mirror of https://gitlab.com/QEF/q-e.git
1533 lines
68 KiB
TeX
1533 lines
68 KiB
TeX
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\begin{document}
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\title{Notes on pseudopotential generation}
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\author{\em Paolo Giannozzi\\
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Universit\`a di Udine and IOM-Democritos, Trieste\\
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URL: {\tt http://www.fisica.uniud.it/$\thicksim$giannozz}}
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\maketitle
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\date
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\tableofcontents
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\section{Introduction}
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When I started to do my first first-principle calculation
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(that is, my first$^2$-principle calculation) with Stefano Baroni
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on CsI under pressure (1985), it became quickly evident that
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available pseudopotentials (PP's) couldn't do the job. So we
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generated our own PP's. Since that first experience I have
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generated a large number of PP's and people keep asking me
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new PP's from time to time. I am happy that "my" PP's are
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appreciated and used by other people. I don't think however
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that the generation of PP's is such a hard task that it requires
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an official (or unofficial) PP wizard to do this. For this reason
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I want to share here my (little) experience.
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These notes are written in general but having in mind the capabilities of
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the {\tt atomic} package, included in the {\sc Quantum ESPRESSO}
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distribution
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(\texttt{http://www.quantum-espresso.org}). {\tt atomic}, mostly written
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and maintained by Andrea Dal Corso and others, is the evolution of
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an older code I maintained for several years. {\tt atomic} can generate
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both Norm-Conserving (NC) \cite{NC} and Ultrasoft (US) \cite{van} PP's,
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plus Projector Augmented Waves (PAW) \cite{PAW} sets.
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It allows multiple projectors, full relativistic calculations,
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spin-split PP's for spin-orbit calculations.
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For the complete description of the input of \texttt{atomic},
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please refer to files \texttt{INPUT\_LD1.txt} and
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\texttt{INPUT\_LD1.html}.
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\subsection{Who needs to generate a pseudopotential?}
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There are at least three well-known published sets of NC-PP's:
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those of Bachelet, Hamann, and Schl\"uter \cite{BHS},
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those of Gonze, Stumpf, and Scheffler \cite{Gonze}, and
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those of Goedecker, Teter, and Hutter \cite{Goedecker}.
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Moreover, all major packages for electronic-structure calculations
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include a downloadable table of PP's. One could then wonder
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what a PP generation code is useful for. The problem is that
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sometimes available PP's will not suit your needs. For instance,
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you may want:
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\begin{enumerate}
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\item[--] a better accuracy;
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\item[--] PP's generated with some exotic or new exchange-correlation
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functional;
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\item[--] a different partition of electrons into valence and core;
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\item[--] ``softer'' PP's (i.e. PP that require a smaller cutoff
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in plane-wave calculations);
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\item[--] PP's with a core-hole for calculations of X-ray Adsorption
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Spectra;
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\item[--] all-electron wavefunctions reconstruction (requires the
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knowledge of atomic all-electron and pseudo-orbitals used in
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the generation of PP's);
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\end{enumerate}
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or you may simply want to know what is a PP, how to produce PP's,
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how reliable they are.
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\subsection{About similar work}
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There are other PP generation packages available on-line.
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Those I am aware of include:
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\begin{itemize}
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\item the code by Jos\'e-Lu{\'\i}s Martins {\em et al.}\cite{TM}:\\
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{\tt http://bohr.inesc-mn.pt/\~{}jlm/pseudo.html}
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\item the {\tt fhi98PP} package\cite{fhi98PP}:\\
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{\tt http://www.fhi-berlin.mpg.de/th/fhi98md/fhi98PP}
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\item the OPIUM code by Andrew Rappe {\em et al.}\cite{RRKJ}:\\
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{\tt http://opium.sourceforge.net/}
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\item David Vanderbilt's US-PP package \cite{van}:\\
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{\tt http://www.physics.rutgers.edu/\~{}dhv/uspp/index.html}.
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\end{itemize}
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Other codes may be available upon request from the authors.
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Years ago, it occurred to me that a web-based PP generation
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tool would have been nice. Being too lazy and too ignorant
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in web-based applications, I did nothing.
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I recently discovered that Miguel Marques {\em et al.} have
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implemented something like this: see
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{\tt http://www.tddft.org/programs/octopus/pseudo.php}.
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\subsection{Pseudopotential generation, in general}
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In the following I am assuming that the basic PP theory
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is known to the reader. Otherwise, see
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Refs.\cite{NC,BHS,TM,fhi98PP,RRKJ} and references quoted
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therein for NC-PP's; Refs.\cite{van,PAW} for US-PP's and PAWsets.
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I am also assuming that the generated PP's are to be used
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in separable form \cite{KB} with a plane-wave (PW) basis set.
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The PP generation is a three-step process. First, one generates
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atomic levels and orbitals with Density-functional theory (DFT).
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Second, from atomic results one generates the PP. Third, one checks
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whether the reesulting PP is actually working. If not, one tries again in
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a different way.
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The first step is invariably done assuming a spherically symmetric
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self-consistent Hamiltonian, so that all elementary quantum mechanics
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results for the atom apply. The atomic state is defined by the
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"electronic configuration", one-electron states are defined by a
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principal quantum number and by the angular momentum and are obtained
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by solving a self-consistent radial Schr\"odinger-like (Kohn-Sham)
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equation.
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The second step exists in many variants. One can generate ``traditional''
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single-projector NC-PP's; multiple-projector NC-PP's; or US-PP's.
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The crucial step is in all cases the generation of smooth
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``pseudo-orbitals'' from atomic all-electron (AE) orbitals.
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Two popular pseudization
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methods are presently implemented: Troullier-Martins \cite{TM}
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and Rappe-Rabe-Kaxiras-Joannopoulos \cite{RRKJ} (RRKJ).
