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Symmetry libraries

Per-topology point-group libraries, the node-centre library, space groups (spglib) and primitive cells: cage_isomer_builder.utils.symmetry.

tet2di4_transformation_library 

tet2di4_transformation_library(fg_anchor)

D4h: 4-fold z rotations and 4 cosets, 15 non-identity operations.

Examples:

>>> from cage_isomer_builder.utils import symmetry
>>> anchors = symmetry.ideal_orbit("D4h")        # a perfectly D4h-symmetric test set
>>> library = symmetry.tet2di4_transformation_library(anchors)
>>> len(library), len(library[0]) + 1              # slots, operations incl. identity
(16, 16)
Source code in cage_isomer_builder/utils/symmetry.py 
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def tet2di4_transformation_library(fg_anchor):
    """
    D4h: 4-fold z rotations and 4 cosets, 15 non-identity operations.

    Examples
    --------
    >>> from cage_isomer_builder.utils import symmetry
    >>> anchors = symmetry.ideal_orbit("D4h")        # a perfectly D4h-symmetric test set
    >>> library = symmetry.tet2di4_transformation_library(anchors)
    >>> len(library), len(library[0]) + 1              # slots, operations incl. identity
    (16, 16)
    """
    return _zrotation_library(fg_anchor, n_fold=4)

tet3_3di3_transformation_library 

tet3_3di3_transformation_library(fg_anchor)

D3h: 3-fold z rotations and 4 cosets, 11 non-identity operations.

Examples:

>>> from cage_isomer_builder.utils import symmetry
>>> anchors = symmetry.ideal_orbit("D3h")        # a perfectly D3h-symmetric test set
>>> library = symmetry.tet3_3di3_transformation_library(anchors)
>>> len(library), len(library[0]) + 1              # slots, operations incl. identity
(12, 12)
Source code in cage_isomer_builder/utils/symmetry.py 
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def tet3_3di3_transformation_library(fg_anchor):
    """
    D3h: 3-fold z rotations and 4 cosets, 11 non-identity operations.

    Examples
    --------
    >>> from cage_isomer_builder.utils import symmetry
    >>> anchors = symmetry.ideal_orbit("D3h")        # a perfectly D3h-symmetric test set
    >>> library = symmetry.tet3_3di3_transformation_library(anchors)
    >>> len(library), len(library[0]) + 1              # slots, operations incl. identity
    (12, 12)
    """
    return _zrotation_library(fg_anchor, n_fold=3)

tet4_4di8_transformation_library 

tet4_4di8_transformation_library(fg_anchor)

D4h: 4-fold z rotations and 4 cosets, 15 non-identity operations.

Examples:

>>> from cage_isomer_builder.utils import symmetry
>>> anchors = symmetry.ideal_orbit("D4h")        # a perfectly D4h-symmetric test set
>>> library = symmetry.tet4_4di8_transformation_library(anchors)
>>> len(library), len(library[0]) + 1              # slots, operations incl. identity
(16, 16)
Source code in cage_isomer_builder/utils/symmetry.py 
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def tet4_4di8_transformation_library(fg_anchor):
    """
    D4h: 4-fold z rotations and 4 cosets, 15 non-identity operations.

    Examples
    --------
    >>> from cage_isomer_builder.utils import symmetry
    >>> anchors = symmetry.ideal_orbit("D4h")        # a perfectly D4h-symmetric test set
    >>> library = symmetry.tet4_4di8_transformation_library(anchors)
    >>> len(library), len(library[0]) + 1              # slots, operations incl. identity
    (16, 16)
    """
    return _zrotation_library(fg_anchor, n_fold=4)

tet5di10_transformation_library 

tet5di10_transformation_library(fg_anchor)

D5h: 5-fold z rotations and 4 cosets, 19 non-identity operations.

Examples:

>>> from cage_isomer_builder.utils import symmetry
>>> anchors = symmetry.ideal_orbit("D5h")        # a perfectly D5h-symmetric test set
>>> library = symmetry.tet5di10_transformation_library(anchors)
>>> len(library), len(library[0]) + 1              # slots, operations incl. identity
(20, 20)
Source code in cage_isomer_builder/utils/symmetry.py 
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def tet5di10_transformation_library(fg_anchor):
    """
    D5h: 5-fold z rotations and 4 cosets, 19 non-identity operations.

    Examples
    --------
    >>> from cage_isomer_builder.utils import symmetry
    >>> anchors = symmetry.ideal_orbit("D5h")        # a perfectly D5h-symmetric test set
    >>> library = symmetry.tet5di10_transformation_library(anchors)
    >>> len(library), len(library[0]) + 1              # slots, operations incl. identity
    (20, 20)
    """
    return _zrotation_library(fg_anchor, n_fold=5)

tet6di12_transformation_library 

tet6di12_transformation_library(fg_anchor)

Oh: 47 non-identity operations.

Examples:

>>> from cage_isomer_builder.utils import symmetry
>>> anchors = symmetry.ideal_orbit("Oh")        # a perfectly Oh-symmetric test set
>>> library = symmetry.tet6di12_transformation_library(anchors)
>>> len(library), len(library[0]) + 1              # slots, operations incl. identity
(48, 48)
Source code in cage_isomer_builder/utils/symmetry.py 
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def tet6di12_transformation_library(fg_anchor):
    """
    Oh: 47 non-identity operations.

