# Continuum solutions for the Dirac equation in Coulomb potential - numerical codes

Following the representation used in [1, pag. 11] the solution of the Dirac equation in polar coordinates for energy $$E$$ is of the type:

$$\psi_{E\kappa m}(\bf{r})= \dfrac{1}{r} \Bigg( \begin{matrix} P_{E\kappa} (r) \chi_{\kappa}^m(\theta,\phi)\\ iQ_{E\kappa} (r) \chi_{-\kappa}^m(\theta,\phi)\\ \end{matrix}\Bigg)\ ,$$

where $$P$$ and $$Q$$ represent the large and small radial components, respectively. While $$\chi_{\kappa}^m(\theta,\phi)$$ is the spherical spinor function.

The book gives the solutions for the radial equation in terms of the solution of the Kummer's confluent hypergeometric equation, that is:

$$\dfrac{d^2Y(\rho)}{d\rho^2}+(b-\rho)\dfrac{dY(\rho)}{d\rho}-aY(\rho)=0$$. Thus, the large and small components are given as:

$$P_{E\kappa}\propto\rho^{\gamma}e^{-\rho/2} [X(\rho)+Y(\rho)]$$

$$Q_{E\kappa}\propto\rho^{\gamma}e^{-\rho/2} [X(\rho)-Y(\rho)]$$

where:

$$X(\rho)\propto \biggl(aY(\rho)+\rho\dfrac{dY(\rho)}{d\rho}\biggr).$$

and $$\rho$$ is proportional to the radius.

Depending from the type of solution searched (bound or continuum) $$P$$ and $$Q$$ assume different shapes. The code GRASP2K [2] provides the possibility to calculate the Dirac bound wavefunctions, implementing the multiconfiguration Dirac-Hartree-Fock method.

I would like to know if anyone knows a code to calculate the continuum wavefunctions for a given atom or I should try to implement the analytical solutions.

References

[1] Relativistic quantum theory of atoms and molecules, by I. Grant.

• Welcome to SciComp.SE. I think that we might need some more details to be able to help you. Apr 25, 2021 at 19:58
• Thanks for your comment. I added further details.
– 081N
Apr 25, 2021 at 22:19