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The third step is closer to cooking than to science. There is a
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large arbitrariness in the preceding step that one would like to
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exploit in order to get the "best" PP, but there is no well-defined
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way to do this. Moreover one is often forced to strike a compromise
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between transferability (thus accuracy) and hardness (i.e. computer
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time). This last step is the main focus of these notes.
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\section{Step-by-step Pseudopotential generation}
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If you want to generate a PP for a given atom, the checklist is the
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following:
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\begin{itemize}
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\item choose the generation parameters:
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\begin{enumerate}
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\item exchange-correlation functional
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\item valence-core partition
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\item electronic reference configuration
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\item nonlinear core correction
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\item type of pseudization
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\item pseudization energies
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\item pseudization radii
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\item local potential
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\end{enumerate}
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\item generate the pseudopotential
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\item check for transferability
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\end{itemize}
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In case of trouble or of unsatisfactory results, one has to
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go back to the first step and change the generation parameters,
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usually in the last four items.
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\subsection{Choosing the generation parameters}
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\subsubsection{Exchange-correlation functional}
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\label{XC}
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PP's must be generated with the {\em same} exchange-correlation
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(XC) functional that will
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be later used in calculations. The use of, for instance, a
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GGA (Generalized Gradient Approximation) functional tegether
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with PP's generated with Local-Density Approximation (LDA)
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is inconsistent. This is why the PP file contains information
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on the DFT level used in their generation: if you or your code
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ignore it, you do it at your own risk.
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The \texttt{atomic} package allows PP generation for a large number of
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functionals, both LDA and GGA. Most of them have
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been extensively tested, but beware: some exotic or seldom-used functionals
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might contain bugs. Currently, \texttt{atomic} does not allow PP generation
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with meta-GGA (TPSS) or hybrid functionals. For the former, an old version
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of \texttt{atomic}, modified by Xiaofei Wang, is available.
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Work is in progress for the latter.
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Some functionals may present numerical problems
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when the charge density goes to zero. For instance, the Becke
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gradient correction to the exchange may diverge for
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$\rho \rightarrow 0$. This does not happen in a free atom
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if the charge density behaves as it should, that is, as
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$\rho(r)\rightarrow exp(-\alpha r)$ for $r \rightarrow \infty$.
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In a pseudoatom, however, a weird behavior may arise
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around the core region, $r\rightarrow 0$, because the
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pseudocharge in that region is very small or sometimes
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vanishing (if there are no filled $s$ states). As a consequence,
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nasty-looking ``spikes'' appear in the unscreened pseudopotential
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very close to the nucleus. This is not nice at all but it is
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usually harmless, because the interested region is really
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very small. However in some unfortunate cases there can be
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convergence problems. If you do not want to see those horrible
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spikes, or if you experience problems, you have the following
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choices:
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\begin{enumerate}
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\item[--] Use a better-behaved GGA, such as PBE
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\item[--] Use the nonlinear core correction, which ensures
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the presence of some charge close to the nucleus.
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\end{enumerate}
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A further possibility would be to cut the gradient correction for small
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$r$ (it used to be implemented, but it isn't any longer).
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% (set variable {\tt rcut} to $\sim 0.001$ or so).
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\subsubsection{Valence-core partition}
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\label{ValCore}
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This seems to be a trivial step, and often it is: valence states
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are those that contribute to bonding, core states are those that
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do not contribute. Things may sometimes be more complicated than
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this. For instance:
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\begin{enumerate}
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\item[--] in transition metals, whose typical outer electronic
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configuration is something like ($n=$ main quantum number)
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$nd^i(n+1)s^j(n+1)p^k$, it is not
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always evident that the $ns$ and $np$ states (``semicore states'')
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can be safely put into the core. The problem is that $nd$ states
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are localized in the same spatial region as $ns$ and $np$ states,
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deeper than $(n+1)s$ and $(n+1)p$ states. This may lead to poor
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transferability. Typically, PP's with semicore states in the core
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work well in solids with weak or metallic bonding, but perform poorly
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in compounds with a stronger (chemical) type of bonding.
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\item[--] Heavy alkali metals (Rb, Cs, maybe also K) have a large
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polarizable core. PP's with just one electron may not always give
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satisfactory results.
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\item[--] In some II-VI and III-V semiconductors, such as ZnSe and
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GaN, the contribution of the $d$ states of the cation to the bonding
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is not negligible and may require explicit inclusion of those $d$
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states into the valence.
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\end{enumerate}
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In all these cases, promoting the highest core states $ns$ and $np$,
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or $nd$, into valence may be a computationally
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expensive but obliged way to improve poor transferability. .
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You should include semicore states into valence only if really needed:
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their inclusion in fact makes your PP harder (unless you resort to
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US pseudization) and increases the number of electrons. In principle
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you should also use more than one projector per angular momentum,
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because the energy range to be covered by the PP with semicore electrons
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is much wider than without. If you don't, the transferability of your PP
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may suffer (a typical phenomenon:
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the error on the lattice parameter of a simple metal gets worse).
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\subsubsection{Electronic reference configuration}
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\label{RefConf}
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This may be any reasonable configuration not too far away from
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the expected configuration in solids or molecules. As a first
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choice, use the atomic ground state, unless you have a reason
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to do otherwise, such as for instance:
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\begin{enumerate}
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\item[--]
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You do not want to deal with unbound states.
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Very often states with highest angular momentum $l$ are not bound
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in the atom (an example: the $3d$ state in Si is not bound on the
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ground state $3s^23p^2$, at least with LDA or GGA). In such a case
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one has the choice between
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\begin{enumerate}
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\item[--] using one configuration for $s$ and $p$, another, more
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ionic one, for $d$, as in Refs.\cite{BHS,Gonze};
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\item[--] choosing a single, more ionic configuration for which
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all desired states are bound;
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\item[--] generate PP's on unbound states: requires to choose
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a suitable reference energy.
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\end{enumerate}
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\item[--]
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The results of your PP are very sensitive to the chosen configuration.
