Skip to content

Decomposition & Candidate Generation

pg_decompose

Referenced by:

_print_to_vector 

_print_to_vector(axes: array, shine: float = 0.2, factor: float = 1.0) -> None

Internal debugging utility: print vectors in a visualization-friendly format.

This helper outputs vectors as formatted lines (prefixed with "V" and "S") intended for use with external visualization tools or custom viewers. It is primarily used to inspect symmetry axes during development.

Parameters:

  • axes (ndarray) –

    Array of vectors to print (shape: (N, 3) or (3,)).

  • shine (float, default: 0.2 ) –

    Visual intensity parameter included in the output format.

  • factor (float, default: 1.0 ) –

    Scaling factor applied to vector length in the output.

Returns:

  • None –
Source code in minimalsym/core/pg_decompose.py 
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
def _print_to_vector(axes:np.array, shine:float=0.2, factor:float=1.0) -> None:
    """
    Internal debugging utility: print vectors in a visualization-friendly format.

    This helper outputs vectors as formatted lines (prefixed with "V" and "S")
    intended for use with external visualization tools or custom viewers.
    It is primarily used to inspect symmetry axes during development.

    Parameters
    ----------
    axes : np.ndarray
        Array of vectors to print (shape: (N, 3) or (3,)).
    shine : float, optional
        Visual intensity parameter included in the output format.
    factor : float, optional
        Scaling factor applied to vector length in the output.

    Returns
    -------
    None
    """
    np.set_printoptions(precision=6)

    axes = np.asarray(axes)
    # If single vector -> wrap it
    if axes.ndim == 1:
        axes = axes.reshape(1, 3)

    color = int(0)
    for axis in axes:
        axis = normalize(axis)
        print("V", end="   ")
        for a in axis:
            print(f"{-a*factor:.6f}", end="   ")
        for a in axis:
            print(f"{a*factor:.6f}", end="   ")
        color +=1
        print(f"S   0.02  0.05 0.{int(color/4)%2*9}    0.{int(color/2)%2*9}   0.{color%2*9}   {shine}")

    print("- - - - - - - -")

_decompose_cyclic 

_decompose_cyclic(family: str, n: int, subfamily: str, paxis: array, saxis: array, taxis: array) -> list[PointGroupResult]

Decompose cyclic and dihedral point groups into symmetry-consistent subgroups.

This function generates a set of candidate point groups derived from a cyclic (Cₙ/Sₙ) or dihedral (Dₙ) parent group, including all compatible subgroups and orientations that preserve the underlying symmetry axes.

Parameters:

  • family (str) –

    Point-group family ("C" or "D").

  • n (int) –

    Order of the principal rotation axis. Special case n = 0 corresponds to linear groups (C∞v / D∞h).

  • subfamily (str or None) –

    Subfamily label ("v", "h", "d", or None).

  • paxis ((ndarray, shape(3))) –

    Principal symmetry axis.

  • saxis ((ndarray, shape(3))) –

    Secondary axis defining the canonical orientation.

  • taxis ((ndarray, shape(3))) –

    Tertiary axis orthogonal to paxis and saxis.

Returns:

  • list[PointGroupResult] –

    List of candidate point groups with associated orientations.

Notes
  • Includes:
    • Parent group (e.g., Cₙ, Dₙ)
    • Subgroups (Cₖ, Dₖ where k | n)
    • Mirror and improper groups (Cs, Ci, Sₙ)
    • All symmetry-equivalent orientations of axes
  • For dihedral groups, additional recursive decomposition is performed through cyclic subgroups (e.g., Cₙᵥ, Cₙₕ).
  • Linear groups (n = 0) are approximated using high-order finite groups.
Source code in minimalsym/core/pg_decompose.py 
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
def _decompose_cyclic(family:str, n:int, subfamily:str, paxis:np.array, saxis:np.array, taxis:np.array) -> list[PointGroupResult]:
    """
    Decompose cyclic and dihedral point groups into symmetry-consistent subgroups.

    This function generates a set of candidate point groups derived from a
    cyclic (Cₙ/Sₙ) or dihedral (Dₙ) parent group, including all compatible
    subgroups and orientations that preserve the underlying symmetry axes.

    Parameters
    ----------
    family : str
        Point-group family ("C" or "D").
    n : int
        Order of the principal rotation axis. Special case n = 0 corresponds
        to linear groups (C∞v / D∞h).
    subfamily : str or None
        Subfamily label ("v", "h", "d", or None).
    paxis : ndarray, shape (3,)
        Principal symmetry axis.
    saxis : ndarray, shape (3,)
        Secondary axis defining the canonical orientation.
    taxis : ndarray, shape (3,)
        Tertiary axis orthogonal to paxis and saxis.

    Returns
    -------
    : list[PointGroupResult]
        List of candidate point groups with associated orientations.

