def get_cn(n):
r = np.clip(np.mod(n - 1, 2), 0, 1)
l = (n - r - 1) // 2
ndim = 1 + l + r
multiplicity = [1] + [2] * l + [1] * r
operations = [Identity(label='E')]
for i in range(n // 2):
j = gcd(i + 1, n)
up = (i + 1) // j
down = n // j
label = 'C{}'.format(down) if up == 1 else 'C^{}_{}'.format(up, down)
operations.append(
Rotation(label=label, axis=[0, 0, 1], order=down, exp=up))
ir_data = {'A': pd.Series([1] * ndim)}
if r == 1:
ir_data.update({'B': pd.Series([(-1)**k for k in range(ndim)])})
for i in range(l):
label = 'E{}'.format(i + 1) if l > 1 else 'E'
ir_data.update({
label:
pd.Series(
[2] +
[real_radical((i + 1) * (k + 1), n) for k in range(ndim - 1)])
})
if n == 1:
rotations = ['A', 'A', 'A']
translations = ['A', 'A', 'A']
elif n == 2:
rotations = ['B', 'B', 'A']
translations = ['B', 'B', 'A']
elif n == 3 or n == 4:
rotations = ['E', 'E', 'A']
translations = ['E', 'E', 'A']
else:
rotations = ['E1', 'E1', 'A']
translations = ['E1', 'E1', 'A']
name = 'C{}'.format(n)
#return name, operations, ir_data, rotations, translations, multiplicity
return CharTable(
'C{}'.format(n),
operations,
ir_data,
rotations=rotations, # x, y, z
translations=translations, # Rx, Ry, Rz
multiplicities=multiplicity)
def get_dnd(n):
ir_data_new = {}
if n < 2:
raise Exception('Group not valid')
if np.mod(n, 2) == 0:
# for even n only
ir_data = get_sn(2 * n)
operations_new = ir_data.operations
ir_data_new['A1'] = pd.Series(list(ir_data['A']) + [1, 1])
ir_data_new['A2'] = pd.Series(list(ir_data['A']) + [-1, -1])
ir_data_new['B1'] = pd.Series(list(ir_data['B']) + [1, -1])
ir_data_new['B2'] = pd.Series(list(ir_data['B']) + [-1, 1])
operations_new += [
Rotation(label="C2'", axis=[1, 0, 0], order=2),
Reflection(label='sd', axis=[1, 0, 0])
]
for data in ir_data.keys():
if data.startswith('E'):
ir_data_new[data] = pd.Series(list(ir_data[data]) + [0, 0])
multiplicites = ir_data.multiplicities + [n, n]
if n == 2:
rotations = ['E', 'E', 'A2']
translations = ['E', 'E', 'B2']
else:
rotations = ['E{}'.format(n - 1), 'E{}'.format(n - 1), 'A2']
translations = ['E1', 'E1', 'B2']
else:
ir_data = get_dn(n)
operations_new = ir_data.operations
ir_data_new_u = {}
ir_data_new["A1g"] = pd.Series(list(ir_data['A1']) + [1, 1])
ir_data_new["A2g"] = pd.Series(list(ir_data['A2']) + [1, -1])
ir_data_new_u["A1u"] = pd.Series(list(ir_data['A1']) + [-1, -1])
ir_data_new_u["A2u"] = pd.Series(list(ir_data['A2']) + [-1, 1])
multiplicites = [1] + [2] * ((n - 1) // 2) + [n, 1] + [2] * (
(n - 1) // 2) + [n]
operations_new += [
Inversion(label='i'),
Reflection(label='sd', axis=[1, 0, 0])
]
for data in ir_data.keys():
if data.startswith('E'):
ir_data_new[data +
"g"] = pd.Series(list(ir_data[data]) + [2, 0])
ir_data_new_u[data +
"u"] = pd.Series(list(ir_data[data]) + [-2, 0])
ir_data_new.update(ir_data_new_u)
for i in range((n - 1) // 2):
