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)

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

property det

Shorthand for the determinant of the SquareTensor

get_scaled(scale_factor)

Scales the tensor by a certain multiplicative scale factor

Parameters:

scale_factor (float) – scalar multiplier to be applied to the SquareTensor object

property inv

Shorthand for matrix inverse on SquareTensor

is_rotation(tol: float = 0.001, include_improper=True)

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

polar_decomposition(side='right')

Calculates matrices for polar decomposition

property principal_invariants

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()

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

property trans

Shorthand for transpose on SquareTensor

class Tensor(input_array, vscale=None, check_rank=None)

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

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)

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)

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)

Calculates the result of an einstein summation expression

fit_to_structure(structure: Structure, symprec: float = 0.1)

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

Instantiate Tensors from dicts (using MSONAble API).

Returns:

hydrated tensor object

Return type:

Tensor

classmethod from_values_indices(values, indices, populate=False, structure=None, voigt_rank=None, vsym=True, verbose=False)

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)

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)

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)

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)

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)

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)

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)

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)

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

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.

1. Find non-zero entries

2. Symmetrize the tensor with respect to crystal symmetry and (optionally) voigt symmetry

3. 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:

Tensor

project(n)

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)

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)

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)

Transforms a tensor from one basis for an original structure into a new basis defined by a new structure.

Parameters:

• original_structure (Structure) – structure corresponding to the basis of the current tensor

• new_structure (Structure) – structure corresponding to the desired basis

• refine_rotation (bool) – whether to refine the rotations generated in get_ieee_rotation

Returns:

Tensor that has been transformed such that its basis corresponds to the new_structure’s basis

symbol = 'T'

property symmetrized

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)

Applies a transformation (via a symmetry operation) to a tensor.

Parameters:

symm_op (SymmOp) – a symmetry operation to apply to the tensor

property voigt

Returns the tensor in Voigt notation

property voigt_symmetrized

Returns a “voigt”-symmetrized tensor, i. e. a voigt-notation tensor such that it is invariant wrt permutation of indices

zeroed(tol: float = 0.001)

Returns the matrix with all entries below a certain threshold (i.e. tol) set to zero.

class TensorCollection(tensor_list, base_class=<class 'pymatgen.core.tensors.Tensor'>)

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)

Parameters:

voigt – Whether to use voight form.

Returns:

Dict representation of TensorCollection.

convert_to_ieee(structure: Structure, initial_fit=True, refine_rotation=True)

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)

Fits all tensors to a Structure.

Parameters:

• structure – Structure

• symprec – symmetry precision.

Returns:

TensorCollection.

classmethod from_dict(d)

Creates TensorCollection from dict.

Parameters:

d – dict

Returns:

TensorCollection

classmethod from_voigt(voigt_input_list, base_class=<class 'pymatgen.core.tensors.Tensor'>)

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)

Parameters:

• structure – Structure

• tol – tolerance

Returns:

Whether all tensors are fitted to Structure.

is_symmetric(tol: float = 1e-05)

Parameters:

tol – tolerance

Returns:

Whether all tensors are symmetric.

is_voigt_symmetric(tol: float = 1e-06)

Parameters:

tol – tolerance

Returns:

Whether all tensors are voigt symmetric.

property ranks

Ranks for all tensors.

Type:

return

rotate(matrix, tol: float = 0.001)

Rotates TensorCollection.

Parameters:

• matrix – Rotation matrix.

• tol – tolerance.

Returns:

TensorCollection.

round(*args, **kwargs)

Round all tensors.

Parameters:

• args – Passthrough to Tensor.round

• kwargs – Passthrough to Tensor.round

Returns:

TensorCollection.

property symmetrized

TensorCollection where all tensors are symmetrized.

Type:

return

transform(symm_op)

Transforms TensorCollection with a symmetry operation.

Parameters:

symm_op – SymmetryOperation.

Returns:

TensorCollection.

property voigt

TensorCollection where all tensors are in voight form.

Type:

return

property voigt_symmetrized

TensorCollection where all tensors are voigt symmetrized.

Type:

return

zeroed(tol: float = 0.001)

Parameters:

tol – Tolerance

Returns:

TensorCollection where small values are set to 0.

class TensorMapping(tensors: Sequence[Tensor] = (), values: Sequence = (), tol: float = 1e-05)

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

items()

Returns:

Items in mapping.

values()

Returns:

Values in mapping.

get_uvec(vec)

Gets a unit vector parallel to input vector

symmetry_reduce(tensors, structure: Structure, tol: float = 1e-08, **kwargs)

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