pyxtal.operations module

Module for generating and analyzing transformation operations. Several functions for working with matrices are provided. The class OperationAnalyzer allows for comparison between pymatgen.core.operations.SymmOp objects, and can be used to identify conjugate operations. The orientation class can be used to identify degrees of freedom for molecules in Wyckoff positions with certain symmetry constraints.

class pyxtal.operations.OperationAnalyzer(op)

Bases: SymmOp

Class for comparing operations. Stores rotation axis and angle, as well as the type and order of operation (identity, inversion, rotation, or rotoinversion). By default, takes a SymmOp as argument. This information can be accessed by calling print(object). The class methods is_conjugate and are_conjugate can be used to determine if two operations are conjugate to each other. That is, whether they represent the same angle of rotation and are either both inverted or both uninverted.

Note: rotoinversions with odd-order rotational parts will have an over-all

even order. For example, the order of (-3) is 6.

Note: reflections are treated as rotoinversions of order 2.

Parameters:

SymmOp – a pymatgen.core.structure.SymmOp object to analyze

affine_matrix

The 4x4 affine matrix of the op

are_conjugate(op2)

Returns whether or not two operations are conjugate (the same operation in a different reference frame). Rotations with the same order will not always return True. For example, a 5/12 and 1/12 rotation will not be considered conjugate.

Parameters:

• op1 – a SymmOp or OperationAnalyzer object

• op2 – a SymmOp or OperationAnalyzer object to compare with op1

Returns:

True if op2 is conjugate to op1, and False otherwise

det

The determinant of self.m

get_order(rotoinversion=False, tol=0.01)

is_conjugate(op2)

Returns whether or not another operation is conjugate (the same operation in a different reference frame). Rotations with the same order will not always return True. For example, a 5/12 and 1/12 rotation will not be considered conjugate.

Parameters:

op2 – a SymmOp or OperationAnalyzer object to compare with

Returns:

True if op2 is conjugate to self.op, and False otherwise

m

The 3x3 rotation (or rotoinversion) matrix, which ignores the translational part of self.op

op

The original SymmOp object being analyzed

order

The order of the operation. This is the number of times the operation must be applied consecutively to return to the identity operation. If no integer number if found, we set this to ‘irrational’.

rotation_order

The order of the rotational (non-inversional) part of the operation. Must be used in conjunction with self.order to determine the properties of the operation.

tol

The numerical tolerance associated with self.op

type

The type of operation. Is one of ‘identity’, ‘inversion’, ‘rotation’, or ‘rotoinversion’.

pyxtal.operations.aa2matrix(axis, angle, radians=True, random=False)

Given an axis and an angle, return a 3x3 rotation matrix. Based on: https://en.wikipedia.org/wiki/Rotation_matrix#Axis_and_angle

Parameters:

• axis – a vector about which to perform a rotation

• angle – the angle of rotation

• radians – whether the supplied angle is in radians (True) or in degrees (False)

• random – whether or not to choose a random rotation matrix. If True, the axis and angle are ignored, and a random orientation is generated

Returns:

a 3x3 numpy array representing a rotation matrix

pyxtal.operations.angle(v1, v2, radians=True)

Calculate the angle (in radians) between two vectors.

Parameters:

• v1 – a 1x3 vector

• v2 – a 1x3 vector

• radians – whether to return angle in radians (default) or degrees

Returns:

the angle in radians between the two vectors

pyxtal.operations.apply_ops(coord, ops)

Apply a list of SymmOps to a single 3-vector and return an array of the generated vectors. This is the inverse of SymmOp.operate_multi.

Parameters:

• coord – a 3-vector (list or numpy array)

• ops – a list, tuple, or array of SymmOp objects

Returns:

an np array of floating-point 3-vectors

pyxtal.operations.apply_ops_diagonal(coords, ops)

Given a list of coordinates and SymmOps, apply the ith op to the ith coord and return the list of transformed coordinates

Parameters:

coords – a list or array of 3-vectors

Returns:

a transformed numpy array of 3-vectors

pyxtal.operations.are_equal(op1, op2, PBC=[1, 1, 1], rtol=0.001, atol=0.001)

Check whether two SymmOp objects are equal up to some numerical tolerance. Allows for optional consideration of periodic boundary conditions. This option is useful for handling the periodicity of crystals.

