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\title{Hardness conservation as a new transferability criterion:
application to fully non-local pseudopotentials}

 
\author{A.~Filippetti,$^1$ A.~Satta,$^1$ David Vanderbilt,$^2$ and
W.~Zhong$^2$}

 
\address{$^1$INFM and Dipartimento di Scienze Fisiche, 
Universita' di Cagliari, Italy}

 
\address{$^2$Department of Physics and Astronomy, 
Rutgers University, Piscataway, NJ, USA}
 
\date{\today}
\maketitle
%..................................................................... 
 
\begin{abstract} 
The concept of chemical hardness has been recently
adopted in the framework of Kohn-Sham theory as a faithful 
{\it ab initio} measure of pseudopotential transferability.
A fully self-consistent hardness theory has been developed and 
employed to evaluate the transferability of semilocal
pseudopotentials. Hardness contains most 
of the relevant physical information determining the transferability
of pseudopotentials, and is an important step forward with respect to 
the logarithmic derivatives analysis.
We discuss the main features of chemical hardness, and the relations
between chemical hardness and the original definitions of
absolute and local hardness. We then apply the new criterion
to investigate the transferability of fully non-local Kleinman-Bylander 
pseudopotentials.
Hardness conservation allows us to obtain a meaningful
comparison between them and the conventional norm-conserving ones
and give us a criterion to improve the pseudopotential tranferability 
of fully non-local pseudopotentials by suitably resetting their local part. 

\end{abstract}
%......................................................................
\pacs{77.80.Bh, 61.50.Lt, 64.60.Cn, 64.70.-p}
\narrowtext
 
\marginparwidth 2.7in
\marginparsep 0.5in
\section{introduction}
In the framework of density-functional theory and the pseudopotential
approximation, many efforts have been made to find physical properties
able to predict {\it a priori} (i.e. just in the atomic reference
configuration) the degree of transferability of pseudopotentials.
Logarithmic derivative analysis has shown to be an important tool to test
pseudopotential transferability, but not a sufficient criterion of
transferability for all atoms of the periodic table.
The chemical hardness defined in the context of Janak theory
has been proposed\cite{teter} as an optimum physical quantity to
test pseudopotentials for all atoms. 
The fully self-consistent hardness has been calculated\cite{me2} and
employed to study transferability of HSC-type semilocal pseudopotentials.


In this work we focus on the relations between hardness and other
two important quantity in the physics of electronic systems, the
absolute hardness,\cite{parr}
and the local hardness.\cite{ghosh,berk}
Then, we employ the hardness conservation criterion to test 
the transferability of norm-conserving pseudopotentials arranged
in the fully non-local structure.\cite{klei}

\section{Theory}

\subsection{Absolute and local hardness}
Chemical hardness, along with the chemical potential, is one of the 
most important quantities determining the physics of many-electron systems. 
{\it Absolute hardness} was introduced by Parr and Pearsons \cite{parr} as,
\begin{equation}
\eta={1 \over 2}\left({\partial^2 E \over \partial N^2}\right)_v=
{1 \over 2}\left({\partial \mu \over \partial N}\right)_v
\end{equation}
where $E$, $N$, $\mu$ and $v$ denote total energy, number of particles,
chemical potential and external potential of the many electron system, 
respectively.

A local expression for the hardness has been introduced in the
framework of DFT,\cite{ghosh,berk}
\begin{equation}
\eta({\bf r})={1 \over 2N}\int\>d{\bf r'}{\delta^2 F \over 
\delta\rho({\bf r})\>\delta\rho({\bf r'})}\>\rho({\bf r'})\>,
\end{equation}
with
\begin{equation}
F[\rho]=E[\rho]-\int\>d{\bf r}\>v({\bf r})\>\rho({\bf r})
\end{equation}