    Notes
    -----
    - Includes:
        * Parent group (e.g., Cₙ, Dₙ)
        * Subgroups (Cₖ, Dₖ where k | n)
        * Mirror and improper groups (Cs, Ci, Sₙ)
        * All symmetry-equivalent orientations of axes
    - For dihedral groups, additional recursive decomposition is performed
      through cyclic subgroups (e.g., Cₙᵥ, Cₙₕ).
    - Linear groups (n = 0) are approximated using high-order finite groups.
    """
    pg_list = []
    zeros = np.zeros(3)

    # Cs(h) Sn Ci
    if subfamily == 'h':
        pg_list.append(PointGroupResult("Cs", paxis, saxis))
        if n != 0 and n % 2 == 0:
            pg_list.append(PointGroupResult(f"S{n}", paxis, saxis))
        if n % 2 == 0:
            pg_list.append(PointGroupResult("Ci", paxis, saxis))

    if n == 0:
        # Ckv
        pg_list.append(PointGroupResult("C16v", paxis, saxis))
        # C0
        pg_list.append(PointGroupResult("C16", paxis, zeros))
        # 0 C2
        pg_list.append(PointGroupResult("Cs", saxis, zeros))
        pg_list.append(PointGroupResult("Cs", taxis, zeros))
        if family == 'D':
            # Dkh
            pg_list.append(PointGroupResult("D16h", paxis, saxis))
            # D0
            pg_list.append(PointGroupResult("D16", paxis, saxis))
            # 0 D2
            pg_list.append(PointGroupResult("D2", saxis, paxis))
            pg_list.append(PointGroupResult("D2", taxis, paxis))
            # 0 C2
            pg_list.append(PointGroupResult("C2", saxis, zeros))
            pg_list.append(PointGroupResult("C2", taxis, zeros))
            # S0
            pg_list.append(PointGroupResult("S2", paxis, zeros))
            # 0 S2
            pg_list.append(PointGroupResult("S2", saxis, zeros))
            pg_list.append(PointGroupResult("S2", taxis, zeros))
            pass
        return pg_list

    # Family(n)
    if subfamily is not None:
        pg_list.append(PointGroupResult(f"{family}{n}", paxis, saxis))

    # Cs(v)
    if subfamily == 'v':
        cyclic_cs_axes = generate_cyclic_axes(paxis, taxis, n)

        for rotated_taxis in cyclic_cs_axes:
            pg_list.append(PointGroupResult("Cs", rotated_taxis, zeros))

    # S2n
    if subfamily == 'd':
        pg_list.append(PointGroupResult(f"S{2*n}", paxis, saxis))

    if family == 'D':
        # Cnv RECURSIVE
        if subfamily == 'h' or subfamily == 'd':
            cnv_pg = PointGroup.from_string(f"C{n}v")
            additional_c2_axis = generate_cyclic_axes(paxis, saxis, n, 2)[1, :]
            dihedral_axis = normalize((saxis + additional_c2_axis))
            if subfamily == 'd': # diherdral mirror planes, bisect angle between 2 nC2 axes
                if n % 2 == 0:
                    # For n == even; all dihedral angle perpendicular to dihedral angle
                    pg_list.extend(_decompose_point_group(cnv_pg, paxis, dihedral_axis))
                else:
                    # For Dnd where n == odd; taxis is a dihedral angle (collinear to dihedral mirror plane)
                    # So for Dnd where n == odd; saxis is norm of dihedral plane
                    # For Cnv where n == odd; taxis is aligned with norm mirror plane
                    # So for Cnv where n == odd; saxis is collinear to mirror plane
                    pg_list.extend(_decompose_point_group(cnv_pg, paxis, taxis))
            if subfamily == 'h': # vertical mirror planes, vertical to nC2 axes
                if n % 2 == 0:
                    pg_list.extend(_decompose_point_group(cnv_pg, paxis, taxis))
                    if n > 2:
                        pg_list.extend(_decompose_point_group(PointGroup.from_string(f"D{n//2}d"), paxis, saxis))
                        pg_list.extend(_decompose_point_group(PointGroup.from_string(f"D{n//2}d"), paxis, additional_c2_axis))
                else:
                    pg_list.extend(_decompose_point_group(cnv_pg, paxis, saxis))
                # Cnh RECURSIVE
                pg_list.extend(_decompose_point_group(PointGroup.from_string(f"C{n}h"), paxis, saxis))
        # Cn RECURSIVE
        else:
            cn_pg = PointGroup.from_string(f"C{n}")
            pg_list.extend(_decompose_point_group(cn_pg, paxis, saxis))

        # n C2 axes
        cyclic_c2_axes = generate_cyclic_axes(paxis, saxis, n)
        for rotated_saxis in cyclic_c2_axes:
            pg_list.append(PointGroupResult("C2", rotated_saxis, zeros))

    # Fam_k Fam_k_sub
    for k in range(2, n):
        if n % k != 0:
            continue

        pg_list.append(PointGroupResult(f"{family}{k}", paxis, saxis))
        if subfamily is not None:
            if subfamily == 'v':
                if n % 2 != k % 2:
                    pg_list.append(PointGroupResult(f"{family}{k}{subfamily}", paxis, taxis))
                else:
                    pg_list.append(PointGroupResult(f"{family}{k}{subfamily}", paxis, saxis))
            elif subfamily != 'd':
                pg_list.append(PointGroupResult(f"{family}{k}{subfamily}", paxis, saxis))

    return pg_list

_decompose_T_family 

_decompose_T_family(subfamily: str, paxis: array, saxis: array, taxis: array) -> list[PointGroupResult]

Decompose tetrahedral point groups into symmetry-consistent subgroups.

