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<LI><A ID="tex2html100"
HREF="node8.html#SECTION00032100000000000000">2.2.1 Pseudization energies</A>
<LI><A ID="tex2html101"
HREF="node8.html#SECTION00032200000000000000">2.2.2 Pseudization radii</A>
<LI><A ID="tex2html102"
HREF="node8.html#SECTION00032300000000000000">2.2.3 Choosing the local potential</A>
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<H2><A ID="SECTION00032000000000000000"></A>
<A ID="pseudization"></A>
<BR>
2.2 Type of pseudization
</H2><EM>
The <TT>atomic</TT> package implements two different NC pseudization
algorithms, both claiming to yield optimally smooth PP's:
</EM>
<UL>
<LI>Troullier-Martins [<A
HREF="node21.html#TM">7</A>] (TM)
</LI>
<LI>Rappe-Rabe-Kaxiras-Joannopoulos [<A
HREF="node21.html#RRKJ">9</A>] (RRKJ).
</LI>
</UL><EM>
Both algorithms replace atomic orbitals in the core region
with smooth nodeless pseudo-orbitals. The TM method uses an
exponential of a polynomial (see Appendix B); the RRKJ method
uses three or four Bessel functions for the pseudo-orbitals in
the core region. The former is very robust. The latter may
occasionally fail to produce the required nodeless pseudo-orbital.
If this happens, first try to force the usage of four Bessel functions
(this is achieved by setting a small nonzero value of
the charge density at the origin, variable <TT>rho0</TT>:
unfortunately it works only for <I>s</I> states).
</EM>
<P>
<EM>Second-row elements N, O, F, 3<I>d</I> transition metals, rare earths,
are typically ``hard'' atoms, i.e. described by NC PP's requiring
a high PW cutoff. These atoms are characterized by 2<I>p</I> (N, O, F),
3<I>d</I> (transition metals), 4<I>f</I> (rare earths) valence states with no
orthogonalization to core states of the same <I>l</I> and no nodes.
In addition, as mentioned in Secs.<A HREF="node7.html#ValCore">2.1.2</A> and <A HREF="node7.html#RefConf">2.1.3</A>,
there are case in which you may be forced to include semicore states
in valence, thus making the PP hard (or even harder).
In all such cases, one should consider
<EM>ultrasoft</EM> pseudization, unless there is a good reason to stick
to NC-PP's. For the specific case of rare earths, however, remember
that the problem of DFT reliability preempts the (tough) problem of
generating a PP. With US-PP's one can give up the NC requirement
and get much softer PP's, at the price of introducing an augmentation
charge that compensates for the missing charge.
</EM>
<P>
<EM>Currently, the <TT>atomic</TT> package generates US-PP's on top of
a ``hard'' NC-PP. In order to ensure sufficient transferability,
at least two states per angular momentum <I>l</I> are required.
</EM>
<P>
<H3><A ID="SECTION00032100000000000000"></A>
<A ID="pseudiz"></A>
<BR>
2.2.1 Pseudization energies
</H3><EM>
If you stick to single-projector PP's (one potential per angular momentum
<I>l</I>, i.e. one projector per <I>l</I> in the separable form), the choice of the
electronic configuration automatically determines the reference states
to pseudize: for each <I>l</I>, the bound valence eigenstate is pseudized
at the corresponding eigenvalue. If no bound valence eigenstate exists,
one has to select a reference energy. The choice is rather arbitrary:
you may try something between than other valence bound state energies
and zero.
</EM>
<P>
<EM>If you have semicore states in valence, remember that for each <I>l</I>
only the state with lowest <I>n</I> can be used to generate a single-projector
PP. The <TT>atomic</TT> package requires that you explicitly specify the
configuration for unscreening in the ``test'' configuration:
see the detailed input documentation.
</EM>
<P>
<EM>It is possible to generate PP's by pseudizing atomic waves,
i.e. regular solutions of the radial Kohn-Sham equation, at any
energy. More than one such atomic waves of different energy can be
pseudized for the same <I>l</I>, resulting in a PP with more than one
projector per <I>l</I> (directly produced in the separable form). Note
however that the implementation of multiple-projector PP's is
correct for US pseudization: NC pseudization is not properly done
(a generalized norm-conservation requirement is not accounted for).
US pseudization is achieved by
setting different NC and US pseudization radii (see Sec.<A HREF="#radii">2.2.2</A>),
</EM>
<P>
<H3><A ID="SECTION00032200000000000000"></A>
<A ID="radii"></A>
<BR>
2.2.2 Pseudization radii
</H3>
<P>
<EM>For NC pseudization, one has to choose, for each state to be pseudized,
a NC pseudization radius <I>r</I><SUB>c</SUB>, at which the AE orbital and the
corresponding NC-PP orbital match, with continuous first derivative
at <I>r</I> = <I>r</I><SUB>c</SUB>. For bound states, <I>r</I><SUB>c</SUB> is typically at the outermost peak or
somewhat larger. The larger the <I>r</I><SUB>c</SUB>, the softer the potential
(less PW needed in the calculations), but also the less transferable.