It has been shown \cite{berk} that 
\begin{equation}
\eta=\int\>d{\bf r}\>\eta({\bf r})\>{\tilde f}({\bf r})
\label{zia}
\end{equation}
where ${\tilde f}$ is the Fukui function, 
\begin{equation}
{\tilde f}({\bf r})= \left[{\partial \rho({\bf r}) \over \partial N}\right]_v=
\left[{\partial \mu \over \partial v({\bf r})}\right]_N
\end{equation}
Local hardness and Fukui function are key quantities to investigate
the reactivity of atomic and molecular systems.\cite{parr2,lee,nale,nale2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Hardness for fractional occupation numbers}
In the framework of Janak theory, Teter\cite{teter} defined 
the chemical hardness to be the matrix
%
\begin{equation}
H_{ij}\>=\>{1\over 2}\>{\partial^2 E[\rho]\over 
\partial f_i\partial f_j}
={1\over 2} {\partial \epsilon_i \over \partial f_j}
\label{zero}
\end{equation}
where $E$ is the Janak functional\cite{janak} and $f_i$ the
occupation number of the $i$th state.
%
Thus, the hardness matrix measures the first-order change of an
energy eigenvalue resulting from a variation of an
occupation number, while allowing the total number of electrons to
vary. The idea of Teter was to employ the hardness matrix in the
pseudopotential theory as a measure of transferability.
Due to the big amount of physical information contained
in the hardness, a comparison between its atomic all-electron and 
pseudo values realizes a meaningful test of transferability. 

Here we give only a very short description of formulation
that has been developed to calculate in practice $H_{ij}$.
All the details and the comments about the physical contents
of hardness can be found in Ref.\ \onlinecite{me2}.   

The atomic hardness in non-relativistic non-spin polarized formulation
is given by, 
\begin{equation}
H_{\vbox{\baselineskip=8pt
    \hbox{$\scriptstyle nlm$}
    \hbox{$\scriptstyle n'l'm'$}}}
= {1 \over 2} \int\int d{\bf r}\>d{\bf r}'\>
n_{nlm}({\bf r})\>w_{\rm hxc}({\bf r},{\bf r}')\>
{\delta \rho({\bf r}')\over \delta f_{n'l'm'}} \; .
\label{zero1}
\end{equation}
where,
\begin{eqnarray}
 n_{nlm}({\bf r}) & = & R_{nl}^2(r)\>|Y_{lm}(\Omega)|^2 \; ,\\
 w_{\rm hxc} ({\bf r},{\bf r}') & = &
 \frac{\partial V_{\rm hxc} ({\bf r})}{\partial \rho({\bf r}')}
\; , \\
\rho({\bf r}) & = & \sum_{nlm}^{\rm occ}\>f_{nlm}\>n_{nlm}({\bf r}) \;.
\end{eqnarray}
The density change due to the variation of occupation numbers
consists of two terms:
\begin{equation}
{\delta \rho({\bf r})\over \delta f_{nlm}}=
n_{nlm}({\bf r})+\Delta n_{nlm}({\bf r}) \; .
\label{zero2}
\end{equation}
The first term arises from the explicit dependence of density on
the occupation numbers, while the second involves relaxation of the
wavefunction with changes of occupation.  We will refer to the
neglect of $\Delta n_{nlm}$ as the ``frozen wavefunction
approximation'' (FWA), while the effect of the $\Delta n_{nlm}$
term will be referred to as the ``self-consistency'' (SC)
correction.
By defining
\begin{equation}
V_{nlm}({\bf r})=\int d{\bf r}'\>w_{\rm hxc}({\bf r},{\bf r}')\>
n_{nlm}({\bf r}') \; .
\label{first}
\end{equation}
We obtain the following expression for the hardness:
\begin{equation}
H_{\vbox{\baselineskip=8pt
    \hbox{$\scriptstyle nlm$}
    \hbox{$\scriptstyle n'l'm'$}}}
={1\over 2}\int d{\bf r}\>
V_{nlm}({\bf r})\>n_{n'l'm'}({\bf r})+
{1\over 2}\int d{\bf r}\>
V_{nlm}({\bf r})\>\Delta n_{n'l'm'}({\bf r})
\label{cagliari}
\end{equation}
The first term is the FWA hardness and it can be easily
calculated. The second is the SC correction
and can be calculated to linear order in
perturbation theory by considering $V_{nlm}$ as a bare
potential perturbation and $\Delta n_{n'l'm'}$  the self-consistent
change of the charge density due to the perturbation,
\begin{equation}
\Delta n_{nlm}({\bf r})=\int d{\bf r}'\>
\chi({\bf r},{\bf r}')
\>V_{nlm}({\bf r}') 
\label{actchi}
\end{equation}
%
where $\chi$ is the linear susceptibility.
The determination of the linear susceptibility
follows closely the modified Stern\-heimer approach discussed
by Mahan and Sub\-ba\-swamy.\cite{mahan,mahan1}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\subsection{Relations between chemical and absolute hardness}
In this section we focus on the relations between the matrix elements
of the fractional occupation hardness and both absolute and local hardness.

