Source code for ase.constraints

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

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

        if mask is not None:
            if indices is not None:
                raise ValueError('Use only one of "indices" and "mask".')
            indices = mask
        indices = np.atleast_1d(indices)
        if np.ndim(indices) > 1:
            raise ValueError('indices has wrong amount of dimensions. '
                             f'Got {np.ndim(indices)}, expected ndim <= 1')

        if indices.dtype == bool:
            indices = np.arange(len(indices))[indices]
        elif not np.issubdtype(indices.dtype, np.integer):
            raise ValueError('Indices must be integers or boolean mask, '
                             f'not dtype={indices.dtype}')

        if len(set(indices)) < len(indices):
            raise ValueError(
                'The indices array contains duplicates. '
                'Perhaps you want to specify a mask instead, but '
                'forgot the mask= keyword.')

        self.index = indices

    def index_shuffle(self, atoms, ind):
        # See docstring of superclass
        index = []

        # Resolve negative indices:
        actual_indices = set(np.arange(len(atoms))[self.index])

        for new, old in slice2enlist(ind, len(atoms)):
            if old in actual_indices:
                index.append(new)
        if len(index) == 0:
            raise IndexError('All indices in FixAtoms not part of slice')
        self.index = np.asarray(index, int)
        # XXX make immutable

    def get_indices(self):
        return self.index.copy()

    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)
        # XXX make immutable
        return self

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

        Required for removing atoms with existing constraint.
        """

        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]
        # XXX make immutable
        if len(self.index) == 0:
            return None
        return self


