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<H2><A ID="SECTION00053000000000000000">
A..3 Scalar-relativistic case</A>
</H2>
<P>
<EM>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: <!-- MATH
 $\hbar=1, m=1/2, e^2=2$
 -->
<IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="img62.png"
 ALT="$\hbar$"> = 1, <I>m</I> = 1/2, <I>e</I><SUP>2</SUP> = 2):
</EM>
<BR>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{eqnarray}
-{d^2 R_{nl}(r)\over dr^2}
+\left( {l(l+1)\over r^2} + M(r)\left(V(r)-\epsilon\right)
\right) R_{nl}(r) \qquad \nonumber \\\mbox{} -
 {\alpha^2\over 4M(r)} {dV(r)\over dr}
                    \left({dR_{nl}(r)\over dr} +
                          \langle\kappa\rangle {R_{nl}(r)\over r}\right)= 0,
\end{eqnarray}
 -->
<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT">- <IMG STYLE="height: 5.07ex; vertical-align: -1.69ex; " SRC="img16.png"
 ALT="$\displaystyle {d^2 R_{nl}(r)\over dr^2}$"> + <IMG STYLE="height: 5.60ex; vertical-align: -2.30ex; " SRC="img63.png"
 ALT="$\displaystyle \left(\vphantom{ {l(l+1)\over r^2} + M(r)\left(V(r)-\epsilon\right)
}\right.$"><IMG STYLE="height: 4.95ex; vertical-align: -1.69ex; " SRC="img30.png"
 ALT="$\displaystyle {l(l+1)\over r^2}$"> + <I>M</I>(<I>r</I>)<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="img64.png"
 ALT="$\displaystyle \left(\vphantom{V(r)-\epsilon}\right.$"><I>V</I>(<I>r</I>) - <I>ε</I><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="img65.png"
 ALT="$\displaystyle \left.\vphantom{V(r)-\epsilon}\right)$"><IMG STYLE="height: 5.60ex; vertical-align: -2.30ex; " SRC="img66.png"
 ALT="$\displaystyle \left.\vphantom{ {l(l+1)\over r^2} + M(r)\left(V(r)-\epsilon\right)
}\right)$"><I>R</I><SUB>nl</SUB>(<I>r</I>)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</TD>
<TD>&nbsp;</TD>
<TD>&nbsp;</TD>
<TD WIDTH=10 ALIGN="RIGHT">
&nbsp;</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT">&nbsp; - <IMG STYLE="height: 5.60ex; vertical-align: -2.27ex; " SRC="img67.png"
 ALT="$\displaystyle {\alpha^2\over 4M(r)}$"><IMG STYLE="height: 4.95ex; vertical-align: -1.69ex; " SRC="img68.png"
 ALT="$\displaystyle {dV(r)\over dr}$"><IMG STYLE="height: 5.60ex; vertical-align: -2.30ex; " SRC="img69.png"
 ALT="$\displaystyle \left(\vphantom{{dR_{nl}(r)\over dr} +
\langle\kappa\rangle {R_{nl}(r)\over r}}\right.$"><IMG STYLE="height: 4.95ex; vertical-align: -1.69ex; " SRC="img70.png"
 ALT="$\displaystyle {dR_{nl}(r)\over dr}$"> + 〈<I>κ</I>〉<IMG STYLE="height: 4.95ex; vertical-align: -1.69ex; " SRC="img11.png"
 ALT="$\displaystyle {R_{nl}(r)\over r}$"><IMG STYLE="height: 5.60ex; vertical-align: -2.30ex; " SRC="img71.png"
 ALT="$\displaystyle \left.\vphantom{{dR_{nl}(r)\over dr} +
\langle\kappa\rangle {R_{nl}(r)\over r}}\right)$"> = 0,</TD>
<TD>&nbsp;</TD>
<TD>&nbsp;</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(12)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL">
<EM>
where <!-- MATH
 $\alpha=1/137.036$
 -->
<I>α</I> = 1/137.036 is the fine-structure constant,
<!-- MATH
 $\langle\kappa\rangle=-1$
 -->
〈<I>κ</I>〉 = - 1 is the degeneracy-weighted average value 
of the Dirac's <I>κ</I> for the two spin-orbit-split levels, <I>M</I>(<I>r</I>) is
defined as
</EM>
<P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{equation}
M(r)= 1 - {\alpha^2\over 4} \left(V(r)-\epsilon\right).
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP>
<I>M</I>(<I>r</I>) = 1 - <IMG STYLE="height: 5.01ex; vertical-align: -1.69ex; " SRC="img72.png"
 ALT="$\displaystyle {\alpha^2\over 4}$"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="img64.png"
 ALT="$\displaystyle \left(\vphantom{V(r)-\epsilon}\right.$"><I>V</I>(<I>r</I>) - <I>ε</I><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="img65.png"
 ALT="$\displaystyle \left.\vphantom{V(r)-\epsilon}\right)$">.
</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(13)</TD></TR>
</TABLE>
</DIV><EM>
The charge density is defined as in the nonrelativistic case:
</EM>
<P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{equation}
n(r) = \sum_{nl} \Theta_{nl}{R^2_{nl}(r)\over 4\pi r^2}.
\end{equation}
 -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP>
<I>n</I>(<I>r</I>) = <IMG STYLE="height: 5.01ex; vertical-align: -3.07ex; " SRC="img36.png"
 ALT="$\displaystyle \sum_{{nl}}^{}$"><I>Θ</I><SUB>nl</SUB><IMG STYLE="height: 5.07ex; vertical-align: -1.69ex; " SRC="img37.png"
 ALT="$\displaystyle {R^2_{nl}(r)\over 4\pi r^2}$">.
</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14)</TD></TR>
</TABLE>
</DIV>
<P>
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