The <I>r</I><SUB>c</SUB> may differ for different <I>l</I>; as a rule, one should avoid large
differences between the <I>r</I><SUB>c</SUB>'s, but this is not always possible. Also,
the <I>r</I><SUB>c</SUB> cannot be smaller than the outermost node.
</EM>
<P>
<EM>A big problem in NC-PP's is how strike a compromise between softness
and transferability, especially for difficult elements. The basic question:
``how much should I push <I>r</I><SUB>c</SUB> outwards in order to have reasonable results
with a reasonable PW cutoff''. has no clear-cut answer. The choice of <I>r</I><SUB>c</SUB>
at the outermost maximum for ``difficult'' elements (those described in
Sec.<A HREF="#pseudiz">2.2.1</A>): typically 0.7-0.8 a.u, even less for 4<I>f</I> electrons,
yields very hard PP's
(more than 100 Ry needed in practical calculations). With a little bit of
experience one can say that for second-row (2<I>p</I>) elements, <!-- MATH
$r_c=1.1-1.2$
-->
<I>r</I><SUB>c</SUB> = 1.1 - 1.2
will yield reasonably good results for 50-70 Ry PW kinetic energy cutoff;
for 3<I>d</I> transition metals, the same <I>r</I><SUB>c</SUB> will require > 80 Ry cutoff
(highest <I>l</I> have slower convergence for the same <I>r</I><SUB>c</SUB>). The above
estimates are for TM pseudization. RRKJ pseudization will yield an
estimate of the required cutoff.
</EM>
<P>
<EM>For multiple-projectors PP's, the <I>r</I><SUB>c</SUB> of unbound states may be chosen
in the same range as for bound states. Use small <I>r</I><SUB>c</SUB> and don't try to
push them outwards: the
US pseudization will take care of softness. US pseudization radii can
be chosen much larger than NC ones (e.g. 1.3÷ 1.5 a.u. for second-row
2<I>p</I> elements, 1.7÷ 2.2 a.u. for 3<I>d</I> transition metals), but do not
forget that the sum of the <I>r</I><SUB>c</SUB> of two atoms should not exceed the typical
bond length of those atoms.
</EM>
<P>
<EM>Note that it is the hardest atom that determines the PW cutoff in a
solid or molecule. Do not waste time trying to find optimally soft
PP's for element X if element Y is harder then element X.
</EM>
<P>
<H3><A ID="SECTION00032300000000000000">
2.2.3 Choosing the local potential</A>
</H3>
<P>
<EM>As explained in Sec. <A HREF="node7.html#RefConf">2.1.3</A>, note 1, one needs in principle
angular momentum channels in PP's up to <I>l</I><SUB>c</SUB> + 1. In the semilocal
form, the choice of a ''local'', <I>l</I>-independent potential is natural
and affects only seldom-important PW components with <I>l</I> > <I>l</I><SUB>c</SUB>.
In PW calculations, however, a separable, fully nonlocal form –
one in which the PP's is written as a local potential plus pr
ojectors – is used.
An arbitrary function can be added to the local potential and
subtracted to all <I>l</I> components. Generally one exploits this
arbitrariness to remove one <I>l</I> component using it as local potential.
The separable form can be either obtained by the Kleinman-Bylander
projection [<A
HREF="node21.html#KB">10</A>] applied to single-projector PP's, or directly
produced using Vanderbilt's procedure [<A
HREF="node21.html#van">2</A>] (for single-projector
PP's the two approaches are equivalent).
</EM>
<P>
<EM>Unfortunately the separable form is not guaranteed to have the
correct ground state (unlike the semilocal form, which, by construction,
has the correct ground states): ``ghost'' states, having the wrong number
of nodes,
can appear among the occupied states or close to them, making the
PP completely useless. This problem may show up in US-PP's as well.
</EM>
<P>
<EM>The freedom in choosing the local part can (and usually must) be used
in order to avoid the appearance of ghosts. For PW calculations it is
convenient to choose as local part the highest <I>l</I>, because this removes
more projectors (2<I>l</I> + 1 per atom) than for low <I>l</I>. According to Murphy's
law, this is also the choice that more often gives raise to problems,
and one is forced to use a different <I>l</I>. Another possibility is to generate
a local potential by pseudizing the AE potential.
</EM>
<P>
<EM>Note that ghosts may not be visible to atomic codes based on radial
integration, since the algorithm discards states with the wrong number
of nodes. Difficult convergence or mysterious errors are almost invariably
a sign tha there is something wrong with our PP.
A simple and safe way to check for the presence of a ghost is to diagonalize
the Kohn-Sham hamiltonian in a basis set of spherical Bessel functions.
This can be done together with transferability tests
(see Sec.<A HREF="node10.html#trans">2.4</A>)
</EM>
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