Wigner-D matrix builders

wignerd module

sphecerix.wignerd.tesseral_wigner_D(l, Robj)[source]

Produce the Wigner D-matrix for tesseral spherical harmonics for a rotation

Parameters:

l (int) – Order of the spherical harmonics
Robj (scipy.spatial.transform.Rotation) – Rotation in \(\mathbb{R}^{3}\)

Returns:

D – Real-valued Wigner-D matrix with dimensions \((2l+1) \times (2l+1)\)

Return type:

numpy.ndarray

Raises:

TypeError – If the Robj object is not of type scipy.spatial.transform.R.

Examples

>>> from sphecerix import tesseral_wigner_D
... from scipy.spatial.transform import Rotation as R
... import numpy as np
...
... # build rotation axis and set angle
... axis = np.ones(3) / np.sqrt(3)
... angle = np.pi
... Robj = R.from_rotvec(axis * angle)
...
... # construct tesseral Wigner D matrix
... D = tesseral_wigner_D(2, Robj)
... print(D)
[[ 5.55555556e-01  2.22222222e-01  7.69800359e-01  2.22222222e-01
   1.89744731e-16]
 [ 2.22222222e-01  5.55555556e-01 -3.84900179e-01  2.22222222e-01
   6.66666667e-01]
 [ 7.69800359e-01 -3.84900179e-01 -3.33333333e-01 -3.84900179e-01
   5.42310034e-16]
 [ 2.22222222e-01  2.22222222e-01 -3.84900179e-01  5.55555556e-01
  -6.66666667e-01]
 [-1.01229242e-16  6.66666667e-01 -4.65653372e-16 -6.66666667e-01
  -3.33333333e-01]]

Construct the Wigner-D matrix for the tesseral p-orbitals for a rotation around the \(\frac{1}{\sqrt{3}}(1,1,1)\) axis by an angle \(\pi\).

sphecerix.wignerd.tesseral_wigner_D_improper(l, Robj)[source]

Produce the Wigner D-matrix for tesseral spherical harmonics under an improper rotation

Parameters:

l (int) – Order of the spherical harmonics
Robj (scipy.spatial.transform.Rotation) – Rotation in \(\mathbb{R}^{3}\)

Returns:

D – Real-valued Wigner-D matrix with dimensions \((2l+1) \times (2l+1)\)

Return type:

numpy.ndarray

Raises:

TypeError – If the Robj object is not of type scipy.spatial.transform.R.

Examples

>>> from sphecerix import tesseral_wigner_D_improper
... from scipy.spatial.transform import Rotation as R
... import numpy as np
...
... # construct (improper) rotation vector
... axis = np.array([1,0,0])
... Robj = R.from_rotvec(axis * np.pi / 2)
...
... # construct wigner D matrix
... D = tesseral_wigner_D_improper(1, Robj)
...
... print(D)
[[ 7.49879891e-33 -1.00000000e+00  1.83697020e-16]
 [ 1.00000000e+00 -2.24963967e-32 -1.83697020e-16]
 [-1.83697020e-16 -1.83697020e-16 -1.00000000e+00]]

Construct the Wigner-D matrix for the tesseral p-orbitals for an improper rotation by 90 degrees around the cartesian x-axis.

sphecerix.wignerd.tesseral_wigner_D_mirror(l, normal)[source]

Produce the Wigner D-matrix for tesseral spherical harmonics for a mirror operation

Parameters:

l (int) – Order of the spherical harmonics
normal (np.array) – Normal vector

Returns:

D – Real-valued Wigner-D matrix with dimensions \((2l+1) \times (2l+1)\)

Return type:

numpy.ndarray

Examples

>>> from sphecerix import tesseral_wigner_D_mirror
... import numpy as np
...
... # construct mirror normal vector
... normal = np.array([-1,1,0]) / np.sqrt(2)
...
... # construct wigner D matrix
... D = tesseral_wigner_D_mirror(1, normal)
...
... print(D)
[[-2.83276945e-16 -0.00000000e+00  1.00000000e+00]
 [ 1.22464680e-16  1.00000000e+00  3.46914204e-32]
 [ 1.00000000e+00 -1.22464680e-16  2.83276945e-16]]

Construct the Wigner-D matrix for the tesseral p-orbitals for a mirror operation with the mirror plane corresponding to the xz plane rotated around the z-axis by 45 degrees.

sphecerix.wignerd.wigner_D(l, Robj)[source]

Produce Wigner D-matrix for canonical spherical harmonics

Parameters:

l (int) – Order of the spherical harmonics
Robj (scipy.spatial.transform.Rotation) – Rotation in \(\mathbb{R}^{3}\)

Returns:

D – Complex-valued Wigner-D matrix with dimensions \((2l+1) \times (2l+1)\)

Return type:

numpy.ndarray

Raises:

TypeError – If the Robj object is not of type scipy.spatial.transform.R.

Examples

>>> from sphecerix import wigner_D
... from scipy.spatial.transform import Rotation as R
... import numpy as np
...
... # build rotation axis and set angle
... axis = np.ones(3) / np.sqrt(3)
... angle = np.pi
... Robj = R.from_rotvec(axis * angle)
...
... # construct tesseral Wigner D matrix
... D = wigner_D(2, Robj)
... print(D)
[[ 1.11111111e-01-1.45486986e-16j -2.22222222e-01+2.22222222e-01j
  -3.29266657e-16-5.44331054e-01j  4.44444444e-01+4.44444444e-01j
  -4.44444444e-01-4.42577444e-17j]
 [-2.22222222e-01-2.22222222e-01j  5.55555556e-01-3.33066907e-16j
  -2.72165527e-01+2.72165527e-01j -5.55111512e-17+2.22222222e-01j
  -4.44444444e-01-4.44444444e-01j]
 [ 3.83471103e-16+5.44331054e-01j -2.72165527e-01-2.72165527e-01j
  -3.33333333e-01+0.00000000e+00j  2.72165527e-01-2.72165527e-01j
   3.83471103e-16-5.44331054e-01j]
 [ 4.44444444e-01-4.44444444e-01j -5.55111512e-17-2.22222222e-01j
   2.72165527e-01+2.72165527e-01j  5.55555556e-01+3.33066907e-16j
   2.22222222e-01-2.22222222e-01j]
 [-4.44444444e-01+4.42577444e-17j -4.44444444e-01+4.44444444e-01j
  -3.29266657e-16+5.44331054e-01j  2.22222222e-01+2.22222222e-01j
   1.11111111e-01+1.45486986e-16j]]

Construct the Wigner-D matrix for the canonical p-orbitals for a rotation around the \(\frac{1}{\sqrt{3}}(1,1,1)\) axis by an angle \(\pi\).