[docs]class FixAtoms(IndexedConstraint): """Fix chosen atoms. Examples -------- Fix all Copper atoms: >>> mask = (atoms.symbols == 'Cu') >>> 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) """ 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 __repr__(self): clsname = type(self).__name__ indices = ints2string(self.index) return f'{clsname}(indices={indices})' def todict(self): return {'name': 'FixAtoms', 'kwargs': {'indices': self.index.tolist()}}
[docs]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] + ', ...]'
[docs]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')
[docs]def FixBondLength(a1, a2): """Fix distance between atoms with indices a1 and a2.""" return FixBondLengths([(a1, a2)])
[docs]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')
[docs]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()
def _normalize(direction): if np.shape(direction) != (3,): raise ValueError("len(direction) is {len(direction)}. Has to be 3") direction = np.asarray(direction) / np.linalg.norm(direction) return direction
[docs]class FixedPlane(IndexedConstraint): """ Constraint object for fixing chosen atoms to only move in a plane. The plane is defined by its normal vector *direction* """ def __init__(self, indices, direction): """Constrain chosen atoms. Parameters ---------- indices : int or list of int Index or indices for atoms that should be constrained direction : list of 3 int Direction of the normal vector Examples -------- Fix all Copper atoms to only move in the yz-plane: >>> from ase.constraints import FixedPlane >>> c = FixedPlane( >>> indices=[atom.index for atom in atoms if atom.symbol == 'Cu'], >>> direction=[1, 0, 0], >>> ) >>> atoms.set_constraint(c) or constrain a single atom with the index 0 to move in the xy-plane: >>> c = FixedPlane(indices=0, direction=[0, 0, 1]) >>> atoms.set_constraint(c) """ super().__init__(indices=indices) self.dir = _normalize(direction) def adjust_positions(self, atoms, newpositions): step = newpositions[self.index] - atoms.positions[self.index] newpositions[self.index] -= _projection(step, self.dir) def adjust_forces(self, atoms, forces): forces[self.index] -= _projection(forces[self.index], self.dir) def get_removed_dof(self, atoms): return len(self.index) def todict(self): return { 'name': 'FixedPlane', 'kwargs': {'indices': self.index.tolist(), 'direction': self.dir.tolist()} } def __repr__(self): return f'FixedPlane(indices={self.index}, {self.dir.tolist()})'
def _projection(vectors, direction): dotprods = vectors @ direction projection = direction[None, :] * dotprods[:, None] return projection
[docs]class FixedLine(IndexedConstraint): """ Constrain an atom index or a list of atom indices to move on a line only. The line is defined by its vector *direction* """ def __init__(self, indices, direction): """Constrain chosen atoms. Parameters ---------- indices : int or list of int Index or indices for atoms that should be constrained direction : list of 3 int Direction of the vector defining the line Examples -------- Fix all Copper atoms to only move in the x-direction: >>> from ase.constraints import FixedLine >>> c = FixedLine( >>> indices=[atom.index for atom in atoms if atom.symbol == 'Cu'], >>> direction=[1, 0, 0], >>> ) >>> atoms.set_constraint(c) or constrain a single atom with the index 0 to move in the z-direction: >>> c = FixedLine(indices=0, direction=[0, 0, 1]) >>> atoms.set_constraint(c) """ super().__init__(indices) self.dir = _normalize(direction) def adjust_positions(self, atoms, newpositions): step = newpositions[self.index] - atoms.positions[self.index] projection = _projection(step, self.dir) newpositions[self.index] = atoms.positions[self.index] + projection def adjust_forces(self, atoms, forces): forces[self.index] = _projection(forces[self.index], self.dir) def get_removed_dof(self, atoms): return 2 * len(self.index) def __repr__(self): return f'FixedLine(indices={self.index}, {self.dir.tolist()})' def todict(self): return { 'name': 'FixedLine', 'kwargs': {'indices': self.index.tolist(), 'direction': self.dir.tolist()} }
class FixCartesian(IndexedConstraint): 'Fix an atom index *a* in the directions of the cartesian coordinates.' def __init__(self, a, mask=(1, 1, 1)): super().__init__(indices=a) self.mask = ~np.asarray(mask, bool) def get_removed_dof(self, atoms): return (3 - self.mask.sum()) * len(self.index) def adjust_positions(self, atoms, new): step = new[self.index] - atoms.positions[self.index] step *= self.mask[None, :] new[self.index] = atoms.positions[self.index] + step def adjust_forces(self, atoms, forces): forces[self.index] *= self.mask[None, :] def __repr__(self): return 'FixCartesian(indices={}, mask={})'.format( self.index.tolist(), list(~self.mask)) def todict(self): return {'name': 'FixCartesian', 'kwargs': {'a': self.index.tolist(), 'mask': (~self.mask).tolist()}} class FixScaled(IndexedConstraint): 'Fix an atom index *a* in the directions of the unit vectors.' def __init__(self, a, mask=(1, 1, 1), cell=None): # XXX The unused cell keyword is there for compatibility # with old trajectory files. super().__init__(a) self.mask = np.array(mask, bool) def get_removed_dof(self, atoms): return self.mask.sum() * len(self.index) def adjust_positions(self, atoms, new): cell = atoms.cell scaled_old = cell.scaled_positions(atoms.positions[self.index]) scaled_new = cell.scaled_positions(new[self.index]) scaled_new[:, self.mask] = scaled_old[:, self.mask] new[self.index] = cell.cartesian_positions(scaled_new) def adjust_forces(self, atoms, forces): # Forces are covariant to the coordinate transformation, # use the inverse transformations cell = atoms.cell scaled_forces = cell.cartesian_positions(forces[self.index]) scaled_forces *= -(self.mask - 1) forces[self.index] = cell.scaled_positions(scaled_forces) def todict(self): return {'name': 'FixScaled', 'kwargs': {'a': self.index.tolist(), 'mask': self.mask.tolist()}} def __repr__(self): return 'FixScaled({}, {})'.format(self.index.tolist(), self.mask) # TODO: Better interface might be to use dictionaries in place of very # nested lists/tuples
[docs]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)
[docs]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). References: Andrew A. Peterson, Topics in Catalysis volume 57, pages40–53 (2014) https://link.springer.com/article/10.1007%2Fs11244-013-0161-8 """ 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
[docs]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}}
[docs]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]]
[docs]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
[docs]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)
[docs]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