index | modules | gitlab

# Source code for ase.constraints

    from math import sqrt
    from warnings import warn

    import numpy as np
    from scipy.linalg import expm, logm
    from ase.calculators.calculator import PropertyNotImplementedError
    from ase.geometry import (find_mic, wrap_positions, get_distances_derivatives,
                              get_angles_derivatives, get_dihedrals_derivatives,
                              conditional_find_mic, get_angles, get_dihedrals)
    from ase.utils.parsemath import eval_expression
    from ase.stress import (full_3x3_to_voigt_6_stress,
                            voigt_6_to_full_3x3_stress)

    __all__ = [
        'FixCartesian', 'FixBondLength', 'FixedMode',
        'FixConstraintSingle', 'FixAtoms', 'UnitCellFilter', 'ExpCellFilter',
        'FixScaled', 'StrainFilter', 'FixCom', 'FixedPlane', 'Filter',
        'FixConstraint', 'FixedLine', 'FixBondLengths', 'FixLinearTriatomic',
        'FixInternals', 'Hookean', 'ExternalForce', 'MirrorForce', 'MirrorTorque',
        "FixScaledParametricRelations", "FixCartesianParametricRelations"]

    def dict2constraint(dct):
        if dct['name'] not in __all__:
            raise ValueError
        return globals()[dct['name']](**dct['kwargs'])

    def slice2enlist(s, n):
        """Convert a slice object into a list of (new, old) tuples."""
        if isinstance(s, slice):
            return enumerate(range(*s.indices(n)))
        return enumerate(s)

    def constrained_indices(atoms, only_include=None):
        """Returns a list of indices for the atoms that are constrained
        by a constraint that is applied.  By setting only_include to a
        specific type of constraint you can make it only look for that
        given constraint.
        """
        indices = []
        for constraint in atoms.constraints:
            if only_include is not None:
                if not isinstance(constraint, only_include):
                    continue
            indices.extend(np.array(constraint.get_indices()))
        return np.array(np.unique(indices))

    class FixConstraint:
        """Base class for classes that fix one or more atoms in some way."""

        def index_shuffle(self, atoms, ind):
            """Change the indices.

            When the ordering of the atoms in the Atoms object changes,
            this method can be called to shuffle the indices of the
            constraints.

            ind -- List or tuple of indices.

            """
            raise NotImplementedError

        def repeat(self, m, n):
            """ basic method to multiply by m, needs to know the length
            of the underlying atoms object for the assignment of
            multiplied constraints to work.
            """
            msg = ("Repeat is not compatible with your atoms' constraints."
                   ' Use atoms.set_constraint() before calling repeat to '
                   'remove your constraints.')
            raise NotImplementedError(msg)

        def adjust_momenta(self, atoms, momenta):
            """Adjusts momenta in identical manner to forces."""
            self.adjust_forces(atoms, momenta)

        def copy(self):
            return dict2constraint(self.todict().copy())

    class FixConstraintSingle(FixConstraint):
        """Base class for classes that fix a single atom."""

        def __init__(self, a):
            self.a = a

        def index_shuffle(self, atoms, ind):
            """The atom index must be stored as self.a."""
            newa = None   # Signal error
            if self.a < 0:
                self.a += len(atoms)
            for new, old in slice2enlist(ind, len(atoms)):
                if old == self.a:
                    newa = new
                    break
            if newa is None:
                raise IndexError('Constraint not part of slice')
            self.a = newa

        def get_indices(self):
            return [self.a]

    class FixAtoms(FixConstraint):
        """Constraint object for fixing some chosen atoms."""

        def __init__(self, indices=None, mask=None):
            """Constrain chosen atoms.

            Parameters
            ----------
            indices : list of int
               Indices for those atoms that should be constrained.
            mask : list of bool
               One boolean per atom indicating if the atom should be
               constrained or not.

            Examples
            --------
            Fix all Copper atoms:

            >>> mask = [s == 'Cu' for s in atoms.get_chemical_symbols()]
            >>> c = FixAtoms(mask=mask)
            >>> atoms.set_constraint(c)

            Fix all atoms with z-coordinate less than 1.0 Angstrom:

            >>> c = FixAtoms(mask=atoms.positions[:, 2] < 1.0)
            >>> atoms.set_constraint(c)
            """

            if indices is None and mask is None:
                raise ValueError('Use "indices" or "mask".')
            if indices is not None and mask is not None:
                raise ValueError('Use only one of "indices" and "mask".')

            if mask is not None:
                indices = np.arange(len(mask))[np.asarray(mask, bool)]
            else:
                # Check for duplicates:
                srt = np.sort(indices)
                if (np.diff(srt) == 0).any():
                    raise ValueError(
                        'FixAtoms: The indices array contained duplicates. '
                        'Perhaps you wanted to specify a mask instead, but '
                        'forgot the mask= keyword.')
            self.index = np.asarray(indices, int)

            if self.index.ndim != 1:
                raise ValueError('Wrong argument to FixAtoms class!')

        def get_removed_dof(self, atoms):
            return 3 * len(self.index)

        def adjust_positions(self, atoms, new):
            new[self.index] = atoms.positions[self.index]

        def adjust_forces(self, atoms, forces):
            forces[self.index] = 0.0

        def index_shuffle(self, atoms, ind):
            # See docstring of superclass
            index = []
            for new, old in slice2enlist(ind, len(atoms)):
                if old in self.index:
                    index.append(new)
            if len(index) == 0:
                raise IndexError('All indices in FixAtoms not part of slice')
            self.index = np.asarray(index, int)

        def get_indices(self):
            return self.index

        def __repr__(self):
            return 'FixAtoms(indices=%s)' % ints2string(self.index)

        def todict(self):
            return {'name': 'FixAtoms',
                    'kwargs': {'indices': self.index.tolist()}}

        def repeat(self, m, n):
            i0 = 0
            natoms = 0
            if isinstance(m, int):
                m = (m, m, m)
            index_new = []
            for m2 in range(m[2]):
                for m1 in range(m[1]):
                    for m0 in range(m[0]):
                        i1 = i0 + n
                        index_new += [i + natoms for i in self.index]
                        i0 = i1
                        natoms += n
            self.index = np.asarray(index_new, int)
            return self

        def delete_atoms(self, indices, natoms):
            """Removes atom number ind from the index array, if present.

            Required for removing atoms with existing FixAtoms constraints.
            """

            i = np.zeros(natoms, int) - 1
            new = np.delete(np.arange(natoms), indices)
            i[new] = np.arange(len(new))
            index = i[self.index]
            self.index = index[index >= 0]
            if len(self.index) == 0:
                return None
            return self

    class FixCom(FixConstraint):
        """Constraint class for fixing the center of mass.

        References

        https://pubs.acs.org/doi/abs/10.1021/jp9722824

        """

        def get_removed_dof(self, atoms):
            return 3

        def adjust_positions(self, atoms, new):
            masses = atoms.get_masses()
            old_cm = atoms.get_center_of_mass()
            new_cm = np.dot(masses, new) / masses.sum()
            d = old_cm - new_cm
            new += d

        def adjust_forces(self, atoms, forces):
            m = atoms.get_masses()
            mm = np.tile(m, (3, 1)).T
            lb = np.sum(mm * forces, axis=0) / sum(m**2)
            forces -= mm * lb

        def todict(self):
            return {'name': 'FixCom',
                    'kwargs': {}}

    def ints2string(x, threshold=None):
        """Convert ndarray of ints to string."""
        if threshold is None or len(x) <= threshold:
            return str(x.tolist())
        return str(x[:threshold].tolist())[:-1] + ', ...]'

    class FixBondLengths(FixConstraint):
        maxiter = 500

        def __init__(self, pairs, tolerance=1e-13,
                     bondlengths=None, iterations=None):
            """iterations:
                    Ignored"""
            self.pairs = np.asarray(pairs)
            self.tolerance = tolerance
            self.bondlengths = bondlengths

        def get_removed_dof(self, atoms):
            return len(self.pairs)

        def adjust_positions(self, atoms, new):
            old = atoms.positions
            masses = atoms.get_masses()

            if self.bondlengths is None:
                self.bondlengths = self.initialize_bond_lengths(atoms)

            for i in range(self.maxiter):
                converged = True
                for j, ab in enumerate(self.pairs):
                    a = ab[0]
                    b = ab[1]
                    cd = self.bondlengths[j]
                    r0 = old[a] - old[b]
                    d0, _ = find_mic(r0, atoms.cell, atoms.pbc)
                    d1 = new[a] - new[b] - r0 + d0
                    m = 1 / (1 / masses[a] + 1 / masses[b])
                    x = 0.5 * (cd**2 - np.dot(d1, d1)) / np.dot(d0, d1)
                    if abs(x) > self.tolerance:
                        new[a] += x * m / masses[a] * d0
                        new[b] -= x * m / masses[b] * d0
                        converged = False
                if converged:
                    break
            else:
                raise RuntimeError('Did not converge')

        def adjust_momenta(self, atoms, p):
            old = atoms.positions
            masses = atoms.get_masses()

            if self.bondlengths is None:
                self.bondlengths = self.initialize_bond_lengths(atoms)

            for i in range(self.maxiter):
                converged = True
                for j, ab in enumerate(self.pairs):
                    a = ab[0]
                    b = ab[1]
                    cd = self.bondlengths[j]
                    d = old[a] - old[b]
                    d, _ = find_mic(d, atoms.cell, atoms.pbc)
                    dv = p[a] / masses[a] - p[b] / masses[b]
                    m = 1 / (1 / masses[a] + 1 / masses[b])
                    x = -np.dot(dv, d) / cd**2
                    if abs(x) > self.tolerance:
                        p[a] += x * m * d
                        p[b] -= x * m * d
                        converged = False
                if converged:
                    break
            else:
                raise RuntimeError('Did not converge')

        def adjust_forces(self, atoms, forces):
            self.constraint_forces = -forces
            self.adjust_momenta(atoms, forces)
            self.constraint_forces += forces

        def initialize_bond_lengths(self, atoms):
            bondlengths = np.zeros(len(self.pairs))

            for i, ab in enumerate(self.pairs):
                bondlengths[i] = atoms.get_distance(ab[0], ab[1], mic=True)

            return bondlengths

        def get_indices(self):
            return np.unique(self.pairs.ravel())

        def todict(self):
            return {'name': 'FixBondLengths',
                    'kwargs': {'pairs': self.pairs.tolist(),
                               'tolerance': self.tolerance}}

        def index_shuffle(self, atoms, ind):
            """Shuffle the indices of the two atoms in this constraint"""
            map = np.zeros(len(atoms), int)
            map[ind] = 1
            n = map.sum()
            map[:] = -1
            map[ind] = range(n)
            pairs = map[self.pairs]
            self.pairs = pairs[(pairs != -1).all(1)]
            if len(self.pairs) == 0:
                raise IndexError('Constraint not part of slice')

    def FixBondLength(a1, a2):
        """Fix distance between atoms with indices a1 and a2."""
        return FixBondLengths([(a1, a2)])

    class FixLinearTriatomic(FixConstraint):
        """Holonomic constraints for rigid linear triatomic molecules."""

        def __init__(self, triples):
            """Apply RATTLE-type bond constraints between outer atoms n and m
               and linear vectorial constraints to the position of central
               atoms o to fix the geometry of linear triatomic molecules of the
               type:

               n--o--m

               Parameters:

               triples: list
                   Indices of the atoms forming the linear molecules to constrain
                   as triples. Sequence should be (n, o, m) or (m, o, n).

               When using these constraints in molecular dynamics or structure
               optimizations, atomic forces need to be redistributed within a
               triple. The function redistribute_forces_optimization implements
               the redistribution of forces for structure optimization, while
               the function redistribute_forces_md implements the redistribution
               for molecular dynamics.

