
            return np.array(data)[:, 1:]

        @property
        def unstable_entries(self):
            """
            Entries that are unstable in the phase diagram. Includes positive
            formation energy entries.
            """
            return [e for e in self.all_entries if e not in self.stable_entries]

        @property
        def stable_entries(self):
            """
            Returns the stable entries in the phase diagram.
            """
            return self._stable_entries

        def get_form_energy(self, entry):
            """
            Returns the formation energy for an entry (NOT normalized) from the
            elemental references.

            Args:
                entry: A PDEntry-like object.

            Returns:
                Formation energy from the elemental references.
            """
            c = entry.composition
            return entry.energy - sum([c[el] * self.el_refs[el].energy_per_atom
                                       for el in c.elements])

        def get_form_energy_per_atom(self, entry):
            """
            Returns the formation energy per atom for an entry from the
            elemental references.

            Args:
                entry: An PDEntry-like object

            Returns:
                Formation energy **per atom** from the elemental references.
            """
            return self.get_form_energy(entry) / entry.composition.num_atoms

        def __repr__(self):
            return self.__str__()

        def __str__(self):
            symbols = [el.symbol for el in self.elements]
            output = ["{} phase diagram".format("-".join(symbols)),
                      "{} stable phases: ".format(len(self.stable_entries)),
                      ", ".join([entry.name
                                 for entry in self.stable_entries])]
            return "\n".join(output)

        def as_dict(self):
            """
            :return: MSONAble dict
            """
            return {"@module": self.__class__.__module__,
                    "@class": self.__class__.__name__,
                    "all_entries": [e.as_dict() for e in self.all_entries],
                    "elements": [e.as_dict() for e in self.elements]}

        @classmethod
        def from_dict(cls, d):
            """
            :param d: Dict representation
            :return: PhaseDiagram
            """
            entries = [MontyDecoder().process_decoded(dd) for dd in d["all_entries"]]
            elements = [Element.from_dict(dd) for dd in d["elements"]]
            return cls(entries, elements)

        @lru_cache(1)
        def _get_facet_and_simplex(self, comp):
            """
            Get any facet that a composition falls into. Cached so successive
            calls at same composition are fast.
            """
            c = self.pd_coords(comp)
            for f, s in zip(self.facets, self.simplexes):
                if s.in_simplex(c, PhaseDiagram.numerical_tol / 10):
                    return f, s
            raise RuntimeError("No facet found for comp = {}".format(comp))

        def _get_facet_chempots(self, facet):
            """
            Calculates the chemical potentials for each element within a facet.

            Args:
                facet: Facet of the phase diagram.

            Returns:
                { element: chempot } for all elements in the phase diagram.
            """
            complist = [self.qhull_entries[i].composition for i in facet]
            energylist = [self.qhull_entries[i].energy_per_atom for i in facet]
            m = [[c.get_atomic_fraction(e) for e in self.elements] for c in
                 complist]
            chempots = np.linalg.solve(m, energylist)
            return dict(zip(self.elements, chempots))

        def get_decomposition(self, comp):
            """
            Provides the decomposition at a particular composition.

            Args:
                comp: A composition

            Returns:
                Decomposition as a dict of {Entry: amount}
            """
            facet, simplex = self._get_facet_and_simplex(comp)
            decomp_amts = simplex.bary_coords(self.pd_coords(comp))
            return {self.qhull_entries[f]: amt
                    for f, amt in zip(facet, decomp_amts)
                    if abs(amt) > PhaseDiagram.numerical_tol}

        def get_hull_energy(self, comp):
            """
            Args:
                comp (Composition): Input composition

            Returns:
                Energy of lowest energy equilibrium at desired composition. Not
                normalized by atoms, i.e. E(Li4O2) = 2 * E(Li2O)
            """
            e = 0
            for k, v in self.get_decomposition(comp).items():
                e += k.energy_per_atom * v
            return e * comp.num_atoms

        def get_decomp_and_e_above_hull(self, entry, allow_negative=False):
            """
            Provides the decomposition and energy above convex hull for an entry.
            Due to caching, can be much faster if entries with the same composition
            are processed together.

            Args:
                entry: A PDEntry like object
                allow_negative: Whether to allow negative e_above_hulls. Used to
                    calculate equilibrium reaction energies. Defaults to False.

            Returns:
                (decomp, energy above convex hull)  Stable entries should have
                energy above hull of 0. The decomposition is provided as a dict of
                {Entry: amount}.
            """
            if entry in self.stable_entries:
                return {entry: 1}, 0

            comp = entry.composition
            facet, simplex = self._get_facet_and_simplex(comp)
            decomp_amts = simplex.bary_coords(self.pd_coords(comp))
            decomp = {self.qhull_entries[f]: amt
                      for f, amt in zip(facet, decomp_amts)
                      if abs(amt) > PhaseDiagram.numerical_tol}
            energies = [self.qhull_entries[i].energy_per_atom for i in facet]
            ehull = entry.energy_per_atom - np.dot(decomp_amts, energies)
            if allow_negative or ehull >= -PhaseDiagram.numerical_tol:
                return decomp, ehull
            raise ValueError("No valid decomp found!")

        def get_e_above_hull(self, entry):
            """
            Provides the energy above convex hull for an entry

            Args:
                entry: A PDEntry like object

            Returns:
                Energy above convex hull of entry. Stable entries should have
                energy above hull of 0.
            """
            return self.get_decomp_and_e_above_hull(entry)[1]

        def get_equilibrium_reaction_energy(self, entry):
            """
            Provides the reaction energy of a stable entry from the neighboring
            equilibrium stable entries (also known as the inverse distance to
            hull).

            Args:
                entry: A PDEntry like object

            Returns:
                Equilibrium reaction energy of entry. Stable entries should have
                equilibrium reaction energy <= 0.
            """
            if entry not in self.stable_entries:
                raise ValueError("Equilibrium reaction energy is available only "
                                 "for stable entries.")
            if entry.is_element:
                return 0
            entries = [e for e in self.stable_entries if e != entry]
            modpd = PhaseDiagram(entries, self.elements)
            return modpd.get_decomp_and_e_above_hull(entry,
                                                     allow_negative=True)[1]

        def get_composition_chempots(self, comp):
            """
            Get the chemical potentials for all elements at a given composition.

            :param comp: Composition
            :return: Dict of chemical potentials.
            """
            facet = self._get_facet_and_simplex(comp)[0]
            return self._get_facet_chempots(facet)

        def get_all_chempots(self, comp):
            """
            Get chemical potentials at a given compositon.

            :param comp: Composition
            :return: Chemical potentials.
            """
            # note the top part takes from format of _get_facet_and_simplex,
            #   but wants to return all facets rather than the first one that meets this criteria
            c = self.pd_coords(comp)
            allfacets = []
            for f, s in zip(self.facets, self.simplexes):
                if s.in_simplex(c, PhaseDiagram.numerical_tol / 10):
                    allfacets.append(f)

            if not len(allfacets):
                raise RuntimeError("No facets found for comp = {}".format(comp))
            else:
                chempots = {}
                for facet in allfacets:
                    facet_elt_list = [self.qhull_entries[j].name for j in facet]
                    facet_name = '-'.join(facet_elt_list)
                    chempots[facet_name] = self._get_facet_chempots(facet)
                return chempots

        def get_transition_chempots(self, element):
            """
            Get the critical chemical potentials for an element in the Phase
            Diagram.

            Args:
                element: An element. Has to be in the PD in the first place.