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This is something that in principle should not happen, but
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I am aware of at least one case in which it does. In III-V
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zincblende semiconductors, the equilibrium lattice parameter
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is rather sensitive to the form of the $d$ potential of the
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cation (due to the presence of $p-d$ coupling between anion
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$p$ states and cation $d$ states \cite{Zunger}). By varying
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the reference configuration, one can change the equilibrium
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lattice parameter by as much as $1-2\%$.
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The problem arises if you want to calculate accurate dynamical
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properties of GaAs/AlAs alloys and superlattices: you need to
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get a good theoretical lattice matching between GaAs and AlAs,
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or otherwise unpleasant spurious effects may arise. When I was
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confronted with this problem, I didn't find any better solution
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than to tweak the $4d$ reference configuration for Ga until I got
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the observed lattice-matching.
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\item[--]
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You know that for the system you are interested in, the atom will
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be in a given configuration and you try to stay close to it.
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This is not very elegant but sometimes it is needed. For instance,
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in transition metals described by a PP with semicore states in the
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core, it is probably wise to chose an electronic configuration for
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$d$ states that is close to what you expect in your system (as a
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hand-waiving argument, consider that the $(n+1)s$ and $(n+1)p$ PP
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have a hard time in reproducing the true potential if the $nd$ state,
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which is much more localized, changes a lot with respect to the
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starting configuration). In Rare-Earth compounds, leaving the $4f$
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electrons in the core with the correct occupancy (if known) may be
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a quick and dirty way to avoid the well-known problems of DFT yielding
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the wrong occupancy in highly correlated materials.
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\item[--]
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You don't manage to build a decent PP with the ground state configuration,
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for whatever reason.
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\end{enumerate}
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NOTE 1: you can calculate PP for a $l$ as high as you want, but you
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are not obliged to use all of them in PW calculations. The general
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rule is that if your atom has states up to $l=l_c$ in the core, you
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need a PP with angular momenta up to $l=l_c+1$. Angular momenta
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$l>l_c+1$ will feel the same potential as $l=l_c+1$, because
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for all of them there is no orthogonalization to core states.
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As a consequence a PP should have projectors on angular momenta up to
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$l_c$; $l=l_c+1$ should be the local reference state for PW
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calculations. This rule is not very strict and may be relaxed: high
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angular momenta are seldom important (but be careful if they are).
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Moreover separable PP pose serious constraints on local reference $l$
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(see below) and the choice is sometimes obliged. Note also that the
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highest the $l$ in the PP, the more expensive the PW calculation will
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be.
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NOTE 2: a completely empty configuration ($s^0p^0d^0$) or
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a configuration with fractional occupation numbers are both
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acceptable. Even if fractional occupation numbers do
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not correspond to a physical atomic state, they correspond to a
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well-defined mathematical object.
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NOTE 3: PP could in principle be generated on a spin-polarized
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configuration, but a spin-unpolarized one is typically used.
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Since PP are constructed to be transferrable, they can describe
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spin-polarized configurations as well. The nonlinear core correction
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is needed if you plan to use PP in spin-polarized (magnetic)
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systems.
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\subsubsection{Nonlinear core correction}
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\label{nlcc}
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The nonlinear core correction\cite{CoreCorr}
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accounts at least partially for the nonlinearity
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in the XC potential. During PP generation one first
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produces a potential yielding the desired pseudo-orbitals and
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pseudoenergies. In order to extract a ``bare'' PP that can be used
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in a self-consistent DFT calculation, one subtracts out the screening
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(Hartree and XC) potential generated by the valence
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charge only. This introduces a trasferability error because the XC
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potential is not
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linear in the charge density. With the nonlinear core correction one
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keeps a pseudized core charge to be added to the valence charge both
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at the unscreening step and when using the PP.
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The nonlinear core correction {\em must} be present in one-electron PP's for
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alkali atoms (especially in ionic compounds) and for PP's to be used in
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spin-polarized (magnetic) systems. It is recommended whenever there is a
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large overlap between valence and core charge: for instance, in transition
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metals if the semicore states are kept into the core. Since it is {\em never}
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harmful, one can take the point of view that it should always be included,
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even in cases where it will not be very useful.
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The pseudized core charge used in practice is equal to the true
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core charge for $r\ge r_{cc}$, differs from it for $r < r_{cc}$
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in such a way as to be much smoother. The parameter $r_{cc}$ is
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typically chosen as the point at which the core charge $\rho_c(r_{cc})$
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is twice as big as the valence charge $\rho_v(r_{cc})$. In fact the
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effect of nonlinearity is important only in regions where
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$\rho_c(r)\sim\rho_v(r)$. Alternatively, $r_{cc}$ can be provided
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in input, Note that the smaller $r_{cc}$, the more accurate the core
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correction, but also the harder the pseudized core charge, and vice versa.
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\subsection{Type of pseudization}
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\label{pseudization}
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The \texttt{atomic} package implements two different NC pseudization
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algorithms, both claiming to yield optimally smooth PP's:
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\begin{itemize}
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\item Troullier-Martins \cite{TM} (TM)
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\item Rappe-Rabe-Kaxiras-Joannopoulos \cite{RRKJ} (RRKJ).
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\end{itemize}
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Both algorithms replace atomic orbitals in the core region
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with smooth nodeless pseudo-orbitals. The TM method uses an
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exponential of a polynomial (see Appendix B); the RRKJ method
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uses three or four Bessel functions for the pseudo-orbitals in
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the core region. The former is very robust. The latter may
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occasionally fail to produce the required nodeless pseudo-orbital.
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If this happens, first try to force the usage of four Bessel functions
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(this is achieved by setting a small nonzero value of
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the charge density at the origin, variable \texttt{rho0}:
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unfortunately it works only for $s$ states).
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Second-row elements N, O, F, $3d$ transition metals, rare earths,
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are typically ``hard'' atoms, i.e. described by NC PP's requiring
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a high PW cutoff. These atoms are characterized by $2p$ (N, O, F),
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$3d$ (transition metals), $4f$ (rare earths) valence states with no
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orthogonalization to core states of the same $l$ and no nodes.