Generates all subgroup candidates of the T family (T, T_d, T_h), including cyclic, dihedral, and improper subgroups derived from tetrahedral symmetry elements.

Parameters:

  • subfamily (str or None) –

    Subfamily label ("d", "h", or None).

  • paxis ((ndarray, shape(3))) –

    Orthogonal axes defining the tetrahedral orientation.

  • saxis ((ndarray, shape(3))) –

    Orthogonal axes defining the tetrahedral orientation.

  • taxis ((ndarray, shape(3))) –

    Orthogonal axes defining the tetrahedral orientation.

Returns:

  • list[PointGroupResult] –

    List of candidate point groups with associated orientations.

Notes
  • Includes:
    • C₂ axes (3)
    • C₃ axes (4, along tetrahedral corners)
    • Derived subgroups (D₂)
  • For T_d:
    • Adds S₄ axes (3) and σ_d planes (6)
    • Includes C₃ᵥ subgroups
  • For T_h:
    • Adds inversion (Ci), σ_h planes (3), and S₆ axes (4)
    • Includes D₂h subgroups
  • Recursive decomposition is applied to composite subgroups.
Source code in minimalsym/core/pg_decompose.py 
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
def _decompose_T_family(subfamily:str, paxis:np.array, saxis:np.array, taxis:np.array) -> list[PointGroupResult]:
    """
    Decompose tetrahedral point groups into symmetry-consistent subgroups.

    Generates all subgroup candidates of the T family (T, T_d, T_h),
    including cyclic, dihedral, and improper subgroups derived from
    tetrahedral symmetry elements.

    Parameters
    ----------
    subfamily : str or None
        Subfamily label ("d", "h", or None).
    paxis, saxis, taxis : ndarray, shape (3,)
        Orthogonal axes defining the tetrahedral orientation.

    Returns
    -------
    list[PointGroupResult]
        List of candidate point groups with associated orientations.

    Notes
    -----
    - Includes:
        * C₂ axes (3)
        * C₃ axes (4, along tetrahedral corners)
        * Derived subgroups (D₂)
    - For T_d:
        * Adds S₄ axes (3) and σ_d planes (6)
        * Includes C₃ᵥ subgroups
    - For T_h:
        * Adds inversion (Ci), σ_h planes (3), and S₆ axes (4)
        * Includes D₂h subgroups
    - Recursive decomposition is applied to composite subgroups.
    """
    # Tetrahedral (n == 3): use two of the three C2 axes.

    pg_list = []
    zeros = np.zeros(3)

    # T
    if subfamily is not None:
        pg_list.append(PointGroupResult('T', paxis, saxis))

    # 3 C2
    pg_list.append(PointGroupResult("C2", paxis, saxis))
    pg_list.append(PointGroupResult("C2", saxis, paxis))
    pg_list.append(PointGroupResult("C2", taxis, saxis))

    # 4 C3
    # Tetrahedron corners
    tetra_corners = [
        normalize(paxis +  saxis +  taxis),
        normalize(paxis +  saxis + -taxis),
        normalize(paxis + -saxis +  taxis),
        normalize(paxis + -saxis + -taxis)]

    for corner in tetra_corners:
        pg_list.append(PointGroupResult("C3", corner, zeros))

    # Combinations
    # D2(=3C2)  RECURSIVE
    d2_pg = PointGroup.from_string("D2")
    pg_list.extend(_decompose_point_group(d2_pg, paxis, saxis))
    pg_list.extend(_decompose_point_group(d2_pg, saxis, paxis))
    pg_list.extend(_decompose_point_group(d2_pg, taxis, paxis))


    if subfamily == 'd':
        # 3 S4 axes (same as C2 axes)
        pg_list.append(PointGroupResult("S4", paxis, saxis))
        pg_list.append(PointGroupResult("S4", saxis, paxis))
        pg_list.append(PointGroupResult("S4", taxis, saxis))
        # 6Cs(d)
        mirror_axes = np.array([
            canonical(paxis + saxis),
            canonical(paxis - saxis),
            canonical(paxis + taxis),
            canonical(paxis - taxis),
            canonical(saxis + taxis),
            canonical(saxis - taxis),
        ])
        for mirror_axis in mirror_axes:
            pg_list.append(PointGroupResult("Cs", mirror_axis, zeros))