            Returns:
                A sorted sequence of critical chemical potentials, from less
                negative to more negative.
            """
            if element not in self.elements:
                raise ValueError("get_transition_chempots can only be called with "
                                 "elements in the phase diagram.")

            critical_chempots = []
            for facet in self.facets:
                chempots = self._get_facet_chempots(facet)
                critical_chempots.append(chempots[element])

            clean_pots = []
            for c in sorted(critical_chempots):
                if len(clean_pots) == 0:
                    clean_pots.append(c)
                else:
                    if abs(c - clean_pots[-1]) > PhaseDiagram.numerical_tol:
                        clean_pots.append(c)
            clean_pots.reverse()
            return tuple(clean_pots)

        def get_critical_compositions(self, comp1, comp2):
            """
            Get the critical compositions along the tieline between two
            compositions. I.e. where the decomposition products change.
            The endpoints are also returned.
            Args:
                comp1, comp2 (Composition): compositions that define the tieline
            Returns:
                [(Composition)]: list of critical compositions. All are of
                    the form x * comp1 + (1-x) * comp2
            """

            n1 = comp1.num_atoms
            n2 = comp2.num_atoms
            pd_els = self.elements

            # the reduced dimensionality Simplexes don't use the
            # first element in the PD
            c1 = self.pd_coords(comp1)
            c2 = self.pd_coords(comp2)

            # none of the projections work if c1 == c2, so just return *copies*
            # of the inputs
            if np.all(c1 == c2):
                return [comp1.copy(), comp2.copy()]

            intersections = [c1, c2]
            for sc in self.simplexes:
                intersections.extend(sc.line_intersection(c1, c2))
            intersections = np.array(intersections)

            # find position along line
            l = (c2 - c1)
            l /= np.sum(l ** 2) ** 0.5
            proj = np.dot(intersections - c1, l)

            # only take compositions between endpoints
            proj = proj[np.logical_and(proj > -self.numerical_tol,
                                       proj < proj[1] + self.numerical_tol)]
            proj.sort()

            # only unique compositions
            valid = np.ones(len(proj), dtype=np.bool)
            valid[1:] = proj[1:] > proj[:-1] + self.numerical_tol
            proj = proj[valid]

            ints = c1 + l * proj[:, None]
            # reconstruct full-dimensional composition array
            cs = np.concatenate([np.array([1 - np.sum(ints, axis=-1)]).T,
                                 ints], axis=-1)
            # mixing fraction when compositions are normalized
            x = proj / np.dot(c2 - c1, l)
            # mixing fraction when compositions are not normalized
            x_unnormalized = x * n1 / (n2 + x * (n1 - n2))
            num_atoms = n1 + (n2 - n1) * x_unnormalized
            cs *= num_atoms[:, None]
            return [Composition((c, v) for c, v in zip(pd_els, m)) for m in cs]

        def get_element_profile(self, element, comp, comp_tol=1e-5):
            """
            Provides the element evolution data for a composition.
            For example, can be used to analyze Li conversion voltages by varying
            uLi and looking at the phases formed. Also can be used to analyze O2
            evolution by varying uO2.

            Args:
                element: An element. Must be in the phase diagram.
                comp: A Composition
                comp_tol: The tolerance to use when calculating decompositions.
                    Phases with amounts less than this tolerance are excluded.
                    Defaults to 1e-5.

            Returns:
                Evolution data as a list of dictionaries of the following format:
                [ {'chempot': -10.487582010000001, 'evolution': -2.0,
                'reaction': Reaction Object], ...]
            """
            element = get_el_sp(element)

            if element not in self.elements:
                raise ValueError("get_transition_chempots can only be called with"
                                 " elements in the phase diagram.")
            gccomp = Composition({el: amt for el, amt in comp.items()
                                  if el != element})
            elref = self.el_refs[element]
            elcomp = Composition(element.symbol)
            evolution = []

            for cc in self.get_critical_compositions(elcomp, gccomp)[1:]:
                decomp_entries = self.get_decomposition(cc).keys()
                decomp = [k.composition for k in decomp_entries]
                rxn = Reaction([comp], decomp + [elcomp])
                rxn.normalize_to(comp)
                c = self.get_composition_chempots(cc + elcomp * 1e-5)[element]
                amt = -rxn.coeffs[rxn.all_comp.index(elcomp)]
                evolution.append({'chempot': c,
                                  'evolution': amt,
                                  'element_reference': elref,
                                  'reaction': rxn, 'entries': decomp_entries})
            return evolution

        def get_chempot_range_map(self, elements, referenced=True, joggle=True):
            """
            Returns a chemical potential range map for each stable entry.

            Args:
                elements: Sequence of elements to be considered as independent
                    variables. E.g., if you want to show the stability ranges
                    of all Li-Co-O phases wrt to uLi and uO, you will supply
                    [Element("Li"), Element("O")]
                referenced: If True, gives the results with a reference being the
                    energy of the elemental phase. If False, gives absolute values.
                joggle (boolean): Whether to joggle the input to avoid precision
                    errors.

            Returns:
                Returns a dict of the form {entry: [simplices]}. The list of
                simplices are the sides of the N-1 dim polytope bounding the
                allowable chemical potential range of each entry.
            """
            all_chempots = []
            pd = self
            facets = pd.facets
            for facet in facets:
                chempots = self._get_facet_chempots(facet)
                all_chempots.append([chempots[el] for el in pd.elements])
            inds = [pd.elements.index(el) for el in elements]
            el_energies = {el: 0.0 for el in elements}
            if referenced:
                el_energies = {el: pd.el_refs[el].energy_per_atom
                               for el in elements}
            chempot_ranges = collections.defaultdict(list)
            vertices = [list(range(len(self.elements)))]
            if len(all_chempots) > len(self.elements):
                vertices = get_facets(all_chempots, joggle=joggle)
            for ufacet in vertices:
                for combi in itertools.combinations(ufacet, 2):
                    data1 = facets[combi[0]]
                    data2 = facets[combi[1]]
                    common_ent_ind = set(data1).intersection(set(data2))
                    if len(common_ent_ind) == len(elements):
                        common_entries = [pd.qhull_entries[i]
                                          for i in common_ent_ind]
                        data = np.array([[all_chempots[i][j]
                                          - el_energies[pd.elements[j]]
                                          for j in inds] for i in combi])
                        sim = Simplex(data)
                        for entry in common_entries:
                            chempot_ranges[entry].append(sim)

            return chempot_ranges

        def getmu_vertices_stability_phase(self, target_comp, dep_elt, tol_en=1e-2):
            """
            returns a set of chemical potentials corresponding to the vertices of
            the simplex in the chemical potential phase diagram.
            The simplex is built using all elements in the target_composition
            except dep_elt.
            The chemical potential of dep_elt is computed from the target
            composition energy.
            This method is useful to get the limiting conditions for
            defects computations for instance.

            Args:
                target_comp: A Composition object
                dep_elt: the element for which the chemical potential is computed
                    from the energy of
                the stable phase at the target composition
                tol_en: a tolerance on the energy to set

            Returns:
                 [{Element:mu}]: An array of conditions on simplex vertices for
                 which each element has a chemical potential set to a given
                 value. "absolute" values (i.e., not referenced to element energies)
            """
            muref = np.array([self.el_refs[e].energy_per_atom
                              for e in self.elements if e != dep_elt])
            chempot_ranges = self.get_chempot_range_map(
                [e for e in self.elements if e != dep_elt])

            for e in self.elements:
                if e not in target_comp.elements:
                    target_comp = target_comp + Composition({e: 0.0})
            coeff = [-target_comp[e] for e in self.elements if e != dep_elt]
            for e in chempot_ranges.keys():
                if e.composition.reduced_composition == \
                        target_comp.reduced_composition:
                    multiplicator = e.composition[dep_elt] / target_comp[dep_elt]
                    ef = e.energy / multiplicator
                    all_coords = []
                    for s in chempot_ranges[e]:
                        for v in s._coords:
                            elts = [e for e in self.elements if e != dep_elt]
                            res = {}
                            for i in range(len(elts)):
                                res[elts[i]] = v[i] + muref[i]
                            res[dep_elt] = (np.dot(v + muref, coeff) + ef) / target_comp[dep_elt]
                            already_in = False
                            for di in all_coords:
                                dict_equals = True
                                for k in di:
                                    if abs(di[k] - res[k]) > tol_en:
                                        dict_equals = False
                                        break
                                if dict_equals:
                                    already_in = True
                                    break
                            if not already_in:
                                all_coords.append(res)
            return all_coords

        def get_chempot_range_stability_phase(self, target_comp, open_elt):
            """
            returns a set of chemical potentials corresponding to the max and min
            chemical potential of the open element for a given composition. It is
            quite common to have for instance a ternary oxide (e.g., ABO3) for
            which you want to know what are the A and B chemical potential leading
            to the highest and lowest oxygen chemical potential (reducing and
            oxidizing conditions). This is useful for defect computations.