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In addition, as mentioned in Secs.\ref{ValCore} and \ref{RefConf},
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there are case in which you may be forced to include semicore states
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in valence, thus making the PP hard (or even harder).
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In all such cases, one should consider
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{\em ultrasoft} pseudization, unless there is a good reason to stick
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to NC-PP's. For the specific case of rare earths, however, remember
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that the problem of DFT reliability preempts the (tough) problem of
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generating a PP. With US-PP's one can give up the NC requirement
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and get much software PP's, at the price of introducing an augmentation
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charge that compensates for the missing charge.
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Currently, the \texttt{atomic} package generates US-PP's on top of
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a ``hard'' NC-PP. In order to ensure sufficient transferability,
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at least two states per angular momentum $l$ are required.
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\subsubsection{Pseudization energies}
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\label{pseudiz}
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If you stick to single-projector PP's (one potential per angular momentum
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$l$, i.e. one projector per $l$ in the separable form), the choice of the
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electronic configuration automatically determines the reference states
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to pseudize: for each $l$, the bound valence eigenstate is pseudized
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at the corresponding eigenvalue. If no bound valence eigenstate exists,
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one has to select a reference energy. The choice is rather arbitrary:
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you may try something between than other valence bound state energies
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and zero.
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If you have semicore states in valence, remember that for each $l$
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only the state with lowest $n$ can be used to generate a single-projector
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PP. The \texttt{atomic} package requires that you explicitly specify the
|
|
configuration for unscreening in the ``test'' configuration:
|
|
see the detailed input documentation.
|
|
|
|
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 $l$, resulting in a PP with more than one
|
|
projector per $l$. Note that the resulting PP is directly produced
|
|
in the separable form.
|
|
|
|
Such possibility considerably extends the transferability, but also the
|
|
number of ``degrees of freedom'' in the generation of a PP. As a rule of
|
|
thumb:
|
|
\begin{itemize}
|
|
\item start with one projector per $l$, at the energy of the bound state;
|
|
\item add a second projector at a second, higher energy not too far above
|
|
the former.
|
|
\end{itemize}
|
|
The latter hint seems to be counterintuitive if you have semicore states
|
|
in valence: the obvious choice would be to include two projectors, using
|
|
both bound states energies. Unfortunately this is currently not possible
|
|
with the \texttt{atomic} package, because the state pseudized at the upper
|
|
bound-state energy will necessarily have a node (in order to be orthogonal
|
|
to the one with lower energy).
|
|
|
|
For US-PP pseudization, you should first find a suitable multi-projector
|
|
NC-PP as above, than turn on the US-PP pseudization. This is done by
|
|
setting different NC and US pseudization radii (see Sec.\ref{radii}),
|
|
|
|
\subsubsection{Pseudization radii}
|
|
\label{radii}
|
|
|
|
For NC pseudization, one has to choose, for each state to be pseudized,
|
|
a NC pseudization radius $r_c$, at which the AE orbital and the
|
|
corresponding NC-PP orbital match, with continuous first derivative
|
|
at $r=r_c$. For bound states, $r_c$ is typically at the outermost peak or
|
|
somewhat larger. The larger the $r_c$, the softer the potential
|
|
(less PW needed in the calculations), but also the less transferable.
|
|
The $r_c$ may differ for different $l$; as a rule, one should avoid large
|
|
differences between the $r_c$'s, but this is not always possible. Also,
|
|
the $r_c$ cannot be smaller than the outermost node.
|
|
|
|
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 $r_c$ outwards in order to have reasonable results
|
|
with a reasonable PW cutoff''. has no clear-cut answer. The choice of $r_c$
|
|
at the outermost maximum for ``difficult'' elements (those described in
|
|
Sec.\ref{pseudiz}): typically 0.7-0.8 a.u, even less for $4f$ 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 ($2p$) elements, $r_c=1.1-1.2$
|
|
will yield reasonably good results for 50-70 Ry PW kinetic energy cutoff;
|
|
for $3d$ transition metals, the same $r_c$ will require $> 80$ Ry cutoff
|
|
(highest $l$ have slower convergence for the same $r_c$). The above
|
|
estimates are for TM pseudization. RRKJ pseudization will yield an
|
|
estimate of the required cutoff.
|
|
|
|
For multiple-projectors NC-PP's, the $r_c$ of unbound states may be chosen
|
|
in the same range as for bound states. If this is the first step for US-PP
|
|
generation, use small $r_c$ 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$\div$ 1.5 a.u. for second-row
|
|
$2p$ elements, 1.7$\div$ 2.2 a.u. for $3d$ transition metals), but do not
|
|
forget that the sum of the $r_c$ of two atoms should not exceed the typical
|
|
bond length of those atoms.
|
|
|
|
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.
|
|
|
|
\subsubsection{Choosing the local potential}
|
|
|
|
As explained in Sec. \ref{RefConf}, note 1, one needs in principle
|
|
angular momentum channels in PP's up to $l_c+1$. In the semilocal
|
|
form, the choice of a ''local'', $l$-independent potential is natural
|
|
and affects only seldom-important PW components with $l> l_c$.
|
|
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 $l$ components. Generally one exploits this
|
|
arbitrariness to remove one $l$ component using it as local potential.
|
|
The separable form can be either obtained by the Kleinman-Bylander
|
|
projection \cite{KB} applied to single-projector PP's, or directly
|
|
produced using Vanderbilt's procedure \cite{van} (for single-projector
|
|
PP's the two approaches are equivalent).
|
|
|
|
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 multiple-projectors
|
|
NC-PP's and in US-PP's as well.
|
|
|
|
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 $l$, because this removes
|
|
more projectors ($2l+1$ per atom) than for low $l$. 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 $l$. Another possibility is to generate
|
|
a local potential by pseudizing the AE potential.
