Definitions
This page defines much of the general notation and conventions used in the code. Where possible, we are consistent with Varshalovich (1988).
Shorthands
The abbreviation $(-)^k\defd(-1)^k$ is used for the roots of negative unity (see also powneg1).
A commonly occurring factor in angular momentum algebra is
\[\angroot{j_1j_2...j_n} \defd[(2j_1+1)(2j_2+1)...(2j_n+1)]^{1/2}. \tag{V13.1.3½}\]
It can be calculated with the unexported AngularMomentumAlgebra.∏ function.
Indices appearing in pairs on only one side of an equation are implicitly summed over.
Spherical harmonics
We assume the following definition of the spherical harmonics
\[Y_{m}^{\ell}(\theta,\varphi) = \sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}} P_{\ell}^m(\cos\theta) \mathrm{e}^{\im m \varphi} \tag{V5.2.1}\]
where $\theta$ and $\varphi$ are the usual spherical coordinates and $P_\ell^m(z)$ are the associated Legendre polynomials. The Condon-Shortley phase $(-)^m$ is included in the definition of the Legendre polynomials, consistent with Varshalovich (1988).
An explicit expression for the Legendre polynomials is given by the Rodrigues formula:
\[P_{\ell}^{m}(x) = \frac{(-)^m}{2^{\ell} \ell !} (1 - x^2)^{m/2} \frac{\mathrm{d}^{\ell+m}}{\mathrm{d}x^{\ell+m}} (x^2 - 1)^\ell\]
Properties. The complex conjugate of a spherical harmonic can be expressed in terms of spherical harmonics:
\[\bar{Y}^{\ell}_{m}(\theta,\varphi) = (-)^m Y^{\ell}_{-m}(\theta,\varphi)\]
The spherical harmonics are normalized
\[\int_{0}^{2\pi} \int_{0}^{\pi} \bar{Y}^{\ell_1}_{m_1}(\theta,\varphi) Y^{\ell_2}_{m_2}(\theta,\varphi) \sin\theta \diff{\theta} \diff{\varphi} = \delta_{\ell_1 \ell_2} \delta_{m_1 m_2} \tag{V5.1.6}\]
and the integral of three spherical harmonics is given by
\[\int_{0}^{2\pi} \int_{0}^{\pi} Y^{\ell_1}_{m_1}(\theta,\varphi) Y^{\ell_2}_{m_2}(\theta,\varphi) Y^{\ell_3}_{m_3}(\theta,\varphi) \sin\theta \diff{\theta} \diff{\varphi} \\= \frac{1}{\sqrt{4\pi}} \angroot{\ell_1,\ell_2,\ell_3} \begin{pmatrix} \ell_1 & \ell_2 & \ell_3 \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} \ell_1 & \ell_2 & \ell_3 \\ m_1 & m_2 & m_3 \end{pmatrix} \tag{V5.9.5}\]
The ISO 80000-2:2009 standard standardizes some mathematical notations and conventions for definitions of some special functions.
Index placement convention Unlike the ISO standard, we put the $\ell$ index on top and $m$ on the bottom, to be consistent with the way the $k$ and $q$ indices are normally written for tensor operators.
Spherical harmonics with negative $m$. The Condon-Shortley phase in the Legendre polynomials is consistent with the ISO standard. However, the definition of spherical harmonics differs slightly. Namely, the standard defines the spherical harmonics as follows
\[Y_{m}^{\ell}(\theta,\varphi) = \sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-|m|)!}{(\ell+|m|)!}} P_{\ell}^{|m|}(\cos\theta) \mathrm{e}^{\im m \varphi} \tag{ISO19.17}\]
which leads to the following relationship for the complex conjugate of $Y_{m}^{\ell}$
\[\bar{Y}_{m}^{\ell}(\theta,\varphi) = Y_{-m}^{\ell}(\theta,\varphi)\]
We opt to follow the (arguably, more common) non-ISO definition to stay consistent with the primary reference of Varshalovich (1988).
Clebsch–Gordan coefficients
The Clebsch–Gordan coefficients are related to the 3j symbols as
\[C_{j_1m_1j_2m_2}^{j_3m_3} \equiv \braket{j_1m_1j_2m_2}{j_3m_3} = (-)^{j_1-j_2+m_3}\angroot{j_3} \begin{pmatrix} j_1&j_2&j_3\\ m_1&m_2&-m_3 \end{pmatrix}. \tag{V8.1.12}\]
They can be calculated with the WignerSymbols.clebschgordan function.
Wigner–Eckart theorem
For the Wigner–Eckart theorem, which defines the reduced matrix elements (RMEs) $\redmatrixel{n' j'}{\tensor{T}^{(k)}}{n j}$ of a tensor operator of rank $k$, the convention is the following
\[\begin{align} \matrixel{n' j' m'}{\tensor{T}^{(k)}_q}{n j m} &\defd (-)^{2k} \frac{1}{\angroot{j'}} C_{jm;kq}^{j'm'} \redmatrixel{n' j'}{\tensor{T}^{(k)}}{n j} \nonumber \\ &= (-)^{j'-m'} \begin{pmatrix} j' & k & j \\ -m' & q & m \end{pmatrix} \redmatrixel{n' j'}{\tensor{T}^{(k)}}{n j} \tag{V13.1.2} \end{align}\]
The second form can be derived by using the relationship between the Clebsch–Gordan coefficients and the Wigner 3j symbols, and the permutation symmetries of the 3j symbol. The $n$ and $n'$ labels represent all non angular momentum quantum numbers.
A simpler convention used by some books, that also generalizes to other symmetry groups, is
\[\matrixel{n' j' m'}{\tensor{T}^{(k)}_q}{n j m} = \braket{j_1 m_1 j_2 m_2}{j_3 m_3} \redmatrixel{n' j'}{\tensor{T}^{(k)}}{n j}\]
However, again to stay consistent with Varshalovich (1988), we shall not use it. But it must be noted that, as the Wigner–Eckart theorem functions as a definition for the reduced matrix elements, this choice will change the values of the RMEs.