            Args:
                target_comp: A Composition object
                open_elt: Element that you want to constrain to be max or min

            Returns:
                 {Element:(mu_min,mu_max)}: Chemical potentials are given in
                 "absolute" values (i.e., not referenced to 0)
            """
            muref = np.array([self.el_refs[e].energy_per_atom
                              for e in self.elements if e != open_elt])
            chempot_ranges = self.get_chempot_range_map(
                [e for e in self.elements if e != open_elt])
            for e in self.elements:
                if e not in target_comp.elements:
                    target_comp = target_comp + Composition({e: 0.0})
            coeff = [-target_comp[e] for e in self.elements if e != open_elt]
            max_open = -float('inf')
            min_open = float('inf')
            max_mus = None
            min_mus = None
            for e in chempot_ranges.keys():
                if e.composition.reduced_composition == \
                        target_comp.reduced_composition:
                    multiplicator = e.composition[open_elt] / target_comp[open_elt]
                    ef = e.energy / multiplicator
                    all_coords = []
                    for s in chempot_ranges[e]:
                        for v in s._coords:
                            all_coords.append(v)
                            if (np.dot(v + muref, coeff) + ef) / target_comp[open_elt] > max_open:
                                max_open = (np.dot(v + muref, coeff) + ef) / target_comp[open_elt]
                                max_mus = v
                            if (np.dot(v + muref, coeff) + ef) / target_comp[open_elt] < min_open:
                                min_open = (np.dot(v + muref, coeff) + ef) / target_comp[open_elt]
                                min_mus = v
            elts = [e for e in self.elements if e != open_elt]
            res = {}
            for i in range(len(elts)):
                res[elts[i]] = (min_mus[i] + muref[i], max_mus[i] + muref[i])
            res[open_elt] = (min_open, max_open)
            return res

    class GrandPotentialPhaseDiagram(PhaseDiagram):
        """
        A class representing a Grand potential phase diagram. Grand potential phase
        diagrams are essentially phase diagrams that are open to one or more
        components. To construct such phase diagrams, the relevant free energy is
        the grand potential, which can be written as the Legendre transform of the
        Gibbs free energy as follows

        Grand potential = G - u_X N_X

        The algorithm is based on the work in the following papers:

        1. S. P. Ong, L. Wang, B. Kang, and G. Ceder, Li-Fe-P-O2 Phase Diagram from
           First Principles Calculations. Chem. Mater., 2008, 20(5), 1798-1807.
           doi:10.1021/cm702327g

        2. S. P. Ong, A. Jain, G. Hautier, B. Kang, G. Ceder, Thermal stabilities
           of delithiated olivine MPO4 (M=Fe, Mn) cathodes investigated using first
           principles calculations. Electrochem. Comm., 2010, 12(3), 427-430.
           doi:10.1016/j.elecom.2010.01.010
        """

        def __init__(self, entries, chempots, elements=None):
            """
            Standard constructor for grand potential phase diagram.

            Args:
                entries ([PDEntry]): A list of PDEntry-like objects having an
                    energy, energy_per_atom and composition.
                chempots {Element: float}: Specify the chemical potentials
                    of the open elements.
                elements ([Element]): Optional list of elements in the phase
                    diagram. If set to None, the elements are determined from
                    the the entries themselves.
            """
            if elements is None:
                elements = set()
                for entry in entries:
                    elements.update(entry.composition.elements)
            self.chempots = {get_el_sp(el): u for el, u in chempots.items()}
            elements = set(elements).difference(self.chempots.keys())
            all_entries = []
            for e in entries:
                if len(set(e.composition.elements).intersection(set(elements))) > 0:
                    all_entries.append(GrandPotPDEntry(e, self.chempots))
            super().__init__(all_entries, elements)

        def __str__(self):
            output = []
            chemsys = "-".join([el.symbol for el in self.elements])
            output.append("{} grand potential phase diagram with ".format(chemsys))
            output[-1] += ", ".join(["u{}={}".format(el, v)
                                     for el, v in self.chempots.items()])
            output.append("{} stable phases: ".format(len(self.stable_entries)))
            output.append(", ".join([entry.name
                                     for entry in self.stable_entries]))
            return "\n".join(output)

        def as_dict(self):
            """
            :return: MSONable dict
            """
            return {"@module": self.__class__.__module__,
                    "@class": self.__class__.__name__,
                    "all_entries": [e.as_dict() for e in self.all_entries],
                    "chempots": self.chempots,
                    "elements": [e.as_dict() for e in self.elements]}

        @classmethod
        def from_dict(cls, d):
            """
            :param d: Dict representation
            :return: GrandPotentialPhaseDiagram
            """
            entries = MontyDecoder().process_decoded(d["all_entries"])
            elements = MontyDecoder().process_decoded(d["elements"])
            return cls(entries, d["chempots"], elements)

    class CompoundPhaseDiagram(PhaseDiagram):
        """
        Generates phase diagrams from compounds as terminations instead of
        elements.
        """

        # Tolerance for determining if amount of a composition is positive.
        amount_tol = 1e-5

        def __init__(self, entries, terminal_compositions,
                     normalize_terminal_compositions=True):
            """
            Initializes a CompoundPhaseDiagram.

            Args:
                entries ([PDEntry]): Sequence of input entries. For example,
                   if you want a Li2O-P2O5 phase diagram, you might have all
                   Li-P-O entries as an input.
                terminal_compositions ([Composition]): Terminal compositions of
                    phase space. In the Li2O-P2O5 example, these will be the
                    Li2O and P2O5 compositions.
                normalize_terminal_compositions (bool): Whether to normalize the
                    terminal compositions to a per atom basis. If normalized,
                    the energy above hulls will be consistent
                    for comparison across systems. Non-normalized terminals are
                    more intuitive in terms of compositional breakdowns.
            """
            self.original_entries = entries
            self.terminal_compositions = terminal_compositions
            self.normalize_terminals = normalize_terminal_compositions
            (pentries, species_mapping) = \
                self.transform_entries(entries, terminal_compositions)
            self.species_mapping = species_mapping
            super().__init__(
                pentries, elements=species_mapping.values())

        def transform_entries(self, entries, terminal_compositions):
            """
            Method to transform all entries to the composition coordinate in the
            terminal compositions. If the entry does not fall within the space
            defined by the terminal compositions, they are excluded. For example,
            Li3PO4 is mapped into a Li2O:1.5, P2O5:0.5 composition. The terminal
            compositions are represented by DummySpecies.

            Args:
                entries: Sequence of all input entries
                terminal_compositions: Terminal compositions of phase space.