|
|
|
|
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.\ref{trans})
|
|
|
|
\subsection{Generating the pseudopotential}
|
|
|
|
As a first step, one can generate AE Kohn-Sham orbitals and one-electron
|
|
levels for the reference configuration. This is done by using executable
|
|
\texttt{ld1.x}. You must specify in the input data:
|
|
\begin{quote}
|
|
atomic symbol,\\
|
|
electronic reference configuration,\\
|
|
exchange-correlation functional (default is LDA).
|
|
\end{quote}
|
|
A complete description of the input is contained in the documentation.
|
|
For accurate AE results in heavy atoms, you may want to specify a denser
|
|
radial grid in $r$-space than the default one. The default grid should
|
|
however be good enough for PP generation.
|
|
|
|
Before you proceed, it is a good idea to verify that the atomic data
|
|
you just produced actually make sense. Some kind souls have posted on
|
|
the web a complete set of reference atomic data :
|
|
\begin{quote}
|
|
{\tt http://physics.nist.gov/PhysRefData/DFTdata/ }
|
|
\end{quote}
|
|
These data have been obtained with the Vosko-Wilk-Nusair functional,
|
|
that for the unpolarized case is very similar to the Perdew-Zunger
|
|
LDA functional (this is hte LDA default).
|
|
|
|
The generation step is also done by program \texttt{ld1.x}.
|
|
One has to supply, in addition to AE data:
|
|
\begin{quote}
|
|
a list of orbitals to be pseudized, with pseudization energies and radii,\\
|
|
the filename where the newly generated PP is written,
|
|
\end{quote}
|
|
plus a number of other optional parameters,
|
|
fully described in the documentation.
|
|
|
|
\subsection{Checking for transferability}
|
|
\label{trans}
|
|
A simple way to check for correctness and to get a feeling for
|
|
the transferability of a PP, with little effort, is to test the
|
|
results of PP and AE atomic calculations on atomic configurations
|
|
differing from the starting one. The error on total energy
|
|
differences between PP and AE results gives a feeling on how
|
|
good the PP is. Just to give an idea: an error $\sim 0.001$ Ry
|
|
is very good, $\sim 0.01$ Ry may still be acceptable.
|
|
The code \texttt{ld1.x} has a ``testing'' mode in which it does
|
|
exactly the above operation. You provide the input PP file and
|
|
a number of test configurations.
|
|
|
|
You are advised to perform also the test with a basis set of spherical
|
|
Bessel functions $j_l (qr)$. In addition to revealing the presence of
|
|
``ghosts'', this test also gives an idea of the smoothness of the
|
|
potential: the dependence of energy levels upon the cutoff in the kinetic
|
|
energy is basically the same for the pseudo-atom in the basis of $j_l (qr)$'s
|
|
and for the same pseudo-atom in a solid-state calculation using PW's.
|
|
|
|
Another way to check for transferability is to compare AE and pseudo (PS)
|
|
logarithmic derivatives, also calculated by \texttt{ld1.x}. Typically
|
|
this comparison is done on the reference configuration,
|
|
but not necessarily so. You should supply on input:
|
|
\begin{enumerate}
|
|
\item[--] the radius $r_d$ at which logarithmic derivatives are
|
|
calculated ($r_d$ should be of the order of the
|
|
ionic or covalent radius, and larger than any of the $r_c$'s)
|
|
\item[--] the energy range $E_{min}, E_{max}$ and the number
|
|
of points for the plot. The energy range
|
|
should cover the typical valence one-electron energy
|
|
range expected in the targeted application of the PP.
|
|
\end{enumerate}
|
|
The files containing logarithmic derivatives can be easily read and
|
|
plotted using for instance the plotting program \texttt{gnuplot}
|
|
or \texttt{xmgrace}.
|
|
Sizable discrepancies between AE and PS logarithmic derivatives
|
|
are a sign of trouble (unless your energy range is too large or
|
|
not centered around the range of pseudization energies, of course).
|
|
|
|
Note that the above checks, based on atomic calculations only,
|
|
do not replace the usual checks (convergence tests, bond lengths,
|
|
etc) one has to perform in at least some simple solid-state or
|
|
molecular systems before starting a serious calculation.
|
|
|
|
\section{A worked example: Ti}
|
|
|
|
Let us consider the Ti atom: $Z=22$, electronic configuration:
|
|
$1s^2 2s^2 2p^6 3s^2 3p^6 3d^2 4s^2$, with PBE XC functional.
|
|
The input data for the AE calculation is simple:
|
|
\begin{verbatim}
|
|
&input
|
|
atom='Ti', dft='PBE', config='[Ar] 3d2 4s2 4p0'
|
|
/
|
|
\end{verbatim}
|
|
and yields the total energy and Kohn-Sham levels. Let us concentrate
|
|
on the outermost states:
|
|
\begin{verbatim}
|
|
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
|
|
\end{verbatim}
|
|
and on their spatial extension:
|
|
\begin{verbatim}
|
|
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
|
|
\end{verbatim}
|
|
Note that the $3d$ state has a small spatial extension, comparable to that of
|
|
$3s$ and $3p$ states and much smaller than for $4s$ and $4p$ states; the
|
|
$3d$ energy is instead comparable to that of $4s$ and $4p$ states and much
|
|
higher than the $3s$ and $3p$ energies.. Much of the chemistry of Ti is
|
|
determined by its $3d$ states. What should we do? We have the choice among
|
|
several possibilities:
|
|
\begin{enumerate}
|
|
\item single-projector NC-PP with 4 electrons in valence ($3d^2 4s^2$),
|
|
with nonlinear core correction;
|
|
\item single-projector NC-PP with 12 electrons in valence
|
|
($3s^2 3p^6 3d^2 4s^2$);
|
|
\item multiple-projector NC-PP with 12 electrons in valence;
|
|
\item US-PP with 12 electrons in valence;
|
|
\item US-PP with 4 electrons in valence and nonlinear core correction;
|
|
\item ...