j = gcd(2 * i + 1, 2 * n)
up = (2 * i + 1) // j
down = 2 * n // j
if up == 1:
operations_new += [
ImproperRotation(label='S{}'.format(down),
axis=[0, 0, 1],
order=down,
exp=up)
]
else:
operations_new += [
ImproperRotation(label='S^{}_{}'.format(up, down),
axis=[0, 0, 1],
order=down,
exp=up)
]
for data in ir_data_new.keys():
sign = np.sign(list(ir_data_new[data])[(n - 1) // 2 + 2])
element = list(ir_data_new[data])[1 + i] * sign
if data.startswith('E'):
try:
k = int(data[1]) - 1
except ValueError:
k = 0
#element = [real_radical((k + 1) * (m + 1), n) for m in range(0, n//2)][-i-1] * sign
element = real_radical(
(k + 1) * (n // 2 + i + 1), n) * sign
ir_data_new[data] = pd.Series(
list(ir_data_new[data]) + [element])
# swap columns for convention
l = 2 + n // 2 + 1
for kk in range(1):
value = operations_new.pop(l)
operations_new.append(value)
for data in ir_data_new.keys():
row = list(ir_data_new[data])
value = row.pop(l)
row.append(value)
ir_data_new[data] = pd.Series(row)
if n == 1:
rotations = ["A1g", "A2g", "A2g"]
translations = ["A1'", "A2'", "A2''"]
elif n == 3:
rotations = ["Eg", "Eg", "A2g"]
translations = ["Eu", "Eu", "A2u"]
else:
rotations = ["E1g", "E1g", "A2g"]
translations = ["E1u", "E1u", "A2u"]
return CharTable(
'D{}d'.format(n),
operations_new,
ir_data_new,
rotations=rotations, # x, y, z
translations=translations, # Rx, Ry, Rz
multiplicities=multiplicites)
def get_dnh(n):
ir_data = get_dn(n)
operations_new = ir_data.operations
ir_data_new = {}
ir_data_new_u = {}
if n == 2:
return CharTable('D2h', [
Identity(label='E'),
Rotation(label='C2', axis=[0, 0, 1], order=2),
Rotation(label="C2'", axis=[0, 1, 0]),
Rotation(label="C2''", axis=[1, 0, 0]),
Inversion(label='i'),
Reflection(label='sh', axis=[0, 0, 1]),
Reflection(label='sv', axis=[0, 1, 0]),
Reflection(label='sd', axis=[1, 0, 0])
], {
'Ag': pd.Series([+1, +1, +1, +1, +1, +1, +1, +1]),
'B1g': pd.Series([+1, +1, -1, -1, +1, +1, -1, -1]),
'B2g': pd.Series([+1, -1, -1, +1, +1, -1, +1, -1]),
'B3g': pd.Series([+1, -1, +1, -1, +1, -1, -1, +1]),
'Au': pd.Series([+1, +1, +1, +1, -1, -1, -1, -1]),
'B1u': pd.Series([+1, +1, -1, -1, -1, -1, +1, +1]),
'B2u': pd.Series([+1, -1, -1, +1, -1, +1, -1, +1]),
'B3u': pd.Series([+1, -1, +1, -1, -1, +1, +1, -1]),
},
rotations=['B3g', 'B2g', 'B1g'],
translations=['B3u', 'B2u', 'B1u'],
multiplicities=[1, 1, 1, 1, 1, 1, 1, 1])
if np.mod(n, 2) == 0:
# for even n only
ir_data_new['A1g'] = pd.Series(list(ir_data['A1']) + [1, 1, 1, 1])
ir_data_new_u['A1u'] = pd.Series(
list(ir_data['A1']) + [-1, -1, -1, -1])
ir_data_new['A2g'] = pd.Series(list(ir_data['A2']) + [1, 1, -1, -1])
ir_data_new_u['A2u'] = pd.Series(list(ir_data['A2']) + [-1, -1, 1, 1])
l = (-1)**(n // 2)
ir_data_new['B1g'] = pd.Series(
list(ir_data['B1']) + [1, 1 * l, 1 * l, -1 * l])
ir_data_new_u['B1u'] = pd.Series(