    Examples
    --------
    >>> from cage_isomer_builder.utils import symmetry
    >>> anchors = symmetry.ideal_orbit("Oh")        # a perfectly Oh-symmetric test set
    >>> library = symmetry.tet6di12_transformation_library(anchors)
    >>> len(library), len(library[0]) + 1              # slots, operations incl. identity
    (48, 48)
    """
    return _oh_library(fg_anchor)

tet8di16_transformation_library 

tet8di16_transformation_library(fg_anchor)

D4d: two sets of 4-fold z rotations, the second offset by a z flip and a 45-degree turn, and a YZ-plane reflection; 15 non-identity operations.

Examples:

>>> from cage_isomer_builder.utils import symmetry
>>> anchors = symmetry.ideal_orbit("D4d")        # a perfectly D4d-symmetric test set
>>> library = symmetry.tet8di16_transformation_library(anchors)
>>> len(library), len(library[0]) + 1              # slots, operations incl. identity
(16, 16)
Source code in cage_isomer_builder/utils/symmetry.py 
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def tet8di16_transformation_library(fg_anchor):
    """
    D4d: two sets of 4-fold z rotations, the second offset by a z flip and a
    45-degree turn, and a YZ-plane reflection; 15 non-identity operations.

    Examples
    --------
    >>> from cage_isomer_builder.utils import symmetry
    >>> anchors = symmetry.ideal_orbit("D4d")        # a perfectly D4d-symmetric test set
    >>> library = symmetry.tet8di16_transformation_library(anchors)
    >>> len(library), len(library[0]) + 1              # slots, operations incl. identity
    (16, 16)
    """
    transformation_lib = []

    t_fg = fg_anchor.copy()
    for deg in range(4):
        if deg != 0:
            t_fg.rotate(90, 'z')
        transformation_lib.append(geometry.project_permutation(fg_anchor, t_fg))

    t_fg.rotate([0, 0, -1], [0, 0, 1], center=[0, 0, 0])
    t_fg.rotate(45, 'z')
    for deg in range(4):
        if deg != 0:
            t_fg.rotate(90, 'z')
        transformation_lib.append(geometry.project_permutation(fg_anchor, t_fg))

    t_fg = geometry.reflect_positions(fg_anchor.copy(), [1, 0, 0])
    for deg in range(4):
        if deg != 0:
            t_fg.rotate(90, 'z')
        transformation_lib.append(geometry.project_permutation(fg_anchor, t_fg))

    t_fg.rotate([0, 0, -1], [0, 0, 1], center=[0, 0, 0])
    t_fg.rotate(45, 'z')
    for deg in range(4):
        if deg != 0:
            t_fg.rotate(90, 'z')
        transformation_lib.append(geometry.project_permutation(fg_anchor, t_fg))

    transformation_lib.pop(0)
    return list(zip(*transformation_lib))

tet16di32_transformation_library 

tet16di32_transformation_library(fg_anchor)

D4h: 4-fold z rotations and 4 cosets, 15 non-identity operations.

Examples:

>>> from cage_isomer_builder.utils import symmetry
>>> anchors = symmetry.ideal_orbit("D4h")        # a perfectly D4h-symmetric test set
>>> library = symmetry.tet16di32_transformation_library(anchors)
>>> len(library), len(library[0]) + 1              # slots, operations incl. identity
(16, 16)
Source code in cage_isomer_builder/utils/symmetry.py 
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def tet16di32_transformation_library(fg_anchor):
    """
    D4h: 4-fold z rotations and 4 cosets, 15 non-identity operations.

    Examples
    --------
    >>> from cage_isomer_builder.utils import symmetry
    >>> anchors = symmetry.ideal_orbit("D4h")        # a perfectly D4h-symmetric test set
    >>> library = symmetry.tet16di32_transformation_library(anchors)
    >>> len(library), len(library[0]) + 1              # slots, operations incl. identity
    (16, 16)
    """
    return _zrotation_library(fg_anchor, n_fold=4)

tet24di48_transformation_library 

tet24di48_transformation_library(fg_anchor)

Oh: 47 non-identity operations.

Examples:

>>> from cage_isomer_builder.utils import symmetry
>>> anchors = symmetry.ideal_orbit("Oh")        # a perfectly Oh-symmetric test set
>>> library = symmetry.tet24di48_transformation_library(anchors)
>>> len(library), len(library[0]) + 1              # slots, operations incl. identity
(48, 48)
Source code in cage_isomer_builder/utils/symmetry.py 
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def tet24di48_transformation_library(fg_anchor):
    """
    Oh: 47 non-identity operations.

    Examples
    --------
    >>> from cage_isomer_builder.utils import symmetry
    >>> anchors = symmetry.ideal_orbit("Oh")        # a perfectly Oh-symmetric test set
    >>> library = symmetry.tet24di48_transformation_library(anchors)
    >>> len(library), len(library[0]) + 1              # slots, operations incl. identity
    (48, 48)
    """
    return _oh_library(fg_anchor)

tri2di3_transformation_library 

tri2di3_transformation_library(fg_anchor)

D3h: 3-fold z rotations and 4 cosets, 11 non-identity operations.