|
|
\end{enumerate}
|
|
The PP of case 1) will be hard due to the presence of $3d$ states, and
|
|
its transferability may turn out not be sufficient for all purposes;
|
|
PP's for 2) and 3) will be even harder due to the presence of $3d$ and
|
|
semicore $3s$ and $3p$ states; PP 4) can be made soft, but generating
|
|
one is not trivial; PP 5) may suffer from insufficient transferability.
|
|
|
|
\subsection {Single-projector, norm-conserving, no semicore}
|
|
|
|
\subsubsection{Generation}
|
|
Let us start from the simplest case with the following input:
|
|
\begin{verbatim}
|
|
&input
|
|
atom='Ti', dft='PBE', config='[Ar] 3d2 4s2 4p0',
|
|
rlderiv=2.90, eminld=-2.0, emaxld=2.0, deld=0.01, nld=3,
|
|
iswitch=3
|
|
/
|
|
&inputp
|
|
pseudotype=1, nlcc=.true., lloc=1,
|
|
file_pseudopw='Ti.pbe-n-rrkj.UPF'
|
|
/
|
|
3
|
|
4S 1 0 2.00 0.00 2.9 2.9
|
|
3D 3 2 2.00 0.00 1.3 1.3
|
|
4P 2 1 0.00 0.00 2.9 2.9
|
|
\end{verbatim}
|
|
In the \texttt{\&input} namelist,
|
|
we specify the we want to generate a PP (\texttt{iswitch=3}) and to
|
|
calculate \texttt{nld=3} logarithmic derivatives at \texttt{rlderiv=2.90} a.u.
|
|
from the origin, in the energy range \texttt{eminld=-2.0} Ry to
|
|
\texttt{emaxld=2.0} Ry, in energy steps \texttt{deld=0.01} Ry
|
|
(note that these values will not affect PP generation).
|
|
In the \texttt{\&inputp} namelist, we specify the we want a single-projector,
|
|
NC-PP (\texttt{pseudotype=1}), with nonlinear core correction
|
|
(\texttt{nlcc=.true.}), using the $l=1$ channel as local (\texttt{lloc=1}).
|
|
The output PP will be written in UPF format to file \texttt{Ti.pbe-n-rrkj.UPF}
|
|
(following the {\sc quantum ESPRESSO} convention for PP names).
|
|
Following the two namelists, there is a list of states used for pseudization:
|
|
the 4S state, with pseudization radius $r_c=2.9$ a.u.; the 3D state,
|
|
$r_c=1.3$ a.u.; the 4P, $r_c=2.9$ a.u., listed as last because it is
|
|
the channel to be chosen as local potential.
|
|
|
|
There is nothing magic or especially deep in the choice of the radius
|
|
and energy range for logarithmic derivatives, of the local
|
|
potential and of pseudization radii: it is just a reasonable guess.
|
|
Running the input, one gets an error:
|
|
\begin{verbatim}
|
|
Wfc 4S rcut= 2.883 Estimated cut-off energy= 14.82 Ry
|
|
l= 0 Node at 0.71997236
|
|
This function has 1 nodes for 0 < r < 2.883
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
from compute_phi : error # 1
|
|
phi has nodes before r_c
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\end{verbatim}
|
|
This means that the 4S pseudized orbitals has one node. With RRKJ
|
|
pseudization (the default), this may occasionally happen. One can
|
|
either choose TM pseudization (\texttt{tm=.true.}) or set a small
|
|
value of $\rho(r=0)$ (e.g. \texttt{rho0=0.001}). Let us do the latter.
|
|
You should carefully look at the output, which will consists in
|
|
an all-electron calculation, followed by the pseudopotential generation
|
|
step, followed by a final test. In particular, notice this message
|
|
about the nonlinear core correction:
|
|
\begin{verbatim}
|
|
Computing core charge for nlcc:
|
|
|
|
r > 1.73 : true rho core
|
|
r < 1.73 : rho core = a sin(br)/r a= 2.40 b= 1.56
|
|
|
|
Integrated core pseudo-charge : 3.43
|
|
\end{verbatim}
|
|
(this is actually not an ideal situation: the pseudization radius for the
|
|
charge density should be smaller than all pseudization radii; in our
|
|
case, smaller than $r_c^{(min)} = r_c^{(l=2)}=1.3$ a.u.).
|
|
Also notice messages on pseudization:
|
|
\begin{verbatim}
|
|
Wfc 4S rcut= 2.883 Estimated cut-off energy= 5.32 Ry
|
|
Using 4 Bessel functions for this wfc, rho(0) = 0.001
|
|
This function has 0 nodes for 0 < r < 2.883
|
|
|
|
Wfc 3D rcut= 1.296 Estimated cut-off energy= 137.82 Ry
|
|
This function has 0 nodes for 0 < r < 1.296
|
|
\end{verbatim}
|
|
(note the large difference between the estimated cutoff for the $s$ and
|
|
the $d$ channel! Of course, it is only the latter the ``problem'' one
|
|
here); and look at the final consistency check:
|
|
\begin{verbatim}
|
|
n l nl e AE (Ry) e PS (Ry) De AE-PS (Ry)
|
|
1 0 4S 1( 2.00) -0.32830 -0.32830 0.00000
|
|
3 2 3D 1( 2.00) -0.31302 -0.31302 0.00000
|
|
2 1 4P 1( 0.00) -0.10777 -0.10777 0.00000
|
|
\end{verbatim}
|
|
You should get exactly 0 (within numerical accuracy) in the columnn
|
|
at the right.
