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<H1><A ID="SECTION00040000000000000000">
3 A worked example: Ti</A>
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<P>
<EM>Let us consider the Ti atom: <I>Z</I> = 22, electronic configuration:
<!-- MATH
$1s^2 2s^2 2p^6 3s^2 3p^6 3d^2 4s^2$
-->
1<I>s</I><SUP>2</SUP>2<I>s</I><SUP>2</SUP>2<I>p</I><SUP>6</SUP>3<I>s</I><SUP>2</SUP>3<I>p</I><SUP>6</SUP>3<I>d</I><SUP>2</SUP>4<I>s</I><SUP>2</SUP>, with PBE XC functional.
The input data for the AE calculation is simple:
</EM><PRE>
&input
atom='Ti', dft='PBE', config='[Ar] 3d2 4s2 4p0'
/
</PRE><EM>
and yields the total energy and Kohn-Sham levels. Let us concentrate
on the outermost states:
</EM><PRE>
3 0 3S 1( 2.00) -4.6035 -2.3017 -62.6334
3 1 3P 1( 6.00) -2.8562 -1.4281 -38.8608
3 2 3D 1( 2.00) -0.3130 -0.1565 -4.2588
4 0 4S 1( 2.00) -0.3283 -0.1641 -4.4667
4 1 4P 1( 0.00) -0.1078 -0.0539 -1.4663
</PRE><EM>
and on their spatial extension:
</EM><PRE>
s(3S/3S) = 1.000000 <r> = 1.0069 <r2> = 1.1699 r(max) = 0.8702
s(3P/3P) = 1.000000 <r> = 1.0860 <r2> = 1.3907 r(max) = 0.8985
s(3D/3D) = 1.000000 <r> = 1.6171 <r2> = 3.5729 r(max) = 0.9811
s(4S/4S) = 1.000000 <r> = 3.5138 <r2> = 14.2491 r(max) = 2.9123
s(4P/4P) = 1.000000 <r> = 4.8653 <r2> = 27.9369 r(max) = 3.8227
</PRE><EM>
Note that the 3<I>d</I> state has a small spatial extension, comparable to that of
3<I>s</I> and 3<I>p</I> states and much smaller than for 4<I>s</I> and 4<I>p</I> states; the
3<I>d</I> energy is instead comparable to that of 4<I>s</I> and 4<I>p</I> states and much
higher than the 3<I>s</I> and 3<I>p</I> energies.. Much of the chemistry of Ti is
determined by its 3<I>d</I> states. What should we do? We have the choice among
several possibilities:
</EM>
<OL>
<LI>single-projector NC-PP with 4 electrons in valence (3<I>d</I><SUP>2</SUP>4<I>s</I><SUP>2</SUP>),
with nonlinear core correction;
</LI>
<LI>single-projector NC-PP with 12 electrons in valence
(<!-- MATH
$3s^2 3p^6 3d^2 4s^2$
-->
3<I>s</I><SUP>2</SUP>3<I>p</I><SUP>6</SUP>3<I>d</I><SUP>2</SUP>4<I>s</I><SUP>2</SUP>);
</LI>
<LI>multiple-projector US-PP with 12 electrons in valence;
</LI>
<LI>multiple-projector US-PP with 4 electrons in valence and
nonlinear core correction;
</LI>
<LI>...
</LI>
</OL><EM>
The PP of case 1) will be hard due to the presence of 3<I>d</I> states, and
its transferability may turn out not be sufficient for all purposes;
PP's for 2) will be even harder due to the presence of 3<I>d</I> and
semicore 3<I>s</I> and 3<I>p</I> states; PP 3) can be made soft, but generating
one is not trivial; PP 4) may suffer from insufficient transferability.
</EM>
<P>
<BR><HR>
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<LI><A ID="tex2html109"
HREF="node12.html">3.1 Single-projector, norm-conserving, no semicore</A>
<UL>
<LI><A ID="tex2html110"
HREF="node12.html#SECTION00041100000000000000">3.1.1 Generation</A>
<LI><A ID="tex2html111"
HREF="node12.html#SECTION00041200000000000000">3.1.2 Testing</A>
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<LI><A ID="tex2html112"
HREF="node13.html">3.2 Single-projector, norm-conserving, with semicore states</A>
<LI><A ID="tex2html113"
HREF="node14.html">3.3 Testing in molecules and solids</A>
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