list(ir_data['B1']) + [-1, -1 * l, -1 * l, 1 * l])
ir_data_new['B2g'] = pd.Series(
list(ir_data['B2']) + [1, 1 * l, -1 * l, 1 * l])
ir_data_new_u['B2u'] = pd.Series(
list(ir_data['B2']) + [-1, -1 * l, 1 * l, -1 * l])
operations_new += [
Inversion(label='i'),
Reflection(label='sh', axis=[0, 0, 1]),
Reflection(label='s_v', axis=[0, 1, 0]),
Reflection(label='sd', axis=[1, 0, 0])
]
for data in ir_data.keys():
if data.startswith('E'):
if len(data[1:]) == 0:
k = int((np.mod(1, 2) - 0.5) * 4)
else:
k = int((np.mod(int(data[1:]), 2) - 0.5) * 4)
# print(data+'u', list(ir_data[data]))
ir_data_new[data + 'g'] = pd.Series(
list(ir_data[data]) + [2, -k, 0, 0])
ir_data_new_u[data + 'u'] = pd.Series(
list(ir_data[data]) + [-2, k, 0, 0])
ir_data_new.update(ir_data_new_u)
for i in range((n - 1) // 2):
j = gcd(i + 1, n)
up = (i + 1) // j
down = n // j
if up == 1:
operations_new += [
ImproperRotation(label='S{}'.format(down),
axis=[0, 0, 1],
order=down,
exp=up)
]
else:
operations_new += [
ImproperRotation(label='S^{}_{}'.format(up, down),
axis=[0, 0, 1],
order=down,
exp=up)
]
for data in ir_data_new.keys():
#print(1+i, (n-1)//2+5)
element = list(ir_data_new[data])[1 + i] * np.sign(
list(ir_data_new[data])[(n - 1) // 2 + 5])
#print(data, len(ir_data_new[data]), element)
ir_data_new[data] = pd.Series(
list(ir_data_new[data]) + [element])
#print('----')
# swap columns for convention
l = 2 + n // 2 + 2
for kk in range(3):
value = operations_new.pop(l)
operations_new.append(value)
for data in ir_data_new.keys():
row = list(ir_data_new[data])
value = row.pop(l)
row.append(value)
ir_data_new[data] = pd.Series(row)
#print(((n-1)//2))
multiplicites = [1] + [2] * (
(n - 1) // 2) + [1] + [n // 2, n // 2] + [1] + [2] * (
(n - 1) // 2) + [1, n // 2, n // 2]
print(multiplicites)
if n == 2:
rotations = ['B3g', 'B2g', 'B1g']
translations = ['B3u', 'B2u', 'B1u']
elif n == 4:
rotations = ['Eg', 'Eg', 'A2g']
translations = ['Eu', 'Eu', 'A2u']
else:
rotations = ['E1g', 'E1g', 'A2g']
translations = ['E1u', 'E1u', 'A2u']
else:
ir_data_new["A1'"] = pd.Series(list(ir_data['A1']) + [1, 1])
ir_data_new["A1''"] = pd.Series(list(ir_data['A1']) + [-1, -1])
ir_data_new["A2'"] = pd.Series(list(ir_data['A2']) + [1, -1])
ir_data_new["A2''"] = pd.Series(list(ir_data['A2']) + [-1, 1])
multiplicites = [1] + [2] * ((n - 1) // 2) + [n, 1] + [2] * (
(n - 1) // 2) + [n]
operations_new += [
Reflection(label='sh', axis=[0, 0, 1]),
Reflection(label='sv', axis=[0, 1, 0])
]
for data in ir_data.keys():
if data.startswith('E'):
ir_data_new[data +
"'"] = pd.Series(list(ir_data[data]) + [2, 0])
ir_data_new[data +
"''"] = pd.Series(list(ir_data[data]) + [-2, 0])
for i in range((n - 1) // 2):
j = gcd(2 * i + 1, n)
up = (2 * i + 1) // j
down = n // j
if up == 1:
operations_new += [
ImproperRotation(label='S_{}'.format(down),
axis=[0, 0, 1],
order=down,