               References:

               Ciccotti et al. Molecular Physics 47 (1982)
               https://doi.org/10.1080/00268978200100942
            """
            self.triples = np.asarray(triples)
            if self.triples.shape[1] != 3:
                raise ValueError('"triples" has wrong size')
            self.bondlengths = None

        def get_removed_dof(self, atoms):
            return 4 * len(self.triples)

        @property
        def n_ind(self):
            return self.triples[:, 0]

        @property
        def m_ind(self):
            return self.triples[:, 2]

        @property
        def o_ind(self):
            return self.triples[:, 1]

        def initialize(self, atoms):
            masses = atoms.get_masses()
            self.mass_n, self.mass_m, self.mass_o = self.get_slices(masses)

            self.bondlengths = self.initialize_bond_lengths(atoms)
            self.bondlengths_nm = self.bondlengths.sum(axis=1)

            C1 = self.bondlengths[:, ::-1] / self.bondlengths_nm[:, None]
            C2 = (C1[:, 0] ** 2 * self.mass_o * self.mass_m +
                  C1[:, 1] ** 2 * self.mass_n * self.mass_o +
                  self.mass_n * self.mass_m)
            C2 = C1 / C2[:, None]
            C3 = self.mass_n * C1[:, 1] - self.mass_m * C1[:, 0]
            C3 = C2 * self.mass_o[:, None] * C3[:, None]
            C3[:, 1] *= -1
            C3 = (C3 + 1) / np.vstack((self.mass_n, self.mass_m)).T
            C4 = (C1[:, 0]**2 + C1[:, 1]**2 + 1)
            C4 = C1 / C4[:, None]

            self.C1 = C1
            self.C2 = C2
            self.C3 = C3
            self.C4 = C4

        def adjust_positions(self, atoms, new):
            old = atoms.positions
            new_n, new_m, new_o = self.get_slices(new)

            if self.bondlengths is None:
                self.initialize(atoms)

            r0 = old[self.n_ind] - old[self.m_ind]
            d0, _ = find_mic(r0, atoms.cell, atoms.pbc)
            d1 = new_n - new_m - r0 + d0
            a = np.einsum('ij,ij->i', d0, d0)
            b = np.einsum('ij,ij->i', d1, d0)
            c = np.einsum('ij,ij->i', d1, d1) - self.bondlengths_nm ** 2
            g = (b - (b**2 - a * c)**0.5) / (a * self.C3.sum(axis=1))
            g = g[:, None] * self.C3
            new_n -= g[:, 0, None] * d0
            new_m += g[:, 1, None] * d0
            if np.allclose(d0, r0):
                new_o = (self.C1[:, 0, None] * new_n
                         + self.C1[:, 1, None] * new_m)
            else:
                v1, _ = find_mic(new_n, atoms.cell, atoms.pbc)
                v2, _ = find_mic(new_m, atoms.cell, atoms.pbc)
                rb = self.C1[:, 0, None] * v1 + self.C1[:, 1, None] * v2
                new_o = wrap_positions(rb, atoms.cell, atoms.pbc)

            self.set_slices(new_n, new_m, new_o, new)

        def adjust_momenta(self, atoms, p):
            old = atoms.positions
            p_n, p_m, p_o = self.get_slices(p)

            if self.bondlengths is None:
                self.initialize(atoms)

            mass_nn = self.mass_n[:, None]
            mass_mm = self.mass_m[:, None]
            mass_oo = self.mass_o[:, None]

            d = old[self.n_ind] - old[self.m_ind]
            d, _ = find_mic(d, atoms.cell, atoms.pbc)
            dv = p_n / mass_nn - p_m / mass_mm
            k = np.einsum('ij,ij->i', dv, d) / self.bondlengths_nm ** 2
            k = self.C3 / (self.C3.sum(axis=1)[:, None]) * k[:, None]
            p_n -= k[:, 0, None] * mass_nn * d
            p_m += k[:, 1, None] * mass_mm * d
            p_o = (mass_oo * (self.C1[:, 0, None] * p_n / mass_nn +
                              self.C1[:, 1, None] * p_m / mass_mm))

            self.set_slices(p_n, p_m, p_o, p)

        def adjust_forces(self, atoms, forces):

            if self.bondlengths is None:
                self.initialize(atoms)

            A = self.C4 * np.diff(self.C1)
            A[:, 0] *= -1
            A -= 1
            B = np.diff(self.C4) / (A.sum(axis=1))[:, None]
            A /= (A.sum(axis=1))[:, None]

            self.constraint_forces = -forces
            old = atoms.positions

            fr_n, fr_m, fr_o = self.redistribute_forces_optimization(forces)

            d = old[self.n_ind] - old[self.m_ind]
            d, _ = find_mic(d, atoms.cell, atoms.pbc)
            df = fr_n - fr_m
            k = -np.einsum('ij,ij->i', df, d) / self.bondlengths_nm ** 2
            forces[self.n_ind] = fr_n + k[:, None] * d * A[:, 0, None]
            forces[self.m_ind] = fr_m - k[:, None] * d * A[:, 1, None]
            forces[self.o_ind] = fr_o + k[:, None] * d * B

            self.constraint_forces += forces

        def redistribute_forces_optimization(self, forces):
            """Redistribute forces within a triple when performing structure
            optimizations.

            The redistributed forces needs to be further adjusted using the
            appropriate Lagrange multipliers as implemented in adjust_forces."""
            forces_n, forces_m, forces_o = self.get_slices(forces)
            C1_1 = self.C1[:, 0, None]
            C1_2 = self.C1[:, 1, None]
            C4_1 = self.C4[:, 0, None]
            C4_2 = self.C4[:, 1, None]

            fr_n = ((1 - C4_1 * C1_1) * forces_n -
                    C4_1 * (C1_2 * forces_m - forces_o))
            fr_m = ((1 - C4_2 * C1_2) * forces_m -
                    C4_2 * (C1_1 * forces_n - forces_o))
            fr_o = ((1 - 1 / (C1_1**2 + C1_2**2 + 1)) * forces_o +
                    C4_1 * forces_n + C4_2 * forces_m)

            return fr_n, fr_m, fr_o

        def redistribute_forces_md(self, atoms, forces, rand=False):
            """Redistribute forces within a triple when performing molecular
            dynamics.

            When rand=True, use the equations for random force terms, as
            used e.g. by Langevin dynamics, otherwise apply the standard
            equations for deterministic forces (see Ciccotti et al. Molecular
            Physics 47 (1982))."""
            if self.bondlengths is None:
                self.initialize(atoms)
            forces_n, forces_m, forces_o = self.get_slices(forces)
            C1_1 = self.C1[:, 0, None]
            C1_2 = self.C1[:, 1, None]
            C2_1 = self.C2[:, 0, None]
            C2_2 = self.C2[:, 1, None]
            mass_nn = self.mass_n[:, None]
            mass_mm = self.mass_m[:, None]
            mass_oo = self.mass_o[:, None]
            if rand:
                mr1 = (mass_mm / mass_nn) ** 0.5
                mr2 = (mass_oo / mass_nn) ** 0.5
                mr3 = (mass_nn / mass_mm) ** 0.5
                mr4 = (mass_oo / mass_mm) ** 0.5
            else:
                mr1 = 1.0
                mr2 = 1.0
                mr3 = 1.0
                mr4 = 1.0

            fr_n = ((1 - C1_1 * C2_1 * mass_oo * mass_mm) * forces_n -
                    C2_1 * (C1_2 * mr1 * mass_oo * mass_nn * forces_m -
                            mr2 * mass_mm * mass_nn * forces_o))

            fr_m = ((1 - C1_2 * C2_2 * mass_oo * mass_nn) * forces_m -
                    C2_2 * (C1_1 * mr3 * mass_oo * mass_mm * forces_n -
                            mr4 * mass_mm * mass_nn * forces_o))

            self.set_slices(fr_n, fr_m, 0.0, forces)

        def get_slices(self, a):
            a_n = a[self.n_ind]
            a_m = a[self.m_ind]
            a_o = a[self.o_ind]

            return a_n, a_m, a_o

        def set_slices(self, a_n, a_m, a_o, a):
            a[self.n_ind] = a_n
            a[self.m_ind] = a_m
            a[self.o_ind] = a_o

        def initialize_bond_lengths(self, atoms):
            bondlengths = np.zeros((len(self.triples), 2))

            for i in range(len(self.triples)):
                bondlengths[i, 0] = atoms.get_distance(self.n_ind[i],
                                                       self.o_ind[i], mic=True)
                bondlengths[i, 1] = atoms.get_distance(self.o_ind[i],
                                                       self.m_ind[i], mic=True)

            return bondlengths

        def get_indices(self):
            return np.unique(self.triples.ravel())

        def todict(self):
            return {'name': 'FixLinearTriatomic',
                    'kwargs': {'triples': self.triples.tolist()}}

        def index_shuffle(self, atoms, ind):
            """Shuffle the indices of the three atoms in this constraint"""
            map = np.zeros(len(atoms), int)
            map[ind] = 1
            n = map.sum()
            map[:] = -1
            map[ind] = range(n)
            triples = map[self.triples]
            self.triples = triples[(triples != -1).all(1)]
            if len(self.triples) == 0:
                raise IndexError('Constraint not part of slice')

    class FixedMode(FixConstraint):
        """Constrain atoms to move along directions orthogonal to
        a given mode only."""

        def __init__(self, mode):
            self.mode = (np.asarray(mode) / np.sqrt((mode**2).sum())).reshape(-1)

        def get_removed_dof(self, atoms):
            return len(atoms)

        def adjust_positions(self, atoms, newpositions):
            newpositions = newpositions.ravel()
            oldpositions = atoms.positions.ravel()
            step = newpositions - oldpositions
            newpositions -= self.mode * np.dot(step, self.mode)

        def adjust_forces(self, atoms, forces):
            forces = forces.ravel()
            forces -= self.mode * np.dot(forces, self.mode)

        def index_shuffle(self, atoms, ind):
            eps = 1e-12
            mode = self.mode.reshape(-1, 3)
            excluded = np.ones(len(mode), dtype=bool)
            excluded[ind] = False
            if (abs(mode[excluded]) > eps).any():
                raise IndexError('All nonzero parts of mode not in slice')
            self.mode = mode[ind].ravel()

        def get_indices(self):
            # This function will never properly work because it works on all
            # atoms and it has no idea how to tell how many atoms it is
            # attached to.  If it is being used, surely the user knows
            # everything is being constrained.
            return []

        def todict(self):
            return {'name': 'FixedMode',
                    'kwargs': {'mode': self.mode.tolist()}}

        def __repr__(self):
            return 'FixedMode(%s)' % self.mode.tolist()

    class FixedPlane(FixConstraintSingle):
        """Constrain an atom index *a* to move in a given plane only.

        The plane is defined by its normal vector *direction*."""

        def __init__(self, a, direction):
            self.a = a
            self.dir = np.asarray(direction) / sqrt(np.dot(direction, direction))

        def get_removed_dof(self, atoms):
            return 1

        def adjust_positions(self, atoms, newpositions):
            step = newpositions[self.a] - atoms.positions[self.a]
            newpositions[self.a] -= self.dir * np.dot(step, self.dir)

        def adjust_forces(self, atoms, forces):
            forces[self.a] -= self.dir * np.dot(forces[self.a], self.dir)

        def todict(self):
            return {'name': 'FixedPlane',
                    'kwargs': {'a': self.a, 'direction': self.dir.tolist()}}

        def __repr__(self):
            return 'FixedPlane(%d, %s)' % (self.a, self.dir.tolist())

    class FixedLine(FixConstraintSingle):
        """Constrain an atom index *a* to move on a given line only.

