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A..3 Scalar-relativistic case

The full relativistic KS equations is be transformed into an equation for the large component only and averaged over spin-orbit components. In atomic units (Rydberg: $ \hbar$ = 1, m = 1/2, e2 = 2):

- $\displaystyle {d^2 R_{nl}(r)\over dr^2}$ + $\displaystyle \left(\vphantom{ {l(l+1)\over r^2} + M(r)\left(V(r)-\epsilon\right)
}\right.$$\displaystyle {l(l+1)\over r^2}$ + M(r)$\displaystyle \left(\vphantom{V(r)-\epsilon}\right.$V(r) - $\displaystyle \epsilon$$\displaystyle \left.\vphantom{V(r)-\epsilon}\right)$$\displaystyle \left.\vphantom{ {l(l+1)\over r^2} + M(r)\left(V(r)-\epsilon\right)
}\right)$Rnl(r)              
  - $\displaystyle {\alpha^2\over 4M(r)}$$\displaystyle {dV(r)\over dr}$$\displaystyle \left(\vphantom{{dR_{nl}(r)\over dr} +
\langle\kappa\rangle {R_{nl}(r)\over r}}\right.$$\displaystyle {dR_{nl}(r)\over dr}$ + $\displaystyle \langle$$\displaystyle \kappa$$\displaystyle \rangle$$\displaystyle {R_{nl}(r)\over r}$$\displaystyle \left.\vphantom{{dR_{nl}(r)\over dr} +
\langle\kappa\rangle {R_{nl}(r)\over r}}\right)$ = 0,     (12)

where $ \alpha$ = 1/137.036 is the fine-structure constant, $ \langle$$ \kappa$$ \rangle$ = - 1 is the degeneracy-weighted average value of the Dirac's $ \kappa$ for the two spin-orbit-split levels, M(r) is defined as

M(r) = 1 - $\displaystyle {\alpha^2\over 4}$$\displaystyle \left(\vphantom{V(r)-\epsilon}\right.$V(r) - $\displaystyle \epsilon$$\displaystyle \left.\vphantom{V(r)-\epsilon}\right)$. (13)
The charge density is defined as in the nonrelativistic case:

n(r) = $\displaystyle \sum_{{nl}}^{}$$\displaystyle \Theta_{{nl}}^{}$$\displaystyle {R^2_{nl}(r)\over 4\pi r^2}$. (14)



paolo giannozzi 2014-05-28