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<LI><A ID="tex2html94"
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<LI><A ID="tex2html95"
HREF="node7.html#SECTION00031200000000000000">2.1.2 Valence-core partition</A>
<LI><A ID="tex2html96"
HREF="node7.html#SECTION00031300000000000000">2.1.3 Electronic reference configuration</A>
<LI><A ID="tex2html97"
HREF="node7.html#SECTION00031400000000000000">2.1.4 Nonlinear core correction</A>
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<H2><A ID="SECTION00031000000000000000">
2.1 Choosing the generation parameters</A>
</H2>
<P>
<H3><A ID="SECTION00031100000000000000"></A>
<A ID="XC"></A>
<BR>
2.1.1 Exchange-correlation functional
</H3><EM>
PP's must be generated with the <EM>same</EM> exchange-correlation
(XC) functional that will
be later used in calculations. The use of, for instance, a
GGA (Generalized Gradient Approximation) functional tegether
with PP's generated with Local-Density Approximation (LDA)
is inconsistent. This is why the PP file contains information
on the DFT level used in their generation: if you or your code
ignore it, you do it at your own risk.
</EM>
<P>
<EM>The <TT>atomic</TT> package allows PP generation for a large number of
functionals, both LDA and GGA. Most of them have
been extensively tested, but beware: some exotic or seldom-used functionals
might contain bugs. Currently, <TT>atomic</TT> does not allow PP generation
with meta-GGA (TPSS) or hybrid functionals. For the former, an old version
of <TT>atomic</TT>, modified by Xiaofei Wang, is available.
Work is in progress for the latter.
</EM>
<P>
<EM>Some functionals may present numerical problems
when the charge density goes to zero. For instance, the Becke
gradient correction to the exchange may diverge for
<!-- MATH
$\rho \rightarrow 0$
-->
<I>ρ</I>→ 0. This does not happen in a free atom
if the charge density behaves as it should, that is, as
<!-- MATH
$\rho(r)\rightarrow exp(-\alpha r)$
-->
<I>ρ</I>(<I>r</I>)→<I>exp</I>(- <I>αr</I>) for <!-- MATH
$r \rightarrow \infty$
-->
<I>r</I>→∞.
In a pseudoatom, however, a weird behavior may arise
around the core region, <!-- MATH
$r\rightarrow 0$
-->
<I>r</I>→ 0, because the
pseudocharge in that region is very small or sometimes
vanishing (if there are no filled <I>s</I> states). As a consequence,
nasty-looking ``spikes'' appear in the unscreened pseudopotential
very close to the nucleus. This is not nice at all but it is
usually harmless, because the interested region is really
very small. However in some unfortunate cases there can be
convergence problems. If you do not want to see those horrible
spikes, or if you experience problems, you have the following
choices:
</EM><DL class="COMPACT">
<DT>–</DT>
<DD>Use a better-behaved GGA, such as PBE
</DD>
<DT>–</DT>
<DD>Use the nonlinear core correction, which ensures
the presence of some charge close to the nucleus.
</DD>
</DL><EM>
A further possibility would be to cut the gradient correction for small
<I>r</I> (it used to be implemented, but it isn't any longer).
</EM>
<P>
<H3><A ID="SECTION00031200000000000000"></A>
<A ID="ValCore"></A>
<BR>
2.1.2 Valence-core partition
</H3><EM>
This seems to be a trivial step, and often it is: valence states
are those that contribute to bonding, core states are those that
do not contribute. Things may sometimes be more complicated than
this. For instance:
</EM><DL class="COMPACT">
<DT>–</DT>
<DD>in transition metals, whose typical outer electronic
configuration is something like (<I>n</I> = main quantum number)
<!-- MATH
$nd^i(n+1)s^j(n+1)p^k$
-->
<I>nd</I><SUP>i</SUP>(<I>n</I> + 1)<I>s</I><SUP>j</SUP>(<I>n</I> + 1)<I>p</I><SUP>k</SUP>, it is not
always evident that the <I>ns</I> and <I>np</I> states (``semicore states'')
can be safely put into the core. The problem is that <I>nd</I> states
are localized in the same spatial region as <I>ns</I> and <I>np</I> states,
deeper than (<I>n</I> + 1)<I>s</I> and (<I>n</I> + 1)<I>p</I> states. This may lead to poor
transferability. Typically, PP's with semicore states in the core
work well in solids with weak or metallic bonding, but perform poorly
in compounds with a stronger (chemical) type of bonding.