|
|
|
|
As a further check, let's have a look at the logaritmic derivatives
|
|
and at pseudized Kohn-Sham orbitals. Logarithmic derivatives are written
|
|
to files \texttt{ld1.dlog} and \texttt{ld1ps.dlog}, for AE and PS
|
|
calculations respectively (file names can be changed using variable
|
|
\texttt{prefix}). They can be plotted using for instance
|
|
\texttt{gnuplot} and the following commands:
|
|
\begin{verbatim}
|
|
plot [-2:2][-20:20] 'ld1.dlog' u 1:2 w l lt 1, 'ld1.dlog' u 1:3 w l lt 2,\
|
|
'ld1.dlog' u 1:4 w l lt 3, 'ld1ps.dlog' u 1:2 lt 1, \
|
|
'ld1ps.dlog' u 1:3 lt 2, 'ld1ps.dlog' u 1:4 lt 3
|
|
\end{verbatim}
|
|
PS orbitals and the corresponding AE ones are written to file
|
|
\texttt{ld1ps.wfc} (PS on the left, AE on the right). They can be
|
|
plotted using the following commands:
|
|
\begin{verbatim}
|
|
plot [0:5] 'ld1ps.wfc' u 1:2 lt 1 , 'ld1ps.wfc' u 1:3 lt 3 , \
|
|
'ld1ps.wfc' u 1:4 lt 2 , 'ld1ps.wfc' u 1:5 lt 1 w l, \
|
|
'ld1ps.wfc' u 1:6 lt 3 w l, 'ld1ps.wfc' u 1:7 lt 2 w l
|
|
\end{verbatim}
|
|
One gets the following plots (AE=lines, PS=points;
|
|
\texttt{lt 1}=red=$s$; \texttt{lt 2}=green=$p$; \texttt{lt 3}=blue=$d$;
|
|
note that in the files, orbitals are ordered as given in input,
|
|
logarithmic derivatives as $s$, $p$, $d$).
|
|
|
|
\includegraphics[width=7.5cm]{pseudo-gen-fig1.pdf}
|
|
\includegraphics[width=7.5cm]{pseudo-gen-fig2.pdf}
|
|
|
|
We observe that our PP seems to reproduce fairly well
|
|
the logarithmic derivatives, with deviations appearing only at
|
|
relatively high ($> 1$ Ry) energies. AE and PS orbitals match
|
|
very well beyond the pseudization radii; the $3d$ orbital is
|
|
slightly deformed, because we have chosen a relatively large
|
|
$r_c^{(l=2)}=1.3$ a.u.. It is easy to verify that a smaller
|
|
$r_c^{(l=2)}$ yields a better $3d$ PS orbital, but also a harder
|
|
$d$ potential: e.g., for $r_c^{(l=2)}=1.0$ a.u., you get
|
|
\begin{verbatim}
|
|
Wfc 3D rcut= 1.009 Estimated cut-off energy= 225.64 Ry
|
|
\end{verbatim}
|
|
Before proceding, it is wise to verify whether our PP has ``ghosts''.
|
|
Let us prepare the following input for the testing phase
|
|
(note the variable \texttt{iswitch=2} and the \texttt{\&test}
|
|
namelist):
|
|
\begin{verbatim}
|
|
&input
|
|
atom='Ti', dft='PBE', config='[Ar] 3d2 4s2 4p0',
|
|
iswitch=2
|
|
/
|
|
&test
|
|
file_pseudo='Ti.pbe-n-rrkj.UPF',
|
|
nconf=1, configts(1)='3d2 4s2 4p0',
|
|
ecutmin=50, ecutmax=200, decut=50
|
|
/
|
|
\end{verbatim}
|
|
This will solve the Kohn-Sham equation for the PP read from
|
|
\texttt{file\_pseudo}, for a single valence configuration
|
|
(\texttt{nconf=1}) listed in \texttt{configts(1)} (the ground state
|
|
in this case), using a base of spherical waves whose cutoff
|
|
(in Ry) ranges from \texttt{ecutmin} to \texttt{ecutmax} in steps of
|
|
\texttt{decut}. The initial part of the output looks good, but let us
|
|
look at the test with spherical waves, towards the end:
|
|
\begin{verbatim}
|
|
Cutoff (Ry) : 200.0
|
|
N = 1 N = 2 N = 3
|
|
E(L=0) = -0.7483 Ry -0.3282 Ry -0.0042 Ry
|
|
E(L=1) = -0.1077 Ry 0.0192 Ry 0.0630 Ry
|
|
E(L=2) = -0.2961 Ry 0.0304 Ry 0.0654 Ry
|
|
\end{verbatim}
|
|
The lowest levels found in this way should be the same\footnote{actually
|
|
there are numerical differences, especially large for localized states
|
|
like $3d$, whose origin is under investigation}
|
|
as those calculated from radial integration (see above).
|
|
This is true for the $4p$ state (-0.1077 Ry),
|
|
for the $3d$ state (-0.2961 Ry vs -0.31302 Ry, see footnote),
|
|
for the $4s$ state (-0.3282 Ry)....but note the spurious $4s$
|
|
level at -0.7483 Ry! Our PP has a ghost and is unusable.
|
|
|
|
What should be do now? we may try to change the definition of the
|
|
local potential. We had chosen $l=1$, let us try $l=2$ and $l=0$.
|
|
The former has the same pathology, the latter has no ghosts.
|
|
So our data for PP generation are as follows:
|
|
\begin{verbatim}
|
|
&input
|
|
atom='Ti', dft='PBE', config='[Ar] 3d2 4s2 4p0',
|
|
rlderiv=2.90, eminld=-2.0, emaxld=2.0, deld=0.01, nld=3,
|
|
iswitch=3
|
|
/
|
|
&inputp
|
|
pseudotype=1, nlcc=.true., lloc=0,
|
|
file_pseudopw='Ti.pbe-n-rrkj.UPF',
|
|
/
|
|
3
|
|
4P 2 1 0.00 0.00 2.9 2.9
|
|
3D 3 2 2.00 0.00 1.3 1.3
|
|
4S 1 0 2.00 0.00 2.9 2.9
|
|
\end{verbatim}
|
|
(note \texttt{lloc=0} and the $4s$ state at the end of the list).
