# Spherical tensors

The spherical tensors are related to the spherical harmonics as

$$$\tensor{C}^{(k)}_q \defd \sqrt{\frac{4\pi}{2k+1}} Y^k_q. \tag{V5.1.7}$$$
AngularMomentumAlgebra.systemMethod
system(::Tensor)

A general tensor acts on the full system, i.e. all coordinates.

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system(::Type{SphericalTensor})

A spherical tensor only acts on the coordinates $\theta$ and $\phi$.

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AngularMomentumAlgebra.rmeMethod
rme(ℓ′,𝐂̂ᵏ,ℓ)

Calculate the reduced matrix element of the spherical tensor of rank k:

\begin{aligned} \redmatrixel{\ell'}{\tensor{C}^{(k)}}{\ell} &= \angroot{\ell} C_{\ell 0;k,0}^{\ell'0} = (-)^{\ell-k} \angroot{\ell\ell'} \wignerthreej{\ell&k&\ell'\\0&0&0}. \end{aligned} \tag{V13.2.107}
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AngularMomentumAlgebra.couplingsMethod
couplings(tensor::SphericalTensor{k}, ℓ) where k

Generate all quantum numbers ℓ′ for which ⟨ℓ′||::SphericalTensor{k}||ℓ⟩ does not vanish.

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AngularMomentumAlgebra.ranksFunction
ranks(a, ::Type{SphericalTensor}, b)

Return which tensor ranks for spherical tensors that fulfill the triangle condition between spin-orbitals a and b.

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ranks(a, ::Type{CoulombTensor}, b)

Return which tensor ranks for Coulomb tensors that fulfill the triangle condition between spin-orbitals a and b.

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### Dipole tensor

The dipole operator is a rank-1 Cartesian tensor that may be expressed using the rank-1 spherical tensor:

$$$\vec{r} \equiv \begin{bmatrix}x\\y\\z\end{bmatrix} \equiv r \begin{bmatrix} \frac{1}{\sqrt{2}}[-\tensor{C}^{(1)}_1 + \tensor{C}^{(1)}_{-1}]\\ \frac{\im}{\sqrt{2}}[\tensor{C}^{(1)}_1 + \tensor{C}^{(1)}_{-1}]\\ \tensor{C}^{(1)}_0 \end{bmatrix}$$$
AngularMomentumAlgebra.RadialOperatorType
RadialOperator()

This represents the matrix element of the radial component of the dipole operator:

$$$\expect{r} = \int_0^\infty\diff{r}r^2 \conj{\Psi}_{n'\ell'}(r) r \Psi_{n\ell}(r)$$$
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#### AngularMomentumAlgebra.Dipoles

This submodule exists only as a shortcut to Cartesian tensor components.

AngularMomentumAlgebra.Dipoles.𝐫̂Constant
𝐫̂

The angular part of the dipole operator; the elements correspond to [x,y,z], i.e. the Cartesian tensor components. Can be entered as \bfr\hat.

Examples

julia> using AngularMomentumAlgebra.Dipoles

julia> z = 𝐫̂
𝐂̂⁽¹⁾₀

julia> dot(SpinOrbital(o"2s", 0, half(1)), z, SpinOrbital(o"2p", 0, half(1)))
0.5773502691896256
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AngularMomentumAlgebra.Dipoles.𝐫Constant
𝐫

The dipole operator; the elements correspond to [x,y,z], i.e. the Cartesian tensor components. Can be entered as \bfr.

Examples

julia> using AngularMomentumAlgebra.Dipoles

julia> z = 𝐫
𝐃̂⁽¹⁾₀

julia> dot(SpinOrbital(o"1s", 0, half(1)), z, SpinOrbital(o"2p", 0, half(1)))
0.57735r

julia> dot(SpinOrbital(o"2s", 0, half(1)), z, SpinOrbital(o"2p", 0, half(1)))
0.57735r

julia> dot(SpinOrbital(o"2p", 0, half(1)), z, SpinOrbital(o"3d", 0, half(1)))
0.516398r
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