The time-independent Dirac equation
In equation \(\eqref{eq:Dirac_equation_emf_alpha_beta}\), we have introduced the time-dependent Dirac equation. If Dirac one-electron Hamiltonian is time-independent, we can separate variables (G. Dyall and Faegri 2007) and write down the time-independent Dirac equation: \[\begin{equation} h^{c, \text{D}} \Psi(\vb{r}) % = E \Psi(\vb{r}) % \thinspace , \end{equation}\] with the Dirac Hamiltonian given by the \((4 \times 4)\)-matrix \[\begin{equation} h^{c, \text{D}} % = c \boldsymbol{\alpha} \vdot \boldsymbol{\pi}^c(\vb{r}) % + \beta m_e c^2 % + q_e \phi_{\text{ext}}^c(\vb{r}) % \thinspace . \end{equation}\] We should note that this separation of variables is only valid in a particular frame of reference, which is naturally the nuclear framework if we employ the usual Born-Oppenheimer approximation. If we would like to obtain results in a particular other frame, we should subsequently Lorentz-transform the results. Note that after the Lorentz transformation, we might even end up in a non-stationary state. (G. Dyall and Faegri 2007)
Let us first subtract the rest energy \(m_e c^2\) (i.e. when the electron is at rest and thus \(\vb{p}=0\)) to align relativistic and non-relativistic energy scales. We then end up with the Hamiltonian \[\begin{equation} h^{\text{c}, D'}(\vb{r}) % = c \boldsymbol{\alpha} \vdot \boldsymbol{\pi}^c(\vb{r}) % + (\beta - \text{I}_4) m_e c^2 % + q_e \phi_{\text{ext}}^c(\vb{r}) \end{equation}\] and the energy \[\begin{equation} E' = E - m_e c^2 % \thinspace . \end{equation}\] It is this form of the Dirac Hamiltonian that we will use in the remainder of these notes, and we will accordingly drop these primes.