        The line is defined by its vector *direction*."""

        def __init__(self, a, direction):
            self.a = a
            self.dir = np.asarray(direction) / sqrt(np.dot(direction, direction))

        def get_removed_dof(self, atoms):
            return 2

        def adjust_positions(self, atoms, newpositions):
            step = newpositions[self.a] - atoms.positions[self.a]
            x = np.dot(step, self.dir)
            newpositions[self.a] = atoms.positions[self.a] + x * self.dir

        def adjust_forces(self, atoms, forces):
            forces[self.a] = self.dir * np.dot(forces[self.a], self.dir)

        def __repr__(self):
            return 'FixedLine(%d, %s)' % (self.a, self.dir.tolist())

        def todict(self):
            return {'name': 'FixedLine',
                    'kwargs': {'a': self.a, 'direction': self.dir.tolist()}}

    class FixCartesian(FixConstraintSingle):
        'Fix an atom index *a* in the directions of the cartesian coordinates.'

        def __init__(self, a, mask=(1, 1, 1)):
            self.a = a
            self.mask = ~np.asarray(mask, bool)

        def get_removed_dof(self, atoms):
            return 3 - self.mask.sum()

        def adjust_positions(self, atoms, new):
            step = new[self.a] - atoms.positions[self.a]
            step *= self.mask
            new[self.a] = atoms.positions[self.a] + step

        def adjust_forces(self, atoms, forces):
            forces[self.a] *= self.mask

        def __repr__(self):
            return 'FixCartesian(a={0}, mask={1})'.format(self.a,
                                                          list(~self.mask))

        def todict(self):
            return {'name': 'FixCartesian',
                    'kwargs': {'a': self.a, 'mask': ~self.mask.tolist()}}

    class FixScaled(FixConstraintSingle):
        'Fix an atom index *a* in the directions of the unit vectors.'

        def __init__(self, cell, a, mask=(1, 1, 1)):
            self.cell = np.asarray(cell)
            self.a = a
            self.mask = np.array(mask, bool)

        def get_removed_dof(self, atoms):
            return self.mask.sum()

        def adjust_positions(self, atoms, new):
            scaled_old = atoms.cell.scaled_positions(atoms.positions)
            scaled_new = atoms.cell.scaled_positions(new)
            for n in range(3):
                if self.mask[n]:
                    scaled_new[self.a, n] = scaled_old[self.a, n]
            new[self.a] = atoms.cell.cartesian_positions(scaled_new)[self.a]

        def adjust_forces(self, atoms, forces):
            # Forces are covarient to the coordinate transformation,
            # use the inverse transformations
            scaled_forces = atoms.cell.cartesian_positions(forces)
            scaled_forces[self.a] *= -(self.mask - 1)
            forces[self.a] = atoms.cell.scaled_positions(scaled_forces)[self.a]

        def todict(self):
            return {'name': 'FixScaled',
                    'kwargs': {'a': self.a,
                               'cell': self.cell.tolist(),
                               'mask': self.mask.tolist()}}

        def __repr__(self):
            return 'FixScaled(%s, %d, %s)' % (repr(self.cell),
                                              self.a,
                                              repr(self.mask))

    # TODO: Better interface might be to use dictionaries in place of very
    # nested lists/tuples
    class FixInternals(FixConstraint):
        """Constraint object for fixing multiple internal coordinates.

        Allows fixing bonds, angles, and dihedrals.
        Please provide angular units in degrees using angles_deg and
        dihedrals_deg.
        """
        def __init__(self, bonds=None, angles=None, dihedrals=None,
                     angles_deg=None, dihedrals_deg=None,
                     bondcombos=None, anglecombos=None, dihedralcombos=None,
                     mic=False, epsilon=1.e-7):

            # deprecate public API using radians; degrees is preferred
            warn_msg = 'Please specify {} in degrees using the {} argument.'
            if angles:
                warn(FutureWarning(warn_msg.format('angles', 'angle_deg')))
                angles = np.asarray(angles)
                angles[:, 0] = angles[:, 0] / np.pi * 180
                angles = angles.tolist()
            else:
                angles = angles_deg
            if dihedrals:
                warn(FutureWarning(warn_msg.format('dihedrals', 'dihedrals_deg')))
                dihedrals = np.asarray(dihedrals)
                dihedrals[:, 0] = dihedrals[:, 0] / np.pi * 180
                dihedrals = dihedrals.tolist()
            else:
                dihedrals = dihedrals_deg

            self.bonds = bonds or []
            self.angles = angles or []
            self.dihedrals = dihedrals or []
            self.bondcombos = bondcombos or []
            self.anglecombos = anglecombos or []
            self.dihedralcombos = dihedralcombos or []
            self.mic = mic
            self.epsilon = epsilon

            self.n = (len(self.bonds) + len(self.angles) + len(self.dihedrals)
                      + len(self.bondcombos) + len(self.anglecombos)
                      + len(self.dihedralcombos))

            # Initialize these at run-time:
            self.constraints = []
            self.initialized = False

        def get_removed_dof(self, atoms):
            return self.n

        def initialize(self, atoms):
            if self.initialized:
                return
            masses = np.repeat(atoms.get_masses(), 3)
            cell = None
            pbc = None
            if self.mic:
                cell = atoms.cell
                pbc = atoms.pbc
            self.constraints = []
            for data, make_constr in [(self.bonds, self.FixBondLengthAlt),
                                      (self.angles, self.FixAngle),
                                      (self.dihedrals, self.FixDihedral),
                                      (self.bondcombos, self.FixBondCombo),
                                      (self.anglecombos, self.FixAngleCombo),
                                      (self.dihedralcombos, self.FixDihedralCombo)]:
                for datum in data:
                    constr = make_constr(datum[0], datum[1], masses, cell, pbc)
                    self.constraints.append(constr)
            self.initialized = True

        def shuffle_definitions(self, shuffle_dic, internal_type):
            dfns = []  # definitions
            for dfn in internal_type:  # e.g. for bond in self.bonds
                append = True
                new_dfn = [dfn[0], list(dfn[1])]
                for old in dfn[1]:
                    if old in shuffle_dic:
                        new_dfn[1][dfn[1].index(old)] = shuffle_dic[old]
                    else:
                        append = False
                        break
                if append:
                    dfns.append(new_dfn)
            return dfns

        def shuffle_combos(self, shuffle_dic, internal_type):
            dfns = []  # definitions
            for dfn in internal_type:  # e.g. for bondcombo in self.bondcombos
                append = True
                all_indices = [idx[0:-1] for idx in dfn[1]]
                new_dfn = [dfn[0], list(dfn[1])]
                for i, indices in enumerate(all_indices):
                    for old in indices:
                        if old in shuffle_dic:
                            new_dfn[1][i][indices.index(old)] = shuffle_dic[old]
                        else:
                            append = False
                            break
                    if not append:
                        break
                if append:
                    dfns.append(new_dfn)
            return dfns

        def index_shuffle(self, atoms, ind):
            # See docstring of superclass
            self.initialize(atoms)
            shuffle_dic = dict(slice2enlist(ind, len(atoms)))
            shuffle_dic = {old: new for new, old in shuffle_dic.items()}
            self.bonds = self.shuffle_definitions(shuffle_dic, self.bonds)
            self.angles = self.shuffle_definitions(shuffle_dic, self.angles)
            self.dihedrals = self.shuffle_definitions(shuffle_dic, self.dihedrals)
            self.bondcombos = self.shuffle_combos(shuffle_dic, self.bondcombos)
            self.anglecombos = self.shuffle_combos(shuffle_dic, self.anglecombos)
            self.dihedralcombos = self.shuffle_combos(shuffle_dic,
                                                      self.dihedralcombos)
            self.initialized = False
            self.initialize(atoms)
            if len(self.constraints) == 0:
                raise IndexError('Constraint not part of slice')

        def get_indices(self):
            cons = []
            for dfn in self.bonds + self.dihedrals + self.angles:
                cons.extend(dfn[1])
            for dfn in self.bondcombos + self.anglecombos + self.dihedralcombos:
                for partial_dfn in dfn[1]:
                    cons.extend(partial_dfn[0:-1])  # last index is the coefficient
            return list(set(cons))

        def todict(self):
            return {'name': 'FixInternals',
                    'kwargs': {'bonds': self.bonds,
                               'angles': self.angles,
                               'dihedrals': self.dihedrals,
                               'bondcombos': self.bondcombos,
                               'anglecombos': self.anglecombos,
                               'dihedralcombos': self.dihedralcombos,
                               'mic': self.mic,
                               'epsilon': self.epsilon}}

        def adjust_positions(self, atoms, new):
            self.initialize(atoms)
            for constraint in self.constraints:
                constraint.prepare_jacobian(atoms.positions)
            for j in range(50):
                maxerr = 0.0
                for constraint in self.constraints:
                    constraint.adjust_positions(atoms.positions, new)
                    maxerr = max(abs(constraint.sigma), maxerr)
                if maxerr < self.epsilon:
                    return
            raise ValueError('Shake did not converge.')

        def adjust_forces(self, atoms, forces):
            """Project out translations and rotations and all other constraints"""
            self.initialize(atoms)
            positions = atoms.positions
            N = len(forces)
            list2_constraints = list(np.zeros((6, N, 3)))
            tx, ty, tz, rx, ry, rz = list2_constraints

            list_constraints = [r.ravel() for r in list2_constraints]

            tx[:, 0] = 1.0
            ty[:, 1] = 1.0
            tz[:, 2] = 1.0
            ff = forces.ravel()

            # Calculate the center of mass
            center = positions.sum(axis=0) / N

            rx[:, 1] = -(positions[:, 2] - center[2])
            rx[:, 2] = positions[:, 1] - center[1]
            ry[:, 0] = positions[:, 2] - center[2]
            ry[:, 2] = -(positions[:, 0] - center[0])
            rz[:, 0] = -(positions[:, 1] - center[1])
            rz[:, 1] = positions[:, 0] - center[0]

            # Normalizing transl., rotat. constraints
            for r in list2_constraints:
                r /= np.linalg.norm(r.ravel())

            # Add all angle, etc. constraint vectors
            for constraint in self.constraints:
                constraint.prepare_jacobian(positions)
                constraint.adjust_forces(positions, forces)
                list_constraints.insert(0, constraint.jacobian)
            # QR DECOMPOSITION - GRAM SCHMIDT

            list_constraints = [r.ravel() for r in list_constraints]
            aa = np.column_stack(list_constraints)
            (aa, bb) = np.linalg.qr(aa)
            # Projection
            hh = []
            for i, constraint in enumerate(self.constraints):
                hh.append(aa[:, i] * np.row_stack(aa[:, i]))

            txx = aa[:, self.n] * np.row_stack(aa[:, self.n])
            tyy = aa[:, self.n + 1] * np.row_stack(aa[:, self.n + 1])
            tzz = aa[:, self.n + 2] * np.row_stack(aa[:, self.n + 2])
            rxx = aa[:, self.n + 3] * np.row_stack(aa[:, self.n + 3])
            ryy = aa[:, self.n + 4] * np.row_stack(aa[:, self.n + 4])
            rzz = aa[:, self.n + 5] * np.row_stack(aa[:, self.n + 5])
            T = txx + tyy + tzz + rxx + ryy + rzz
            for vec in hh:
                T += vec
            ff = np.dot(T, np.row_stack(ff))
            forces[:, :] -= np.dot(T, np.row_stack(ff)).reshape(-1, 3)

        def __repr__(self):
            constraints = repr(self.constraints)
            return 'FixInternals(_copy_init=%s, epsilon=%s)' % (constraints,
                                                                repr(self.epsilon))

        def __str__(self):
            return '\n'.join([repr(c) for c in self.constraints])