We start considering a system with a zero-temperature Fermi-Dirac
distribution of occupation
numbers. For atomic and molecular systems, the Fukui function and the chemical 
potential are discontinuous functionals of $N$ for any integer value. 
Also, for insulators, a discontinuity arises at the gap.
At $T=0$, the Fukui function can be immediately
related to the charge density change given by Eq.(\ref{zero2}).

The left derivative (i.e., derivative from below) of the charge
density with respect to $N$
($\delta N=\delta f_{\rm homo}$, where HOMO stands for the Highest 
Occupied Molecular Orbital) is given by
\begin{equation}
{\tilde f^-}({\bf r})=|\psi_{\rm homo}({\bf r})|^2\>+
\>\sum_i^{1,M}\>f_i\>{\delta |\psi_i({\bf r})|^2 \over \delta f_{\rm homo}} ,
\end{equation}
where $M$ is the number of occupied states, $f_i=1$ for $i=1,M-1$
and $f_i=f_{\rm homo}$ for $i=M$.
The right derivative ($\delta N=\delta f_{\rm lumo}$, where LUMO stands
for Lowest Unoccupied Molecular Orbital) is,
\begin{equation}
{\tilde f^+}({\bf r})=|\psi_{\rm lumo}({\bf r})|^2\>+
\>\sum_i^{1,M}\>f_i\>{\delta |\psi_i({\bf r})|^2 \over \delta f_{\rm lumo}}
\end{equation}
where $f_i=1$ for $i=1,M$.

Perdew {\it et al.}\cite{perd} proved the equalities
$\mu^+=\epsilon_{\rm lumo}$ and $\mu^-=\epsilon_{\rm homo}$, where $\mu^+$ 
and $\mu^-$ are the right and 
left derivatives of the energy with respect to $N$.
The chemical potential is defined as 
$\mu=1/2(\epsilon_{\rm homo}+\epsilon_{\rm lumo})$. Obviously, 
the absolute hardness is discontinuous for integer $N$: 

\begin{equation}
\eta^-\>=\>{1 \over 2}\left({\delta \epsilon_{\rm homo} \over 
\delta f_{\rm homo}}\right)
\end{equation} 