</DD>
<DT>–</DT>
<DD>Heavy alkali metals (Rb, Cs, maybe also K) have a large
polarizable core. PP's with just one electron may not always give
satisfactory results.
</DD>
<DT>–</DT>
<DD>In some II-VI and III-V semiconductors, such as ZnSe and
GaN, the contribution of the <I>d</I> states of the cation to the bonding
is not negligible and may require explicit inclusion of those <I>d</I>
states into the valence.
</DD>
</DL><EM>
In all these cases, promoting the highest core states <I>ns</I> and <I>np</I>,
or <I>nd</I>, into valence may be a computationally
expensive but obliged way to improve poor transferability. .
</EM>
<P>
<EM>You should include semicore states into valence only if really needed:
their inclusion in fact makes your PP harder (unless you resort to
US pseudization) and increases the number of electrons. In principle
you should also use more than one projector per angular momentum,
because the energy range to be covered by the PP with semicore electrons
is much wider than without. For instance, it may happen that the error
on the lattice parameter of a simple metal is larger
with a semicore PP than with a valence-only PP.
</EM>
<P>
<H3><A ID="SECTION00031300000000000000"></A>
<A ID="RefConf"></A>
<BR>
2.1.3 Electronic reference configuration
</H3><EM>
This may be any reasonable configuration not too far away from
the expected configuration in solids or molecules. As a first
choice, use the atomic ground state, unless you have a reason
to do otherwise, such as for instance:
</EM><DL class="COMPACT">
<DT>–</DT>
<DD>You do not want to deal with unbound states.
Very often states with highest angular momentum <I>l</I> are not bound
in the atom (an example: the 3<I>d</I> state in Si is not bound on the
ground state 3<I>s</I><SUP>2</SUP>3<I>p</I><SUP>2</SUP>, at least with LDA or GGA). In such a case
one has the choice between
<DL class="COMPACT">
<DT>–</DT>
<DD>using one configuration for <I>s</I> and <I>p</I>, another, more
ionic one, for <I>d</I>, as in Refs.[<A
HREF="node21.html#BHS">4</A>,<A
HREF="node21.html#Gonze">5</A>];
</DD>
<DT>–</DT>
<DD>choosing a single, more ionic configuration for which
all desired states are bound;
</DD>
<DT>–</DT>
<DD>generate PP's on unbound states: requires to choose
a suitable reference energy.
</DD>
</DL>
</DD>
<DT>–</DT>
<DD>The results of your PP are very sensitive to the chosen configuration.
This is something that in principle should not happen, but
I am aware of at least one case in which it does. In III-V
zincblende semiconductors, the equilibrium lattice parameter
is rather sensitive to the form of the <I>d</I> potential of the
cation (due to the presence of <I>p</I> - <I>d</I> coupling between anion
<I>p</I> states and cation <I>d</I> states [<A
HREF="node21.html#Zunger">12</A>]). By varying
the reference configuration, one can change the equilibrium
lattice parameter by as much as 1 - 2%.
The problem arises if you want to calculate accurate dynamical
properties of GaAs/AlAs alloys and superlattices: you need to
get a good theoretical lattice matching between GaAs and AlAs,
or otherwise unpleasant spurious effects may arise. When I was
confronted with this problem, I didn't find any better solution
than to tweak the 4<I>d</I> reference configuration for Ga until I got
the observed lattice-matching.
</DD>
<DT>–</DT>
<DD>You know that for the system you are interested in, the atom will
be in a given configuration and you try to stay close to it.
This is not very elegant but sometimes it is needed. For instance,
in transition metals described by a PP with semicore states in the
core, it is probably wise to chose an electronic configuration for
<I>d</I> states that is close to what you expect in your system (as a
hand-waiving argument, consider that the (<I>n</I> + 1)<I>s</I> and (<I>n</I> + 1)<I>p</I> PP
have a hard time in reproducing the true potential if the <I>nd</I> state,
which is much more localized, changes a lot with respect to the
starting configuration). In Rare-Earth compounds, leaving the 4<I>f</I>
electrons in the core with the correct occupancy (if known) may be
a quick and dirty way to avoid the well-known problems of DFT yielding
the wrong occupancy in highly correlated materials.
</DD>
<DT>–</DT>
<DD>You don't manage to build a decent PP with the ground state configuration,
for whatever reason.