Examples:

>>> from cage_isomer_builder.utils import symmetry
>>> anchors = symmetry.ideal_orbit("D3h")        # a perfectly D3h-symmetric test set
>>> library = symmetry.tri2di3_transformation_library(anchors)
>>> len(library), len(library[0]) + 1              # slots, operations incl. identity
(12, 12)
Source code in cage_isomer_builder/utils/symmetry.py 
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def tri2di3_transformation_library(fg_anchor):
    """
    D3h: 3-fold z rotations and 4 cosets, 11 non-identity operations.

    Examples
    --------
    >>> from cage_isomer_builder.utils import symmetry
    >>> anchors = symmetry.ideal_orbit("D3h")        # a perfectly D3h-symmetric test set
    >>> library = symmetry.tri2di3_transformation_library(anchors)
    >>> len(library), len(library[0]) + 1              # slots, operations incl. identity
    (12, 12)
    """
    return _zrotation_library(fg_anchor, n_fold=3)

tri4di6_transformation_library 

tri4di6_transformation_library(fg_anchor, rotation_axes)

Td: the full tetrahedral group, from the four node-centre positions.

All 24 permutations of the node centres are tried; SVD gives the matching 3x3 transformation, which is applied to the FG anchors. No determinant correction is applied, so the improper operations (sigma_d, S4) are included and the full Td group is obtained.

Parameters:

Name Type Description Default
fg_anchor Atoms
required
rotation_axes (array - like, shape(4, 3))

Positions of the four node centres.

 required

Returns:

Type Description
list of tuple

Slot-major transformation library, 23 non-identity operations.

Examples:

>>> import numpy as np
>>> from cage_isomer_builder.utils import symmetry
>>> anchors = symmetry.ideal_orbit("Td")        # a perfectly Td-symmetric test set
>>> library = symmetry.tri4di6_transformation_library(anchors, 5 * np.array([[1, 1, 1], [1, -1, -1], [-1, 1, -1], [-1, -1, 1]]))
>>> len(library), len(library[0]) + 1              # slots, operations incl. identity
(24, 24)
Source code in cage_isomer_builder/utils/symmetry.py 
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def tri4di6_transformation_library(fg_anchor, rotation_axes):
    """
    Td: the full tetrahedral group, from the four node-centre positions.

    All 24 permutations of the node centres are tried; SVD gives the matching
    3x3 transformation, which is applied to the FG anchors. No determinant
    correction is applied, so the improper operations (sigma_d, S4) are included
    and the full Td group is obtained.

    Parameters
    ----------
    fg_anchor : ase.Atoms
    rotation_axes : array-like, shape (4, 3)
        Positions of the four node centres.

    Returns
    -------
    list of tuple
        Slot-major transformation library, 23 non-identity operations.

    Examples
    --------
    >>> import numpy as np
    >>> from cage_isomer_builder.utils import symmetry
    >>> anchors = symmetry.ideal_orbit("Td")        # a perfectly Td-symmetric test set
    >>> library = symmetry.tri4di6_transformation_library(anchors, 5 * np.array([[1, 1, 1], [1, -1, -1], [-1, 1, -1], [-1, -1, 1]]))
    >>> len(library), len(library[0]) + 1              # slots, operations incl. identity
    (24, 24)
    """
    node_positions = np.array(rotation_axes, dtype=float)
    node_center = node_positions.mean(axis=0)
    node_centered = node_positions - node_center

    identity = tuple(range(len(fg_anchor)))
    seen = set()
    transformation_lib = []

    for perm in permutations(range(4)):
        target = node_centered[list(perm)]
        H = node_centered.T @ target
        U, _S, Vt = np.linalg.svd(H)
        Rt = U @ Vt

        t_fg = fg_anchor.copy()
        t_fg.positions = (t_fg.positions - node_center) @ Rt + node_center

        p = tuple(geometry.project_permutation(fg_anchor, t_fg))
        if p not in seen:
            seen.add(p)
            if p != identity:
                transformation_lib.append(p)

    if not transformation_lib:
        return [() for _ in range(len(fg_anchor))]
    return list(zip(*transformation_lib))

tri4_2di6_transformation_library 

tri4_2di6_transformation_library(fg_anchor)

D2h: 2-fold z rotations and 4 cosets, 7 non-identity operations.

Examples:

>>> from cage_isomer_builder.utils import symmetry
>>> anchors = symmetry.ideal_orbit("D2h")        # a perfectly D2h-symmetric test set
>>> library = symmetry.tri4_2di6_transformation_library(anchors)
>>> len(library), len(library[0]) + 1              # slots, operations incl. identity
(8, 8)
Source code in cage_isomer_builder/utils/symmetry.py 
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def tri4_2di6_transformation_library(fg_anchor):
    """
    D2h: 2-fold z rotations and 4 cosets, 7 non-identity operations.

    Examples
    --------
    >>> from cage_isomer_builder.utils import symmetry
    >>> anchors = symmetry.ideal_orbit("D2h")        # a perfectly D2h-symmetric test set
    >>> library = symmetry.tri4_2di6_transformation_library(anchors)
    >>> len(library), len(library[0]) + 1              # slots, operations incl. identity
    (8, 8)
    """
    return _zrotation_library(fg_anchor, n_fold=2)

tri6di9_transformation_library 

tri6di9_transformation_library(fg_anchor)

D3h: 3-fold z rotations and 4 cosets, 11 non-identity operations.