        # Classes for internal use in FixInternals
        class FixInternalsBase:
            """Base class for subclasses of FixInternals."""
            def __init__(self, targetvalue, indices, masses, cell, pbc):
                self.targetvalue = targetvalue  # constant target value
                self.indices = [defin[0:-1] for defin in indices]  # indices, defs
                self.coefs = np.asarray([defin[-1] for defin in indices])  # coefs
                self.masses = masses
                self.jacobian = []  # geometric Jacobian matrix, Wilson B-matrix
                self.sigma = 1.  # difference between current and target value
                self.projected_force = None  # helps optimizers scan along constr.
                self.cell = cell
                self.pbc = pbc

            def finalize_jacobian(self, pos, n_internals, n, derivs):
                """Populate jacobian with derivatives for `n_internals` defined
                internals. n = 2 (bonds), 3 (angles), 4 (dihedrals)."""
                jacobian = np.zeros((n_internals, *pos.shape))
                for i, idx in enumerate(self.indices):
                    for j in range(n):
                        jacobian[i, idx[j]] = derivs[i, j]
                jacobian = jacobian.reshape((n_internals, 3 * len(pos)))
                self.jacobian = self.coefs @ jacobian

            def finalize_positions(self, newpos):
                jacobian = self.jacobian / self.masses
                lamda = -self.sigma / np.dot(jacobian, self.jacobian)
                dnewpos = lamda * jacobian
                newpos += dnewpos.reshape(newpos.shape)

            def adjust_forces(self, positions, forces):
                self.projected_force = np.dot(self.jacobian, forces.ravel())
                self.jacobian /= np.linalg.norm(self.jacobian)

        class FixBondCombo(FixInternalsBase):
            """Constraint subobject for fixing linear combination of bond lengths
            within FixInternals.

            sum_i( coef_i * bond_length_i ) = constant
            """
            def prepare_jacobian(self, pos):
                bondvectors = [pos[k] - pos[h] for h, k in self.indices]
                derivs = get_distances_derivatives(bondvectors, cell=self.cell,
                                                   pbc=self.pbc)
                self.finalize_jacobian(pos, len(bondvectors), 2, derivs)

            def adjust_positions(self, oldpos, newpos):
                bondvectors = [newpos[k] - newpos[h] for h, k in self.indices]
                (_, ), (dists, ) = conditional_find_mic([bondvectors],
                                                        cell=self.cell,
                                                        pbc=self.pbc)
                value = np.dot(self.coefs, dists)
                self.sigma = value - self.targetvalue
                self.finalize_positions(newpos)

            def __repr__(self):
                return 'FixBondCombo({}, {}, {})'.format(repr(self.targetvalue),
                                                         self.indices, self.coefs)

        class FixBondLengthAlt(FixBondCombo):
            """Constraint subobject for fixing bond length within FixInternals.
            Fix distance between atoms with indices a1, a2."""
            def __init__(self, targetvalue, indices, masses, cell, pbc):
                indices = [list(indices) + [1.]]  # bond definition with coef 1.
                super().__init__(targetvalue, indices, masses, cell=cell, pbc=pbc)

            def __repr__(self):
                return 'FixBondLengthAlt({}, {})'.format(self.targetvalue,
                                                         *self.indices)

        class FixAngleCombo(FixInternalsBase):
            """Constraint subobject for fixing linear combination of angles
            within FixInternals.

            sum_i( coef_i * angle_i ) = constant
            """
            def gather_vectors(self, pos):
                v0 = [pos[h] - pos[k] for h, k, l in self.indices]
                v1 = [pos[l] - pos[k] for h, k, l in self.indices]
                return v0, v1

            def prepare_jacobian(self, pos):
                v0, v1 = self.gather_vectors(pos)
                derivs = get_angles_derivatives(v0, v1, cell=self.cell,
                                                pbc=self.pbc)
                self.finalize_jacobian(pos, len(v0), 3, derivs)

            def adjust_positions(self, oldpos, newpos):
                v0, v1 = self.gather_vectors(newpos)
                value = get_angles(v0, v1, cell=self.cell, pbc=self.pbc)
                value = np.dot(self.coefs, value)
                self.sigma = value - self.targetvalue
                self.finalize_positions(newpos)

            def __repr__(self):
                return 'FixAngleCombo({}, {}, {})'.format(self.targetvalue,
                                                          self.indices, self.coefs)

        class FixAngle(FixAngleCombo):
            """Constraint object for fixing an angle within
            FixInternals using the SHAKE algorithm.

            SHAKE convergence is potentially problematic for angles very close to
            0 or 180 degrees as there is a singularity in the Cartesian derivative.
            """
            def __init__(self, targetvalue, indices, masses, cell, pbc):
                """Fix atom movement to construct a constant angle."""
                indices = [list(indices) + [1.]]  # angle definition with coef 1.
                super().__init__(targetvalue, indices, masses, cell=cell, pbc=pbc)

            def __repr__(self):
                return 'FixAngle({}, {})'.format(self.targetvalue, *self.indices)

        class FixDihedralCombo(FixInternalsBase):
            """Constraint subobject for fixing linear combination of dihedrals
            within FixInternals.

            sum_i( coef_i * dihedral_i ) = constant
            """
            def gather_vectors(self, pos):
                v0 = [pos[k] - pos[h] for h, k, l, m in self.indices]
                v1 = [pos[l] - pos[k] for h, k, l, m in self.indices]
                v2 = [pos[m] - pos[l] for h, k, l, m in self.indices]
                return v0, v1, v2

            def prepare_jacobian(self, pos):
                v0, v1, v2 = self.gather_vectors(pos)
                derivs = get_dihedrals_derivatives(v0, v1, v2, cell=self.cell,
                                                   pbc=self.pbc)
                self.finalize_jacobian(pos, len(v0), 4, derivs)

            def adjust_positions(self, oldpos, newpos):
                v0, v1, v2 = self.gather_vectors(newpos)
                value = get_dihedrals(v0, v1, v2, cell=self.cell, pbc=self.pbc)
                value = np.dot(self.coefs, value)
                self.sigma = value - self.targetvalue
                self.finalize_positions(newpos)

            def __repr__(self):
                return 'FixDihedralCombo({}, {}, {})'.format(self.targetvalue,
                                                             self.indices,
                                                             self.coefs)

        class FixDihedral(FixDihedralCombo):
            """Constraint object for fixing a dihedral angle using
            the SHAKE algorithm. This one allows also other constraints.

            SHAKE convergence is potentially problematic for near-undefined
            dihedral angles (i.e. when one of the two angles a012 or a123
            approaches 0 or 180 degrees).
            """
            def __init__(self, targetvalue, indices, masses, cell, pbc):
                indices = [list(indices) + [1.]]  # dihedral def. with coef 1.
                super().__init__(targetvalue, indices, masses, cell=cell, pbc=pbc)

            def adjust_positions(self, oldpos, newpos):
                v0, v1, v2 = self.gather_vectors(newpos)
                value = get_dihedrals(v0, v1, v2, cell=self.cell, pbc=self.pbc)
                # apply minimum dihedral difference 'convention': (diff <= 180)
                self.sigma = (value - self.targetvalue + 180) % 360 - 180
                self.finalize_positions(newpos)

            def __repr__(self):
                return 'FixDihedral({}, {})'.format(self.targetvalue, *self.indices)

    class FixParametricRelations(FixConstraint):

        def __init__(
            self,
            indices,
            Jacobian,
            const_shift,
            params=None,
            eps=1e-12,
            use_cell=False,
        ):
            """Constrains the degrees of freedom to act in a reduced parameter space defined by the Jacobian

            These constraints are based off the work in: https://arxiv.org/abs/1908.01610

            The constraints linearly maps the full 3N degrees of freedom, where N is number of active
            lattice vectors/atoms onto a reduced subset of M free parameters, where M <= 3*N. The
            Jacobian matrix and constant shift vector map the full set of degrees of freedom onto the
            reduced parameter space.

            Currently the constraint is set up to handle either atomic positions or lattice vectors
            at one time, but not both. To do both simply add a two constraints for each set. This is
            done to keep the mathematics behind the operations separate.

            It would be possible to extend these constraints to allow non-linear transformations
            if functionality to update the Jacobian at each position update was included. This would
            require passing an update function evaluate it every time adjust_positions is callled.
            This is currently NOT supported, and there are no plans to implement it in the future.

            Args:
                indices (list of int): indices of the constrained atoms
                    (if not None or empty then cell_indices must be None or Empty)
                Jacobian (np.ndarray(shape=(3*len(indices), len(params)))): The Jacobian describing
                    the parameter space transformation
                const_shift (np.ndarray(shape=(3*len(indices)))): A vector describing the constant term
                    in the transformation not accounted for in the Jacobian
                params (list of str): parameters used in the parametric representation
                    if None a list is generated based on the shape of the Jacobian
                eps (float): a small number to compare the similarity of numbers and set the precision used
                    to generate the constraint expressions
                use_cell (bool): if True then act on the cell object
            """
            self.indices = np.array(indices)
            self.Jacobian = np.array(Jacobian)
            self.const_shift = np.array(const_shift)

            assert self.const_shift.shape[0] == 3*len(self.indices)
            assert self.Jacobian.shape[0] == 3*len(self.indices)

            self.eps = eps
            self.use_cell = use_cell

            if params is None:
                params = []
                if self.Jacobian.shape[1] > 0:
                    int_fmt_str = "{:0" + str(int(np.ceil(np.log10(self.Jacobian.shape[1])))) + "d}"
                    for param_ind in range(self.Jacobian.shape[1]):
                        params.append("param_" + int_fmt_str.format(param_ind))
            else:
                assert len(params) == self.Jacobian.shape[-1]

            self.params = params

            self.Jacobian_inv = np.linalg.inv(self.Jacobian.T @ self.Jacobian) @ self.Jacobian.T

        @classmethod
        def from_expressions(cls, indices, params, expressions, eps=1e-12, use_cell=False):
            """Converts the expressions into a Jacobian Matrix/const_shift vector and constructs a FixParametricRelations constraint

            The expressions must be a list like object of size 3*N and elements must be ordered as:
            [n_0,i; n_0,j; n_0,k; n_1,i; n_1,j; .... ; n_N-1,i; n_N-1,j; n_N-1,k],
            where i, j, and k are the first, second and third component of the atomic position/lattice
            vector. Currently only linear operations are allowed to be included in the expressions so
            only terms like:
                - const * param_0
                - sqrt[const] * param_1
                - const * param_0 +/- const * param_1 +/- ... +/- const * param_M
            where const is any real number and param_0, param_1, ..., param_M are the parameters passed in
            params, are allowed.

            For example, the fractional atomic position constraints for wurtzite are:
            params = ["z1", "z2"]
            expressions = [
                "1.0/3.0", "2.0/3.0", "z1",
                "2.0/3.0", "1.0/3.0", "0.5 + z1",
                "1.0/3.0", "2.0/3.0", "z2",
                "2.0/3.0", "1.0/3.0", "0.5 + z2",
            ]

            For diamond are:
            params = []
            expressions = [
                "0.0", "0.0", "0.0",
                "0.25", "0.25", "0.25",
            ],

            and for stannite are
            params=["x4", "z4"]
            expressions = [
                "0.0", "0.0", "0.0",
                "0.0", "0.5", "0.5",
                "0.75", "0.25", "0.5",
                "0.25", "0.75", "0.5",
                "x4 + z4", "x4 + z4", "2*x4",
                "x4 - z4", "x4 - z4", "-2*x4",
                 "0.0", "-1.0 * (x4 + z4)", "x4 - z4",
                 "0.0", "x4 - z4", "-1.0 * (x4 + z4)",
            ]

            Args:
                indices (list of int): indices of the constrained atoms
                    (if not None or empty then cell_indices must be None or Empty)
                params (list of str): parameters used in the parametric representation
                expressions (list of str): expressions used to convert from the parametric to the real space
                    representation
                eps (float): a small number to compare the similarity of numbers and set the precision used
                    to generate the constraint expressions
                use_cell (bool): if True then act on the cell object

            Returns:
                cls(
                    indices,
                    Jacobian generated from expressions,
                    const_shift generated from expressions,
                    params,
                    eps-12,
                    use_cell,
                )
            """
            Jacobian = np.zeros((3*len(indices), len(params)))
            const_shift = np.zeros(3*len(indices))

            for expr_ind, expression in enumerate(expressions):
                expression = expression.strip()

                # Convert subtraction to addition
                expression = expression.replace("-", "+(-1.0)*")
                if "+" == expression[0]:
                    expression = expression[1:]
                elif "(+" == expression[:2]:
                    expression = "(" + expression[2:]