Parameters:

• op1 – a SymmOp object

• op2 – another SymmOp object

• PBC – A periodic boundary condition list.

• rtol – the relative numerical tolerance for equivalence

• atol – the absolute numerical tolerance for equivalence

Returns:

True if op1 and op2 are equivalent, False otherwise

pyxtal.operations.check_distance(coord1, coord2, species1, species2, lattice, PBC=[1, 1, 1], tm=<pyxtal.tolerance.Tol_matrix object>, d_factor=1.0)

Check the distances between two atom set. Only distances between points from different sets are checked.

Parameters:

• coord1 – a list of fractional coordinates e.g. [[.1,.6,.4] [.3,.8,.2]]

• coord2 – a list of new fractional coordinates e.g. [[.7,.8,.9], [.4,.5,.6]]

• species1 – a list of atomic species or numbers for coord1

• species2 – a list of atomic species or numbers for coord2

• lattice – matrix describing the unit cell vectors

• PBC – A periodic boundary condition list, where 1 means periodic, 0 means not periodic. [1,1,1] -> full 3d periodicity, [0,0,1] -> periodicity along the z axis

• tm – a Tol_matrix object, or a string representing Tol_matrix

• d_factor – the tolerance is multiplied by this amount. Larger values mean atoms must be farther apart

Returns:

a bool for whether or not the atoms are sufficiently far enough apart

pyxtal.operations.check_images(coords, species, lattice, PBC=[1, 1, 1], tm=<pyxtal.tolerance.Tol_matrix object>, tol=None, d_factor=1.0)

Given a set of (unfiltered) frac coordinates, checks if the periodic images are too close.

Parameters:

• coords – a list of fractional coordinates

• species – the atomic species of each coordinate

• lattice – a 3x3 lattice matrix

• PBC – the periodic boundary conditions

• tm – a Tol_matrix object

• tol – a single override value for the distance tolerances

• d_factor – the tolerance is multiplied by this amount. Larger values mean atoms must be farther apart

Returns:

False if distances are too close. True if distances are not too close

pyxtal.operations.create_matrix(PBC=[1, 1, 1], omit=False)

Used for calculating distances in lattices with periodic boundary conditions. When multiplied with a set of points, generates additional points in cells adjacent to and diagonal to the original cell

Parameters:

PBC – A periodic boundary condition list (1: periodic; 0: nonperiodic).

Returns:

A numpy array which can be multiplied by a set of coordinates

pyxtal.operations.distance(xyz, lattice, PBC=[1, 1, 1])

Returns the Euclidean distance from the origin for a fractional displacement vector. Takes into account the lattice metric and periodic boundary conditions, including up to one non-periodic axis.

Parameters:

• xyz – a fractional 3d displacement vector. Can be obtained by subtracting one fractional vector from another

• lattice – a 3x3 matrix describing a unit cell’s lattice vectors

• PBC – A periodic boundary condition list, where 1 means periodic, 0 means

• Ex (not periodic.) – [1,1,1] -> full 3d periodicity, [0,0,1] -> 1d

• axis (periodicity along the z)

Returns:

a scalar for the distance of the point from the origin

pyxtal.operations.distance_matrix(pts1, pts2, lattice, PBC=[1, 1, 1], single=False, metric='euclidean')

Returns the distances between two sets of fractional coordinates. Takes into account the lattice metric and periodic boundary conditions.

Parameters:

• pts1 – a list of fractional coordinates

• pts2 – another list of fractional coordinates

• lattice – a 3x3 matrix describing a unit cell’s lattice vectors

• PBC – A periodic boundary condition list, where 1 means periodic, 0 means not periodic. Ex: [1,1,1] -> full 3d periodicity, [0,0,1] -> 1d periodicity along the z axis

• single – return a scalor and matrix?