Examples:

>>> from cage_isomer_builder.utils import symmetry
>>> anchors = symmetry.ideal_orbit("D3h")        # a perfectly D3h-symmetric test set
>>> library = symmetry.tri6di9_transformation_library(anchors)
>>> len(library), len(library[0]) + 1              # slots, operations incl. identity
(12, 12)
Source code in cage_isomer_builder/utils/symmetry.py 
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def tri6di9_transformation_library(fg_anchor):
    """
    D3h: 3-fold z rotations and 4 cosets, 11 non-identity operations.

    Examples
    --------
    >>> from cage_isomer_builder.utils import symmetry
    >>> anchors = symmetry.ideal_orbit("D3h")        # a perfectly D3h-symmetric test set
    >>> library = symmetry.tri6di9_transformation_library(anchors)
    >>> len(library), len(library[0]) + 1              # slots, operations incl. identity
    (12, 12)
    """
    return _zrotation_library(fg_anchor, n_fold=3)

tri8di12_transformation_library 

tri8di12_transformation_library(fg_anchor)

Oh: 47 non-identity operations.

Examples:

>>> from cage_isomer_builder.utils import symmetry
>>> anchors = symmetry.ideal_orbit("Oh")        # a perfectly Oh-symmetric test set
>>> library = symmetry.tri8di12_transformation_library(anchors)
>>> len(library), len(library[0]) + 1              # slots, operations incl. identity
(48, 48)
Source code in cage_isomer_builder/utils/symmetry.py 
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def tri8di12_transformation_library(fg_anchor):
    """
    Oh: 47 non-identity operations.

    Examples
    --------
    >>> from cage_isomer_builder.utils import symmetry
    >>> anchors = symmetry.ideal_orbit("Oh")        # a perfectly Oh-symmetric test set
    >>> library = symmetry.tri8di12_transformation_library(anchors)
    >>> len(library), len(library[0]) + 1              # slots, operations incl. identity
    (48, 48)
    """
    return _oh_library(fg_anchor)

tri20di30_transformation_library 

tri20di30_transformation_library(fg_anchor)

Ih: the icosahedral group (order 120), 119 non-identity operations.

Generated from two 72-degree rotations about adjacent C5 axes of the icosahedron, (0, phi, 1) and (0, -phi, 1), and the inversion, and closed explicitly.

Examples:

>>> from cage_isomer_builder.utils import symmetry
>>> anchors = symmetry.ideal_orbit("Ih")        # a perfectly Ih-symmetric test set
>>> library = symmetry.tri20di30_transformation_library(anchors)
>>> len(library), len(library[0]) + 1              # slots, operations incl. identity
(120, 120)
Source code in cage_isomer_builder/utils/symmetry.py 
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def tri20di30_transformation_library(fg_anchor):
    """
    Ih: the icosahedral group (order 120), 119 non-identity operations.

    Generated from two 72-degree rotations about adjacent C5 axes of the
    icosahedron, (0, phi, 1) and (0, -phi, 1), and the inversion, and closed
    explicitly.

    Examples
    --------
    >>> from cage_isomer_builder.utils import symmetry
    >>> anchors = symmetry.ideal_orbit("Ih")        # a perfectly Ih-symmetric test set
    >>> library = symmetry.tri20di30_transformation_library(anchors)
    >>> len(library), len(library[0]) + 1              # slots, operations incl. identity
    (120, 120)
    """
    phi = (1 + 5 ** 0.5) / 2
    return _matrix_group_library(fg_anchor, [
        _axis_rotation([0, phi, 1], 72),
        _axis_rotation([0, -phi, 1], 72),
        -np.eye(3),
    ])

tet6tri8_transformation_library 

tet6tri8_transformation_library(fg_anchor)

Oh: same symmetry as tri8di12.

Examples:

>>> from cage_isomer_builder.utils import symmetry
>>> anchors = symmetry.ideal_orbit("Oh")        # a perfectly Oh-symmetric test set
>>> library = symmetry.tet6tri8_transformation_library(anchors)
>>> len(library), len(library[0]) + 1              # slots, operations incl. identity
(48, 48)
Source code in cage_isomer_builder/utils/symmetry.py 
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def tet6tri8_transformation_library(fg_anchor):
    """
    Oh: same symmetry as tri8di12.

    Examples
    --------
    >>> from cage_isomer_builder.utils import symmetry
    >>> anchors = symmetry.ideal_orbit("Oh")        # a perfectly Oh-symmetric test set
    >>> library = symmetry.tet6tri8_transformation_library(anchors)
    >>> len(library), len(library[0]) + 1              # slots, operations incl. identity
    (48, 48)
    """
    return _oh_library(fg_anchor)

tri4tri4_transformation_library 

tri4tri4_transformation_library(fg_anchor, rotation_axes)

Td: same symmetry as tri4di6.

Examples:

>>> import numpy as np
>>> from cage_isomer_builder.utils import symmetry
>>> anchors = symmetry.ideal_orbit("Td")        # a perfectly Td-symmetric test set
>>> library = symmetry.tri4tri4_transformation_library(anchors, 5 * np.array([[1, 1, 1], [1, -1, -1], [-1, 1, -1], [-1, -1, 1]]))
>>> len(library), len(library[0]) + 1              # slots, operations incl. identity
(24, 24)
Source code in cage_isomer_builder/utils/symmetry.py 
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def tri4tri4_transformation_library(fg_anchor, rotation_axes):
    """
    Td: same symmetry as tri4di6.