                # Explicitly add leading zeros so when replacing param_1 with 0.0 param_11 does not become 0.01
                int_fmt_str = "{:0" + str(int(np.ceil(np.log10(len(params)+1)))) + "d}"

                param_dct = dict()
                param_map = dict()

                # Construct a standardized param template for A/B filling
                for param_ind, param in enumerate(params):
                    param_str = "param_" + int_fmt_str.format(param_ind)
                    param_map[param] = param_str
                    param_dct[param_str] = 0.0

                # Replace the parameters according to the map
                # Sort by string length (long to short) to prevent cases like x11 becoming f"{param_map["x1"]}1"
                for param in sorted(params, key=lambda s: -1.0*len(s)):
                    expression = expression.replace(param, param_map[param])

                # Partial linearity check
                for express_sec in expression.split("+"):
                    in_sec = [param in express_sec for param in param_dct]
                    n_params_in_sec = len(np.where(np.array(in_sec))[0])
                    if n_params_in_sec > 1:
                        raise ValueError("The FixParametricRelations expressions must be linear.")

                const_shift[expr_ind] = float(eval_expression(expression, param_dct))

                for param_ind in range(len(params)):
                    param_str = "param_" + int_fmt_str.format(param_ind)
                    if param_str not in expression:
                        Jacobian[expr_ind, param_ind] = 0.0
                        continue
                    param_dct[param_str] = 1.0
                    test_1 = float(eval_expression(expression, param_dct))
                    test_1 -= const_shift[expr_ind]
                    Jacobian[expr_ind, param_ind] = test_1

                    param_dct[param_str] = 2.0
                    test_2 = float(eval_expression(expression, param_dct))
                    test_2 -= const_shift[expr_ind]
                    if abs(test_2 / test_1 - 2.0) > eps:
                        raise ValueError("The FixParametricRelations expressions must be linear.")
                    param_dct[param_str] = 0.0

            args = [
                indices,
                Jacobian,
                const_shift,
                params,
                eps,
                use_cell,
            ]
            if cls is FixScaledParametricRelations:
                args = args[:-1]
            return cls(*args)

        @property
        def expressions(self):
            """Generate the expressions represented by the current self.Jacobian and self.const_shift objects"""
            expressions = []
            per = int(round(-1 * np.log10(self.eps)))
            fmt_str = "{:." + str(per + 1) + "g}"
            for index, shift_val in enumerate(self.const_shift):
                exp = ""
                if np.all(np.abs(self.Jacobian[index]) < self.eps) or np.abs(shift_val) > self.eps:
                    exp += fmt_str.format(shift_val)

                param_exp = ""
                for param_index, jacob_val in enumerate(self.Jacobian[index]):
                    abs_jacob_val = np.round(np.abs(jacob_val), per + 1)
                    if abs_jacob_val < self.eps:
                        continue

                    param = self.params[param_index]
                    if param_exp or exp:
                        if jacob_val > -1.0*self.eps:
                            param_exp += " + "
                        else:
                            param_exp += " - "
                    elif (not exp) and (not param_exp) and (jacob_val < -1.0*self.eps):
                        param_exp += "-"

                    if np.abs(abs_jacob_val-1.0) <= self.eps:
                        param_exp += "{:s}".format(param)
                    else:
                        param_exp += (fmt_str + "*{:s}").format(abs_jacob_val, param)

                exp += param_exp

                expressions.append(exp)
            return np.array(expressions).reshape((-1, 3))

        def todict(self):
            """Create a dictionary representation of the constraint"""
            return {
                "name": type(self).__name__,
                "kwargs": {
                    "indices": self.indices,
                    "params": self.params,
                    "Jacobian": self.Jacobian,
                    "const_shift": self.const_shift,
                    "eps": self.eps,
                    "use_cell": self.use_cell,
                }
            }

        def __repr__(self):
            """The str representation of the constraint"""
            if len(self.indices) > 1:
                indices_str = "[{:d}, ..., {:d}]".format(self.indices[0], self.indices[-1])
            else:
                indices_str = "[{:d}]".format(self.indices[0])

            if len(self.params) > 1:
                params_str = "[{:s}, ..., {:s}]".format(self.params[0], self.params[-1])
            elif len(self.params) == 1:
                params_str = "[{:s}]".format(self.params[0])
            else:
                params_str = "[]"

            return '{:s}({:s}, {:s}, ..., {:e})'.format(
                type(self).__name__,
                indices_str,
                params_str,
                self.eps
            )

    class FixScaledParametricRelations(FixParametricRelations):

        def __init__(
            self,
            indices,
            Jacobian,
            const_shift,
            params=None,
            eps=1e-12,
        ):
            """The fractional coordinate version of FixParametricRelations

            All arguments are the same, but since this is for fractional coordinates use_cell is false
            """
            super(FixScaledParametricRelations, self).__init__(
                indices,
                Jacobian,
                const_shift,
                params,
                eps,
                False,
            )

        def adjust_contravariant(self, cell, vecs, B):
            """Adjust the values of a set of vectors that are contravariant with the unit transformation"""
            scaled = cell.scaled_positions(vecs).flatten()
            scaled = self.Jacobian_inv @ (scaled - B)
            scaled = ((self.Jacobian @ scaled) + B).reshape((-1, 3))

            return cell.cartesian_positions(scaled)

        def adjust_positions(self, atoms, positions):
            """Adjust positions of the atoms to match the constraints"""
            positions[self.indices] = self.adjust_contravariant(
                atoms.cell,
                positions[self.indices],
                self.const_shift,
            )
            positions[self.indices] = self.adjust_B(atoms.cell, positions[self.indices])

        def adjust_B(self, cell, positions):
            """Wraps the positions back to the unit cell and adjust B to keep track of this change"""
            fractional = cell.scaled_positions(positions)
            wrapped_fractional = (fractional % 1.0) % 1.0
            self.const_shift += np.round(wrapped_fractional - fractional).flatten()
            return cell.cartesian_positions(wrapped_fractional)

        def adjust_momenta(self, atoms, momenta):
            """Adjust momenta of the atoms to match the constraints"""
            momenta[self.indices] = self.adjust_contravariant(
                atoms.cell,
                momenta[self.indices],
                np.zeros(self.const_shift.shape),
            )

        def adjust_forces(self, atoms, forces):
            """Adjust forces of the atoms to match the constraints"""
            # Forces are coavarient to the coordinate transformation, use the inverse transformations
            cart2frac_jacob = np.zeros(2*(3*len(atoms),))
            for i_atom in range(len(atoms)):
                cart2frac_jacob[3*i_atom:3*(i_atom+1), 3*i_atom:3*(i_atom+1)] = atoms.cell.T

            jacobian = cart2frac_jacob @ self.Jacobian
            jacobian_inv = np.linalg.inv(jacobian.T @ jacobian) @ jacobian.T

            reduced_forces = jacobian.T @ forces.flatten()
            forces[self.indices] = (jacobian_inv.T @ reduced_forces).reshape(-1, 3)

        def todict(self):
            """Create a dictionary representation of the constraint"""
            dct = super(FixScaledParametricRelations, self).todict()
            del(dct["kwargs"]["use_cell"])
            return dct

    class FixCartesianParametricRelations(FixParametricRelations):

        def __init__(
            self,
            indices,
            Jacobian,
            const_shift,
            params=None,
            eps=1e-12,
            use_cell=False,
        ):
            """The Cartesian coordinate version of FixParametricRelations"""
            super(FixCartesianParametricRelations, self).__init__(
                indices,
                Jacobian,
                const_shift,
                params,
                eps,
                use_cell,
            )

        def adjust_contravariant(self, vecs, B):
            """Adjust the values of a set of vectors that are contravariant with the unit transformation"""
            vecs = self.Jacobian_inv @ (vecs.flatten() - B)
            vecs = ((self.Jacobian @ vecs) + B).reshape((-1, 3))
            return vecs

        def adjust_positions(self, atoms, positions):
            """Adjust positions of the atoms to match the constraints"""
            if self.use_cell:
                return
            positions[self.indices] = self.adjust_contravariant(
                positions[self.indices],
                self.const_shift,
            )

        def adjust_momenta(self, atoms, momenta):
            """Adjust momenta of the atoms to match the constraints"""
            if self.use_cell:
                return
            momenta[self.indices] = self.adjust_contravariant(
                momenta[self.indices],
                np.zeros(self.const_shift.shape),
            )

        def adjust_forces(self, atoms, forces):
            """Adjust forces of the atoms to match the constraints"""
            if self.use_cell:
                return

            forces_reduced = self.Jacobian.T @ forces[self.indices].flatten()
            forces[self.indices] = (self.Jacobian_inv.T @ forces_reduced).reshape(-1, 3)

        def adjust_cell(self, atoms, cell):
            """Adjust the cell of the atoms to match the constraints"""
            if not self.use_cell:
                return
            cell[self.indices] = self.adjust_contravariant(
                cell[self.indices],
                np.zeros(self.const_shift.shape),
            )

        def adjust_stress(self, atoms, stress):
            """Adjust the stress of the atoms to match the constraints"""
            if not self.use_cell:
                return

            stress_3x3 = voigt_6_to_full_3x3_stress(stress)
            stress_reduced = self.Jacobian.T @ stress_3x3[self.indices].flatten()
            stress_3x3[self.indices] = (self.Jacobian_inv.T @ stress_reduced).reshape(-1, 3)

            stress[:] = full_3x3_to_voigt_6_stress(stress_3x3)

    class Hookean(FixConstraint):
        """Applies a Hookean restorative force between a pair of atoms, an atom
        and a point, or an atom and a plane."""

        def __init__(self, a1, a2, k, rt=None):
            """Forces two atoms to stay close together by applying no force if
            they are below a threshold length, rt, and applying a Hookean
            restorative force when the distance between them exceeds rt. Can
            also be used to tether an atom to a fixed point in space or to a
            distance above a plane.

            a1 : int
               Index of atom 1
            a2 : one of three options
               1) index of atom 2
               2) a fixed point in cartesian space to which to tether a1
               3) a plane given as (A, B, C, D) in A x + B y + C z + D = 0.
            k : float
               Hooke's law (spring) constant to apply when distance
               exceeds threshold_length. Units of eV A^-2.
            rt : float
               The threshold length below which there is no force. The
               length is 1) between two atoms, 2) between atom and point.
               This argument is not supplied in case 3. Units of A.