        # Combinations
        # D2d(3C2) RECURSIVE
        d2d_pg = PointGroup.from_string("D2d")
        pg_list.extend(_decompose_point_group(d2d_pg, paxis, saxis))
        pg_list.extend(_decompose_point_group(d2d_pg, saxis, paxis))
        pg_list.extend(_decompose_point_group(d2d_pg, taxis, paxis))
        # C3v(4C3, 6Cs)
        pg_list.append(PointGroupResult("C3v", tetra_corners[0], canonical(np.cross(mirror_axes[1],  tetra_corners[0]))))
        pg_list.append(PointGroupResult("C3v", tetra_corners[1], canonical(np.cross(mirror_axes[1],  tetra_corners[1]))))
        pg_list.append(PointGroupResult("C3v", tetra_corners[2], canonical(np.cross(mirror_axes[0],  tetra_corners[2]))))
        pg_list.append(PointGroupResult("C3v", tetra_corners[3], canonical(np.cross(mirror_axes[0],  tetra_corners[3]))))

    if subfamily == 'h':
        # Ci
        pg_list.append(PointGroupResult("Ci", paxis, saxis))
        # 3Cs(h)
        pg_list.append(PointGroupResult("Cs", paxis, saxis))
        pg_list.append(PointGroupResult("Cs", paxis, taxis))
        pg_list.append(PointGroupResult("Cs", saxis, taxis))
        # 4 S6 (same as C3 axes)
        for corner in tetra_corners:
            pg_list.append(PointGroupResult("S6", corner, zeros))
        # Combinations
        # D2h(3C2) RECURSIVE
        d2h_pg = PointGroup.from_string("D2h")
        pg_list.extend(_decompose_point_group(d2h_pg, paxis, saxis))
        pg_list.extend(_decompose_point_group(d2h_pg, saxis, paxis))
        pg_list.extend(_decompose_point_group(d2h_pg, taxis, paxis))

    return pg_list

_decompose_O_family 

_decompose_O_family(subfamily: str, paxis: array, saxis: array, taxis: array) -> list[PointGroupResult]

Decompose octahedral point groups into symmetry-consistent subgroups.

Generates subgroup candidates of the O family (O, O_h), including tetrahedral, cyclic, and dihedral subgroups derived from cubic symmetry.

Parameters:

  • subfamily (str or None) –

    Subfamily label ("h" for O_h, or None for O).

  • paxis ((ndarray, shape(3))) –

    Orthogonal C₄ axes defining the cubic orientation.

  • saxis ((ndarray, shape(3))) –

    Orthogonal C₄ axes defining the cubic orientation.

  • taxis ((ndarray, shape(3))) –

    Orthogonal C₄ axes defining the cubic orientation.

Returns:

  • list[PointGroupResult] –

    List of candidate point groups with associated orientations.

Notes
  • Includes:
    • C₄ axes (3), C₃ axes (4, cube diagonals), C₂ axes (6)
    • Subgroups: D₄, D₃, D₂, T
  • For O_h:
    • Adds inversion, mirror planes, and improper rotations (Ci, S₄, S₆)
    • Subgroups: D₄h, D₃d, D₂h, T, Th, Td
    • Includes recursive decomposition into D₄h, D₃d, D₂h
  • Axis orientations are explicitly constructed from cube geometry.
Source code in minimalsym/core/pg_decompose.py 
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
def _decompose_O_family(subfamily:str, paxis:np.array, saxis:np.array, taxis:np.array) -> list[PointGroupResult]:
    """
    Decompose octahedral point groups into symmetry-consistent subgroups.

    Generates subgroup candidates of the O family (O, O_h), including
    tetrahedral, cyclic, and dihedral subgroups derived from cubic symmetry.

    Parameters
    ----------
    subfamily : str or None
        Subfamily label ("h" for O_h, or None for O).
    paxis, saxis, taxis : ndarray, shape (3,)
        Orthogonal C₄ axes defining the cubic orientation.

    Returns
    -------
    list[PointGroupResult]
        List of candidate point groups with associated orientations.

    Notes
    -----
    - Includes:
        * C₄ axes (3), C₃ axes (4, cube diagonals), C₂ axes (6)
        * Subgroups: D₄, D₃, D₂, T
    - For O_h:
        * Adds inversion, mirror planes, and improper rotations (Ci, S₄, S₆)
        * Subgroups: D₄h, D₃d, D₂h, T, Th, Td
        * Includes recursive decomposition into D₄h, D₃d, D₂h
    - Axis orientations are explicitly constructed from cube geometry.
    """
    # Octahedral: paxis and saxis are two orthogonal C4 axes.

    pg_list = []
    zeros = np.zeros(3)

    # O
    if subfamily is not None:
        pg_list.append(PointGroupResult('O', paxis, saxis))

    # T
    pg_list.append(PointGroupResult('T', paxis, saxis))

    # 3 C4 RECURSIVE
    c4_pg = PointGroup.from_string("C4")
    pg_list.extend(_decompose_point_group(c4_pg, paxis, saxis))
    pg_list.extend(_decompose_point_group(c4_pg, saxis, paxis))
    pg_list.extend(_decompose_point_group(c4_pg, taxis, saxis))

    # 4 C3
    # Cube diagonals
    cube_corners = [
        normalize(paxis +  saxis +  taxis),
        normalize(paxis +  saxis + -taxis),
        normalize(paxis + -saxis +  taxis),
        normalize(paxis + -saxis + -taxis)]

    for corner in cube_corners:
        pg_list.append(PointGroupResult("C3", corner, zeros))