\begin{equation}
\eta^+\>=\>{1 \over 2}\left({\delta \epsilon_{\rm lumo} 
\over \delta f_{\rm lumo}}\right)
\label{etam}
\end{equation}
so we take $\eta={1 \over 2}\>(\eta^++\eta^-)$, where
\begin{equation}
\eta^- = {1 \over 2} \int\int d{\bf r}\>d{\bf r}'\>
n_{\rm homo}({\bf r})\>w_{\rm hxc}({\bf r},{\bf r}')\>
{\delta \rho({\bf r}')\over \delta f_{\rm homo}} \; .
\label{etap}
\end{equation}
\begin{equation}
\eta^+ = {1 \over 2} \int\int d{\bf r}\>d{\bf r}'\>
n_{\rm lumo}({\bf r})\>w_{\rm hxc}({\bf r},{\bf r}')\>
{\delta \rho({\bf r}')\over \delta f_{\rm lumo}} \; .
\end{equation}
%
where $n_{homo(lumo)}=|\psi_{homo(lumo)}|^2$. 
Thus, we have related the absolute hardnesses $\eta^-$ and $\eta^+$
back to the diagonal elements $H_{\rm homo,homo}$ and $H_{\rm
lumo,lumo}$ of the hardness matrix, respectively.
%
Also, taking
\begin{equation}
\eta^{\pm}=\int\>d{\bf r}\>{\tilde f}^{\pm}({\bf r})\>
\eta^{\pm}({\bf r})
\end{equation}
from Eq. (\ref{etam}) and (\ref{etap}) we obtain an expression for the
local hardness,
\begin{equation}
\eta^-({\bf r})\>=\>\int d{\bf r'}\>
w_{\rm hxc}({\bf r},{\bf r'})\>n_{\rm homo}({\bf r'})
\end{equation}
\begin{equation}
\eta^+({\bf r})\>=\>\int d{\bf r'}\>
w_{\rm hxc}({\bf r},{\bf r'})\>n_{\rm lumo}({\bf r'})
\end{equation}
It should be noted that calculation of ${\tilde f^+}$, $\eta^+$
and $\eta^+({\bf r})$ of atoms and molecules will be strongly 
affected by LDA error.

For a metallic system there is no discontinuity in the chemical
potential. The various HOMO- and LUMO-related quantities
coincide and all the previous expressions involving left and right
limits simplify to just one. 

It is an artifact of the LDA that for many open-shell systems,
the ground state has degenerate eigenvalues
with fractional occupation numbers.  In such a case,
a relation between the change of electron number and the occupation of
eigenstates is difficult to obtain. The added fraction of electron
number will be redistributed among one-electron states to minimize the total
energy, and we can argue $\delta N$ should be a linear combination
of $\delta f$ for all fractionally occupied states, 
the weights being generally hard to predict. 

For open-shell atoms, the fractional occupations of the degenerate
HOMOs are typically equal, so that the spherical symmetry of the
potential is preserved.
In such a case, the occupation numbers are simply linearly related
to the electron number, and there are no discontinuities.
Letting $l$ be the angular momentum of the HOMO (suppressing
the $n$ index and spin), the Fukui function is given by
\begin{equation}
{\tilde f} = {1 \over 2l+1}\>\sum_m^{-l,+l}\>
{\delta \rho \over \delta f_{lm}}
\label{fukui}
\end{equation}
having taken
\begin{equation}
{\delta f_{lm} \over \delta N}={1 \over 2l+1} \;.
\end{equation}
Thus we have
\begin{equation}
\eta={1 \over 2}\>{1 \over 2l+1}\> \sum_{m'}^{-l,+l}\>
{\delta \epsilon_{lm} \over \delta f_{lm'}} \;.
\label{sabri}
\end{equation}
Then, from Eq. (\ref{fukui}) we obtain,
\begin{equation}
\eta={1 \over 2}\>\int\int d{\bf r}\>d{\bf r}'\>
n_{lm}({\bf r})\>w_{\rm hxc}({\bf r},{\bf r}')\>
{\tilde f}({\bf r'}) \; .
\label{eta}
\end{equation}
It should be noted that $\eta$ in Eq.\ (\ref{eta}) does not really depend 
on $m$, because the Fukui function has spherical symmetry.
The above formulation can easily be extended to closed-shell atoms,
for which the discontinuities will clearly reappear.