    Examples
    --------
    >>> import numpy as np
    >>> from cage_isomer_builder.utils import symmetry
    >>> anchors = symmetry.ideal_orbit("Td")        # a perfectly Td-symmetric test set
    >>> library = symmetry.tri4tri4_transformation_library(anchors, 5 * np.array([[1, 1, 1], [1, -1, -1], [-1, 1, -1], [-1, -1, 1]]))
    >>> len(library), len(library[0]) + 1              # slots, operations incl. identity
    (24, 24)
    """
    return tri4di6_transformation_library(fg_anchor, rotation_axes)

tri1tri1_transformation_library 

tri1tri1_transformation_library(fg_anchor)

C3v: 3-fold z rotations and 2 cosets (E and sigma_v), 5 non-identity operations.

Examples:

>>> from cage_isomer_builder.utils import symmetry
>>> library = symmetry.tri1tri1_transformation_library(symmetry.ideal_orbit("D3h"))
>>> len(library[0]) + 1                    # operations incl. identity
6
Source code in cage_isomer_builder/utils/symmetry.py 
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def tri1tri1_transformation_library(fg_anchor):
    """
    C3v: 3-fold z rotations and 2 cosets (E and sigma_v), 5 non-identity operations.

    Examples
    --------
    >>> from cage_isomer_builder.utils import symmetry
    >>> library = symmetry.tri1tri1_transformation_library(symmetry.ideal_orbit("D3h"))
    >>> len(library[0]) + 1                    # operations incl. identity
    6
    """
    return _zrotation_library(fg_anchor, n_fold=3, cosets=_CNV_COSETS)

tri2_2tri2_transformation_library 

tri2_2tri2_transformation_library(fg_anchor)

D2h: same symmetry as tri4_2di6.

Examples:

>>> from cage_isomer_builder.utils import symmetry
>>> anchors = symmetry.ideal_orbit("D2h")        # a perfectly D2h-symmetric test set
>>> library = symmetry.tri2_2tri2_transformation_library(anchors)
>>> len(library), len(library[0]) + 1              # slots, operations incl. identity
(8, 8)
Source code in cage_isomer_builder/utils/symmetry.py 
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def tri2_2tri2_transformation_library(fg_anchor):
    """
    D2h: same symmetry as tri4_2di6.

    Examples
    --------
    >>> from cage_isomer_builder.utils import symmetry
    >>> anchors = symmetry.ideal_orbit("D2h")        # a perfectly D2h-symmetric test set
    >>> library = symmetry.tri2_2tri2_transformation_library(anchors)
    >>> len(library), len(library[0]) + 1              # slots, operations incl. identity
    (8, 8)
    """
    return _zrotation_library(fg_anchor, n_fold=2)

ideal_orbit 

ideal_orbit(group, radius=12.0, seed=0)

A perfectly symmetric test structure: the orbit of one generic point under an ideal point group, as X atoms, in the orientation the per-topology libraries of this module assume (principal axis z, a vertical mirror with normal x; Td/Oh/Ih in their standard cubic setting). Every point is at least 2.5 Angstrom from every other, so anchor matching is unambiguous.

Use it to check a transformation library: on the orbit, a correct library returns exactly |G| - 1 non-identity operations.

Parameters:

Name Type Description Default
group (D2h, D3h, D4h, D5h, D4d, Td, Oh, Ih)
"D2h"
radius float

Distance of the points from the origin (Angstrom).

 12.0
seed int

Chooses the generic point.

 0

Returns:

Type Description
Atoms

|G| atoms, one per group element.

Examples:

>>> from cage_isomer_builder.utils.symmetry import ideal_orbit
>>> len(ideal_orbit("D4h")), len(ideal_orbit("Ih"))
(16, 120)
Source code in cage_isomer_builder/utils/symmetry.py 
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def ideal_orbit(group, radius=12.0, seed=0):
    """
    A perfectly symmetric test structure: the orbit of one generic point
    under an ideal point group, as ``X`` atoms, in the orientation the
    per-topology libraries of this module assume (principal axis z, a
    vertical mirror with normal x; Td/Oh/Ih in their standard cubic
    setting). Every point is at least 2.5 Angstrom from every other, so
    anchor matching is unambiguous.

    Use it to check a transformation library: on the orbit, a correct
    library returns exactly ``|G| - 1`` non-identity operations.

    Parameters
    ----------
    group : {"D2h", "D3h", "D4h", "D5h", "D4d", "Td", "Oh", "Ih"}
    radius : float, default 12.0
        Distance of the points from the origin (Angstrom).
    seed : int, default 0
        Chooses the generic point.

    Returns
    -------
    ase.Atoms
        ``|G|`` atoms, one per group element.