    # 6 C2
    face_diagonals = [
        normalize(paxis + saxis),
        normalize(paxis - saxis),
        normalize(paxis + taxis),
        normalize(paxis - taxis),
        normalize(saxis + taxis),
        normalize(saxis - taxis),
    ]

    for diagonal in face_diagonals:
        pg_list.append(PointGroupResult("C2", diagonal, zeros))

    # Combinations
    # D4(=3C4)8
    d4_axis_pairs = [
        (paxis, saxis),
        (saxis, paxis),
        (taxis, paxis),
    ]
    for axis_pair in d4_axis_pairs:
        pg_list.append(PointGroupResult("D4", axis_pair[0], axis_pair[1]))

    # D3(=4C3,6C2)
    d3_axis_pairs = [
        (cube_corners[0], face_diagonals[1]),
        (cube_corners[1], face_diagonals[1]),
        (cube_corners[2], face_diagonals[0]),
        (cube_corners[3], face_diagonals[0]),
    ]
    for axis_pair in d3_axis_pairs:
        pg_list.append(PointGroupResult("D3", axis_pair[0], axis_pair[1]))

    # D2(=6C2)
    d2_axis_pairs = [
        (face_diagonals[0], taxis),
        (face_diagonals[1], taxis),
        (face_diagonals[2], saxis),
        (face_diagonals[3], saxis),
        (face_diagonals[4], paxis),
        (face_diagonals[5], paxis),
    ]
    for axis_pair in d2_axis_pairs:
        pg_list.append(PointGroupResult("D2", axis_pair[0], axis_pair[1]))

    if subfamily == 'h':
        # Th
        pg_list.append(PointGroupResult("Th", paxis, saxis))
        # Td
        pg_list.append(PointGroupResult("Td", paxis, saxis))

        # Ci
        pg_list.append(PointGroupResult("Ci", paxis, saxis))
        # 3 Cs(h)
        pg_list.append(PointGroupResult("Cs", paxis, saxis))
        pg_list.append(PointGroupResult("Cs", saxis, paxis))
        pg_list.append(PointGroupResult("Cs", taxis, paxis))

        # 6 Cs(d)
        for diagonal in face_diagonals:
            pg_list.append(PointGroupResult("Cs", diagonal, zeros))

        # 4 S6 (same as C3 axes)
        for corner in cube_corners:
            pg_list.append(PointGroupResult("S6", corner, zeros))

        # 3 S4 (same as C4 axes)
        pg_list.append(PointGroupResult("S4", paxis, saxis))
        pg_list.append(PointGroupResult("S4", saxis, paxis))
        pg_list.append(PointGroupResult("S4", taxis, paxis))

        # Combinations
        # D4h(=3C4) RECURSIVE
        d4h_pg = PointGroup.from_string("D4h")
        for axis_pair in d4_axis_pairs:
            pg_list.extend(_decompose_point_group(d4h_pg, axis_pair[0], axis_pair[1]))

        # D3d(=4C3,6C2) RECURSIVE
        d3d_pg = PointGroup.from_string("D3d")
        for axis_pair in d3_axis_pairs:
            pg_list.extend(_decompose_point_group(d3d_pg, axis_pair[0], axis_pair[1]))
        # D2h(=6C2) RECURSIVE
        d2h_pg = PointGroup.from_string("D2h")
        for axis_pair in d2_axis_pairs:
            pg_list.extend(_decompose_point_group(d2h_pg, axis_pair[0], axis_pair[1]))

    return pg_list

_decompose_I_family 

_decompose_I_family(subfamily: str, paxis: array, saxis: array, taxis: array) -> list[PointGroupResult]

Decompose icosahedral point groups into symmetry-consistent subgroups.

Generates subgroup candidates of the I family (I, I_h), including all cyclic and dihedral subgroups derived from icosahedral symmetry.

Parameters:

  • subfamily (str or None) –

    Subfamily label ("h" for I_h, or None for I).

  • paxis ((ndarray, shape(3))) –

    Principal C₅ axis.

  • saxis ((ndarray, shape(3))) –

    Secondary C₂ axis.

  • taxis ((ndarray, shape(3))) –

    Orthogonal axis completing the reference frame.

Returns:

  • list[PointGroupResult] –

    List of candidate point groups with associated orientations.

Notes
  • Includes:
    • C₅ axes (6), C₃ axes (10), C₂ axes (15)
    • Subgroups: D₅, D₃, D₂, T
  • Axes are constructed from icosahedral geometry using the golden ratio.
  • Subgroup axes are selected by enforcing orthogonality constraints.
  • For I_h:
    • Adds inversion, mirror planes, and improper rotations (Ci, S₁₀ (6), S₆ (10))
    • Includes recursive decomposition into D₂h, D₃d, D₅d
Source code in minimalsym/core/pg_decompose.py 
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
def _decompose_I_family(subfamily:str, paxis:np.array, saxis:np.array, taxis:np.array) -> list[PointGroupResult]:
    """
    Decompose icosahedral point groups into symmetry-consistent subgroups.