In Table 1 we showed the absolute hardness values for some
atoms calculated by means of Eq.(\ref{sabri}), and compared
with experimental values\cite{parr} calculated in according to the
finite difference expression $\eta^{exp}=(1/2)(I-A)$,
being $I$ and $A$ the ionization potential and the electronic
affinity, respectively. For the open-shell atoms (C,O and Si)
the agreement is very good. For $Ti^{2+}$ $\eta$ is computed
as the average between its left and right values. In such a case the
deviation from the experimental value is reasonable at all, because
for closed-shell atoms the difference between HOMO and LUMO eigenvalues
contributes to the finite-difference expression of hardness as well.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Hardness generalization to non-diagonal occupation numbers}
The hardness matrix has been further generalized to non diagonal
occupation numbers.\cite{me2} This extension allows one
to consider the change of
occupation with respect to any specified angular component of the density.
In a perturbation theory framework, the density of the perturbed system 
in terms of the eigenstates of the unperturbed one will be,    
\begin{equation}
n({\bf r})=\sum_{ij}\>f_{ij}\>\psi_i^*({\bf r})\psi_j({\bf r})
\label{latina}
\end{equation}
Instead of working in the explicit atomic representation
$f_{ij}=f_{nlm,n'l'm'}$, we find it convenient to work in a
representation $(nn'll'LM)$ in which $LM$ are labels of total
angular momentum, and $L= |l-l'|, |l-l'|+2,...,l+l'$ following the
usual angular-momentum addition rules.  Introducing the
index $\alpha=nn'll'$, we thus have
%
\begin{equation}
n({\bf r})=\sum_{\alpha LM}\>f_{\alpha LM}\>n_{\alpha}(r)\>
Y_{LM}(\Omega)
\label{roma}
\end{equation}
%
where $n_{\alpha}=R_{nl}\>R_{n'l'}$ and
%
\begin{equation}
f_{\alpha LM}=\sum_{mm'} C(LM;lm,l'm) f_{nlm,n'l'm'}
\label{ftof}
\end{equation}
The $C(LM;lm,l'm)$ are Clebsch-Gordon coefficients.
The generalized Kohn-Sham eigenvalues are given by
\begin{equation}
\epsilon_{\alpha LM}={\partial E\over \partial f_{\alpha LM}} \>,
\label{milano}
\end{equation}
and the generalized hardness matrix, because of the radial
symmetry of unperturbed configuration, is 
\begin{equation}
H_{\alpha \beta LL'MM'}={1\over 2}{\partial^2 E\over
\partial f_{\alpha LM}\>\partial f_{\beta L'M'}}=
H_{\alpha \beta L} \, \delta_{LL'} \delta_{M,-M'}  \; .
\label{firenze}
\end{equation}
where
\begin{equation}
H_{\alpha \beta L} = {1 \over 2} \int\int dr\>dr'\>\>r^2\>r'^2\>
n_{\alpha}(r)\>w_{\rm hxc}^{(L)} (r,r')\>
{\delta \rho(r')\over \delta f_{\beta}} \; .
\end{equation}
As in the diagonal case, the density variation due to the change 
of occupation numbers consists of the FWA contribution and the SC one.
Calculation of $H_{\alpha \beta L}$ has been performed with the same
procedure used before for the diagonal case.\cite{me2}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
Hardness conservation can be considered as an important
step forward with respect to the logarithmic derivative criterion.
It contains all the most important ``first principles'' properties
needed to characterize pseudopotential transferability. 
It gives information about changes of both the electrostatic 
and exchange-correlation part of pseudopotential with respect to 
occupation number changes of any angular character.

Moreover, the fully self consistent hardness
takes into account wavefunction relaxations caused by charge
rearrangement. This feature gives us a sound guide to analyze 
pseudopotential transferability.
Analysis of the hardness matrix enables us not only to carry out  
meaningful transferability tests in a large ensemble of physical 
environments, but also to detect what kind of atomic characteristic 
can damage the transferability of pseudopotentials. Also, 
it inserts norm-conservation and the Louie-Froyen-Cohen correction 
in a more rigorous theoretical framework.     
Hardness is a linear-order property. Its range of
accuracy might be be insufficient to test pseudopotential features
involving large deviation from the atomic reference configuration.
For instance, considering fully non-local pseudopotentials,
hardness is generally insensitive to the presence of possible ghost
states below the reference atomic eigenvalue. In fact, wavefunction
relaxation needed to pinpoint a ghost level\cite{gonze} can't be
achieved by means of a linear-order approximation.
Nonetheless, many atomic species show
a quite extended linearity interval around eigenvalues, and 
in those cases, hardness is meaningful and accurate.
Detailed hardness analysis characteristics for a set of 
representative atoms with HSC-type pseudopotentials\cite{hsc} can be
found in Ref.\ \onlinecite{me2}. In what follows, we present a hardness
comparison between semilocal HSC and fully non local KB\cite{klei}
 pseudopotentials employing the non-diagonal hardness formulation.
To this end, its useful to consider the {\it average hardness error}. 
\cite{me2} This quantity consists of a weighted average of hardness 
matrix elements that represent at best the typical changes 
occurring in the occupation numbers when we pass from reference 
to hybridized configurations.
In such a way, the large amount of physical information of the whole matrix 
is compressed into a single number. 