    Examples
    --------
    >>> from cage_isomer_builder.utils.symmetry import ideal_orbit
    >>> len(ideal_orbit("D4h")), len(ideal_orbit("Ih"))
    (16, 120)
    """
    from ase import Atoms

    if group not in _IDEAL_GENERATORS:
        raise ValueError(f"group={group!r}; choose from {sorted(_IDEAL_GENERATORS)}.")
    generators = _IDEAL_GENERATORS[group]()
    elements = [np.eye(3)]
    frontier = list(elements)
    while frontier:
        new = []
        for a in frontier:
            for g in generators:
                b = g @ a
                if not any(np.allclose(b, c, atol=1e-8) for c in elements):
                    elements.append(b)
                    new.append(b)
        frontier = new
    rng = np.random.default_rng(seed)
    while True:
        p = rng.normal(size=3)
        p = radius * p / np.linalg.norm(p)
        pts = np.array([g @ p for g in elements])
        gaps = np.linalg.norm(pts[:, None] - pts[None], axis=2) + 1e9 * np.eye(len(pts))
        if gaps.min() > 2.5:
            return Atoms(numbers=np.zeros(len(pts), dtype=int), positions=pts)

universal_transformation_library 

universal_transformation_library(fg_anchor, node_centers, tol=0.5)

Transformation library from the node-centre positions.

Works for any topology whose point group acts faithfully on the node centres, i.e. no non-identity operation fixes every centre:

  • 4 nodes at tetrahedral vertices (Td, 23 non-identity operations)
  • 6 nodes at octahedral vertices (Oh, 47 non-identity operations)
  • 8 nodes at cube vertices (Oh, 47 non-identity operations)

Prism and bipyramid topologies have operations that fix all node centres (e.g. the C4 rotation of a D4h bipyramid), which this function cannot see; use their topology-specific library.

Algorithm

For every permutation of the N node centres:

  1. Compute the best-fit orthogonal transformation by SVD.
  2. Keep it only if every transformed centre lands within tol of its target.
  3. Apply it to the FG anchors and record the permutation.
  4. Remove duplicates and the identity.

Parameters:

Name Type Description Default
fg_anchor Atoms
required
node_centers (array - like, shape(N, 3))

Node centres, N at most 8.

 required
tol float

Matching tolerance (Angstrom).

 0.5

Returns:

Type Description
list of tuple

Slot-major transformation library.

Raises:

Type Description
ValueError

If the centres lie on a line or in a plane, or there are more than 8.

Examples:

>>> import numpy as np
>>> from cage_isomer_builder.utils import symmetry
>>> anchors = symmetry.ideal_orbit("Oh")
>>> octahedron = 5 * np.vstack([np.eye(3), -np.eye(3)])     # six node centres
>>> library = symmetry.universal_transformation_library(anchors, octahedron)
>>> len(library[0]) + 1
48
Source code in cage_isomer_builder/utils/symmetry.py 
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def universal_transformation_library(fg_anchor, node_centers, tol=0.5):
    """
    Transformation library from the node-centre positions.

    Works for any topology whose point group acts faithfully on the node
    centres, i.e. no non-identity operation fixes every centre:

    * 4 nodes at tetrahedral vertices (Td, 23 non-identity operations)
    * 6 nodes at octahedral vertices (Oh, 47 non-identity operations)
    * 8 nodes at cube vertices (Oh, 47 non-identity operations)

    Prism and bipyramid topologies have operations that fix all node centres
    (e.g. the C4 rotation of a D4h bipyramid), which this function cannot see;
    use their topology-specific library.

    Algorithm
    ---------
    For every permutation of the N node centres:

    1. Compute the best-fit orthogonal transformation by SVD.
    2. Keep it only if every transformed centre lands within ``tol`` of its
       target.
    3. Apply it to the FG anchors and record the permutation.
    4. Remove duplicates and the identity.

    Parameters
    ----------
    fg_anchor : ase.Atoms
    node_centers : array-like, shape (N, 3)
        Node centres, N at most 8.
    tol : float, default 0.5
        Matching tolerance (Angstrom).

    Returns
    -------
    list of tuple
        Slot-major transformation library.

    Raises
    ------
    ValueError
        If the centres lie on a line or in a plane, or there are more than 8.

    Examples
    --------
    >>> import numpy as np
    >>> from cage_isomer_builder.utils import symmetry
    >>> anchors = symmetry.ideal_orbit("Oh")
    >>> octahedron = 5 * np.vstack([np.eye(3), -np.eye(3)])     # six node centres
    >>> library = symmetry.universal_transformation_library(anchors, octahedron)
    >>> len(library[0]) + 1
    48
    """
    node_positions = np.array(node_centers, dtype=float)
    n_nodes = len(node_positions)

    spread = np.linalg.svd(node_positions - node_positions.mean(axis=0), compute_uv=False)
    if n_nodes < 4 or spread[2] < 0.1 * spread[0]:
        raise ValueError(
            "universal_transformation_library: the node centres lie on a line or "
            "in a plane, so operations that fix every node (e.g. the mirror "
            "through that plane, or rotations about the line) are invisible here "
            "and the result would be missing operations. Use point-group "
            "detection (CageBuilder.symmetry_method = 'detect') instead."
        )
    if n_nodes > 8:
        raise ValueError(
            f"universal_transformation_library: {n_nodes} nodes give "
            f"{math.factorial(n_nodes):,} permutations (the limit is 8 nodes, "
            "40,320 permutations). Use point-group detection "
            "(symmetry_method='detect') for larger cages."
        )