    Generates subgroup candidates of the I family (I, I_h), including all
    cyclic and dihedral subgroups derived from icosahedral symmetry.

    Parameters
    ----------
    subfamily : str or None
        Subfamily label ("h" for I_h, or None for I).
    paxis : ndarray, shape (3,)
        Principal C₅ axis.
    saxis : ndarray, shape (3,)
        Secondary C₂ axis.
    taxis : ndarray, shape (3,)
        Orthogonal axis completing the reference frame.

    Returns
    -------
    list[PointGroupResult]
        List of candidate point groups with associated orientations.

    Notes
    -----
    - Includes:
        * C₅ axes (6), C₃ axes (10), C₂ axes (15)
        * Subgroups: D₅, D₃, D₂, T
    - Axes are constructed from icosahedral geometry using the golden ratio.
    - Subgroup axes are selected by enforcing orthogonality constraints.
    - For I_h:
        * Adds inversion, mirror planes, and improper rotations (Ci, S₁₀ (6), S₆ (10))
        * Includes recursive decomposition into D₂h, D₃d, D₅d
    """
    pg_list = []
    zeros = np.zeros(3)

    # Icosahedral: paxis = C5 axis, saxis = C2 axis.

    # I
    if subfamily is not None:
        pg_list.append(PointGroupResult('I', paxis, saxis))

    phi = (1 + np.sqrt(5)) / 2
    rphi = 1/phi

    # 6 C5
    raw_dirs_faces = np.array([
        [0,  1,  phi],
        [0, -1,  phi],
        [1,  phi, 0],
        [-1, phi, 0],
        [phi, 0, 1],
        [-phi, 0, 1],
    ])

    B = np.column_stack((saxis, taxis, paxis))
    raw_dirs_faces = (B @ raw_dirs_faces.T).T

    raw_dirs_faces = np.array([normalize(raw_axis) for raw_axis in raw_dirs_faces])

    theta = np.arccos(abs(np.dot(paxis, raw_dirs_faces[1])))
    R_align = rotation_matrix(saxis, -theta)
    c5_axes = transform(raw_dirs_faces, R_align)

    for c5_axis in c5_axes:
        pg_list.append(PointGroupResult("C5", c5_axis, zeros))

    # 15 C2
    c2_axes = []
    base_c2_axes = generate_cyclic_axes(c5_axes[-1], saxis, 5, 3)

    for c2_axis in base_c2_axes:
        c2_axes.extend(generate_cyclic_axes(paxis, c2_axis, 5))

    c2_axes = np.array(c2_axes)

    for c2_axis in c2_axes:
        pg_list.append(PointGroupResult("C2", c2_axis, zeros))

    # 10 C3
    c3_axes = []

    base_c3_axes = [normalize(saxis + base_c2_axes[1] - c2_axes[3])]
    base_c3_axes = generate_cyclic_axes(c5_axes[-1], base_c3_axes[0], 5, 2)
    base_c3_axes = np.array(base_c3_axes)

    for c3_axis in base_c3_axes:
        c3_axes.extend(generate_cyclic_axes(paxis, c3_axis, 5))

    c3_axes = np.array(c3_axes)

    for c3_axis in c3_axes:
        pg_list.append(PointGroupResult("C3", c3_axis, zeros))

    # Combinations
    # D2(=15C2)
    ortho_c2_axes = []
    global_ortho_list=[]
    for ii in range(15):
        i_axis = c2_axes[ii]
        ortho_list=[]
        for jj in range(15):
            if ii ==  jj: continue
            j_axis = c2_axes[jj]
            if float_isclose(np.dot(i_axis, j_axis), 0.):
                ortho_list.append(jj)
        ortho_c2_axes.append([i_axis])
        for ortho_id in ortho_list:
            ortho_c2_axes[ii].append(c2_axes[ortho_id])
        ortho_list.append(ii)
        global_ortho_list.append(ortho_list)

    for axis_pair in ortho_c2_axes:
        pg_list.append(PointGroupResult("D2", axis_pair[0], axis_pair[1]))

    # T(=15C2)
    t_axis_pairs=[]
    present_c2_axes = set()
    for ii, axes in enumerate(ortho_c2_axes):
        if global_ortho_list[ii][0] not in present_c2_axes:
            t_axis_pairs.append(axes)
            for id in global_ortho_list[ii]:
                present_c2_axes.add(id)

    for axis_pair in t_axis_pairs:
        pg_list.append(PointGroupResult("T", axis_pair[0], axis_pair[1]))

    # D5(=6C5,15C2)
    d5_axis_pairs=[]
    for c5_axis in c5_axes:
        for c2_axis in c2_axes:
            if float_isclose(np.dot(c5_axis, c2_axis), 0.):
                d5_axis_pairs.append((c5_axis, c2_axis))
                break
    for axis_pair in d5_axis_pairs:
        pg_list.append(PointGroupResult("D5", axis_pair[0], axis_pair[1]))