The average hardness error $X$ is defined as
\begin{equation}
X^2 \>=\>\sum_{\alpha\beta L}\>w_{\alpha\beta L}\>
(\Delta H_{\alpha\beta L})^2 \; .
\label{muzio}
\end{equation}
$\Delta H$ is the difference between all-electron and pseudopotential
 hardness matrix elements, and $w_{\alpha\beta L}$ is a weight.
Thus, $X$ is just a weighted RMS average of the errors in
the hardness matrix elements.
The weights are taken to be
\begin{equation}
w_{\alpha \beta L}\>=\>(2L+1)
\> \eta_{l_\alpha} \> \eta_{l'_\alpha}
\> \eta_{l_\beta} \> \eta_{l'_\beta} \;,
\label{weights}
\end{equation}
and
\begin{equation}
\eta_l \> = \> (2l+1) \> \sqrt{2f_l(1-f_l)}
\label{etas}
\end{equation}
We refer the reader to Ref.\ \onlinecite{me2} for the discussion of this choice.
It's clear from Eq.\ (\ref{weights}) that $X$ tends to weight more 
shells which are partially occupied than empty or fully occupied ones.

In general, the fully non-local modification of semilocal pseudopotentials
 should produce an overall deterioration of its transferability. 
KB-like pseudopotentials are constructed as to be  
equal to semilocals in the reference atomic configuration. Such
a correspondence is lost once we movie out of that. Thus, the FWA hardness
of semilocal and KB potentials will be the same, but the SC part will not.
The arbitrariness contained in the choice of local part of KB 
potentials should be employed to optimize at best the transferability.
In practice this is taken to be equal to the non-local component of 
pseudopotentials allowing us to get rid of ghost states. 
Actually, the guideline to reduce
both the loss of transferability and the risk to get ghost states
is to choose the local part so as to minimize as much as possible
the non-local contributions. 
Optimization of the local part by means of a hardness criterion satisfies 
this request. 

Here, we parameterize the local part of the KB pseudopotentials as 
a linear combination of the $l$-dependent radial potentials
for $s$, $p$, and $d$
\begin{equation}
V_{loc}^{KB}=\alpha\>V_s+\beta\>V_p+\gamma\>V_d \;.
\end{equation}
%
and determine the coefficients $\alpha$, $\beta$, and $\gamma$
in order to minimize the average hardness error
(under the constraint $\alpha+\beta+\gamma=1$
to ensure the correct behavior of the pseudopotential tail).
The hardness error is monitored as function of pseudopotential core radii
to focus on the loss of transferability occurring when we enlarge them.

We present results for Ti, Rh and Ge. The first two are characterized
(as are most of the transition metals) by strong pseudopotential non-locality.
Obviously these are the most interesting cases
to investigate (the more the pseudopotential is local, the less 
important is the choice of local part). 

In Figs.\ 1 and 2, the average errors for Ti are shown.
$\langle R_c \rangle$ is the
average value of the three core radii $R_s$, $R_p$ and $R_d$. 
We varied $\langle R_c \rangle$ equally enlarging $R_s$ and $R_p$ 
while $R_d$ has 
been taken fixed (being the $d$ wavefunction without nodes).  
In Fig.\ 1 we see the configuration $(\alpha,\beta,\gamma)=(0,0,1)$ 
(i.e. $V_{loc}=V_d$) gives an error close to that of the semilocal 
pseudopotential for close $\langle R_c \rangle$.  

