def _create_mappings(self, rotations, translations):
structure_analyzer = StructureAnalyzer(self._atoms)
mappings = []
for r, t in zip(rotations, translations):
mapping = structure_analyzer.extract_mapping_for_symopr(r, t)
mappings.append(mapping)
self._mappings = np.array(mappings)
self._mappings_modifier = MappingsModifier(mappings)
def test_expand_mappings2(self):
mappings = [
[
[0, 1, 2, 3],
[0, 2, 1, 3],
],
[
[1, 2, 3, 0],
[2, 3, 0, 1],
],
]
n = 3
expanded_mappings_expected = [
[
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11],
[0, 1, 2, 6, 7, 8, 3, 4, 5, 9, 10, 11],
],
[
[3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], # Good
[6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], # Good
],
]
expanded_mappings = MappingsModifier(mappings).expand_mappings(n)
self.assertTrue(
np.all(expanded_mappings == expanded_mappings_expected))
def reduce_vectors_to_primitive(self, vectors, primitive):
"""
Reduce size of vectors to primitive.
Parameters
----------
vectors : (..., natoms_u * ndims, nbands) array
Vectors which will be reduced.
Phase factors must be removed in advance.
primitive : Phonopy Primitive object.
Returns
-------
reduced_vectors : (..., natoms_p, ndims, nbands) array
Reduced vectors.
"""
ndim = 3
p2s_map = primitive.get_primitive_to_supercell_map()
indices = MappingsModifier(p2s_map).expand_mappings(ndim)
reduced_vectors = vectors[..., indices, :]
# Renormalization of the reduced vectors
relative_size = vectors.shape[-2] / reduced_vectors.shape[-2]
reduced_vectors *= np.sqrt(relative_size)
return reduced_vectors
def test_expand_mappings_inv(self):
mappings = [
[0, 1, 2, 3],
[0, 2, 1, 3],
[1, 2, 3, 0],
]
n = 3
expanded_mappings_inv_expected = [
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11],
[0, 1, 2, 6, 7, 8, 3, 4, 5, 9, 10, 11],
[9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8],
]
mappings_modifier = MappingsModifier(mappings)
expanded_mappings_inv = mappings_modifier.expand_mappings(
n, is_inverse=True)
self.assertTrue(
np.all(expanded_mappings_inv == expanded_mappings_inv_expected))
def test_expand_mappings_inv(self):
mappings = [
[0, 1, 2, 3],
[0, 2, 1, 3],
[1, 2, 3, 0],
]
n = 3
expanded_mappings_inv_expected = [
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11],
[0, 1, 2, 6, 7, 8, 3, 4, 5, 9, 10, 11],
[9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8],
]
mappings_modifier = MappingsModifier(mappings)
expanded_mappings_inv = mappings_modifier.expand_mappings(
n, is_inverse=True)
self.assertTrue(
np.all(expanded_mappings_inv == expanded_mappings_inv_expected))
def test_invert_mappings(self):
mappings = [
[0, 1, 2, 3],
[0, 2, 1, 3],
[1, 2, 3, 0],
]
mappings_inv_expected = [
[0, 1, 2, 3],
[0, 2, 1, 3],
[3, 0, 1, 2],
]
mappings_inv = MappingsModifier(mappings).invert_mappings()
self.assertTrue(np.all(mappings_inv == mappings_inv_expected))
def project_vectors(self, vectors, kpoint):
"""Project vectors onto kpoint
Parameters
----------
vectors : (..., natoms * ndim, nbands) array
Vectors for SC at kpoint. Each "column" vector is an eigenvector.
kpoint : (ndim) array
Reciprocal space point in fractional coordinates for SC.
Returns
-------
projected_vectors : (..., natoms_primitive * ndim, nbands) array
Projection of the given vectors.
This is reduced into the primitive cell.
