Background
Wigner-D matrices
The canonical spherical harmonics \(Y_{lm}\) subject to a rotation in \(\mathbb{R}^{3}\) will always mix among each other. As such, we can represent the act of rotating the spherical harmonics by the following matrix-vector equation
wherein \(\mathbf{D}\) is the Wigner D-matrix and \(\vec{Y}_{l}\) a vector composed of the canonical spherical harmonics of order \(l\). The vector \(\vec{Y}_{l}\prime\) is the linear combination that represents the result of the rotation upon the (linear combination) of spherical harmonics prior to the rotation.
Tesseral transformation
The canonical spherical harmonics are complex-valued by nature, yet a transformation among them exists which casts the canonical spherical harmonics into real-valued so-called tesseral spherical harmonics. A similar transformation can be applied to the Wigner-D matrix such that it represents the effect of a rotation among tesseral spherical harmonics.
Given the tesseral transformation that converts canonical spherical harmonics into tesseral ones as given by \(\mathbf{T}\) and the Wigner-D matrix \(\mathbf{D}\) built for canonical spherical harmonics, we can readily compute the Wigner-D matrix \(\mathbf{D}\prime\) for tesseral spherical harmonics by means of a basis transformation.
Note
In Sphecerix one can choose to construct the Wigner-D matrix for either canonical or tesseral spherical harmonics.
Mirror operations among tesseral spherical harmonics
To construct the transformation of a mirror operation \(\hat{M}\), we can note that any mirror operation can be decomposed into a rotation and an inversion. Given a mirror operation through a mirror plane as represented by its normal vector \(\vec{n}\).
The matrix representation of this mirror operation is given by
This matrix has a negative determinant and thus contains a reflection. We can get rid of this reflection by multiplying this matrix by -1 which is equivalent to an extraction of an inversion operation. In other words, we can decompose the matrix \(\mathbf{M}\) such that
The reflection operation in the space spanned by the tesseral spherical harmonics of order \(l\) is known and corresponds to the character for the inversion operation which is equal to \((-1)^{l}\). As such, the transformation matrix corresponding to a mirror operation among tesseral spherical harmonics of order \(l\) is given by
wherein \(\mathbf{D}\prime\) is the tesseral Wigner-D matrix for the rotation \(\hat{R}\) as extracted from the mirror operation \(\hat{M}\).
Labels tesseral spherical harmonics
For the real-valued spherical harmonics, i.e. the spherical harmonics after a tesseral transformation, their canonical names are derived from the mathematical equation that describes the angular part of the hydrogen-like wave function. In the table below, an overview is given how the labels for the tesseral spherical harmonics are associated to the values for \(m\).
Tip
For a detailed description how these labels are constructed, have a look at this publication of Ashkenazi.
\(-4\) |
\(-3\) |
\(-2\) |
\(-1\) |
\(0\) |
\(1\) |
\(2\) |
\(3\) |
\(4\) |
|---|---|---|---|---|---|---|---|---|
\(-\) |
\(-\) |
\(-\) |
\(-\) |
\(s\) |
\(-\) |
\(-\) |
\(-\) |
\(-\) |
\(-\) |
\(-\) |
\(-\) |
\(p_{y}\) |
\(p_{z}\) |
\(p_{x}\) |
\(-\) |
\(-\) |
\(-\) |
\(-\) |
\(-\) |
\(d_{xy}\) |
\(d_{yz}\) |
\(d_{z^{2}}\) |
\(d_{xz}\) |
\(d_{x^2-y^2}\) |
\(-\) |
\(-\) |
\(-\) |
\(f_{y(3x^2-y^2)}\) |
\(f_{xyz}\) |
\(f_{yz^2}\) |
\(f_{z^3}\) |
\(f_{xz^2}\) |
\(f_{z(x^2-y^2)}\) |
\(f_{x(x^2-3y^2)}\) |
\(-\) |
\(g_{xy(x^2-y^2)}\) |
\(g_{zy^3}\) |
\(g_{xyz^2}\) |
\(g_{yz^3}\) |
\(g_{z^4}\) |
\(g_{xz^3}\) |
\(g_{z^2(x^2-y^2)}\) |
\(g_{zx^3}\) |
\(g_{x^4+y^4}\) |