    # D3(=10C3,15C2)
    d3_axis_pairs=[]
    for c3_axis in c3_axes:
        for c2_axis in c2_axes:
            if float_isclose(np.dot(c3_axis, c2_axis), 0.):
                d3_axis_pairs.append((c3_axis, c2_axis))
                break
    for axis_pair in d3_axis_pairs:
        pg_list.append(PointGroupResult("D3", axis_pair[0], axis_pair[1]))

    if subfamily == 'h':
        # Ci
        pg_list.append(PointGroupResult("Ci", paxis, saxis))
        # 15Cs(v for C5, C2, C3. h for C2)
        for c2_axis in c2_axes:
            pg_list.append(PointGroupResult("Cs", c2_axis, zeros))
        # 6S10 (same as C5 axes)
        for c5_axis in c5_axes:
            pg_list.append(PointGroupResult("S10", c5_axis, zeros))
        # 10S6 (same as C3 axes)
        for c3_axis in c3_axes:
            pg_list.append(PointGroupResult("S6", c3_axis, zeros))

        # Combinations
        # D2h(=15C2) RECURSIVE
        d2h_pg = PointGroup.from_string("D2h")
        for axis_pair in ortho_c2_axes:
            pg_list.extend(_decompose_point_group(d2h_pg, axis_pair[0], axis_pair[1]))
        # Th(=15C2)
        for axis_pair in t_axis_pairs:
            pg_list.append(PointGroupResult("Th", axis_pair[0], axis_pair[1]))
        # D5d(=6C5,15C2) RECURSIVE
        d5d_pg = PointGroup.from_string("D5d")
        for axis_pair in d5_axis_pairs:
            pg_list.extend(_decompose_point_group(d5d_pg, axis_pair[0], axis_pair[1]))
        # D3d(=10C3,15C2) RECURSIVE
        d3d_pg = PointGroup.from_string("D3d")
        for axis_pair in d3_axis_pairs:
            pg_list.extend(_decompose_point_group(d3d_pg, axis_pair[0], axis_pair[1]))

    return pg_list

_decompose_point_group 

_decompose_point_group(pg: PointGroup, paxis: array = array([0.0, 0.0, 1.0]), saxis: array = array([1.0, 0.0, 0.0])) -> list[PointGroupResult]

Decompose a point group into its symmetry elements represented as PointGroupResult objects.

Parameters:

  • pg (PointGroup) –

    Point group to decompose

  • paxis (array, default: array([0.0, 0.0, 1.0]) ) –

    Principal axis of point group

  • saxis (array, default: array([1.0, 0.0, 0.0]) ) –

    Secondary axis of point group

Returns:

Source code in minimalsym/core/pg_decompose.py 
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
def _decompose_point_group(pg: PointGroup, paxis:np.array = np.array([0., 0., 1.0]), saxis:np.array = np.array([1.0, 0., 0.])) -> list[PointGroupResult]:
    """
    Decompose a point group into its symmetry elements represented
    as PointGroupResult objects.

    Parameters
    ----------
    pg: PointGroup
        Point group to decompose
    paxis: np.array
        Principal axis of point group
    saxis: np.array
        Secondary axis of point group
    Returns
    -------
    list[PointGroupResult]
        Internal symmetry elements of given point group
    """
    family = pg.family
    n = pg.n
    subfamily = pg.subfamily

    taxis = normalize(np.cross(paxis, saxis))

    pg_list = [PointGroupResult(pg=pg.str, paxis=paxis, saxis=saxis)]

    if subfamily == 's' or subfamily == 'i':
        return pg_list

    if family == 'S':
        # Cn/2 RECURSIVE
        cn2_pg = PointGroup.from_string(f"C{n // 2}")
        pg_list.extend(_decompose_point_group(cn2_pg, paxis, saxis))

        if n == 2:
            pg_list.append(PointGroupResult("Ci", paxis, saxis))

    if family == 'C' or family == 'D':
        pg_list.extend(_decompose_cyclic(family, n, subfamily, paxis, saxis, taxis))

    elif family == 'T':
        pg_list.extend(_decompose_T_family(subfamily, paxis, saxis, taxis))

    elif family == 'O':
        pg_list.extend(_decompose_O_family(subfamily, paxis, saxis, taxis))

    elif family == 'I':
        pg_list.extend(_decompose_I_family(subfamily, paxis, saxis, taxis))

    return pg_list

_check_O_point_group 

_check_O_point_group(mol, invertable: bool) -> list[PointGroupResult]

Checks if a given mol has O family symmetry. If it does returns the decomposition of the found O point group.

Parameters:

  • mol –

    Molecule to check for O family symmetry.

  • invertable (bool) –

    True if the molecule has an inversion center

Returns:

  • list[PointGroupResult] –

    The internal decompositon of the found O point group. Empty if no O point group was found.

Source code in minimalsym/core/pg_decompose.py 
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
def _check_O_point_group(mol, invertable:bool) -> list[PointGroupResult]:
    """
    Checks if a given mol has O family symmetry.
    If it does returns the decomposition of the found O point group.