            Returns:
                Sequence of TransformedPDEntries falling within the phase space.
            """
            new_entries = []
            if self.normalize_terminals:
                fractional_comp = [c.fractional_composition
                                   for c in terminal_compositions]
            else:
                fractional_comp = terminal_compositions

            # Map terminal compositions to unique dummy species.
            sp_mapping = collections.OrderedDict()
            for i, comp in enumerate(fractional_comp):
                sp_mapping[comp] = DummySpecie("X" + chr(102 + i))

            for entry in entries:
                try:
                    rxn = Reaction(fractional_comp, [entry.composition])
                    rxn.normalize_to(entry.composition)
                    # We only allow reactions that have positive amounts of
                    # reactants.
                    if all([rxn.get_coeff(comp) <= CompoundPhaseDiagram.amount_tol
                            for comp in fractional_comp]):
                        newcomp = {sp_mapping[comp]: -rxn.get_coeff(comp)
                                   for comp in fractional_comp}
                        newcomp = {k: v for k, v in newcomp.items()
                                   if v > CompoundPhaseDiagram.amount_tol}
                        transformed_entry = \
                            TransformedPDEntry(Composition(newcomp), entry)
                        new_entries.append(transformed_entry)
                except ReactionError:
                    # If the reaction can't be balanced, the entry does not fall
                    # into the phase space. We ignore them.
                    pass
            return new_entries, sp_mapping

        def as_dict(self):
            """
            :return: MSONable dict
            """
            return {
                "@module": self.__class__.__module__,
                "@class": self.__class__.__name__,
                "original_entries": [e.as_dict() for e in self.original_entries],
                "terminal_compositions": [c.as_dict()
                                          for c in self.terminal_compositions],
                "normalize_terminal_compositions":
                    self.normalize_terminals}

        @classmethod
        def from_dict(cls, d):
            """
            :param d: Dict Representation
            :return: CompoundPhaseDiagram
            """
            dec = MontyDecoder()
            entries = dec.process_decoded(d["original_entries"])
            terminal_compositions = dec.process_decoded(d["terminal_compositions"])
            return cls(entries, terminal_compositions,
                       d["normalize_terminal_compositions"])

    class ReactionDiagram:
        """
        Analyzes the possible reactions between a pair of compounds, e.g.,
        an electrolyte and an electrode.
        """

        def __init__(self, entry1, entry2, all_entries, tol=1e-4,
                     float_fmt="%.4f"):
            """
            Args:
                entry1 (ComputedEntry): Entry for 1st component. Note that
                    corrections, if any, must already be pre-applied. This is to
                    give flexibility for different kinds of corrections, e.g.,
                    if a particular entry is fitted to an experimental data (such
                    as EC molecule).
                entry2 (ComputedEntry): Entry for 2nd component. Note that
                    corrections must already be pre-applied. This is to
                    give flexibility for different kinds of corrections, e.g.,
                    if a particular entry is fitted to an experimental data (such
                    as EC molecule).
                all_entries ([ComputedEntry]): All other entries to be
                    considered in the analysis. Note that corrections, if any,
                    must already be pre-applied.
                tol (float): Tolerance to be used to determine validity of reaction.
                float_fmt (str): Formatting string to be applied to all floats.
                    Determines number of decimal places in reaction string.
            """
            elements = set()
            for e in [entry1, entry2]:
                elements.update([el.symbol for el in e.composition.elements])

            elements = tuple(elements)  # Fix elements to ensure order.

            comp_vec1 = np.array([entry1.composition.get_atomic_fraction(el)
                                  for el in elements])
            comp_vec2 = np.array([entry2.composition.get_atomic_fraction(el)
                                  for el in elements])
            r1 = entry1.composition.reduced_composition
            r2 = entry2.composition.reduced_composition

            logger.debug("%d total entries." % len(all_entries))

            pd = PhaseDiagram(all_entries + [entry1, entry2])
            terminal_formulas = [entry1.composition.reduced_formula,
                                 entry2.composition.reduced_formula]

            logger.debug("%d stable entries" % len(pd.stable_entries))
            logger.debug("%d facets" % len(pd.facets))
            logger.debug("%d qhull_entries" % len(pd.qhull_entries))

            rxn_entries = []
            done = []

            def fmt(fl):
                return float_fmt % fl

            for facet in pd.facets:
                for face in itertools.combinations(facet, len(facet) - 1):
                    face_entries = [pd.qhull_entries[i] for i in face]

                    if any([e.composition.reduced_formula in terminal_formulas
                            for e in face_entries]):
                        continue

                    try:

                        m = []
                        for e in face_entries:
                            m.append([e.composition.get_atomic_fraction(el)
                                      for el in elements])
                        m.append(comp_vec2 - comp_vec1)
                        m = np.array(m).T
                        coeffs = np.linalg.solve(m, comp_vec2)

                        x = coeffs[-1]

                        if all([c >= -tol for c in coeffs]) and \
                                (abs(sum(coeffs[:-1]) - 1) < tol) and \
                                (tol < x < 1 - tol):

                            c1 = x / r1.num_atoms
                            c2 = (1 - x) / r2.num_atoms
                            factor = 1 / (c1 + c2)

                            c1 *= factor
                            c2 *= factor

                            # Avoid duplicate reactions.
                            if any([np.allclose([c1, c2], cc) for cc in done]):
                                continue

                            done.append((c1, c2))

                            rxn_str = "%s %s + %s %s -> " % (
                                fmt(c1), r1.reduced_formula,
                                fmt(c2), r2.reduced_formula)
                            products = []
                            product_entries = []

                            energy = - (x * entry1.energy_per_atom +
                                        (1 - x) * entry2.energy_per_atom)

                            for c, e in zip(coeffs[:-1], face_entries):
                                if c > tol:
                                    r = e.composition.reduced_composition
                                    products.append("%s %s" % (
                                        fmt(c / r.num_atoms * factor),
                                        r.reduced_formula))
                                    product_entries.append((c, e))
                                    energy += c * e.energy_per_atom

                            rxn_str += " + ".join(products)
                            comp = x * comp_vec1 + (1 - x) * comp_vec2
                            entry = PDEntry(
                                Composition(dict(zip(elements, comp))),
                                energy=energy, attribute=rxn_str)
                            entry.decomposition = product_entries
                            rxn_entries.append(entry)
                    except np.linalg.LinAlgError:
                        logger.debug("Reactants = %s" % (", ".join([
                            entry1.composition.reduced_formula,
                            entry2.composition.reduced_formula])))
                        logger.debug("Products = %s" % (
                            ", ".join([e.composition.reduced_formula
                                       for e in face_entries])))

            rxn_entries = sorted(rxn_entries, key=lambda e: e.name, reverse=True)

            self.entry1 = entry1
            self.entry2 = entry2
            self.rxn_entries = rxn_entries
            self.labels = collections.OrderedDict()
            for i, e in enumerate(rxn_entries):
                self.labels[str(i + 1)] = e.attribute
                e.name = str(i + 1)
            self.all_entries = all_entries
            self.pd = pd

        def get_compound_pd(self):
            """
            Get the CompoundPhaseDiagram object, which can then be used for
            plotting.

            Returns:
                (CompoundPhaseDiagram)
            """
            # For this plot, since the reactions are reported in formation
            # energies, we need to set the energies of the terminal compositions
            # to 0. So we make create copies with 0 energy.
            entry1 = PDEntry(self.entry1.composition, 0)
            entry2 = PDEntry(self.entry2.composition, 0)

            cpd = CompoundPhaseDiagram(
                self.rxn_entries + [entry1, entry2],
                [Composition(entry1.composition.reduced_formula),
                 Composition(entry2.composition.reduced_formula)],
                normalize_terminal_compositions=False)
            return cpd

    class PhaseDiagramError(Exception):
        """
        An exception class for Phase Diagram generation.
        """
        pass

    def get_facets(qhull_data, joggle=False):
        """
        Get the simplex facets for the Convex hull.

