pymatgen.core.tensors module
This module provides a base class for tensor-like objects and methods for basic tensor manipulation. It also provides a class, SquareTensor, that provides basic methods for creating and manipulating rank 2 tensors
- class SquareTensor(input_array, vscale=None)[source]
Bases:
Tensor
Base class for doing useful general operations on second rank tensors (stress, strain etc.).
Create a SquareTensor object. Note that the constructor uses __new__ rather than __init__ according to the standard method of subclassing numpy ndarrays. Error is thrown when the class is initialized with non-square matrix.
- Parameters:
input_array (3x3 array-like) – the 3x3 array-like representing the content of the tensor
vscale (6x1 array-like) – 6x1 array-like scaling the voigt-notation vector with the tensor entries
- get_scaled(scale_factor)[source]
Scales the tensor by a certain multiplicative scale factor
- Parameters:
scale_factor (float) – scalar multiplier to be applied to the SquareTensor object
- is_rotation(tol: float = 0.001, include_improper=True)[source]
Test to see if tensor is a valid rotation matrix, performs a test to check whether the inverse is equal to the transpose and if the determinant is equal to one within the specified tolerance
- Parameters:
tol (float) – tolerance to both tests of whether the the determinant is one and the inverse is equal to the transpose
include_improper (bool) – whether to include improper rotations in the determination of validity
- property principal_invariants[source]
Returns a list of principal invariants for the tensor, which are the values of the coefficients of the characteristic polynomial for the matrix
- refine_rotation()[source]
Helper method for refining rotation matrix by ensuring that second and third rows are perpendicular to the first. Gets new y vector from an orthogonal projection of x onto y and the new z vector from a cross product of the new x and y
- Parameters:
rotation (tol to test for) –
- Returns:
new rotation matrix
- class Tensor(input_array, vscale=None, check_rank=None)[source]
Bases:
ndarray
,MSONable
Base class for doing useful general operations on Nth order tensors, without restrictions on the type (stress, elastic, strain, piezo, etc.)
Create a Tensor object. Note that the constructor uses __new__ rather than __init__ according to the standard method of subclassing numpy ndarrays.
- Parameters:
input_array – (array-like with shape 3^N): array-like representing a tensor quantity in standard (i. e. non-voigt) notation
vscale – (N x M array-like): a matrix corresponding to the coefficients of the voigt-notation tensor
check_rank – (int): If not None, checks that input_array’s rank == check_rank. Defaults to None.
- as_dict(voigt: bool = False) dict [source]
Serializes the tensor object
- Parameters:
voigt (bool) – flag for whether to store entries in voigt-notation. Defaults to false, as information may be lost in conversion.
- Returns (dict):
serialized format tensor object
- average_over_unit_sphere(quad=None)[source]
Method for averaging the tensor projection over the unit with option for custom quadrature.
- Parameters:
quad (dict) – quadrature for integration, should be dictionary with “points” and “weights” keys defaults to quadpy.sphere.Lebedev(19) as read from file
- Returns:
Average of tensor projected into vectors on the unit sphere
- convert_to_ieee(structure: Structure, initial_fit=True, refine_rotation=True)[source]
Given a structure associated with a tensor, attempts a calculation of the tensor in IEEE format according to the 1987 IEEE standards.
- Parameters:
structure (Structure) – a structure associated with the tensor to be converted to the IEEE standard
initial_fit (bool) – flag to indicate whether initial tensor is fit to the symmetry of the structure. Defaults to true. Note that if false, inconsistent results may be obtained due to symmetrically equivalent, but distinct transformations being used in different versions of spglib.
refine_rotation (bool) – whether to refine the rotation produced by the ieee transform generator, default True
- einsum_sequence(other_arrays, einsum_string=None)[source]
Calculates the result of an einstein summation expression
- fit_to_structure(structure: Structure, symprec: float = 0.1)[source]
Returns a tensor that is invariant with respect to symmetry operations corresponding to a structure
- Parameters:
structure (Structure) – structure from which to generate symmetry operations
symprec (float) – symmetry tolerance for the Spacegroup Analyzer used to generate the symmetry operations
- classmethod from_dict(d) Tensor [source]
Instantiate Tensors from dicts (using MSONAble API).
- Returns:
hydrated tensor object
- Return type:
- classmethod from_values_indices(values, indices, populate=False, structure=None, voigt_rank=None, vsym=True, verbose=False)[source]
Creates a tensor from values and indices, with options for populating the remainder of the tensor.