    node_center = node_positions.mean(axis=0)
    node_centered = node_positions - node_center
    identity = tuple(range(len(fg_anchor)))
    seen = set()
    transformation_lib = []

    for perm in permutations(range(n_nodes)):
        target = node_centered[list(perm)]
        H = node_centered.T @ target
        U, _S, Vt = np.linalg.svd(H)
        Rt = U @ Vt  # no determinant correction, so improper operations are kept

        if not np.allclose(node_centered @ Rt, target, atol=tol):
            continue  # not a valid symmetry operation

        t_fg = fg_anchor.copy()
        t_fg.positions = (t_fg.positions - node_center) @ Rt + node_center

        p = tuple(geometry.project_permutation(fg_anchor, t_fg))
        if p not in seen:
            seen.add(p)
            if p != identity:
                transformation_lib.append(p)

    if not transformation_lib:
        return [() for _ in range(len(fg_anchor))]
    return list(zip(*transformation_lib))

spglib_dataset 

spglib_dataset(atoms, symprec=0.01)

spglib symmetry dataset of a periodic structure, or None if spglib finds none at this tolerance.

Examples:

>>> from cage_isomer_builder import example_data
>>> from cage_isomer_builder.utils.symmetry import spglib_dataset
>>> spglib_dataset(example_data.read("RUBTAK01")).international
'Fm-3m'
Source code in cage_isomer_builder/utils/symmetry.py 
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def spglib_dataset(atoms, symprec=1e-2):
    """
    spglib symmetry dataset of a periodic structure, or None if spglib
    finds none at this tolerance.

    Examples
    --------
    >>> from cage_isomer_builder import example_data
    >>> from cage_isomer_builder.utils.symmetry import spglib_dataset
    >>> spglib_dataset(example_data.read("RUBTAK01")).international
    'Fm-3m'
    """
    cell = (np.asarray(atoms.cell[:], dtype=float), atoms.get_scaled_positions(), atoms.numbers)
    return _spglib_call("get_symmetry_dataset", cell, symprec=symprec)

primitive_cell 

primitive_cell(atoms, symprec=0.01)

The primitive cell of a periodic structure: the smallest cell that repeats to give the whole crystal (via spglib).

The atom positions are kept as they are (not moved to ideal symmetric positions), so the geometry is unchanged. Only the cell, positions and elements are kept; other per-atom arrays and info are dropped.

Parameters:

Name Type Description Default
atoms Atoms

Periodic structure.

 required
symprec float

spglib symmetry tolerance, the same default as :func:spacegroup_transformation_library.

 1e-2

Returns:

Type Description
Atoms

The primitive cell. If atoms is already primitive, a copy with the same number of atoms.

Raises:

Type Description
ValueError

If atoms is not periodic, or spglib finds no symmetry at this tolerance.

Examples:

>>> from cage_isomer_builder import example_data
>>> from cage_isomer_builder.utils.symmetry import primitive_cell
>>> uio66 = example_data.read("RUBTAK01")            # conventional fcu cell
>>> len(uio66), len(primitive_cell(uio66))           # four lattice points per cell
(440, 110)
Source code in cage_isomer_builder/utils/symmetry.py 
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def primitive_cell(atoms, symprec=1e-2):
    """
    The primitive cell of a periodic structure: the smallest cell that
    repeats to give the whole crystal (via spglib).

    The atom positions are kept as they are (not moved to ideal symmetric
    positions), so the geometry is unchanged. Only the cell, positions and
    elements are kept; other per-atom arrays and ``info`` are dropped.

    Parameters
    ----------
    atoms : ase.Atoms
        Periodic structure.
    symprec : float, default 1e-2
        spglib symmetry tolerance, the same default as
        :func:`spacegroup_transformation_library`.

    Returns
    -------
    ase.Atoms
        The primitive cell. If ``atoms`` is already primitive, a copy with
        the same number of atoms.

    Raises
    ------
    ValueError
        If ``atoms`` is not periodic, or spglib finds no symmetry at this
        tolerance.

    Examples
    --------
    >>> from cage_isomer_builder import example_data
    >>> from cage_isomer_builder.utils.symmetry import primitive_cell
    >>> uio66 = example_data.read("RUBTAK01")            # conventional fcu cell
    >>> len(uio66), len(primitive_cell(uio66))           # four lattice points per cell
    (440, 110)
    """
    from ase import Atoms

    if not atoms.pbc.all():
        raise ValueError("primitive_cell needs a periodic structure (pbc in all directions).")
    cell = (np.asarray(atoms.cell[:], dtype=float), atoms.get_scaled_positions(), atoms.numbers)
    reduced = _spglib_call("standardize_cell", cell, to_primitive=True, no_idealize=True,
                           symprec=symprec)
    if reduced is None:
        raise ValueError("spglib could not find the symmetry of this structure; try a looser symprec.")
    lattice, scaled, numbers = reduced
    return Atoms(numbers=numbers, scaled_positions=scaled, cell=lattice, pbc=True)

spacegroup_transformation_library 

spacegroup_transformation_library(cage_atoms, fg_anchor_indices, symprec=0.01, atom_tol=0.5)

Transformation library of a periodic structure from its space group (spglib).

A unit cell's FG anchors are related by the full space group, including translations, screw axes and glide planes. A point group about one centre misses most of these operations, so a periodic structure must use this function.