exp=up)
]
else:
operations_new += [
ImproperRotation(label='S^{}_{}'.format(up, down),
axis=[0, 0, 1],
order=down,
exp=up)
]
for data in ir_data_new.keys():
sign = np.sign(list(ir_data_new[data])[(n - 1) // 2 + 2])
element = list(ir_data_new[data])[1 + i] * sign
if data.startswith('E'):
try:
k = int(data[1]) - 1
except ValueError:
k = 0
# print('E{}'.format(k+1), [real_radical((k + 1) * (m + 1), n) for m in range(0, 7)])
# element = [real_radical((k + 1) * (m + 1), n) for m in range(0, n,2)][i] * sign
element = real_radical((k + 1) * (2 * i + 1), n) * sign
ir_data_new[data] = pd.Series(
list(ir_data_new[data]) + [element])
# swap columns for convention
l = 2 + n // 2 + 1
for kk in range(1):
value = operations_new.pop(l)
operations_new.append(value)
for data in ir_data_new.keys():
row = list(ir_data_new[data])
value = row.pop(l)
row.append(value)
ir_data_new[data] = pd.Series(row)
if n == 1:
rotations = ["A1''", "A2''", "A2'"]
translations = ["A1'", "A2'", "A2''"]
elif n == 3:
rotations = ["E''", "E''", "A2'"]
translations = ["E'", "E'", "A2''"]
else:
rotations = ["E1''", "E1''", "A2'"]
translations = ["E1'", "E1'", "A2''"]
return CharTable(
'D{}h'.format(n),
operations_new,
ir_data_new,
rotations=rotations, # x, y, z
translations=translations, # Rx, Ry, Rz
multiplicities=multiplicites)
def get_sn(n):
if np.mod(n, 2) != 0:
raise Exception('Order of Sn group must have even')
if np.mod(n, 4) == 0:
r = np.clip(np.mod(n - 1, 2), 0, 1)
l = (n - r - 1) // 2
ndim = 1 + l + r
multiplicity = [1] + [2] * l + [1] * r
operations = [Identity(label='E')]
for i in range(n // 2):
j = gcd(i + 1, n)
up = (i + 1) // j
down = n // j
if np.mod(i, 2) == 0:
label = 'S{}'.format(down) if up == 1 else 'S^{}_{}'.format(
up, down)
operations.append(
ImproperRotation(label=label,
axis=[0, 0, 1],
order=down,
exp=up))
else:
label = 'C{}'.format(down) if up == 1 else 'C^{}_{}'.format(
up, down)
operations.append(
Rotation(label=label, axis=[0, 0, 1], order=down, exp=up))
ir_data = {'A': pd.Series([1] * ndim)}
if r == 1:
ir_data.update({'B': pd.Series([(-1)**k for k in range(ndim)])})
for i in range(l):
label = 'E{}'.format(i + 1) if l > 1 else 'E'
ir_data.update({
label:
pd.Series([2] + [
real_radical((i + 1) * (k + 1), n) for k in range(ndim - 1)
])
})
if n == 4:
rotations = ['E', 'E', 'A']
translations = ['E', 'E', 'B']
else:
rotations = ['E1', 'E1', 'A']
translations = ['E{}'.format(l // 2), 'E{}'.format(l // 2), 'B']
else:
r = np.clip(np.mod(n - 1, 2), 0, 1)
l = (n - r - 1) // 2
ndim = 1 + l + r
multiplicity = [1] + [2] * l + [1] * r
operations = [Identity(label='E')]
for i in range(n // 2):
j = gcd(i + 1, n)
up = (i + 1) // j
down = n // j
if np.mod(i, 2) == 0:
if up == 1 and down == 2:
label = 'i'.format(down)
operations.append(Inversion(label=label))
else:
label = 'S{}'.format(