        Args:
            qhull_data (np.ndarray): The data from which to construct the convex
                hull as a Nxd array (N being number of data points and d being the
                dimension)
            joggle (boolean): Whether to joggle the input to avoid precision
                errors.

        Returns:
            List of simplices of the Convex Hull.
        """
        if joggle:
            return ConvexHull(qhull_data, qhull_options="QJ i").simplices
        else:
            return ConvexHull(qhull_data, qhull_options="Qt i").simplices

    class PDPlotter:
        """
        A plotter class for phase diagrams.
        """

        def __init__(self, phasediagram, show_unstable=0, **plotkwargs):
            r"""

            Args:
                phasediagram: PhaseDiagram object.
                show_unstable (float): Whether unstable phases will be plotted as
                    well as red crosses. If a number > 0 is entered, all phases with
                    ehull < show_unstable will be shown.
                **plotkwargs: Keyword args passed to matplotlib.pyplot.plot. Can
                    be used to customize markers etc. If not set, the default is
                    {
                        "markerfacecolor": (0.2157, 0.4941, 0.7216),
                        "markersize": 10,
                        "linewidth": 3
                    }
            """
            # note: palettable imports matplotlib
            from palettable.colorbrewer.qualitative import Set1_3

            self._pd = phasediagram
            self._dim = len(self._pd.elements)
            if self._dim > 4:
                raise ValueError("Only 1-4 components supported!")
            self.lines = uniquelines(self._pd.facets) if self._dim > 1 else \
                [[self._pd.facets[0][0], self._pd.facets[0][0]]]
            self.show_unstable = show_unstable
            colors = Set1_3.mpl_colors
            self.plotkwargs = plotkwargs or {
                "markerfacecolor": colors[2],
                "markersize": 10,
                "linewidth": 3
            }

        @property
        def pd_plot_data(self):
            """
            Plot data for phase diagram.
            2-comp - Full hull with energies
            3/4-comp - Projection into 2D or 3D Gibbs triangle.

            Returns:
                (lines, stable_entries, unstable_entries):
                - lines is a list of list of coordinates for lines in the PD.
                - stable_entries is a {coordinate : entry} for each stable node
                in the phase diagram. (Each coordinate can only have one
                stable phase)
                - unstable_entries is a {entry: coordinates} for all unstable
                nodes in the phase diagram.
            """
            pd = self._pd
            entries = pd.qhull_entries
            data = np.array(pd.qhull_data)
            lines = []
            stable_entries = {}
            for line in self.lines:
                entry1 = entries[line[0]]
                entry2 = entries[line[1]]
                if self._dim < 3:
                    x = [data[line[0]][0], data[line[1]][0]]
                    y = [pd.get_form_energy_per_atom(entry1),
                         pd.get_form_energy_per_atom(entry2)]
                    coord = [x, y]
                elif self._dim == 3:
                    coord = triangular_coord(data[line, 0:2])
                else:
                    coord = tet_coord(data[line, 0:3])
                lines.append(coord)
                labelcoord = list(zip(*coord))
                stable_entries[labelcoord[0]] = entry1
                stable_entries[labelcoord[1]] = entry2

            all_entries = pd.all_entries
            all_data = np.array(pd.all_entries_hulldata)
            unstable_entries = dict()
            stable = pd.stable_entries
            for i in range(0, len(all_entries)):
                entry = all_entries[i]
                if entry not in stable:
                    if self._dim < 3:
                        x = [all_data[i][0], all_data[i][0]]
                        y = [pd.get_form_energy_per_atom(entry),
                             pd.get_form_energy_per_atom(entry)]
                        coord = [x, y]
                    elif self._dim == 3:
                        coord = triangular_coord([all_data[i, 0:2],
                                                  all_data[i, 0:2]])
                    else:
                        coord = tet_coord([all_data[i, 0:3], all_data[i, 0:3],
                                           all_data[i, 0:3]])
                    labelcoord = list(zip(*coord))
                    unstable_entries[entry] = labelcoord[0]

            return lines, stable_entries, unstable_entries

        def get_plot(self, label_stable=True, label_unstable=True, ordering=None,
                     energy_colormap=None, process_attributes=False, plt=None):
            """
            :param label_stable: Whether to label stable compounds.
            :param label_unstable: Whether to label unstable compounds.
            :param ordering: Ordering of vertices.
            :param energy_colormap: Colormap for coloring energy.
            :param process_attributes: Whether to process the attributes.
            :param plt: Existing plt object if plotting multiple phase diagrams.
            :return: matplotlib.pyplot.
            """
            if self._dim < 4:
                plt = self._get_2d_plot(label_stable, label_unstable, ordering,
                                        energy_colormap, plt=plt,
                                        process_attributes=process_attributes)
            elif self._dim == 4:
                plt = self._get_3d_plot(label_stable)

            return plt

        def plot_element_profile(self, element, comp, show_label_index=None,
                                 xlim=5):
            """
            Draw the element profile plot for a composition varying different
            chemical potential of an element.
            X value is the negative value of the chemical potential reference to
            elemental chemical potential. For example, if choose Element("Li"),
            X= -(µLi-µLi0), which corresponds to the voltage versus metal anode.
            Y values represent for the number of element uptake in this composition
            (unit: per atom). All reactions are printed to help choosing the
            profile steps you want to show label in the plot.

            Args:
             element (Element): An element of which the chemical potential is
                considered. It also must be in the phase diagram.
             comp (Composition): A composition.
             show_label_index (list of integers): The labels for reaction products
                you want to show in the plot. Default to None (not showing any
                annotation for reaction products). For the profile steps you want
                to show the labels, just add it to the show_label_index. The
                profile step counts from zero. For example, you can set
                show_label_index=[0, 2, 5] to label profile step 0,2,5.
             xlim (float): The max x value. x value is from 0 to xlim. Default to
                5 eV.

            Returns:
                Plot of element profile evolution by varying the chemical potential
                of an element.
            """
            plt = pretty_plot(12, 8)
            pd = self._pd
            evolution = pd.get_element_profile(element, comp)
            num_atoms = evolution[0]["reaction"].reactants[0].num_atoms
            element_energy = evolution[0]['chempot']
            x1, x2, y1 = None, None, None
            for i, d in enumerate(evolution):
                v = -(d["chempot"] - element_energy)
                if i != 0:
                    plt.plot([x2, x2], [y1, d["evolution"] / num_atoms],
                             'k', linewidth=2.5)
                x1 = v
                y1 = d["evolution"] / num_atoms

                if i != len(evolution) - 1:
                    x2 = - (evolution[i + 1]["chempot"] - element_energy)
                else:
                    x2 = 5.0
                if show_label_index is not None and i in show_label_index:
                    products = [re.sub(r"(\d+)", r"$_{\1}$", p.reduced_formula)
                                for p in d["reaction"].products
                                if p.reduced_formula != element.symbol]
                    plt.annotate(", ".join(products), xy=(v + 0.05, y1 + 0.05),
                                 fontsize=24, color='r')
                    plt.plot([x1, x2], [y1, y1], 'r', linewidth=3)
                else:
                    plt.plot([x1, x2], [y1, y1], 'k', linewidth=2.5)

            plt.xlim((0, xlim))
            plt.xlabel("-$\\Delta{\\mu}$ (eV)")
            plt.ylabel("Uptake per atom")

            return plt

        def show(self, *args, **kwargs):
            r"""
            Draws the phase diagram using Matplotlib and show it.