            If a plane is specified, the Hooke's law force is applied if the atom
            is on the normal side of the plane. For instance, the plane with
            (A, B, C, D) = (0, 0, 1, -7) defines a plane in the xy plane with a z
            intercept of +7 and a normal vector pointing in the +z direction.
            If the atom has z > 7, then a downward force would be applied of
            k * (atom.z - 7). The same plane with the normal vector pointing in
            the -z direction would be given by (A, B, C, D) = (0, 0, -1, 7).
            """

            if isinstance(a2, int):
                self._type = 'two atoms'
                self.indices = [a1, a2]
            elif len(a2) == 3:
                self._type = 'point'
                self.index = a1
                self.origin = np.array(a2)
            elif len(a2) == 4:
                self._type = 'plane'
                self.index = a1
                self.plane = a2
            else:
                raise RuntimeError('Unknown type for a2')
            self.threshold = rt
            self.spring = k

        def get_removed_dof(self, atoms):
            return 0

        def todict(self):
            dct = {'name': 'Hookean'}
            dct['kwargs'] = {'rt': self.threshold,
                             'k': self.spring}
            if self._type == 'two atoms':
                dct['kwargs']['a1'] = self.indices[0]
                dct['kwargs']['a2'] = self.indices[1]
            elif self._type == 'point':
                dct['kwargs']['a1'] = self.index
                dct['kwargs']['a2'] = self.origin
            elif self._type == 'plane':
                dct['kwargs']['a1'] = self.index
                dct['kwargs']['a2'] = self.plane
            else:
                raise NotImplementedError('Bad type: %s' % self._type)
            return dct

        def adjust_positions(self, atoms, newpositions):
            pass

        def adjust_momenta(self, atoms, momenta):
            pass

        def adjust_forces(self, atoms, forces):
            positions = atoms.positions
            if self._type == 'plane':
                A, B, C, D = self.plane
                x, y, z = positions[self.index]
                d = ((A * x + B * y + C * z + D) /
                     np.sqrt(A**2 + B**2 + C**2))
                if d < 0:
                    return
                magnitude = self.spring * d
                direction = - np.array((A, B, C)) / np.linalg.norm((A, B, C))
                forces[self.index] += direction * magnitude
                return
            if self._type == 'two atoms':
                p1, p2 = positions[self.indices]
            elif self._type == 'point':
                p1 = positions[self.index]
                p2 = self.origin
            displace, _ = find_mic(p2 - p1, atoms.cell, atoms.pbc)
            bondlength = np.linalg.norm(displace)
            if bondlength > self.threshold:
                magnitude = self.spring * (bondlength - self.threshold)
                direction = displace / np.linalg.norm(displace)
                if self._type == 'two atoms':
                    forces[self.indices[0]] += direction * magnitude
                    forces[self.indices[1]] -= direction * magnitude
                else:
                    forces[self.index] += direction * magnitude

        def adjust_potential_energy(self, atoms):
            """Returns the difference to the potential energy due to an active
            constraint. (That is, the quantity returned is to be added to the
            potential energy.)"""
            positions = atoms.positions
            if self._type == 'plane':
                A, B, C, D = self.plane
                x, y, z = positions[self.index]
                d = ((A * x + B * y + C * z + D) /
                     np.sqrt(A**2 + B**2 + C**2))
                if d > 0:
                    return 0.5 * self.spring * d**2
                else:
                    return 0.
            if self._type == 'two atoms':
                p1, p2 = positions[self.indices]
            elif self._type == 'point':
                p1 = positions[self.index]
                p2 = self.origin
            displace, _ = find_mic(p2 - p1, atoms.cell, atoms.pbc)
            bondlength = np.linalg.norm(displace)
            if bondlength > self.threshold:
                return 0.5 * self.spring * (bondlength - self.threshold)**2
            else:
                return 0.

        def get_indices(self):
            if self._type == 'two atoms':
                return self.indices
            elif self._type == 'point':
                return self.index
            elif self._type == 'plane':
                return self.index

        def index_shuffle(self, atoms, ind):
            # See docstring of superclass
            if self._type == 'two atoms':
                newa = [-1, -1]  # Signal error
                for new, old in slice2enlist(ind, len(atoms)):
                    for i, a in enumerate(self.indices):
                        if old == a:
                            newa[i] = new
                if newa[0] == -1 or newa[1] == -1:
                    raise IndexError('Constraint not part of slice')
                self.indices = newa
            elif (self._type == 'point') or (self._type == 'plane'):
                newa = -1   # Signal error
                for new, old in slice2enlist(ind, len(atoms)):
                    if old == self.index:
                        newa = new
                        break
                if newa == -1:
                    raise IndexError('Constraint not part of slice')
                self.index = newa

        def __repr__(self):
            if self._type == 'two atoms':
                return 'Hookean(%d, %d)' % tuple(self.indices)
            elif self._type == 'point':
                return 'Hookean(%d) to cartesian' % self.index
            else:
                return 'Hookean(%d) to plane' % self.index

    class ExternalForce(FixConstraint):
        """Constraint object for pulling two atoms apart by an external force.

        You can combine this constraint for example with FixBondLength but make
        sure that *ExternalForce* comes first in the list if there are overlaps
        between atom1-2 and atom3-4:

        >>> con1 = ExternalForce(atom1, atom2, f_ext)
        >>> con2 = FixBondLength(atom3, atom4)
        >>> atoms.set_constraint([con1, con2])

        see ase/test/external_force.py"""

        def __init__(self, a1, a2, f_ext):
            self.indices = [a1, a2]
            self.external_force = f_ext

        def get_removed_dof(self, atoms):
            return 0

        def adjust_positions(self, atoms, new):
            pass

        def adjust_forces(self, atoms, forces):
            dist = np.subtract.reduce(atoms.positions[self.indices])
            force = self.external_force * dist / np.linalg.norm(dist)
            forces[self.indices] += (force, -force)

        def adjust_potential_energy(self, atoms):
            dist = np.subtract.reduce(atoms.positions[self.indices])
            return -np.linalg.norm(dist) * self.external_force

        def index_shuffle(self, atoms, ind):
            """Shuffle the indices of the two atoms in this constraint"""
            newa = [-1, -1]  # Signal error
            for new, old in slice2enlist(ind, len(atoms)):
                for i, a in enumerate(self.indices):
                    if old == a:
                        newa[i] = new
            if newa[0] == -1 or newa[1] == -1:
                raise IndexError('Constraint not part of slice')
            self.indices = newa

        def __repr__(self):
            return 'ExternalForce(%d, %d, %f)' % (self.indices[0],
                                                  self.indices[1],
                                                  self.external_force)

        def todict(self):
            return {'name': 'ExternalForce',
                    'kwargs': {'a1': self.indices[0], 'a2': self.indices[1],
                               'f_ext': self.external_force}}

    class MirrorForce(FixConstraint):
        """Constraint object for mirroring the force between two atoms.

        This class is designed to find a transition state with the help of a
        single optimization. It can be used if the transition state belongs to a
        bond breaking reaction. First the given bond length will be fixed until
        all other degrees of freedom are optimized, then the forces of the two
        atoms will be mirrored to find the transition state. The mirror plane is
        perpendicular to the connecting line of the atoms. Transition states in
        dependence of the force can be obtained by stretching the molecule and
        fixing its total length with *FixBondLength* or by using *ExternalForce*
        during the optimization with *MirrorForce*.

        Parameters
        ----------
        a1: int
            First atom index.
        a2: int
            Second atom index.
        max_dist: float
            Upper limit of the bond length interval where the transition state
            can be found.
        min_dist: float
            Lower limit of the bond length interval where the transition state
            can be found.
        fmax: float
            Maximum force used for the optimization.

        Notes
        -----
        You can combine this constraint for example with FixBondLength but make
        sure that *MirrorForce* comes first in the list if there are overlaps
        between atom1-2 and atom3-4:

        >>> con1 = MirrorForce(atom1, atom2)
        >>> con2 = FixBondLength(atom3, atom4)
        >>> atoms.set_constraint([con1, con2])

        """

        def __init__(self, a1, a2, max_dist=2.5, min_dist=1., fmax=0.1):
            self.indices = [a1, a2]
            self.min_dist = min_dist
            self.max_dist = max_dist
            self.fmax = fmax

        def adjust_positions(self, atoms, new):
            pass

        def adjust_forces(self, atoms, forces):
            dist = np.subtract.reduce(atoms.positions[self.indices])
            d = np.linalg.norm(dist)
            if (d < self.min_dist) or (d > self.max_dist):
                # Stop structure optimization
                forces[:] *= 0
                return
            dist /= d
            df = np.subtract.reduce(forces[self.indices])
            f = df.dot(dist)
            con_saved = atoms.constraints
            try:
                con = [con for con in con_saved
                       if not isinstance(con, MirrorForce)]
                atoms.set_constraint(con)
                forces_copy = atoms.get_forces()
            finally:
                atoms.set_constraint(con_saved)
            df1 = -1 / 2. * f * dist
            forces_copy[self.indices] += (df1, -df1)
            # Check if forces would be converged if the bond with mirrored forces
            # would also be fixed
            if (forces_copy**2).sum(axis=1).max() < self.fmax**2:
                factor = 1.
            else:
                factor = 0.
            df1 = -(1 + factor) / 2. * f * dist
            forces[self.indices] += (df1, -df1)

        def index_shuffle(self, atoms, ind):
            """Shuffle the indices of the two atoms in this constraint

            """
            newa = [-1, -1]  # Signal error
            for new, old in slice2enlist(ind, len(atoms)):
                for i, a in enumerate(self.indices):
                    if old == a:
                        newa[i] = new
            if newa[0] == -1 or newa[1] == -1:
                raise IndexError('Constraint not part of slice')
            self.indices = newa

        def __repr__(self):
            return 'MirrorForce(%d, %d, %f, %f, %f)' % (
                self.indices[0], self.indices[1], self.max_dist, self.min_dist,
                self.fmax)

        def todict(self):
            return {'name': 'MirrorForce',
                    'kwargs': {'a1': self.indices[0], 'a2': self.indices[1],
                               'max_dist': self.max_dist,
                               'min_dist': self.min_dist, 'fmax': self.fmax}}

    class MirrorTorque(FixConstraint):
        """Constraint object for mirroring the torque acting on a dihedral
        angle defined by four atoms.

        This class is designed to find a transition state with the help of a
        single optimization. It can be used if the transition state belongs to a
        cis-trans-isomerization with a change of dihedral angle. First the given
        dihedral angle will be fixed until all other degrees of freedom are
        optimized, then the torque acting on the dihedral angle will be mirrored
        to find the transition state. Transition states in
        dependence of the force can be obtained by stretching the molecule and
        fixing its total length with *FixBondLength* or by using *ExternalForce*
        during the optimization with *MirrorTorque*.

        This constraint can be used to find
        transition states of cis-trans-isomerization.

        a1    a4
        |      |
        a2 __ a3

        Parameters
        ----------
        a1: int
            First atom index.
        a2: int
            Second atom index.
        a3: int
            Third atom index.
        a4: int
            Fourth atom index.
        max_angle: float
            Upper limit of the dihedral angle interval where the transition state
            can be found.
        min_angle: float
            Lower limit of the dihedral angle interval where the transition state
            can be found.
        fmax: float
            Maximum force used for the optimization.

        Notes
        -----
        You can combine this constraint for example with FixBondLength but make
        sure that *MirrorTorque* comes first in the list if there are overlaps
        between atom1-4 and atom5-6:

        >>> con1 = MirrorTorque(atom1, atom2, atom3, atom4)
        >>> con2 = FixBondLength(atom5, atom6)
        >>> atoms.set_constraint([con1, con2])