- Parameters:
values (floats) – numbers to place at indices
indices (array-likes) – indices to place values at
populate (bool) – whether to populate the tensor
structure (Structure) – structure to base population or fit_to_structure on
voigt_rank (int) – full tensor rank to indicate the shape of the resulting tensor. This is necessary if one provides a set of indices more minimal than the shape of the tensor they want, e.g. Tensor.from_values_indices((0, 0), 100)
vsym (bool) – whether to voigt symmetrize during the optimization procedure
verbose (bool) – whether to populate verbosely
- classmethod from_voigt(voigt_input)[source]
Constructor based on the voigt notation vector or matrix.
- Parameters:
voigt_input (array-like) – voigt input for a given tensor
- get_grouped_indices(voigt=False, **kwargs)[source]
Gets index sets for equivalent tensor values
- Parameters:
voigt (bool) – whether to get grouped indices of voigt or full notation tensor, defaults to false
**kwargs –
keyword args for np.isclose. Can take atol and rtol for absolute and relative tolerance, e. g.
>>> tensor.group_array_indices(atol=1e-8)
or
>>> tensor.group_array_indices(rtol=1e-5)
- Returns:
list of index groups where tensor values are equivalent to within tolerances
- static get_ieee_rotation(structure, refine_rotation=True)[source]
Given a structure associated with a tensor, determines the rotation matrix for IEEE conversion according to the 1987 IEEE standards.
- Parameters:
structure (Structure) – a structure associated with the tensor to be converted to the IEEE standard
refine_rotation (bool) – whether to refine the rotation using SquareTensor.refine_rotation
- get_symbol_dict(voigt=True, zero_index=False, **kwargs)[source]
Creates a summary dict for tensor with associated symbol
- Parameters:
voigt (bool) – whether to get symbol dict for voigt notation tensor, as opposed to full notation, defaults to true
zero_index (bool) – whether to set initial index to zero, defaults to false, since tensor notations tend to use one-indexing, rather than zero indexing like python
**kwargs –
keyword args for np.isclose. Can take atol and rtol for absolute and relative tolerance, e. g.
>>> tensor.get_symbol_dict(atol=1e-8)
or
>>> tensor.get_symbol_dict(rtol=1e-5)
- Returns:
list of index groups where tensor values are equivalent to within tolerances
- static get_voigt_dict(rank)[source]
Returns a dictionary that maps indices in the tensor to those in a voigt representation based on input rank
- Parameters:
rank (int) – Tensor rank to generate the voigt map
- is_fit_to_structure(structure: Structure, tol: float = 0.01)[source]
Tests whether a tensor is invariant with respect to the symmetry operations of a particular structure by testing whether the residual of the symmetric portion is below a tolerance
- Parameters:
structure (Structure) – structure to be fit to
tol (float) – tolerance for symmetry testing
- is_symmetric(tol: float = 1e-05)[source]
Tests whether a tensor is symmetric or not based on the residual with its symmetric part, from self.symmetrized
- Parameters:
tol (float) – tolerance to test for symmetry
- is_voigt_symmetric(tol: float = 1e-06)[source]
Tests symmetry of tensor to that necessary for voigt-conversion by grouping indices into pairs and constructing a sequence of possible permutations to be used in a tensor transpose
- populate(structure: Structure, prec: float = 1e-05, maxiter: int = 200, verbose: bool = False, precond: bool = True, vsym: bool = True) Tensor [source]
Takes a partially populated tensor, and populates the non-zero entries according to the following procedure, iterated until the desired convergence (specified via prec) is achieved.
Find non-zero entries
Symmetrize the tensor with respect to crystal symmetry and (optionally) voigt symmetry
Reset the non-zero entries of the original tensor
- Parameters:
structure (structure object) –
prec (float) – precision for determining a non-zero value
maxiter (int) – maximum iterations for populating the tensor
verbose (bool) – whether to populate verbosely
precond (bool) – whether to precondition by cycling through all symmops and storing new nonzero values, default True
vsym (bool) – whether to enforce voigt symmetry, defaults to True
- Returns:
Populated tensor
- Return type:
- project(n)[source]
Convenience method for projection of a tensor into a vector. Returns the tensor dotted into a unit vector along the input n.
- Parameters:
n (3x1 array-like) – direction to project onto
- Returns (float):
scalar value corresponding to the projection of the tensor into the vector
- rotate(matrix, tol: float = 0.001)[source]
Applies a rotation directly, and tests input matrix to ensure a valid rotation.
- Parameters:
matrix (3x3 array-like) – rotation matrix to be applied to tensor
tol (float) – tolerance for testing rotation matrix validity
- round(decimals=0)[source]
Wrapper around numpy.round to ensure object of same type is returned
- Parameters:
decimals – Number of decimal places to round to (default: 0). If decimals is negative, it specifies the number of positions to the left of the decimal point.