Configurations $(1,0,0)$ and $(0,1,0)$ strongly worsen 
the error. In Fig.\ 2 we see that a nearly equally weighted mixture 
of pseudopotentials $(0.3,0.3,0.4)$ doesn't give good results, 
while a combination of the d component with a small fraction
of $p$ improves significantly the behavior of the $(0,0,1)$ arrangement.

In Fig.\ 3 we compare for rhodium 
semilocal and KB pseudopotentials with local part $(0,1,0)$, $(0,0,1)$ and 
$(0.8,0.1,0.1)$. The first two give small errors, but the third 
configuration makes the KB errors almost indistinguishable from 
the semilocal one. Here change of $\langle R_c \rangle$ is obtained 
by varying all the core radii.

Finally, results for germanium are showed in Fig.\ 4. 
Non-locality is weaker here than in the two first cases. 
We found configurations $(0,1,0)$ and $(0.2,0.4,0.4)$ are the
best suited to fit with semilocal error.

Unfortunately, we do not have any rigorous argument nor a practical
recipe to predict the best configuration of the local part. 
We have to employ qualitative considerations about localization 
of pseudopotentials, and then to try a good guess. 
Nonetheless, a few trials are usually sufficient to find
mixed configurations giving KB pseudopotentials with improved
transferability. 

Of course, the ($\alpha,\beta,\gamma)$ configurations
found for the three atoms treated here are not necessarily optimal,
and with further adjustments the
errors could probably be further reduced.  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
In this work we presented a theoretical analysis of the connection 
between our hardness matrix and the absolute hardness.

Also we showed an application of the hardness-conservation
criterion to compare the transferability of semilocal and KB pseudopotentials.
As could be expected, use of the KB form produces a general decrease of 
transferability, which however can be strongly limited by a proper
choice of local part.
We showed that the parameterization of the local part in terms of non-local 
components gives us a sufficient variational freedom to generate  
KB pseudopotentials with high transferability as measured by
hardness conservation.

\begin{table}
\caption{Table1:  calculated absolute hardness values (in Rydberg units)
compared with the experimental ones for open- and closed-shell atoms.
In cases where the discontinuity doesn't occur, the agreement is
quite good}
\begin{tabular}{lrrrr}
atom & $\eta^{exp}$ & $\eta$ & $\eta^+$ & $\eta^-$ \\
\hline
C &0.3674  &0.3644 & & \\
\hline
O &0.4470  &0.4953 & &  \\
\hline
Si&0.2484  &0.2477 & &  \\
\hline
$Ti^{2+}(4s^2,3d^0)$ &0.5144 &0.4439 &0.5984&0.2894 \\
\hline
\end{tabular}
\end{table}

\begin{figure}
\caption{ Average hardness errors of semilocal and relative KB
pseudopotentials arranged with different local parts as functions
of the core radii average $\langle R_C \rangle$. Numbers in the
legend correspond to the parameters $\alpha,\beta,\gamma$.
Taking the $d$ component as local part of KB pseudopotential
makes its average error very close to that of the semilocal one
for small $\langle R_C \rangle$.
\label{ti}}
\end{figure}

\begin{figure}
\caption{The choice of a suitable mixed configuration for the local
part of KB pseudopotential consistently improves the transferability
with respect to take the $d$ component of semilocal as local part.
\label{ti1}}
\end{figure}

\begin{figure}
\caption{The configuration (0.8,0.1,0.1) makes the error for KB
pseudopotential almost indistinguishable with respect to that
of the semilocal one.
\label{rh}}
\end{figure}

\begin{figure}
\caption{The two local parts considereds both make the KB pseudopotential
in optimum agreement with the semilocal one.
\label{ge}}
\end{figure}


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\bibitem{me2} A. Filippetti, D. Vanderbilt, W. Zhong, Y. Cai and
              G. B. Bachelet (to be published in Phys. Rev. B)

\bibitem{parr} R. G. Parr and R. G. Pearson, J. Am. Chem. Soc. {\bf 105}, 
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\bibitem{ghosh} S. K. Ghosh and M. Berkowitz, J. Chem. Phys. {\bf 83}, 
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\end{document}