• metric – the metric to use with cdist, such as ‘euclidean’, ‘sqeuclidean’, ‘minkowski’, and others

Returns:

a scalor or distance matrix

pyxtal.operations.distance_matrix_no_PBC(pts1, pts2, lattice, single=False, metric='euclidean')

Returns the distances between two sets of fractional coordinates. Without periodic boundary conditions.

Parameters:

• pts1 – a list of fractional coordinates (N1*3)

• pts2 – another list of fractional coordinates (N2*3)

• lattice – a 3x3 matrix describing a unit cell’s lattice vectors

• single – return the minimum distance or the matrix

• metric – the metric to use with cdist. e.g. euclidean, sqeuclidean, minkowski, and others

Returns:

a scalor or distance matrix

pyxtal.operations.filtered_coords(coords, PBC=[1, 1, 1])

Transform all coordinates to [0, 1] interval if PBC is allowed For example, [1.2, 1.6, -.4] becomes [0.2, 0.6, 0.6] when PBC=[1,1,1] [0.2, 1.6, 0.6] when PBC=[1,0,1]

Parameters:

• coords – an array of real 3d vectors.

• PBC – A periodic boundary condition list (1: periodic; 0: nonperiodic).

Returns:

an array of filtered coords with the same shape as coords

pyxtal.operations.filtered_coords_euclidean(coords, PBC=[1, 1, 1])

Given an array of fractional 3-vectors, filters coordinates to between 0 and 1. Then, values which are greater than 0.5 are converted to 1 minus their value. This is used for converting displacement vectors with a Euclidean lattice.

Parameters:

• coords – an array of real 3d vectors. The shape does not matter

• PBC – A periodic boundary condition list (1: periodic; 0: nonperiodic).

Returns:

an array of filtered coords with the same shape as coords

pyxtal.operations.find_ids(coords, ref, tol=0.001)

find the refernce ids that can match

pyxtal.operations.get_best_match(positions, ref, cell)

find the best match with the reference from a set of positions

Parameters:

• positions – N*3 array

• ref – 1*3 array

• cell – cell matrix 3*3 array

Returns:

matched position id: matched id

Return type:

position

pyxtal.operations.get_inverse(op)

Given a SymmOp object, returns its inverse.

Parameters:

op – a Symmop object

Returns:

the inverse

pyxtal.operations.get_inverse_ops(ops)

Given a inverse list of SymmOp objects

Parameters:

ops – a list of Symmop’s

Returns:

a list of equal shape to ops, with the inverse operations

pyxtal.operations.is_orthogonal(m, tol=0.001)

Check whether or not a 3x3 matrix is orthogonal. An orthogonal matrix has the property that it is equal to its transpose.

Parameters:

• m – a 3x3 matrix (list or numpy array)

• tol – the numerical tolerance for checking if two matrices are equal

Returns:

True if the matrix is orthogonal, False if it is not

pyxtal.operations.rotate_vector(v1, v2, rtol=0.0001)

Rotates a vector v1 to v2 about an axis perpendicular to both. Returns the 3x3 rotation matrix used to do so.

Parameters:

• v1 – a 1x3 vector (list or array) of floats

• v2 – a 1x3 vector (list or array) of floats

Returns:

a 3x3 matrix corresponding to a rotation which can be applied to v1 to obtain v2

pyxtal.operations.verify_distances(coordinates, species, lattice, factor=1.0, PBC=[1, 1, 1])

Checks the inter-atomic distance between all pairs of atoms in a crystal.

Parameters:

• coordinates – a 1x3 list of fractional coordinates

• species – a list of atomic symbols for each coordinate

• lattice – a 3x3 matrix representing the lattice vectors of the unit cell

• factor – a tolerance factor for checking distances. A larger value means atoms must be farther apart

• PBC – A periodic boundary condition list, where 1 means periodic, 0 means not periodic. Ex: [1,1,1] -> full 3d periodicity, [0,0,1] -> 1d periodicity along the z axis

Returns:

True if no atoms are too close together, False if any pair is too close