            Args:
                *args: Passed to get_plot.
                **kwargs: Passed to get_plot.
            """
            self.get_plot(*args, **kwargs).show()

        def _get_2d_plot(self, label_stable=True, label_unstable=True,
                         ordering=None, energy_colormap=None, vmin_mev=-60.0,
                         vmax_mev=60.0, show_colorbar=True,
                         process_attributes=False, plt=None):
            """
            Shows the plot using pylab.  Usually I won't do imports in methods,
            but since plotting is a fairly expensive library to load and not all
            machines have matplotlib installed, I have done it this way.
            """
            if plt is None:
                plt = pretty_plot(8, 6)
            from matplotlib.font_manager import FontProperties
            if ordering is None:
                (lines, labels, unstable) = self.pd_plot_data
            else:
                (_lines, _labels, _unstable) = self.pd_plot_data
                (lines, labels, unstable) = order_phase_diagram(
                    _lines, _labels, _unstable, ordering)
            if energy_colormap is None:
                if process_attributes:
                    for x, y in lines:
                        plt.plot(x, y, "k-", linewidth=3, markeredgecolor="k")
                    # One should think about a clever way to have "complex"
                    # attributes with complex processing options but with a clear
                    # logic. At this moment, I just use the attributes to know
                    # whether an entry is a new compound or an existing (from the
                    #  ICSD or from the MP) one.
                    for x, y in labels.keys():
                        if labels[(x, y)].attribute is None or \
                                labels[(x, y)].attribute == "existing":
                            plt.plot(x, y, "ko", **self.plotkwargs)
                        else:
                            plt.plot(x, y, "k*", **self.plotkwargs)
                else:
                    for x, y in lines:
                        plt.plot(x, y, "ko-", **self.plotkwargs)
            else:
                from matplotlib.colors import Normalize, LinearSegmentedColormap
                from matplotlib.cm import ScalarMappable
                for x, y in lines:
                    plt.plot(x, y, "k-", markeredgecolor="k")
                vmin = vmin_mev / 1000.0
                vmax = vmax_mev / 1000.0
                if energy_colormap == 'default':
                    mid = - vmin / (vmax - vmin)
                    cmap = LinearSegmentedColormap.from_list(
                        'my_colormap', [(0.0, '#005500'), (mid, '#55FF55'),
                                        (mid, '#FFAAAA'), (1.0, '#FF0000')])
                else:
                    cmap = energy_colormap
                norm = Normalize(vmin=vmin, vmax=vmax)
                _map = ScalarMappable(norm=norm, cmap=cmap)
                _energies = [self._pd.get_equilibrium_reaction_energy(entry)
                             for coord, entry in labels.items()]
                energies = [en if en < 0.0 else -0.00000001 for en in _energies]
                vals_stable = _map.to_rgba(energies)
                ii = 0
                if process_attributes:
                    for x, y in labels.keys():
                        if labels[(x, y)].attribute is None or \
                                labels[(x, y)].attribute == "existing":
                            plt.plot(x, y, "o", markerfacecolor=vals_stable[ii],
                                     markersize=12)
                        else:
                            plt.plot(x, y, "*", markerfacecolor=vals_stable[ii],
                                     markersize=18)
                        ii += 1
                else:
                    for x, y in labels.keys():
                        plt.plot(x, y, "o", markerfacecolor=vals_stable[ii],
                                 markersize=15)
                        ii += 1

            font = FontProperties()
            font.set_weight("bold")
            font.set_size(24)

            # Sets a nice layout depending on the type of PD. Also defines a
            # "center" for the PD, which then allows the annotations to be spread
            # out in a nice manner.
            if len(self._pd.elements) == 3:
                plt.axis("equal")
                plt.xlim((-0.1, 1.2))
                plt.ylim((-0.1, 1.0))
                plt.axis("off")
                center = (0.5, math.sqrt(3) / 6)
            else:
                all_coords = labels.keys()
                miny = min([c[1] for c in all_coords])
                ybuffer = max(abs(miny) * 0.1, 0.1)
                plt.xlim((-0.1, 1.1))
                plt.ylim((miny - ybuffer, ybuffer))
                center = (0.5, miny / 2)
                plt.xlabel("Fraction", fontsize=28, fontweight='bold')
                plt.ylabel("Formation energy (eV/atom)", fontsize=28,
                           fontweight='bold')

            for coords in sorted(labels.keys(), key=lambda x: -x[1]):
                entry = labels[coords]
                label = entry.name

                # The follow defines an offset for the annotation text emanating
                # from the center of the PD. Results in fairly nice layouts for the
                # most part.
                vec = (np.array(coords) - center)
                vec = vec / np.linalg.norm(vec) * 10 if np.linalg.norm(vec) != 0 \
                    else vec
                valign = "bottom" if vec[1] > 0 else "top"
                if vec[0] < -0.01:
                    halign = "right"
                elif vec[0] > 0.01:
                    halign = "left"
                else:
                    halign = "center"
                if label_stable:
                    if process_attributes and entry.attribute == 'new':
                        plt.annotate(latexify(label), coords, xytext=vec,
                                     textcoords="offset points",
                                     horizontalalignment=halign,
                                     verticalalignment=valign,
                                     fontproperties=font,
                                     color='g')
                    else:
                        plt.annotate(latexify(label), coords, xytext=vec,
                                     textcoords="offset points",
                                     horizontalalignment=halign,
                                     verticalalignment=valign,
                                     fontproperties=font)

            if self.show_unstable:
                font = FontProperties()
                font.set_size(16)
                energies_unstable = [self._pd.get_e_above_hull(entry)
                                     for entry, coord in unstable.items()]
                if energy_colormap is not None:
                    energies.extend(energies_unstable)
                    vals_unstable = _map.to_rgba(energies_unstable)
                ii = 0
                for entry, coords in unstable.items():
                    ehull = self._pd.get_e_above_hull(entry)
                    if ehull < self.show_unstable:
                        vec = (np.array(coords) - center)
                        vec = vec / np.linalg.norm(vec) * 10 \
                            if np.linalg.norm(vec) != 0 else vec
                        label = entry.name
                        if energy_colormap is None:
                            plt.plot(coords[0], coords[1], "ks", linewidth=3,
                                     markeredgecolor="k", markerfacecolor="r",
                                     markersize=8)
                        else:
                            plt.plot(coords[0], coords[1], "s", linewidth=3,
                                     markeredgecolor="k",
                                     markerfacecolor=vals_unstable[ii],
                                     markersize=8)
                        if label_unstable:
                            plt.annotate(latexify(label), coords, xytext=vec,
                                         textcoords="offset points",
                                         horizontalalignment=halign, color="b",
                                         verticalalignment=valign,
                                         fontproperties=font)
                        ii += 1
            if energy_colormap is not None and show_colorbar:
                _map.set_array(energies)
                cbar = plt.colorbar(_map)
                cbar.set_label(
                    'Energy [meV/at] above hull (in red)\nInverse energy ['
                    'meV/at] above hull (in green)',
                    rotation=-90, ha='left', va='center')
            f = plt.gcf()
            f.set_size_inches((8, 6))
            plt.subplots_adjust(left=0.09, right=0.98, top=0.98, bottom=0.07)
            return plt

        def _get_3d_plot(self, label_stable=True):
            """
            Shows the plot using pylab.  Usually I won"t do imports in methods,
            but since plotting is a fairly expensive library to load and not all
            machines have matplotlib installed, I have done it this way.
            """
            import matplotlib.pyplot as plt
            import mpl_toolkits.mplot3d.axes3d as p3
            from matplotlib.font_manager import FontProperties
            fig = plt.figure()
            ax = p3.Axes3D(fig)
            font = FontProperties()
            font.set_weight("bold")
            font.set_size(20)
            (lines, labels, unstable) = self.pd_plot_data
            count = 1
            newlabels = list()
            for x, y, z in lines:
                ax.plot(x, y, z, "bo-", linewidth=3, markeredgecolor="b",
                        markerfacecolor="r", markersize=10)
            for coords in sorted(labels.keys()):
                entry = labels[coords]
                label = entry.name
                if label_stable:
                    if len(entry.composition.elements) == 1:
                        ax.text(coords[0], coords[1], coords[2], label)
                    else:
                        ax.text(coords[0], coords[1], coords[2], str(count))
                        newlabels.append("{} : {}".format(count, latexify(label)))
                        count += 1
            plt.figtext(0.01, 0.01, "\n".join(newlabels))
            ax.axis("off")
            return plt

        def write_image(self, stream, image_format="svg", **kwargs):
            r"""
            Writes the phase diagram to an image in a stream.

