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A..2 Fully relativistic case

The relativistic KS equations are Dirac-like equations for a spinor with a ``large'' Rnlj(r) and a ``small'' Snlj(r) component:

c$\displaystyle \left(\vphantom{{d \over dr} + {\kappa\over r}}\right.$$\displaystyle {d \over dr}$ + $\displaystyle {\kappa\over r}$$\displaystyle \left.\vphantom{{d \over dr} + {\kappa\over r}}\right)$Rnlj(r) = $\displaystyle \left(\vphantom{2mc^2 - V(r) + \epsilon }\right.$2mc2 - V(r) + $\displaystyle \epsilon$$\displaystyle \left.\vphantom{2mc^2 - V(r) + \epsilon }\right)$Snlj(r) (9)
c$\displaystyle \left(\vphantom{{d \over dr} - {\kappa\over r}}\right.$$\displaystyle {d \over dr}$ - $\displaystyle {\kappa\over r}$$\displaystyle \left.\vphantom{{d \over dr} - {\kappa\over r}}\right)$Snlj(r) = $\displaystyle \left(\vphantom{ V(r) + \epsilon }\right.$V(r) + $\displaystyle \epsilon$$\displaystyle \left.\vphantom{ V(r) + \epsilon }\right)$Rnlj(r) (10)

where j is the total angular momentum (j = 1/2 if l = 0 , j = l + 1/2, l - 1/2 otherwise); $ \kappa$ = - 2(j - l )(j + 1/2) is the Dirac quantum number ($ \kappa$ = - 1 is l = 0 , $ \kappa$ = - l - 1, l otherwise); and the charge density is given by

n(r) = $\displaystyle \sum_{{nlj}}^{}$$\displaystyle \Theta_{{nlj}}^{}$$\displaystyle {R^2_{nlj}(r)+S^2_{nlj}(r)\over 4\pi r^2}$. (11)



Layla Martin-Samos Colomer 2012-05-10