down) if up == 1 else 'S^{}_{}'.format(up, down)
operations.append(
ImproperRotation(label=label,
axis=[0, 0, 1],
order=down,
exp=up))
else:
label = 'C{}'.format(down) if up == 1 else 'C^{}_{}'.format(
up, down)
operations.append(
Rotation(label=label, axis=[0, 0, 1], order=down, exp=up))
ir_data = {'Ag': pd.Series([1] * ndim)}
traces = [1]
for i in range(n // 2):
if np.mod(i, 2) == 0:
traces.append(-1)
else:
traces.append(1)
for i in range(l // 2):
subtrace = np.roll([k + 1 for k in range(n // 4)], -i)
label = 'E{}g'.format(i + 1) if l > 2 else 'Eg'
ir_data.update({
label:
pd.Series(
[2] +
[real_radical(k, n) * (-1)**(k + 2 * i)
for k in subtrace] + [
real_radical(k, n) * (-1)**(k + 2 * i)
for k in subtrace[::-1]
] + [2])
})
ir_data.update({'Au': pd.Series(traces)})
for i in range(l // 2):
subtrace = np.roll([k + 1 for k in range(n // 4)], -i)
label = 'E{}u'.format(i + 1) if l > 2 else 'Eu'
# print(i, '-', [[2] , [(-1)**(m+1) for m, k in enumerate(subtrace)] , [(-1)**(m+l//2+1) for m, k in enumerate(subtrace[::-1])] , [-2]])
ir_data.update({
label:
pd.Series([2] + [
real_radical(k, n) * (-1)**(k + 2 * i) * (-1)**(m + 1)
for m, k in enumerate(subtrace)
] + [
real_radical(k, n) * (-1)**(k + 2 * i) *
(-1)**(m + l // 2 + 1)
for m, k in enumerate(subtrace[::-1])
] + [-2])
})
# reorder columns to follow convention
new_indexing = [0] + list(range(2, l + 1, 2)) + [ndim - 1] + list(
range(1, l + 1, 2))
operations = np.array(operations)[new_indexing].tolist()
multiplicity = np.array(multiplicity)[new_indexing].tolist()
for key in ir_data.keys():
ir_data[key] = ir_data[key][new_indexing]
if n == 2:
rotations = ['Ag', 'Ag', 'Ag']
translations = ['Au', 'Au', 'Au']
elif n == 6:
rotations = ['Eg', 'Eg', 'Ag']
translations = ['Eu', 'Eu', 'Au']
else:
rotations = ['E1g', 'E1g', 'Ag']
translations = ['E1u', 'E1u', 'Au']
return CharTable(
'S{}'.format(n),
operations,
ir_data,
rotations=rotations, # x, y, z
translations=translations, # Rx, Ry, Rz
multiplicities=multiplicity)
def get_cnh(n):
ir_data = get_cn(n)
operations_new = ir_data.operations
ir_data_new = {}
ir_data_new_u = {}
if np.mod(n, 2) == 0:
# for even n only
ir_data_new['Ag'] = pd.Series(list(ir_data['A']) + [1, 1])
ir_data_new_u['Au'] = pd.Series(list(ir_data['A']) + [-1, -1])
l = (-1)**(n // 2)
ir_data_new['Bg'] = pd.Series(list(ir_data['B']) + [1, 1 * l])
ir_data_new_u['Bu'] = pd.Series(list(ir_data['B']) + [-1, -1 * l])
operations_new += [
Inversion(label='i'),
Reflection(label='sh', axis=[0, 0, 1])
]
for data in ir_data.keys():
if data.startswith('E'):
if len(data[1:]) == 0:
k = int((np.mod(1, 2) - 0.5) * 4)
else:
k = int((np.mod(int(data[1:]), 2) - 0.5) * 4)
# print(data+'u', list(ir_data[data]))
ir_data_new[data +
'g'] = pd.Series(list(ir_data[data]) + [2, -k])
ir_data_new_u[data +
'u'] = pd.Series(list(ir_data[data]) + [-2, k])