        """

        def __init__(self, a1, a2, a3, a4, max_angle=2 * np.pi, min_angle=0.,
                     fmax=0.1):
            self.indices = [a1, a2, a3, a4]
            self.min_angle = min_angle
            self.max_angle = max_angle
            self.fmax = fmax

        def adjust_positions(self, atoms, new):
            pass

        def adjust_forces(self, atoms, forces):
            angle = atoms.get_dihedral(self.indices[0], self.indices[1],
                                       self.indices[2], self.indices[3])
            angle *= np.pi / 180.
            if (angle < self.min_angle) or (angle > self.max_angle):
                # Stop structure optimization
                forces[:] *= 0
                return
            p = atoms.positions[self.indices]
            f = forces[self.indices]

            f0 = (f[1] + f[2]) / 2.
            ff = f - f0
            p0 = (p[2] + p[1]) / 2.
            m0 = np.cross(p[1] - p0, ff[1]) / (p[1] - p0).dot(p[1] - p0)
            fff = ff - np.cross(m0, p - p0)
            d1 = np.cross(np.cross(p[1] - p0, p[0] - p[1]), p[1] - p0) / \
                (p[1] - p0).dot(p[1] - p0)
            d2 = np.cross(np.cross(p[2] - p0, p[3] - p[2]), p[2] - p0) / \
                (p[2] - p0).dot(p[2] - p0)
            omegap1 = (np.cross(d1, fff[0]) / d1.dot(d1)).dot(p[1] - p0) / \
                np.linalg.norm(p[1] - p0)
            omegap2 = (np.cross(d2, fff[3]) / d2.dot(d2)).dot(p[2] - p0) / \
                np.linalg.norm(p[2] - p0)
            omegap = omegap1 + omegap2
            con_saved = atoms.constraints
            try:
                con = [con for con in con_saved
                       if not isinstance(con, MirrorTorque)]
                atoms.set_constraint(con)
                forces_copy = atoms.get_forces()
            finally:
                atoms.set_constraint(con_saved)
            df1 = -1 / 2. * omegap * np.cross(p[1] - p0, d1) / \
                np.linalg.norm(p[1] - p0)
            df2 = -1 / 2. * omegap * np.cross(p[2] - p0, d2) / \
                np.linalg.norm(p[2] - p0)
            forces_copy[self.indices] += (df1, [0., 0., 0.], [0., 0., 0.], df2)
            # Check if forces would be converged if the dihedral angle with
            # mirrored torque would also be fixed
            if (forces_copy**2).sum(axis=1).max() < self.fmax**2:
                factor = 1.
            else:
                factor = 0.
            df1 = -(1 + factor) / 2. * omegap * np.cross(p[1] - p0, d1) / \
                np.linalg.norm(p[1] - p0)
            df2 = -(1 + factor) / 2. * omegap * np.cross(p[2] - p0, d2) / \
                np.linalg.norm(p[2] - p0)
            forces[self.indices] += (df1, [0., 0., 0.], [0., 0., 0.], df2)

        def index_shuffle(self, atoms, ind):
            # See docstring of superclass
            indices = []
            for new, old in slice2enlist(ind, len(atoms)):
                if old in self.indices:
                    indices.append(new)
            if len(indices) == 0:
                raise IndexError('All indices in MirrorTorque not part of slice')
            self.indices = np.asarray(indices, int)

        def __repr__(self):
            return 'MirrorTorque(%d, %d, %d, %d, %f, %f, %f)' % (
                self.indices[0], self.indices[1], self.indices[2],
                self.indices[3], self.max_angle, self.min_angle, self.fmax)

        def todict(self):
            return {'name': 'MirrorTorque',
                    'kwargs': {'a1': self.indices[0], 'a2': self.indices[1],
                               'a3': self.indices[2], 'a4': self.indices[3],
                               'max_angle': self.max_angle,
                               'min_angle': self.min_angle, 'fmax': self.fmax}}

    class Filter:
        """Subset filter class."""

        def __init__(self, atoms, indices=None, mask=None):
            """Filter atoms.

            This filter can be used to hide degrees of freedom in an Atoms
            object.

            Parameters
            ----------
            indices : list of int
               Indices for those atoms that should remain visible.
            mask : list of bool
               One boolean per atom indicating if the atom should remain
               visible or not.

            If a Trajectory tries to save this object, it will instead
            save the underlying Atoms object.  To prevent this, override
            the iterimages method.
            """

            self.atoms = atoms
            self.constraints = []
            # Make self.info a reference to the underlying atoms' info dictionary.
            self.info = self.atoms.info

            if indices is None and mask is None:
                raise ValueError('Use "indices" or "mask".')
            if indices is not None and mask is not None:
                raise ValueError('Use only one of "indices" and "mask".')

            if mask is not None:
                self.index = np.asarray(mask, bool)
                self.n = self.index.sum()
            else:
                self.index = np.asarray(indices, int)
                self.n = len(self.index)

        def iterimages(self):
            # Present the real atoms object to Trajectory and friends
            return self.atoms.iterimages()

        def get_cell(self):
            """Returns the computational cell.

            The computational cell is the same as for the original system.
            """
            return self.atoms.get_cell()

        def get_pbc(self):
            """Returns the periodic boundary conditions.

            The boundary conditions are the same as for the original system.
            """
            return self.atoms.get_pbc()

        def get_positions(self):
            'Return the positions of the visible atoms.'
            return self.atoms.get_positions()[self.index]

        def set_positions(self, positions, **kwargs):
            'Set the positions of the visible atoms.'
            pos = self.atoms.get_positions()
            pos[self.index] = positions
            self.atoms.set_positions(pos, **kwargs)

        positions = property(get_positions, set_positions,
                             doc='Positions of the atoms')

        def get_momenta(self):
            'Return the momenta of the visible atoms.'
            return self.atoms.get_momenta()[self.index]

        def set_momenta(self, momenta, **kwargs):
            'Set the momenta of the visible atoms.'
            mom = self.atoms.get_momenta()
            mom[self.index] = momenta
            self.atoms.set_momenta(mom, **kwargs)

        def get_atomic_numbers(self):
            'Return the atomic numbers of the visible atoms.'
            return self.atoms.get_atomic_numbers()[self.index]

        def set_atomic_numbers(self, atomic_numbers):
            'Set the atomic numbers of the visible atoms.'
            z = self.atoms.get_atomic_numbers()
            z[self.index] = atomic_numbers
            self.atoms.set_atomic_numbers(z)

        def get_tags(self):
            'Return the tags of the visible atoms.'
            return self.atoms.get_tags()[self.index]

        def set_tags(self, tags):
            'Set the tags of the visible atoms.'
            tg = self.atoms.get_tags()
            tg[self.index] = tags
            self.atoms.set_tags(tg)

        def get_forces(self, *args, **kwargs):
            return self.atoms.get_forces(*args, **kwargs)[self.index]

        def get_stress(self, *args, **kwargs):
            return self.atoms.get_stress(*args, **kwargs)

        def get_stresses(self, *args, **kwargs):
            return self.atoms.get_stresses(*args, **kwargs)[self.index]

        def get_masses(self):
            return self.atoms.get_masses()[self.index]

        def get_potential_energy(self, **kwargs):
            """Calculate potential energy.

            Returns the potential energy of the full system.
            """
            return self.atoms.get_potential_energy(**kwargs)

        def get_chemical_symbols(self):
            return self.atoms.get_chemical_symbols()

        def get_initial_magnetic_moments(self):
            return self.atoms.get_initial_magnetic_moments()

        def get_calculator(self):
            """Returns the calculator.

            WARNING: The calculator is unaware of this filter, and sees a
            different number of atoms.
            """
            return self.atoms.calc

        @property
        def calc(self):
            return self.atoms.calc

        def get_celldisp(self):
            return self.atoms.get_celldisp()

        def has(self, name):
            'Check for existence of array.'
            return self.atoms.has(name)

        def __len__(self):
            'Return the number of movable atoms.'
            return self.n

        def __getitem__(self, i):
            'Return an atom.'
            return self.atoms[self.index[i]]

    class StrainFilter(Filter):
        """Modify the supercell while keeping the scaled positions fixed.

        Presents the strain of the supercell as the generalized positions,
        and the global stress tensor (times the volume) as the generalized
        force.

        This filter can be used to relax the unit cell until the stress is
        zero.  If MDMin is used for this, the timestep (dt) to be used
        depends on the system size. 0.01/x where x is a typical dimension
        seems like a good choice.

        The stress and strain are presented as 6-vectors, the order of the
        components follow the standard engingeering practice: xx, yy, zz,
        yz, xz, xy.

        """

        def __init__(self, atoms, mask=None, include_ideal_gas=False):
            """Create a filter applying a homogeneous strain to a list of atoms.

            The first argument, atoms, is the atoms object.

            The optional second argument, mask, is a list of six booleans,
            indicating which of the six independent components of the
            strain that are allowed to become non-zero.  It defaults to
            [1,1,1,1,1,1].

            """

            self.strain = np.zeros(6)
            self.include_ideal_gas = include_ideal_gas

            if mask is None:
                mask = np.ones(6)
            else:
                mask = np.array(mask)

            Filter.__init__(self, atoms, mask=mask)
            self.mask = mask
            self.origcell = atoms.get_cell()

        def get_positions(self):
            return self.strain.reshape((2, 3)).copy()

        def set_positions(self, new):
            new = new.ravel() * self.mask
            eps = np.array([[1.0 + new[0], 0.5 * new[5], 0.5 * new[4]],
                            [0.5 * new[5], 1.0 + new[1], 0.5 * new[3]],
                            [0.5 * new[4], 0.5 * new[3], 1.0 + new[2]]])

            self.atoms.set_cell(np.dot(self.origcell, eps), scale_atoms=True)
            self.strain[:] = new

        def get_forces(self, **kwargs):
            stress = self.atoms.get_stress(include_ideal_gas=self.include_ideal_gas)
            return -self.atoms.get_volume() * (stress * self.mask).reshape((2, 3))

        def has(self, x):
            return self.atoms.has(x)

        def __len__(self):
            return 2

    class UnitCellFilter(Filter):
        """Modify the supercell and the atom positions. """
        def __init__(self, atoms, mask=None,
                     cell_factor=None,
                     hydrostatic_strain=False,
                     constant_volume=False,
                     scalar_pressure=0.0):
            """Create a filter that returns the atomic forces and unit cell
            stresses together, so they can simultaneously be minimized.

            The first argument, atoms, is the atoms object. The optional second
            argument, mask, is a list of booleans, indicating which of the six
            independent components of the strain are relaxed.

            - True = relax to zero
            - False = fixed, ignore this component

            Degrees of freedom are the positions in the original undeformed cell,
            plus the deformation tensor (extra 3 "atoms"). This gives forces
            consistent with numerical derivatives of the potential energy
            with respect to the cell degreees of freedom.

            For full details see:
                E. B. Tadmor, G. S. Smith, N. Bernstein, and E. Kaxiras,
                Phys. Rev. B 59, 235 (1999)

            You can still use constraints on the atoms, e.g. FixAtoms, to control
            the relaxation of the atoms.

            >>> # this should be equivalent to the StrainFilter
            >>> atoms = Atoms(...)
            >>> atoms.set_constraint(FixAtoms(mask=[True for atom in atoms]))
            >>> ucf = UnitCellFilter(atoms)

            You should not attach this UnitCellFilter object to a
            trajectory. Instead, create a trajectory for the atoms, and
            attach it to an optimizer like this:

            >>> atoms = Atoms(...)
            >>> ucf = UnitCellFilter(atoms)
            >>> qn = QuasiNewton(ucf)
            >>> traj = Trajectory('TiO2.traj', 'w', atoms)
            >>> qn.attach(traj)
            >>> qn.run(fmax=0.05)

            Helpful conversion table:

            - 0.05 eV/A^3   = 8 GPA
            - 0.003 eV/A^3  = 0.48 GPa
            - 0.0006 eV/A^3 = 0.096 GPa
            - 0.0003 eV/A^3 = 0.048 GPa
            - 0.0001 eV/A^3 = 0.02 GPa

            Additional optional arguments:

            cell_factor: float (default float(len(atoms)))
                Factor by which deformation gradient is multiplied to put
                it on the same scale as the positions when assembling
                the combined position/cell vector. The stress contribution to
                the forces is scaled down by the same factor. This can be thought
                of as a very simple preconditioners. Default is number of atoms
                which gives approximately the correct scaling.

            hydrostatic_strain: bool (default False)
                Constrain the cell by only allowing hydrostatic deformation.
                The virial tensor is replaced by np.diag([np.trace(virial)]*3).

            constant_volume: bool (default False)
                Project out the diagonal elements of the virial tensor to allow
                relaxations at constant volume, e.g. for mapping out an
                energy-volume curve. Note: this only approximately conserves
                the volume and breaks energy/force consistency so can only be
                used with optimizers that do require do a line minimisation
                (e.g. FIRE).

            scalar_pressure: float (default 0.0)
                Applied pressure to use for enthalpy pV term. As above, this
                breaks energy/force consistency.
            """

            Filter.__init__(self, atoms, indices=range(len(atoms)))
            self.atoms = atoms
            self.orig_cell = atoms.get_cell()
            self.stress = None

            if mask is None:
                mask = np.ones(6)
            mask = np.asarray(mask)
            if mask.shape == (6,):
                self.mask = voigt_6_to_full_3x3_stress(mask)
            elif mask.shape == (3, 3):
                self.mask = mask
            else:
                raise ValueError('shape of mask should be (3,3) or (6,)')

            if cell_factor is None:
                cell_factor = float(len(atoms))
            self.hydrostatic_strain = hydrostatic_strain
            self.constant_volume = constant_volume
            self.scalar_pressure = scalar_pressure
            self.cell_factor = cell_factor
            self.copy = self.atoms.copy
            self.arrays = self.atoms.arrays

        def deform_grad(self):
            return np.linalg.solve(self.orig_cell, self.atoms.cell).T

        def get_positions(self):
            """
            this returns an array with shape (natoms + 3,3).