    Parameters
    ----------
    mol: ase.Atoms
        Molecule to check for O family symmetry.
    invertable: bool
        True if the molecule has an inversion center

    Returns
    -------
    list[PointGroupResult]
        The internal decompositon of the found O point group.
        Empty if no O point group was found.
    """
    try:
        c4_axes = _find_C4s_for_Oh(mol)
        paxis = c4_axes[0]
        saxis = c4_axes[1]
        if invertable:
            pg_str = "Oh"
        else:
            pg_str = "O"
        return _decompose_point_group(PointGroup.from_string(pg_str), paxis, saxis)
    except RuntimeError:
        return []

_check_general_point_group 

_check_general_point_group(mol) -> list[PointGroupResult]

Finds a general symmetry PointGroupResult for a molecule and returns the internal decomposition of the found point group.

Parameters:

  • mol –

    Molecule to get general point group from.

Returns:

  • list[PointGroupResult] –

    The internal decompositon of the found point group.

Source code in minimalsym/core/pg_decompose.py 
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
def _check_general_point_group(mol) -> list[PointGroupResult]:
    """
    Finds a general symmetry PointGroupResult for a molecule and
    returns the internal decomposition of the found point group.

    Parameters
    ----------
    mol: ase.Atoms
        Molecule to get general point group from.

    Returns
    -------
    list[PointGroupResult]
        The internal decompositon of the found point group.
    """
    general_pgr = _classify_general(mol, mol.positions, mol.get_masses(), mol.info["geom_tol"])
    return _decompose_point_group(PointGroup.from_string(general_pgr.pg), general_pgr.paxis, general_pgr.saxis)

_unique_point_group 

_unique_point_group(pg_list: list[PointGroupResult]) -> list[PointGroupResult]

Returns a list of PointGroupResult without repetitions.

Parameters:

  • pg_list (list[PointGroupResult]) –

    PointGroupResult list to be deduplicated.

Returns:

Source code in minimalsym/core/pg_decompose.py 
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
def _unique_point_group(pg_list: list[PointGroupResult]) -> list[PointGroupResult]:
    """
    Returns a list of PointGroupResult without repetitions.

    Parameters
    ----------
    pg_list : list[PointGroupResult]
        PointGroupResult list to be deduplicated.

    Returns
    -------
    list[PointGroupResult]
        Deduplicated PointGroupResult list.
    """
    unique_pg_list_per_type = {}
    unique_index = []
    for idx, pgr in enumerate(pg_list):
        pg_str = pgr.pg.lower()
        paxis = pgr.paxis
        saxis = pgr.saxis
        pgr_is_unique = True

        if pg_str not in unique_pg_list_per_type:
            unique_pg_list_per_type[pg_str] = []

        ignore_saxis = False
        if pg_str[0] == 'c' or pg_str[0] == 'd':
            ignore_saxis = ((pg_str[0] == 'c' and pg_str[-1] != 'v') or (pg_str[1] == '0'))

        for existent_pgr in unique_pg_list_per_type[pg_str]:
            if pg_str == "ci":
                pgr_is_unique = False

            if ignore_saxis:
                if issame_axis(paxis, existent_pgr.paxis):
                    pgr_is_unique = False
                    break
                continue

            if issame_axis(paxis, existent_pgr.paxis) and issame_axis(saxis, existent_pgr.saxis):
                pgr_is_unique = False
                break

        if pgr_is_unique:
            unique_pg_list_per_type[pg_str].append(pgr)
            unique_index.append(idx)

    unique_pg_list = []
    for idx in unique_index:
        unique_pg_list.append(pg_list[idx])
    return unique_pg_list

_find_pg_score 

_find_pg_score(pg_str: str) -> int

Given a string of the Schoenflies symbol of a point group, returns a score from the number of symmetry operations it possess.

Parameters:

  • pg_str (str) –

    Schoenflies symbol of a point group.

Returns:

  • int –

    Score.

Source code in minimalsym/core/pg_decompose.py 
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
def _find_pg_score(pg_str: str) -> int:
    """
    Given a string of the Schoenflies symbol of a point group,
    returns a score from the number of symmetry operations it possess.

    Parameters
    ----------
    pg_str: str
        Schoenflies symbol of a point group.

    Returns
    -------
    : int
        Score.
    """
    pg = PointGroup.from_string(pg_str)
    family = pg.family
    n = pg.n
    sub = pg.subfamily

    # Linear groups
    if n == 0:
        if family == 'C':   # C∞v
            return float('inf')
        elif family == 'D': # D∞h
            return float('inf')

    # Simple groups
    if family == 'C':
        if sub is None:
            return n
        elif sub in ('v', 'h'):
            return 2 * n

    elif family == 'D':
        if sub is None:
            return 2 * n
        elif sub in ('h', 'd'):
            return 4 * n

    elif family == 'S':
        return n

    elif family == 'T':
        if sub is None:
            return 12
        elif sub in ('d', 'h'):
            return 24.1

    elif family == 'O':
        if sub is None:
            return 24.2
        elif sub == 'h':
            return 48

    elif family == 'I':
        if sub is None:
            return 60
        elif sub == 'h':
            return 120

    # Special cases
    if pg.str == 'C1':
        return 1
    if pg.str in ('Cs', 'Ci'):
        return 2

    return 0  # fallback