ir_data_new.update(ir_data_new_u)
for i in range((n - 1) // 2):
j = gcd(i + 1, n)
up = (i + 1) // j
down = n // j
if up == 1:
operations_new += [
ImproperRotation(label='S_{}'.format(down),
axis=[0, 0, 1],
order=down,
exp=up)
]
else:
operations_new += [
ImproperRotation(label='S^{}_{}'.format(up, down),
axis=[0, 0, 1],
order=down,
exp=up)
]
for data in ir_data_new.keys():
element = list(ir_data_new[data])[1 + i] * np.sign(
list(ir_data_new[data])[(n - 1) // 2 + 3])
ir_data_new[data] = pd.Series(
list(ir_data_new[data]) + [element])
l = 2 + n // 2
# swap columns for convention
value = operations_new.pop(l)
operations_new.append(value)
for data in ir_data_new.keys():
row = list(ir_data_new[data])
value = row.pop(l)
row.append(value)
ir_data_new[data] = pd.Series(row)
multiplicites = [1] + [2] * ((n - 1) // 2) + [1, 1] + [2] * (
(n - 1) // 2) + [1]
if n == 2:
rotations = ['Bu', 'Bu', 'Au']
translations = ['Bg', 'Bg', 'Ag']
elif n == 4:
rotations = ['Eu', 'Eu', 'Au']
translations = ['Eg', 'Eg', 'Ag']
else:
rotations = ['E1u', 'E1u', 'Au']
translations = ['E1g', 'E1g', 'Ag']
else:
ir_data_new["A'"] = pd.Series(list(ir_data['A']) + [1])
ir_data_new["A''"] = pd.Series(list(ir_data['A']) + [-1])
multiplicites = [1] + [2] * ((n - 1) // 2) + [1] + [2] * ((n - 1) // 2)
operations_new += [Reflection(label='sh', axis=[0, 0, 1])]
for data in ir_data.keys():
if data.startswith('E'):
ir_data_new[data + "'"] = pd.Series(list(ir_data[data]) + [2])
ir_data_new[data +
"''"] = pd.Series(list(ir_data[data]) + [-2])
for i in range((n - 1) // 2):
j = gcd(2 * i + 1, n)
up = (2 * i + 1) // j
#print('----------------', n, i*2+1, ':', (2*i+1)//j, n//j)
down = n // j
if up == 1:
operations_new += [
ImproperRotation(label='S_{}'.format(down),
axis=[0, 0, 1],
order=down,
exp=up)
]
else:
operations_new += [
ImproperRotation(label='S^{}_{}'.format(up, down),
axis=[0, 0, 1],
order=down,
exp=up)
]
for l, data in enumerate(ir_data_new.keys()):
# real_radical((i + 1) * (k + 1), n)
sign = np.sign(list(ir_data_new[data])[(n - 1) // 2 + 1])
element = list(ir_data_new[data])[1 + i] * sign
if data.startswith('E'):
try:
k = int(data[1]) - 1
except ValueError:
k = 0
# print([real_radical((k + 1) * (m + 1), n) for m in range(0, n,2)])
# element = [real_radical((k + 1) * (m + 1), n) for m in range(0, n, 2)][i] * sign
element = real_radical((k + 1) * (2 * i + 1), n) * sign
ir_data_new[data] = pd.Series(
list(ir_data_new[data]) + [element])
if n == 1:
rotations = ["A'", "A'", "A''"]
translations = ["A''", "A''", "A'"]
elif n == 3:
rotations = ["E'", "E'", "A''"]
translations = ["E''", "E''", "A'"]
else:
rotations = ["E1'", "E1'", "A''"]
translations = ["E1''", "E1''", "A'"]
return CharTable(
'C{}h'.format(n),
operations_new,
ir_data_new,
rotations=rotations, # x, y, z
translations=translations, # Rx, Ry, Rz
multiplicities=multiplicites)