"""
ncells = self._ncells
ndim = self._ndim
primitive = self._primitive
p2s_map = primitive.get_primitive_to_supercell_map()
indices = MappingsModifier(p2s_map).expand_mappings(ndim)
expanded_mappings = self._expanded_mappings
shape = list(vectors.shape)
shape[-2] //= ncells
projected_vectors = np.zeros(shape, dtype=vectors.dtype)
for expanded_mapping in expanded_mappings:
jndices = expanded_mapping.take(indices)
projected_vectors += vectors.take(jndices, axis=-2)
# The following intend;
# # Definition of projection operators
# projected_vectors /= ncells
# # Reduction into primitive
# projected_vectors *= np.sqrt(ncells)
projected_vectors /= np.sqrt(ncells)
return projected_vectors
def project_vectors(self, vectors, ndims=3):
map_atoms_u2p = self._map_atoms_u2p
map_elements = self._map_elements
natoms_p = len(map_atoms_u2p)
num_elements = len(map_elements)
tmp = np.zeros_like(vectors[None, None]) # Add two dimensions
projected_vectors = (np.repeat(np.repeat(tmp, natoms_p, axis=0),
num_elements,
axis=1))
for ip, lp in enumerate(map_atoms_u2p):
for ie, le in enumerate(map_elements):
indices_tmp = sorted(set(lp) & set(le))
indices = MappingsModifier(indices_tmp).expand_mappings(ndims)
if len(indices
) > 0: # The element "le" exists on the sublattice.
projected_vectors[ip, ie, indices] = vectors[indices]
return projected_vectors
class RotationalProjector(object):
def __init__(self, atoms):
"""
Decomposer of vectors according to rotations
Parameters
----------
atoms : Phonopy Atoms object.
"""
self._atoms = atoms
self._symmetry = UnfolderSymmetry(atoms)
def create_standard_rotations(self, kpoint):
"""
Create standard rotations for IR labels
Parameters
----------
kpoint : 1d array
Reciprocal space point in fractional coordinates for PC.
This is the representative of the star.
IR labels are determined using this.
TODO
----
To be modified for nonsymmorphic space groups.
"""
symmetry = self._symmetry
rotations, translations = symmetry.get_group_of_wave_vector(kpoint)
self._create_irreps(rotations)
self._standard_rotations = rotations
# self.print_symmetry_operations(rotations, translations)
@staticmethod
def calculate_factor_system(rotations, translations, kpoint):
"""Calculate factor system
Here the factor system is defined as
.. math::
e^{-i (\mathbf{R}_i^T \mathbf{k} - \mathbf{k}) \cdot \mathbf{w}_j,
"""
nsyms = len(rotations)
factor_system = np.zeros((nsyms, nsyms), dtype=complex)
for i, rotation in enumerate(rotations):
for j, translation in enumerate(translations):
kdiff = np.dot(rotation.T, kpoint) - kpoint
factor_system[i, j] = np.exp(-2.0j * np.pi *
np.dot(kdiff, translation))
return factor_system
@staticmethod
def print_factor_system(factor_system):
nsyms = factor_system.shape[0]
print('Factor system:')
for i in range(nsyms):
for j in range(nsyms):
v = factor_system[i, j]
print('({:10.6f}{:+10.6f}i) '.format(v.real, v.imag), end='')
print()
@staticmethod
def is_trivial_factor_system(factor_system, prec=1e-6):
if np.any(np.abs(factor_system - 1.0) > prec):
return False
return True
@staticmethod
def print_symmetry_operations(rotations, translations):
print('Symmetry operations:')
for r, t in zip(rotations, translations):
for i in range(3):
for j in range(3):
print('{:10.6f}'.format(r[i, j]), end='')
print(' {:10.6f}'.format(t[i]))
print()
def _assign_characters_to_rotations(self, rotations, arm_transformation):
irreps = self._irreps
rotation_labels = self._assign_class_labels_to_rotations(
rotations, arm_transformation)
characters = irreps.assign_characters_to_rotations(rotation_labels)
return characters
def _assign_class_labels_to_rotations(self, rotations, arm_transformation):
standard_rotation_labels = self._standard_rotation_labels
modulated_standard_rotations = (
self._modulate_standard_rotations(arm_transformation))
rotation_labels = []
for r in rotations:
for i, sr in enumerate(modulated_standard_rotations):
if np.all(r == sr):
rotation_labels.append(standard_rotation_labels[i])
break
if len(rotation_labels) != len(standard_rotation_labels):
raise ValueError("Rotation labels cannot be correctly assigned.")
