forte2.symmetry.sph_harm_utils#
Module Contents#
- forte2.symmetry.sph_harm_utils.sph_real_to_complex(l)#
Conversion matrix from real spherical harmonics to complex spherical harmonics for a given l.
- Parameters:
- lint
The angular momentum quantum number.
- Returns:
- NDArray
The conversion matrix of shape (2*l+1, 2*l+1).
Notes
See https://en.wikipedia.org/wiki/Spherical_harmonics#Real_form Y^m_l are the complex spherical harmonics, and Y_{lm} are the real spherical harmonics.
- forte2.symmetry.sph_harm_utils.sph_complex_to_real(l)#
Conversion matrix from complex spherical harmonics to real spherical harmonics for a given l.
- Parameters:
- lint
The angular momentum quantum number.
- Returns:
- NDArray
The conversion matrix of shape (2*l+1, 2*l+1).
- forte2.symmetry.sph_harm_utils.clebsh_gordan_spin_half(l, msdouble, jdouble, mjdouble)#
Clebsch Gordon coefficient specialized for coupling orbital angular momentum l with spin 1/2 to form total angular momentum j = l + 1/2, |l - 1/2|
\[C^{j,m_j}_{l,m_l;1/2,m_s} = \langle l,m_l;\frac{1}{2},m_s|j,m_j\rangle\]Since m_s + m_l must equal m_j, specifying m_j and m_s uniquely determines m_l.
- Parameters:
- lint
The orbital angular momentum quantum number.
- jdoubleint
The total angular momentum quantum number j is either l + 1/2 or l - 1/2.
- mjdoubleint
The magnetic quantum number of the total angular momentum. m_j can take a value in {j, j-1, …, -j}.
- msdoubleint
The magnetic quantum number of the z-component of the total angular momentum. mz is either 1/2 or -1/2.
Notes
Ch. 4.1.2, Atkins, P. W. (2011). Molecular Quantum Mechanics Fifth Ed.
https://en.wikipedia.org/wiki/Clebsch%E2%80%93Gordan_coefficients#Special_cases
- forte2.symmetry.sph_harm_utils.real_sph_to_j_adapted_per_l(l)#
Transformation matrix that transforms real spherical harmonics of a given angular momentum to (j-adapted) spinor basis.
- Parameters:
- lint
The angular momentum quantum number.
- Returns:
- tuple[NDArray]
The transformation matrices for alpha and beta spinors. Each matrix has shape (2*l+1, 4*l+2).
Notes
The transformation is based on the Clebsch-Gordan coefficients for coupling the orbital angular momentum l with spin 1/2 to form total angular momentum j = l + 1/2, l - 1/2. l = 0 is a special case where the transformation is trivial.
\[\begin{split}|j, m_j> = \sum_{m_l=-l}^l\sum_{m_s=-1/2}^{1/2} |l, m_l; 1/2, m_s\rangle\langle l, m_l; 1/2, m_s|j, m_j\rangle\\ = \sum_{m_l=-l}^l\sum_{m_s=-1/2}^{1/2} |l, m_l; 1/2, m_s\rangle * C^{j,m_j}_{l,m_l;1/2,m_s}.\end{split}\]
- forte2.symmetry.sph_harm_utils.real_sph_to_j_adapted(basis)#
Transformation matrix that transforms real-spherical GTOs to spinor GTOs for all basis functions