Recurrence and Differential Relations for Spherical Spinors

Abstract

We present a comprehensive table of recurrence and differential relations obeyed by spin one-half spherical spinors (spinor spherical harmonics) $\Omega_{\kappa\mu}(\mathbf{n})$ used in relativistic atomic, molecular, and solid state physics, as well as in relativistic quantum chemistry. First, we list finite expansions in the spherical spinor basis of the expressions $\mathbf{A}\cdot\mathbf{B}\,\Omega_{\kappa\mu}(\mathbf{n})$ and {$\mathbf{A}\cdot(\mathbf{B}\times\mathbf{C})\, \Omega_{\kappa\mu}(\mathbf{n})$}, where $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$ are either of the following vectors or vector operators: $\mathbf{n}=\mathbf{r}/r$ (the radial unit vector), $\mathbf{e}_{0}$, $\mathbf{e}_{\pm1}$ (the spherical, or cyclic, versors), $\boldsymbol{\sigma}$ (the $2\times2$ Pauli matrix vector), $\hat{\mathbf{L}}=-i\mathbf{r}\times\boldsymbol{\nabla}I$ (the dimensionless orbital angular momentum operator; $I$ is the $2\times2$ unit matrix), $\hat{\mathbf{J}}=\hat{\mathbf{L}}+1/2\boldsymbol{\sigma}$ (the dimensionless total angular momentum operator). Then, we list finite expansions in the spherical spinor basis of the expressions $\mathbf{A}\cdot\mathbf{B}\,F(r)\Omega_{\kappa\mu}(\mathbf{n})$ and $\mathbf{A}\cdot(\mathbf{B}\times\mathbf{C})\, F(r)\Omega_{\kappa\mu}(\mathbf{n})$, where at least one of the objects $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ is the nabla operator $\boldsymbol{\nabla}$, while the remaining ones are chosen from the set $\mathbf{n}$, $\mathbf{e}_{0}$, $\mathbf{e}_{\pm1}$, $\boldsymbol{\sigma}$, $\hat{\mathbf{L}}$, $\hat{\mathbf{J}}$.