return rotation_labels
def _modulate_standard_rotations(self, arm_transformation):
standard_rotations = self._standard_rotations
modulated_standard_rotations = np.zeros_like(standard_rotations)
for i, r in enumerate(standard_rotations):
modulated_standard_rotations[i] = (np.dot(
np.dot(np.linalg.inv(arm_transformation), r),
arm_transformation))
return np.array(modulated_standard_rotations, dtype=int)
def _create_mappings(self, rotations, translations):
structure_analyzer = StructureAnalyzer(self._atoms)
mappings = []
for r, t in zip(rotations, translations):
mapping = structure_analyzer.extract_mapping_for_symopr(r, t)
mappings.append(mapping)
self._mappings = np.array(mappings)
self._mappings_modifier = MappingsModifier(mappings)
def _invert_mappings(self):
return self._mappings_modifier.invert_mappings()
def _create_irreps(self, rotations):
irreps = Irreps(rotations)
character_table_data = irreps.get_character_table_data()
self._ir_labels = character_table_data["ir_labels"]
character_table = np.array(character_table_data["character_table"])
self._ir_dimensions = character_table[:, 0]
self._standard_rotation_labels = irreps.get_rotation_labels()
self._irreps = irreps
def _create_rotations_cart(self, rotations):
rotations_cart = []
cell = self._atoms.get_cell()
for r in rotations:
rotation_cart = np.dot(np.dot(cell.T, r), np.linalg.inv(cell.T))
rotations_cart.append(rotation_cart)
return np.array(rotations_cart)
def project_vectors(self, vectors, kpoint, arm_transformation):
"""
Parameters
----------
vectors : Vectors without phase factors.
kpoint : K point in fractional coordinates for SC.
arm_transformation : 3 x 3 array
Matrix to get "kpoint" from the representative of the star.
TODO
----
To be modified for nonsymmorphic space groups.
"""
rotations, translations = self._symmetry.get_group_of_wave_vector(
kpoint)
factor_system = self.calculate_factor_system(rotations, translations,
kpoint)
if not self.is_trivial_factor_system(factor_system):
errmsg = (
"The current factor system is not trivial.\n"
"So far the code cannot treat nontrivial factor system correctly\n"
"(even if it is p-equivalent to the trivial system).")
raise ValueError(errmsg)
self._create_mappings(rotations, translations)
characters = self._assign_characters_to_rotations(
rotations, arm_transformation)
rotations_cart = self._create_rotations_cart(rotations)
ir_dimensions = self._ir_dimensions
natoms = self._atoms.get_number_of_atoms()
ndim = kpoint.shape[0] # The number of dimensions of space
order = rotations.shape[0]
expanded_mappings_inv = self._mappings_modifier.expand_mappings(
ndim, is_inverse=True)
# scaled_positions = self._atoms.get_scaled_positions()
# phases = np.exp(2.0j * np.pi * np.dot(scaled_positions, kpoint))
# phases = np.repeat(phases, ndim)
shape = (len(ir_dimensions), ) + vectors.shape
projected_vectors = np.zeros(shape, dtype=vectors.dtype)
for i, (r, expanded_mapping_inv) in enumerate(
zip(rotations_cart, expanded_mappings_inv)):
tmp = vectors[..., expanded_mapping_inv, :]
tmp2 = np.zeros_like(vectors)
for iatom in range(natoms):
tmp2[..., (3 * iatom):(3 * (iatom + 1)), :] = np.einsum(
'ij,...jk->...ik', r,
tmp[..., (3 * iatom):(3 * (iatom + 1)), :])
projected_vectors += (tmp2[None].T * np.conj(characters[i, :])).T
# projected_vectors *= phases[None, :, None]
projected_vectors = (projected_vectors.T * ir_dimensions[:]).T
projected_vectors /= order
return projected_vectors
def get_ir_labels(self):
return self._ir_labels
def get_num_irs(self):
return len(self._ir_labels)
def get_pointgroup_symbol(self):
return self._irreps.get_pointgroup_symbol()
def _create_expanded_mappings(self, mappings, ndim):
mappings_modifier = MappingsModifier(mappings)
self._expanded_mappings = mappings_modifier.expand_mappings(ndim)
def _create_expanded_mappings(self, mappings, ndim):
mappings_modifier = MappingsModifier(mappings)
self._expanded_mappings = mappings_modifier.expand_mappings(ndim)
