Gradients
The gradient is a rank-1 tensor, the reduced matrix element of which is given by
\[\begin{equation} \label{eqn:gradient-rme} \tag{V13.2.22} \redmatrixel{n'\ell'}{\tensor{\nabla}^{(1)}}{n\ell} = \sqrt{\ell+1} A_{n'\ell'n\ell}\delta_{\ell',\ell+1} - \sqrt{\ell} B_{n'\ell'n\ell}\delta_{\ell',\ell-1}, \end{equation}\]
where
\[\begin{equation} \begin{aligned} A_{n'\ell'n\ell} &\defd \int_0^\infty\diff{r}r^2 \conj{\Psi}_{n'\ell'}(r) \left(\partial_r-\frac{\ell}{r}\right) \Psi_{n\ell}(r),\\ B_{n'\ell'n\ell} &\defd \int_0^\infty\diff{r}r^2 \conj{\Psi}_{n'\ell'}(r) \left(\partial_r+\frac{\ell+1}{r}\right) \Psi_{n\ell}(r). \end{aligned} \tag{V13.2.23} \label{eqn:gradient-rme-terms} \end{equation}\]
When working in spherical coordinates, it is common to use the reduced wavefunction, since that simplifies the Laplacian and enforces vanishing Dirichlet boundary conditions at $r=0$:
\[\Phi(\vec{r}) \defd r\Psi(\vec{r}).\]
This must be taken into account when computing matrix elements of differential operators with respect to the partial waves of the reduced wavefunction:
\[\nabla \Psi = \nabla \frac{\Phi}{r} = \left(\nabla\frac{1}{r}\right)\Phi + \frac{1}{r}(\nabla\Phi) = -\frac{\tensor{n}_1}{r^2}\Phi + \frac{1}{r}\nabla\Phi = \frac{1}{r}\left(\nabla - \frac{\tensor{n}_1}{r}\right)\Phi,\]
which means that if $\ket{n'\ell'}$ and $\ket{n\ell}$ are partial waves of $\Phi$, instead of $\eqref{eqn:gradient-rme}$, we need to use
\[\begin{equation} \tag{\ref{eqn:gradient-rme}*} \redmatrixel{n'\ell'}{\tensor{\nabla}^{(1)}-\frac{\tensor{n}_1}{r}}{n\ell} = \sqrt{\ell+1} \tilde{A}_{n'\ell'n\ell}\delta_{\ell',\ell+1} - \sqrt{\ell} \tilde{B}_{n'\ell'n\ell}\delta_{\ell',\ell-1}, \end{equation}\]
where
\[\begin{equation} \begin{aligned} \tilde{A}_{n'\ell'n\ell}\delta_{\ell',\ell+1} &\defd \int_0^\infty\diff{r}r^2 \conj{\Psi}_{n'\ell'}(r) \left(\partial_r-\frac{\ell+1}{r}\right) \Psi_{n\ell}(r),\\ \tilde{B}_{n'\ell'n\ell}\delta_{\ell',\ell-1} &\defd \int_0^\infty\diff{r}r^2 \conj{\Psi}_{n'\ell'}(r) \left(\partial_r+\frac{\ell}{r}\right) \Psi_{n\ell}(r), \end{aligned} \tag{\ref{eqn:gradient-rme-terms}*} \end{equation}\]
since
\[\begin{equation} \tag{V13.2.11} \redmatrixel{\ell'}{\tensor{n}_1}{\ell} = \angroot{\ell} C_{\ell 0;10}^{\ell'0} \end{equation}\]
(cf Spherical tensors since $\tensor{n}_1\equiv\tensor{C}^{(1)}$) and
\[\begin{equation} \tag{V8.5.33,34} C_{\ell 0;10}^{\ell'0} = % (\ell+1) % \sqrt{\frac{(2\ell)!2!}{(2\ell+2)!}} % \delta_{\ell',\ell+1} % - \ell % \sqrt{\frac{2!(2\ell-1)!}{(2\ell+1)!}} % \delta_{\ell',\ell-1} \sqrt{\frac{(\ell+1)}{(2\ell+1)}} \delta_{\ell',\ell+1} - \sqrt{\frac{\ell}{(2\ell+1)}} \delta_{\ell',\ell-1} \end{equation}\]
\[\begin{equation} \tag{V13.2.11*} \implies\redmatrixel{\ell'}{\tensor{n}_1}{\ell} = \sqrt{\ell+1} \delta_{\ell',\ell+1} - \sqrt{\ell} \delta_{\ell',\ell-1}, \end{equation}\]
i.e. $r^{-1}$ is subtracted from the centrifugal terms of $A$ and $B$ in $\eqref{eqn:gradient-rme-terms}$.