Algorithm
  1. Get every (rotation, translation) operation of the space group from spglib, in fractional coordinates.
  2. Apply each operation to the anchors' fractional positions and match every image to the anchor it coincides with (minimum image).
  3. Remove the identity and duplicate permutations.

Parameters:

Name Type Description Default
cage_atoms Atoms

The full periodic structure (with pbc and a cell); the space group is computed from all its atoms.

 required
fg_anchor_indices sequence of int

Indices of the FG anchors in cage_atoms, in slot order.

 required
symprec float

spglib tolerance. Real, slightly noisy structures usually need a looser value than spglib's default of 1e-5.

 1e-2
atom_tol float

Distance (Angstrom) within which an image matches an anchor. Anchors are always much further apart than this.

 0.5

Returns:

Type Description
list of tuple

Slot-major transformation library, in the same format as universal_transformation_library.

Raises:

Type Description
RuntimeError

If spglib finds no space group, or an operation does not map the anchors onto themselves within atom_tol (usually symprec is too loose for the structure).

Examples:

>>> from cage_isomer_builder import example_data
>>> from cage_isomer_builder.utils.symmetry import primitive_cell, spacegroup_transformation_library
>>> cell = primitive_cell(example_data.read("RUBTAK01"))
>>> zr = [a.index for a in cell if a.symbol == "Zr"]
>>> library = spacegroup_transformation_library(cell, zr)    # how the 6 Zr permute
>>> len(library), len(library[0]) + 1                         # Fm-3m: 48 point operations
(6, 48)
Source code in cage_isomer_builder/utils/symmetry.py 
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def spacegroup_transformation_library(cage_atoms, fg_anchor_indices, symprec=1e-2, atom_tol=0.5):
    """
    Transformation library of a periodic structure from its space group
    (spglib).

    A unit cell's FG anchors are related by the full space group, including
    translations, screw axes and glide planes. A point group about one centre
    misses most of these operations, so a periodic structure must use this
    function.

    Algorithm
    ---------
    1. Get every (rotation, translation) operation of the space group from
       spglib, in fractional coordinates.
    2. Apply each operation to the anchors' fractional positions and match every
       image to the anchor it coincides with (minimum image).
    3. Remove the identity and duplicate permutations.

    Parameters
    ----------
    cage_atoms : ase.Atoms
        The full periodic structure (with ``pbc`` and a cell); the space group
        is computed from all its atoms.
    fg_anchor_indices : sequence of int
        Indices of the FG anchors in ``cage_atoms``, in slot order.
    symprec : float, default 1e-2
        spglib tolerance. Real, slightly noisy structures usually need a looser
        value than spglib's default of 1e-5.
    atom_tol : float, default 0.5
        Distance (Angstrom) within which an image matches an anchor. Anchors are
        always much further apart than this.

    Returns
    -------
    list of tuple
        Slot-major transformation library, in the same format as
        universal_transformation_library.

    Raises
    ------
    RuntimeError
        If spglib finds no space group, or an operation does not map the
        anchors onto themselves within ``atom_tol`` (usually ``symprec`` is too
        loose for the structure).

    Examples
    --------
    >>> from cage_isomer_builder import example_data
    >>> from cage_isomer_builder.utils.symmetry import primitive_cell, spacegroup_transformation_library
    >>> cell = primitive_cell(example_data.read("RUBTAK01"))
    >>> zr = [a.index for a in cell if a.symbol == "Zr"]
    >>> library = spacegroup_transformation_library(cell, zr)    # how the 6 Zr permute
    >>> len(library), len(library[0]) + 1                         # Fm-3m: 48 point operations
    (6, 48)
    """
    lattice = np.asarray(cage_atoms.cell[:], dtype=float)
    dataset = spglib_dataset(cage_atoms, symprec=symprec)
    if dataset is None:
        raise RuntimeError(
            "spglib could not determine a space group for this structure - "
            "try a looser symprec."
        )

    anchor_frac = cage_atoms.get_scaled_positions()[list(fg_anchor_indices)]
    n_anchors = len(anchor_frac)
    identity = tuple(range(n_anchors))
    seen = set()
    transformation_lib = []

    for rotation, translation in zip(dataset.rotations, dataset.translations):
        image_frac = anchor_frac @ rotation.T + translation

        perm = []
        for i in range(n_anchors):
            diff_frac = anchor_frac - image_frac[i]
            diff_frac -= np.round(diff_frac)   # minimum image, fractional
            diff_cart = diff_frac @ lattice
            dists = np.linalg.norm(diff_cart, axis=1)
            j = int(np.argmin(dists))
            if dists[j] > atom_tol:
                raise RuntimeError(
                    "A space-group operation did not map the FG anchor set "
                    "onto itself (closest match "
                    f"{dists[j]:.3f} Å > atom_tol={atom_tol}). The structure "
                    "doesn't satisfy the symmetry spglib detected at this "
                    "symprec; try tightening symprec or loosening atom_tol."
                )
            perm.append(j)

        p = tuple(perm)
        if p not in seen:
            seen.add(p)
            if p != identity:
                transformation_lib.append(p)

    if not transformation_lib:
        return [() for _ in range(n_anchors)]
    return list(zip(*transformation_lib))