            Args:
                stream:
                    stream to write to. Can be a file stream or a StringIO stream.
                image_format
                    format for image. Can be any of matplotlib supported formats.
                    Defaults to svg for best results for vector graphics.
                **kwargs: Pass through to get_plot functino.
            """
            plt = self.get_plot(**kwargs)

            f = plt.gcf()
            f.set_size_inches((12, 10))

            plt.savefig(stream, format=image_format)

        def plot_chempot_range_map(self, elements, referenced=True):
            """
            Plot the chemical potential range _map. Currently works only for
            3-component PDs.

            Args:
                elements: Sequence of elements to be considered as independent
                    variables. E.g., if you want to show the stability ranges of
                    all Li-Co-O phases wrt to uLi and uO, you will supply
                    [Element("Li"), Element("O")]
                referenced: if True, gives the results with a reference being the
                            energy of the elemental phase. If False, gives absolute values.
            """
            self.get_chempot_range_map_plot(elements, referenced=referenced).show()

        def get_chempot_range_map_plot(self, elements, referenced=True):
            """
            Returns a plot of the chemical potential range _map. Currently works
            only for 3-component PDs.

            Args:
                elements: Sequence of elements to be considered as independent
                    variables. E.g., if you want to show the stability ranges of
                    all Li-Co-O phases wrt to uLi and uO, you will supply
                    [Element("Li"), Element("O")]
                referenced: if True, gives the results with a reference being the
                            energy of the elemental phase. If False, gives absolute values.

            Returns:
                A matplotlib plot object.
            """

            plt = pretty_plot(12, 8)
            chempot_ranges = self._pd.get_chempot_range_map(
                elements, referenced=referenced)
            missing_lines = {}
            excluded_region = []
            for entry, lines in chempot_ranges.items():
                comp = entry.composition
                center_x = 0
                center_y = 0
                coords = []
                contain_zero = any([comp.get_atomic_fraction(el) == 0
                                    for el in elements])
                is_boundary = (not contain_zero) and sum([comp.get_atomic_fraction(el) for el in elements]) == 1
                for line in lines:
                    (x, y) = line.coords.transpose()
                    plt.plot(x, y, "k-")

                    for coord in line.coords:
                        if not in_coord_list(coords, coord):
                            coords.append(coord.tolist())
                            center_x += coord[0]
                            center_y += coord[1]
                    if is_boundary:
                        excluded_region.extend(line.coords)

                if coords and contain_zero:
                    missing_lines[entry] = coords
                else:
                    xy = (center_x / len(coords), center_y / len(coords))
                    plt.annotate(latexify(entry.name), xy, fontsize=22)

            ax = plt.gca()
            xlim = ax.get_xlim()
            ylim = ax.get_ylim()

            # Shade the forbidden chemical potential regions.
            excluded_region.append([xlim[1], ylim[1]])
            excluded_region = sorted(excluded_region, key=lambda c: c[0])
            (x, y) = np.transpose(excluded_region)
            plt.fill(x, y, "0.80")

            # The hull does not generate the missing horizontal and vertical lines.
            # The following code fixes this.
            el0 = elements[0]
            el1 = elements[1]
            for entry, coords in missing_lines.items():
                center_x = sum([c[0] for c in coords])
                center_y = sum([c[1] for c in coords])
                comp = entry.composition
                is_x = comp.get_atomic_fraction(el0) < 0.01
                is_y = comp.get_atomic_fraction(el1) < 0.01
                n = len(coords)
                if not (is_x and is_y):
                    if is_x:
                        coords = sorted(coords, key=lambda c: c[1])
                        for i in [0, -1]:
                            x = [min(xlim), coords[i][0]]
                            y = [coords[i][1], coords[i][1]]
                            plt.plot(x, y, "k")
                            center_x += min(xlim)
                            center_y += coords[i][1]
                    elif is_y:
                        coords = sorted(coords, key=lambda c: c[0])
                        for i in [0, -1]:
                            x = [coords[i][0], coords[i][0]]
                            y = [coords[i][1], min(ylim)]
                            plt.plot(x, y, "k")
                            center_x += coords[i][0]
                            center_y += min(ylim)
                    xy = (center_x / (n + 2), center_y / (n + 2))
                else:
                    center_x = sum(coord[0] for coord in coords) + xlim[0]
                    center_y = sum(coord[1] for coord in coords) + ylim[0]
                    xy = (center_x / (n + 1), center_y / (n + 1))

                plt.annotate(latexify(entry.name), xy,
                             horizontalalignment="center",
                             verticalalignment="center", fontsize=22)

            plt.xlabel("$\\mu_{{{0}}} - \\mu_{{{0}}}^0$ (eV)"
                       .format(el0.symbol))
            plt.ylabel("$\\mu_{{{0}}} - \\mu_{{{0}}}^0$ (eV)"
                       .format(el1.symbol))
            plt.tight_layout()
            return plt

        def get_contour_pd_plot(self):
            """
            Plot a contour phase diagram plot, where phase triangles are colored
            according to degree of instability by interpolation. Currently only
            works for 3-component phase diagrams.

            Returns:
                A matplotlib plot object.
            """
            from scipy import interpolate
            from matplotlib import cm

            pd = self._pd
            entries = pd.qhull_entries
            data = np.array(pd.qhull_data)

            plt = self._get_2d_plot()
            data[:, 0:2] = triangular_coord(data[:, 0:2]).transpose()
            for i, e in enumerate(entries):
                data[i, 2] = self._pd.get_e_above_hull(e)

            gridsize = 0.005
            xnew = np.arange(0, 1., gridsize)
            ynew = np.arange(0, 1, gridsize)

            f = interpolate.LinearNDInterpolator(data[:, 0:2], data[:, 2])
            znew = np.zeros((len(ynew), len(xnew)))
            for (i, xval) in enumerate(xnew):
                for (j, yval) in enumerate(ynew):
                    znew[j, i] = f(xval, yval)

            plt.contourf(xnew, ynew, znew, 1000, cmap=cm.autumn_r)

            plt.colorbar()
            return plt

    def uniquelines(q):
        """
        Given all the facets, convert it into a set of unique lines.  Specifically
        used for converting convex hull facets into line pairs of coordinates.

        Args:
            q: A 2-dim sequence, where each row represents a facet. E.g.,
                [[1,2,3],[3,6,7],...]

        Returns:
            setoflines:
                A set of tuple of lines.  E.g., ((1,2), (1,3), (2,3), ....)
        """
        setoflines = set()
        for facets in q:
            for line in itertools.combinations(facets, 2):
                setoflines.add(tuple(sorted(line)))
        return setoflines

    def triangular_coord(coord):
        """
        Convert a 2D coordinate into a triangle-based coordinate system for a
        prettier phase diagram.

        Args:
            coordinate: coordinate used in the convex hull computation.