</DD>
</DL>
<P>
<EM>NOTE 1: you can calculate PP for a <I>l</I> as high as you want, but you
are not obliged to use all of them in PW calculations. The general
rule is that if your atom has states up to <I>l</I> = <I>l</I><SUB>c</SUB> in the core, you
need a PP with angular momenta up to <I>l</I> = <I>l</I><SUB>c</SUB> + 1. Angular momenta
<I>l</I> > <I>l</I><SUB>c</SUB> + 1 will feel the same potential as <I>l</I> = <I>l</I><SUB>c</SUB> + 1, because
for all of them there is no orthogonalization to core states.
As a consequence a PP should have projectors on angular momenta up to
<I>l</I><SUB>c</SUB>; <I>l</I> = <I>l</I><SUB>c</SUB> + 1 should be the local reference state for PW
calculations. This rule is not very strict and may be relaxed: high
angular momenta are seldom important (but be careful if they are).
Moreover separable PP pose serious constraints on local reference <I>l</I>
(see below) and the choice is sometimes obliged. Note also that the
highest the <I>l</I> in the PP, the more expensive the PW calculation will
be.
</EM>
<P>
<EM>NOTE 2: a completely empty configuration (<I>s</I><SUP>0</SUP><I>p</I><SUP>0</SUP><I>d</I><SUP>0</SUP>) or
a configuration with fractional occupation numbers are both
acceptable. Even if fractional occupation numbers do
not correspond to a physical atomic state, they correspond to a
well-defined mathematical object.
</EM>
<P>
<EM>NOTE 3: PP could in principle be generated on a spin-polarized
configuration, but a spin-unpolarized one is typically used.
Since PP are constructed to be transferrable, they can describe
spin-polarized configurations as well. The nonlinear core correction
is needed if you plan to use PP in spin-polarized (magnetic)
systems.
</EM>
<P>
<H3><A ID="SECTION00031400000000000000"></A>
<A ID="nlcc"></A>
<BR>
2.1.4 Nonlinear core correction
</H3><EM>
The nonlinear core correction[<A
HREF="node21.html#CoreCorr">11</A>]
accounts at least partially for the nonlinearity
in the XC potential. During PP generation one first
produces a potential yielding the desired pseudo-orbitals and
pseudoenergies. In order to extract a ``bare'' PP that can be used
in a self-consistent DFT calculation, one subtracts out the screening
(Hartree and XC) potential generated by the valence
charge only. This introduces a trasferability error because the XC
potential is not
linear in the charge density. With the nonlinear core correction one
keeps a pseudized core charge to be added to the valence charge both
at the unscreening step and when using the PP.
</EM>
<P>
<EM>The nonlinear core correction <EM>must</EM> be present in one-electron PP's for
alkali atoms (especially in ionic compounds) and for PP's to be used in
spin-polarized (magnetic) systems. It is recommended whenever there is a
large overlap between valence and core charge: for instance, in transition
metals if the semicore states are kept into the core. Since it is <EM>never</EM>
harmful, one can take the point of view that it should always be included,
even in cases where it will not be very useful.
</EM>
<P>
<EM>The pseudized core charge used in practice is equal to the true
core charge for <!-- MATH
$r\ge r_{cc}$
-->
<I>r</I>≥<I>r</I><SUB>cc</SUB>, differs from it for <!-- MATH
$r < r_{cc}$
-->
<I>r</I> < <I>r</I><SUB>cc</SUB>
in such a way as to be much smoother. The parameter <I>r</I><SUB>cc</SUB> is
typically chosen as the point at which the core charge <!-- MATH
$\rho_c(r_{cc})$
-->
<I>ρ</I><SUB>c</SUB>(<I>r</I><SUB>cc</SUB>)
is twice as big as the valence charge <!-- MATH
$\rho_v(r_{cc})$
-->
<I>ρ</I><SUB>v</SUB>(<I>r</I><SUB>cc</SUB>). In fact the
effect of nonlinearity is important only in regions where
<!-- MATH
$\rho_c(r)\sim\rho_v(r)$
-->
<I>ρ</I><SUB>c</SUB>(<I>r</I>)∼<I>ρ</I><SUB>v</SUB>(<I>r</I>). Alternatively, <I>r</I><SUB>cc</SUB> can be provided
in input, Note that the smaller <I>r</I><SUB>cc</SUB>, the more accurate the core
correction, but also the harder the pseudized core charge, and vice versa.
</EM>
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