For convenience, the ReducedGradient tensor is provided, to simplify working with reduced wavefunctions.
Example
julia> using AngularMomentumAlgebra, AtomicLevels
julia> orbitals = sos"k[s-d]"
18-element Array{SpinOrbital{Orbital{Symbol},Tuple{Int64,HalfIntegers.Half{Int64}}},1}:
ks₀α
ks₀β
kp₋₁α
kp₋₁β
kp₀α
kp₀β
kp₁α
kp₁β
kd₋₂α
kd₋₂β
kd₋₁α
kd₋₁β
kd₀α
kd₀β
kd₁α
kd₁β
kd₂α
kd₂β
julia> a,b,c = orbitals[[3,9,13]]
3-element Array{SpinOrbital{Orbital{Symbol},Tuple{Int64,HalfIntegers.Half{Int64}}},1}:
kp₋₁α
kd₋₂α
kd₀α
julia> ∂x = cartesian_tensor_component(Gradient(), :x)
- 0.707107 𝛁̂⁽¹⁾₁ + 0.707107 𝛁̂⁽¹⁾₋₁
julia> dot(a, ∂x, b)
0.447214(∂ᵣ + 3/r)
julia> dot(b, ∂x, a)
0.447214(∂ᵣ - 1/r)
julia> dot(a, ∂x, c)
- 0.182574(∂ᵣ + 3/r)
julia> dot(c, ∂x, a)
- 0.182574(∂ᵣ - 1/r)Reference
Gradient
AngularMomentumAlgebra.Gradient — TypeGradient()Construct a gradient tensor.
AngularMomentumAlgebra.system — Methodsystem(::Type{Gradient})The gradient only acts on the coordinates $r$, $\theta$, and $\phi$.
AngularMomentumAlgebra.RadialGradientOperator — TypeRadialGradientOperator(k)This represents the matrix element of the radial component of the gradient operator:
\[\begin{equation} \tag{V13.2.23} \int_0^\infty\diff{r}r^2 \conj{\Psi}_{n'\ell'}(r) \left(\partial_r+\frac{k}{r}\right) \Psi_{n\ell}(r) \end{equation}\]
AngularMomentumAlgebra.rme — Methodrme((n′,ℓ′), ::Gradient, (n,ℓ))Computes the reduced matrix element of ∇ in terms of RadialGradientOperator.
AngularMomentumAlgebra.couplings — Methodcouplings(tensor::Gradient, (n, ℓ))Generate all quantum numbers n′ℓ′ for which ⟨n′ℓ′||::Gradient||nℓ⟩ does not vanish.
Reduced gradient
AngularMomentumAlgebra.ReducedGradient — TypeReducedGradient()Construct a gradient tensor acting on reduced wavefunctions.
AngularMomentumAlgebra.system — Methodsystem(::Type{ReducedGradient})The reduced gradient only acts on the coordinates $r$, $\theta$, and $\phi$.
AngularMomentumAlgebra.rme — Methodrme((n′,ℓ′), ::ReducedGradient, (n,ℓ))Computes the reduced matrix element of ∂ in terms of RadialGradientOperator.
AngularMomentumAlgebra.couplings — Methodcouplings(tensor::ReducedGradient, (n, ℓ))Generate all quantum numbers n′ℓ′ for which ⟨n′ℓ′||::ReducedGradient||nℓ⟩ does not vanish.
AngularMomentumAlgebra.LinearMomenta
AngularMomentumAlgebra.LinearMomenta.𝐩 — Constant𝐩The linear momentum operator $\vec{p}=-\im\nabla$; the elements correspond to [px,py,pz], i.e. the Cartesian tensor components. Can be entered as \bfp.
AngularMomentumAlgebra.LinearMomenta.𝐩̃ — Constant𝐩̃The linear momentum operator $\vec{p}=-\im\nabla$, but evaluated in the basis of reduced wavefunctions; the elements correspond to [px,py,pz], i.e. the Cartesian tensor components. Can be entered as \bfp\tilde.