- Returns (Tensor):
rounded tensor of same type
- structure_transform(original_structure, new_structure, refine_rotation=True)[source]
Transforms a tensor from one basis for an original structure into a new basis defined by a new structure.
- Parameters:
- Returns:
Tensor that has been transformed such that its basis corresponds to the new_structure’s basis
- property symmetrized[source]
Returns a generally symmetrized tensor, calculated by taking the sum of the tensor and its transpose with respect to all possible permutations of indices
- transform(symm_op)[source]
Applies a transformation (via a symmetry operation) to a tensor.
- Parameters:
symm_op (SymmOp) – a symmetry operation to apply to the tensor
- class TensorCollection(tensor_list, base_class=<class 'pymatgen.core.tensors.Tensor'>)[source]
Bases:
Sequence
,MSONable
A sequence of tensors that can be used for fitting data or for having a tensor expansion
- Parameters:
tensor_list – List of tensors.
base_class – Class to be used.
- as_dict(voigt=False)[source]
- Parameters:
voigt – Whether to use voight form.
- Returns:
Dict representation of TensorCollection.
- convert_to_ieee(structure: Structure, initial_fit=True, refine_rotation=True)[source]
Convert all tensors to IEEE.
- Parameters:
structure – Structure
initial_fit – Whether to perform an initial fit.
refine_rotation – Whether to refine the rotation.
- Returns:
TensorCollection.
- fit_to_structure(structure: Structure, symprec: float = 0.1)[source]
Fits all tensors to a Structure.
- Parameters:
structure – Structure
symprec – symmetry precision.
- Returns:
TensorCollection.
- classmethod from_dict(d)[source]
Creates TensorCollection from dict.
- Parameters:
d – dict
- Returns:
TensorCollection
- classmethod from_voigt(voigt_input_list, base_class=<class 'pymatgen.core.tensors.Tensor'>)[source]
Creates TensorCollection from voigt form.
- Parameters:
voigt_input_list – List of voigt tensors
base_class – Class for tensor.
- Returns:
TensorCollection.
- is_fit_to_structure(structure: Structure, tol: float = 0.01)[source]
- Parameters:
structure – Structure
tol – tolerance
- Returns:
Whether all tensors are fitted to Structure.
- is_symmetric(tol: float = 1e-05)[source]
- Parameters:
tol – tolerance
- Returns:
Whether all tensors are symmetric.
- is_voigt_symmetric(tol: float = 1e-06)[source]
- Parameters:
tol – tolerance
- Returns:
Whether all tensors are voigt symmetric.
- rotate(matrix, tol: float = 0.001)[source]
Rotates TensorCollection.
- Parameters:
matrix – Rotation matrix.
tol – tolerance.
- Returns:
TensorCollection.
- round(*args, **kwargs)[source]
Round all tensors.
- Parameters:
args – Passthrough to Tensor.round
kwargs – Passthrough to Tensor.round
- Returns:
TensorCollection.
- transform(symm_op)[source]
Transforms TensorCollection with a symmetry operation.
- Parameters:
symm_op – SymmetryOperation.
- Returns:
TensorCollection.
- class TensorMapping(tensors: Sequence[Tensor] = (), values: Sequence = (), tol: float = 1e-05)[source]
Bases:
MutableMapping
Base class for tensor mappings, which function much like a dictionary, but use numpy routines to determine approximate equality to keys for getting and setting items.
This is intended primarily for convenience with things like stress-strain pairs and fitting data manipulation. In general, it is significantly less robust than a typical hashing and should be used with care.
Initialize a TensorMapping
- Parameters:
tensors (Sequence[Tensor], optional) – Defaults to (,).
values (Sequence, optional) – Values to be associated with tensors. Defaults to (,).
tol (float, optional) – an absolute tolerance for getting and setting items in the mapping. Defaults to 1e-5.
- Raises:
ValueError – if tensors and values are not the same length
- symmetry_reduce(tensors, structure: Structure, tol: float = 1e-08, **kwargs)[source]
Function that converts a list of tensors corresponding to a structure and returns a dictionary consisting of unique tensor keys with symmop values corresponding to transformations that will result in derivative tensors from the original list
- Parameters:
tensors (list of tensors) – list of Tensor objects to test for symmetrically-equivalent duplicates
structure (Structure) – structure from which to get symmetry
tol (float) – tolerance for tensor equivalence
kwargs – keyword arguments for the SpacegroupAnalyzer
- Returns:
dictionary consisting of unique tensors with symmetry operations corresponding to those which will reconstruct the remaining tensors as values