        Returns:
            coordinates in a triangular-based coordinate system.
        """
        unitvec = np.array([[1, 0], [0.5, math.sqrt(3) / 2]])
        result = np.dot(np.array(coord), unitvec)
        return result.transpose()

    def tet_coord(coord):
        """
        Convert a 3D coordinate into a tetrahedron based coordinate system for a
        prettier phase diagram.

        Args:
            coordinate: coordinate used in the convex hull computation.

        Returns:
            coordinates in a tetrahedron-based coordinate system.
        """
        unitvec = np.array([[1, 0, 0], [0.5, math.sqrt(3) / 2, 0],
                            [0.5, 1.0 / 3.0 * math.sqrt(3) / 2, math.sqrt(6) / 3]])
        result = np.dot(np.array(coord), unitvec)
        return result.transpose()

    def order_phase_diagram(lines, stable_entries, unstable_entries, ordering):
        """
        Orders the entries (their coordinates) in a phase diagram plot according
        to the user specified ordering.
        Ordering should be given as ['Up', 'Left', 'Right'], where Up,
        Left and Right are the names of the entries in the upper, left and right
        corners of the triangle respectively.

        Args:
            lines: list of list of coordinates for lines in the PD.
            stable_entries: {coordinate : entry} for each stable node in the
                phase diagram. (Each coordinate can only have one stable phase)
            unstable_entries: {entry: coordinates} for all unstable nodes in the
                phase diagram.
            ordering: Ordering of the phase diagram, given as a list ['Up',
                'Left','Right']

        Returns:
            (newlines, newstable_entries, newunstable_entries):
            - newlines is a list of list of coordinates for lines in the PD.
            - newstable_entries is a {coordinate : entry} for each stable node
            in the phase diagram. (Each coordinate can only have one
            stable phase)
            - newunstable_entries is a {entry: coordinates} for all unstable
            nodes in the phase diagram.
        """
        yup = -1000.0
        xleft = 1000.0
        xright = -1000.0

        for coord in stable_entries:
            if coord[0] > xright:
                xright = coord[0]
                nameright = stable_entries[coord].name
            if coord[0] < xleft:
                xleft = coord[0]
                nameleft = stable_entries[coord].name
            if coord[1] > yup:
                yup = coord[1]
                nameup = stable_entries[coord].name

        if (nameup not in ordering) or (nameright not in ordering) or (nameleft not in ordering):
            raise ValueError(
                'Error in ordering_phase_diagram : \n"{up}", "{left}" and "{'
                'right}"'
                ' should be in ordering : {ord}'.format(up=nameup, left=nameleft,
                                                        right=nameright,
                                                        ord=ordering))

        cc = np.array([0.5, np.sqrt(3.0) / 6.0], np.float)

        if nameup == ordering[0]:
            if nameleft == ordering[1]:
                # The coordinates were already in the user ordering
                return lines, stable_entries, unstable_entries
            else:
                newlines = [[np.array(1.0 - x), y] for x, y in lines]
                newstable_entries = {(1.0 - c[0], c[1]): entry
                                     for c, entry in stable_entries.items()}
                newunstable_entries = {entry: (1.0 - c[0], c[1])
                                       for entry, c in
                                       unstable_entries.items()}
                return newlines, newstable_entries, newunstable_entries
        elif nameup == ordering[1]:
            if nameleft == ordering[2]:
                c120 = np.cos(2.0 * np.pi / 3.0)
                s120 = np.sin(2.0 * np.pi / 3.0)
                newlines = []
                for x, y in lines:
                    newx = np.zeros_like(x)
                    newy = np.zeros_like(y)
                    for ii, xx in enumerate(x):
                        newx[ii] = c120 * (xx - cc[0]) - s120 * (y[ii] - cc[1]) + cc[0]
                        newy[ii] = s120 * (xx - cc[0]) + c120 * (y[ii] - cc[1]) + cc[1]
                    newlines.append([newx, newy])
                newstable_entries = {
                    (c120 * (c[0] - cc[0]) - s120 * (c[1] - cc[1]) + cc[0],
                     s120 * (c[0] - cc[0]) + c120 * (c[1] - cc[1]) + cc[1]): entry
                    for c, entry in stable_entries.items()}
                newunstable_entries = {
                    entry: (c120 * (c[0] - cc[0]) - s120 * (c[1] - cc[1]) + cc[0],
                            s120 * (c[0] - cc[0]) + c120 * (c[1] - cc[1]) + cc[1])
                    for entry, c in unstable_entries.items()}
                return newlines, newstable_entries, newunstable_entries
            else:
                c120 = np.cos(2.0 * np.pi / 3.0)
                s120 = np.sin(2.0 * np.pi / 3.0)
                newlines = []
                for x, y in lines:
                    newx = np.zeros_like(x)
                    newy = np.zeros_like(y)
                    for ii, xx in enumerate(x):
                        newx[ii] = -c120 * (xx - 1.0) - s120 * y[ii] + 1.0
                        newy[ii] = -s120 * (xx - 1.0) + c120 * y[ii]
                    newlines.append([newx, newy])
                newstable_entries = {(-c120 * (c[0] - 1.0) - s120 * c[1] + 1.0,
                                      -s120 * (c[0] - 1.0) + c120 * c[1]): entry
                                     for c, entry in stable_entries.items()}
                newunstable_entries = {
                    entry: (-c120 * (c[0] - 1.0) - s120 * c[1] + 1.0,
                            -s120 * (c[0] - 1.0) + c120 * c[1])
                    for entry, c in unstable_entries.items()}
                return newlines, newstable_entries, newunstable_entries
        elif nameup == ordering[2]:
            if nameleft == ordering[0]:
                c240 = np.cos(4.0 * np.pi / 3.0)
                s240 = np.sin(4.0 * np.pi / 3.0)
                newlines = []
                for x, y in lines:
                    newx = np.zeros_like(x)
                    newy = np.zeros_like(y)
                    for ii, xx in enumerate(x):
                        newx[ii] = c240 * (xx - cc[0]) - s240 * (y[ii] - cc[1]) + cc[0]
                        newy[ii] = s240 * (xx - cc[0]) + c240 * (y[ii] - cc[1]) + cc[1]
                    newlines.append([newx, newy])
                newstable_entries = {
                    (c240 * (c[0] - cc[0]) - s240 * (c[1] - cc[1]) + cc[0],
                     s240 * (c[0] - cc[0]) + c240 * (c[1] - cc[1]) + cc[1]): entry
                    for c, entry in stable_entries.items()}
                newunstable_entries = {
                    entry: (c240 * (c[0] - cc[0]) - s240 * (c[1] - cc[1]) + cc[0],
                            s240 * (c[0] - cc[0]) + c240 * (c[1] - cc[1]) + cc[1])
                    for entry, c in unstable_entries.items()}
                return newlines, newstable_entries, newunstable_entries
            else:
                c240 = np.cos(4.0 * np.pi / 3.0)
                s240 = np.sin(4.0 * np.pi / 3.0)
                newlines = []
                for x, y in lines:
                    newx = np.zeros_like(x)
                    newy = np.zeros_like(y)
                    for ii, xx in enumerate(x):
                        newx[ii] = -c240 * xx - s240 * y[ii]
                        newy[ii] = -s240 * xx + c240 * y[ii]
                    newlines.append([newx, newy])
                newstable_entries = {(-c240 * c[0] - s240 * c[1],
                                      -s240 * c[0] + c240 * c[1]): entry
                                     for c, entry in stable_entries.items()}
                newunstable_entries = {entry: (-c240 * c[0] - s240 * c[1],
                                               -s240 * c[0] + c240 * c[1])
                                       for entry, c in unstable_entries.items()}
                return newlines, newstable_entries, newunstable_entries