            the first natoms rows are the positions of the atoms, the last
            three rows are the deformation tensor associated with the unit cell,
            scaled by self.cell_factor.
            """

            cur_deform_grad = self.deform_grad()
            natoms = len(self.atoms)
            pos = np.zeros((natoms + 3, 3))
            # UnitCellFilter's positions are the self.atoms.positions but without
            # the applied deformation gradient
            pos[:natoms] = np.linalg.solve(cur_deform_grad,
                                           self.atoms.positions.T).T
            # UnitCellFilter's cell DOFs are the deformation gradient times a
            # scaling factor
            pos[natoms:] = self.cell_factor * cur_deform_grad
            return pos

        def set_positions(self, new, **kwargs):
            """
            new is an array with shape (natoms+3,3).

            the first natoms rows are the positions of the atoms, the last
            three rows are the deformation tensor used to change the cell shape.

            the new cell is first set from original cell transformed by the new
            deformation gradient, then the positions are set with respect to the
            current cell by transforming them with the same deformation gradient
            """

            natoms = len(self.atoms)
            new_atom_positions = new[:natoms]
            new_deform_grad = new[natoms:] / self.cell_factor
            # Set the new cell from the original cell and the new
            # deformation gradient.  Both current and final structures should
            # preserve symmetry, so if set_cell() calls FixSymmetry.adjust_cell(),
            # it should be OK
            self.atoms.set_cell(self.orig_cell @ new_deform_grad.T,
                                scale_atoms=True)
            # Set the positions from the ones passed in (which are without the
            # deformation gradient applied) and the new deformation gradient.
            # This should also preserve symmetry, so if set_positions() calls
            # FixSymmetyr.adjust_positions(), it should be OK
            self.atoms.set_positions(new_atom_positions @ new_deform_grad.T,
                                     **kwargs)

        def get_potential_energy(self, force_consistent=True):
            """
            returns potential energy including enthalpy PV term.
            """
            atoms_energy = self.atoms.get_potential_energy(
                force_consistent=force_consistent)
            return atoms_energy + self.scalar_pressure * self.atoms.get_volume()

        def get_forces(self, **kwargs):
            """
            returns an array with shape (natoms+3,3) of the atomic forces
            and unit cell stresses.

            the first natoms rows are the forces on the atoms, the last
            three rows are the forces on the unit cell, which are
            computed from the stress tensor.
            """

            stress = self.atoms.get_stress(**kwargs)
            atoms_forces = self.atoms.get_forces(**kwargs)

            volume = self.atoms.get_volume()
            virial = -volume * (voigt_6_to_full_3x3_stress(stress) +
                                np.diag([self.scalar_pressure] * 3))
            cur_deform_grad = self.deform_grad()
            atoms_forces = atoms_forces @ cur_deform_grad
            virial = np.linalg.solve(cur_deform_grad, virial.T).T

            if self.hydrostatic_strain:
                vtr = virial.trace()
                virial = np.diag([vtr / 3.0, vtr / 3.0, vtr / 3.0])

            # Zero out components corresponding to fixed lattice elements
            if (self.mask != 1.0).any():
                virial *= self.mask

            if self.constant_volume:
                vtr = virial.trace()
                np.fill_diagonal(virial, np.diag(virial) - vtr / 3.0)

            natoms = len(self.atoms)
            forces = np.zeros((natoms + 3, 3))
            forces[:natoms] = atoms_forces
            forces[natoms:] = virial / self.cell_factor

            self.stress = -full_3x3_to_voigt_6_stress(virial) / volume
            return forces

        def get_stress(self):
            raise PropertyNotImplementedError

        def has(self, x):
            return self.atoms.has(x)

        def __len__(self):
            return (len(self.atoms) + 3)

    class ExpCellFilter(UnitCellFilter):
        """Modify the supercell and the atom positions."""
        def __init__(self, atoms, mask=None,
                     cell_factor=None,
                     hydrostatic_strain=False,
                     constant_volume=False,
                     scalar_pressure=0.0):
            r"""Create a filter that returns the atomic forces and unit cell
            stresses together, so they can simultaneously be minimized.

            The first argument, atoms, is the atoms object. The optional second
            argument, mask, is a list of booleans, indicating which of the six
            independent components of the strain are relaxed.

            - True = relax to zero
            - False = fixed, ignore this component

            Degrees of freedom are the positions in the original undeformed cell,
            plus the log of the deformation tensor (extra 3 "atoms"). This gives
            forces consistent with numerical derivatives of the potential energy
            with respect to the cell degrees of freedom.

            For full details see:
                E. B. Tadmor, G. S. Smith, N. Bernstein, and E. Kaxiras,
                Phys. Rev. B 59, 235 (1999)

            You can still use constraints on the atoms, e.g. FixAtoms, to control
            the relaxation of the atoms.

            >>> # this should be equivalent to the StrainFilter
            >>> atoms = Atoms(...)
            >>> atoms.set_constraint(FixAtoms(mask=[True for atom in atoms]))
            >>> ecf = ExpCellFilter(atoms)

            You should not attach this ExpCellFilter object to a
            trajectory. Instead, create a trajectory for the atoms, and
            attach it to an optimizer like this:

            >>> atoms = Atoms(...)
            >>> ecf = ExpCellFilter(atoms)
            >>> qn = QuasiNewton(ecf)
            >>> traj = Trajectory('TiO2.traj', 'w', atoms)
            >>> qn.attach(traj)
            >>> qn.run(fmax=0.05)

            Helpful conversion table:

            - 0.05 eV/A^3   = 8 GPA
            - 0.003 eV/A^3  = 0.48 GPa
            - 0.0006 eV/A^3 = 0.096 GPa
            - 0.0003 eV/A^3 = 0.048 GPa
            - 0.0001 eV/A^3 = 0.02 GPa

            Additional optional arguments:

            cell_factor: (DEPRECATED)
                Retained for backwards compatibility, but no longer used.

            hydrostatic_strain: bool (default False)
                Constrain the cell by only allowing hydrostatic deformation.
                The virial tensor is replaced by np.diag([np.trace(virial)]*3).

            constant_volume: bool (default False)
                Project out the diagonal elements of the virial tensor to allow
                relaxations at constant volume, e.g. for mapping out an
                energy-volume curve.

            scalar_pressure: float (default 0.0)
                Applied pressure to use for enthalpy pV term. As above, this
                breaks energy/force consistency.

            Implementation details:

            The implementation is based on that of Christoph Ortner in JuLIP.jl:
            https://github.com/libAtoms/JuLIP.jl/blob/expcell/src/Constraints.jl#L244

            We decompose the deformation gradient as

                F = exp(U) F0
                x =  F * F0^{-1} z  = exp(U) z

            If we write the energy as a function of U we can transform the
            stress associated with a perturbation V into a derivative using a
            linear map V -> L(U, V).

            \phi( exp(U+tV) (z+tv) ) ~ \phi'(x) . (exp(U) v) + \phi'(x) .
                                                        ( L(U, V) exp(-U) exp(U) z )

            where

                   \nabla E(U) : V  =  [S exp(-U)'] : L(U,V)
                                    =  L'(U, S exp(-U)') : V
                                    =  L(U', S exp(-U)') : V
                                    =  L(U, S exp(-U)) : V     (provided U = U')

            where the : operator represents double contraction,
            i.e. A:B = trace(A'B), and

              F = deformation tensor - 3x3 matrix
              F0 = reference deformation tensor - 3x3 matrix, np.eye(3) here
              U = cell degrees of freedom used here - 3x3 matrix
              V = perturbation to cell DoFs - 3x3 matrix
              v = perturbation to position DoFs
              x = atomic positions in deformed cell
              z = atomic positions in original cell
              \phi = potential energy
              S = stress tensor [3x3 matrix]
              L(U, V) = directional derivative of exp at U in direction V, i.e
              d/dt exp(U + t V)|_{t=0} = L(U, V)

            This means we can write

              d/dt E(U + t V)|_{t=0} = L(U, S exp (-U)) : V

            and therefore the contribution to the gradient of the energy is

              \nabla E(U) / \nabla U_ij =  [L(U, S exp(-U))]_ij
            """

            Filter.__init__(self, atoms, indices=range(len(atoms)))
            UnitCellFilter.__init__(self, atoms, mask, cell_factor,
                                    hydrostatic_strain,
                                    constant_volume, scalar_pressure)
            if cell_factor is not None:
                warn("cell_factor is deprecated")
            self.cell_factor = 1.0

        def get_positions(self):
            pos = UnitCellFilter.get_positions(self)
            natoms = len(self.atoms)
            pos[natoms:] = logm(self.deform_grad())
            return pos

        def set_positions(self, new, **kwargs):
            natoms = len(self.atoms)
            new2 = new.copy()
            new2[natoms:] = expm(new[natoms:])
            UnitCellFilter.set_positions(self, new2, **kwargs)

        def get_forces(self, **kwargs):
            forces = UnitCellFilter.get_forces(self, **kwargs)

            # forces on atoms are same as UnitCellFilter, we just
            # need to modify the stress contribution
            stress = self.atoms.get_stress(**kwargs)
            volume = self.atoms.get_volume()
            virial = -volume * (voigt_6_to_full_3x3_stress(stress) +
                                np.diag([self.scalar_pressure] * 3))

            cur_deform_grad = self.deform_grad()
            cur_deform_grad_log = logm(cur_deform_grad)

            if self.hydrostatic_strain:
                vtr = virial.trace()
                virial = np.diag([vtr / 3.0, vtr / 3.0, vtr / 3.0])

            # Zero out components corresponding to fixed lattice elements
            if (self.mask != 1.0).any():
                virial *= self.mask

            deform_grad_log_force_naive = virial.copy()
            Y = np.zeros((6, 6))
            Y[0:3, 0:3] = cur_deform_grad_log
            Y[3:6, 3:6] = cur_deform_grad_log
            Y[0:3, 3:6] = - virial @ expm(-cur_deform_grad_log)
            deform_grad_log_force = -expm(Y)[0:3, 3:6]
            for (i1, i2) in [(0, 1), (0, 2), (1, 2)]:
                ff = 0.5 * (deform_grad_log_force[i1, i2] +
                            deform_grad_log_force[i2, i1])
                deform_grad_log_force[i1, i2] = ff
                deform_grad_log_force[i2, i1] = ff

            # check for reasonable alignment between naive and
            # exact search directions
            all_are_equal = np.all(np.isclose(deform_grad_log_force,
                                              deform_grad_log_force_naive))
            if all_are_equal or \
                (np.sum(deform_grad_log_force * deform_grad_log_force_naive) /
                 np.sqrt(np.sum(deform_grad_log_force**2) *
                         np.sum(deform_grad_log_force_naive**2)) > 0.8):
                deform_grad_log_force = deform_grad_log_force_naive

            # Cauchy stress used for convergence testing
            convergence_crit_stress = -(virial / volume)
            if self.constant_volume:
                # apply constraint to force
                dglf_trace = deform_grad_log_force.trace()
                np.fill_diagonal(deform_grad_log_force,
                                 np.diag(deform_grad_log_force) - dglf_trace / 3.0)
                # apply constraint to Cauchy stress used for convergence testing
                ccs_trace = convergence_crit_stress.trace()
                np.fill_diagonal(convergence_crit_stress,
                                 np.diag(convergence_crit_stress) - ccs_trace / 3.0)

            # pack gradients into vector
            natoms = len(self.atoms)
            forces[natoms:] = deform_grad_log_force
            self.stress = full_3x3_to_voigt_6_stress(convergence_crit_stress)
            return forces