|
|
Let us plot again logarithmic derivatives and orbitals (they look
|
|
quite the same as before) and run again the test with spherical
|
|
waves. We get (see the last section in the output):
|
|
\begin{verbatim}
|
|
Cutoff (Ry) : 50.0
|
|
N = 1 N = 2 N = 3
|
|
E(L=0) = -0.3282 Ry -0.0049 Ry 0.0361 Ry
|
|
E(L=1) = -0.1077 Ry 0.0192 Ry 0.0630 Ry
|
|
E(L=2) = -0.1469 Ry 0.0311 Ry 0.0682 Ry
|
|
|
|
Cutoff (Ry) : 100.0
|
|
N = 1 N = 2 N = 3
|
|
E(L=0) = -0.3282 Ry -0.0049 Ry 0.0361 Ry
|
|
E(L=1) = -0.1077 Ry 0.0192 Ry 0.0630 Ry
|
|
E(L=2) = -0.2959 Ry 0.0303 Ry 0.0652 Ry
|
|
Cutoff (Ry) : 150.0
|
|
N = 1 N = 2 N = 3
|
|
E(L=0) = -0.3282 Ry -0.0049 Ry 0.0361 Ry
|
|
E(L=1) = -0.1077 Ry 0.0192 Ry 0.0630 Ry
|
|
E(L=2) = -0.2961 Ry 0.0303 Ry 0.0652 Ry
|
|
\end{verbatim}
|
|
This time the first column yields (with a small discrepancy for $3d$)
|
|
the expected levels, and only those levels. It is wise to inspect
|
|
the second column as well for absence of suspiciously low levels:
|
|
ghosts may appear also as spurious excited states close to occupied
|
|
states. Note how bad the energy for the $3d$ level is at 50 Ry.
|
|
At 100 Ry however we are close to convergence and at 150 Ry
|
|
well converged, in agreement with the estimate given during
|
|
the PP generation (138 Ry).
|
|
|
|
We have now our first candidate (i.e. not surely wrong) PP.
|
|
In order to 1) verify if it really does the job, 2) quantify
|
|
its transferability, 3) quantify its hardness, and 4) improve
|
|
it, if possible, we need to perform some more testing.
|
|
|
|
\subsubsection{Testing}
|
|
|
|
As a first idea of how good our PP is, let us verify how it
|
|
behaves on differente electronic configuration. The code
|
|
allows to test several configurations in the following way:
|
|
\begin{verbatim}
|
|
&input
|
|
atom='Ti', dft='PBE', config='[Ar] 3d2 4s2 4p0',
|
|
iswitch=2
|
|
/
|
|
&test
|
|
file_pseudo='Ti.pbe-n-rrkj.UPF',
|
|
nconf=9
|
|
configts(1)='3d2 4s2 4p0'
|
|
configts(2)='3d2 4s1 4p1'
|
|
configts(3)='3d2 4s1 4p0'
|
|
configts(4)='3d2 4s0 4p0'
|
|
configts(5)='3d1 4s2 4p1'
|
|
configts(6)='3d1 4s2 4p0'
|
|
configts(7)='3d1 4s1 4p0'
|
|
configts(8)='3d1 4s0 4p0'
|
|
configts(9)='3d0 4s0 4p0'
|
|
/
|
|
\end{verbatim}
|
|
here we have chosen 9 different valence configurations
|
|
(the corresponding AE configurations are obtained by
|
|
superimposing \texttt{configts} to core states in \texttt{config}).
|
|
Some of them are neutral, some are ionic, the first five leave
|
|
the $3d$ states unchanged, the last one is a completely ionized
|
|
Ti$^{4+}$. For each configuration, the code writes results
|
|
(e.g. orbitals) into files \texttt{ld1}$N.*$ and \texttt{ld1ps}$N.*$,
|
|
where $N$ is the index of the configuration. A summary is written to
|
|
file \texttt{ld1.test}. For the first configuration, AE and PS
|
|
eigenvalues and total energies are written:
|
|
\begin{verbatim}
|
|
3 2 3D 1( 2.00) -0.31302 -0.31302 0.00000
|
|
1 0 4S 1( 2.00) -0.32830 -0.32830 0.00000
|
|
2 1 4P 1( 0.00) -0.10777 -0.10777 0.00000
|
|
Etot = -1707.131006 Ry, -853.565503 Ha, -23226.698556 eV
|
|
Etotps = -9.748745 Ry, -4.874372 Ha, -132.638416 eV
|
|
\end{verbatim}
|
|
(AE and PS eigenvalues are in this case the same, since this is the
|
|
reference configuration used to build the PP). For the following
|
|
configurations, AE and PS eigenvalues, plus total energy
|
|
{\em differences}\footnote{Reminder: absolute PS total energies
|
|
depend upon the specific PP! Only energy differences are significant.}
|
|
wrt configuration 1 are printed:
|
|
\begin{verbatim}
|
|
3 2 3D 1( 2.00) -0.40319 -0.40457 0.00138
|
|
1 0 4S 1( 1.00) -0.38394 -0.38420 0.00026
|
|
2 1 4P 1( 1.00) -0.15248 -0.15237 -0.00011
|
|
dEtot_ae = 0.226061 Ry
|
|
dEtot_ps = 0.226250 Ry, Delta E= -0.000189 Ry
|
|
\end{verbatim}
|
|
The discrepancy between AE and PS energy differences (in this case,
|
|
wrt the ground state) as well as the discrepancies in AE and PS
|
|
eigenvalues, are a measure of how transferrable a PP is. In this case,
|
|
the AE-PS discrepancy on $\delta E = E(4s^14p^13d^2) - E(4s^24p^03d^2)$
|
|
(look at \texttt{Delta E}) is quite small, $<0.2$ mRy, while the
|
|
maximum discrepancy of the eigenvalues (rightmost column) $\sim 1$ mRy.
|
|
These are very good results.
|
|
Unfortunately this is also a configuration that doesn't differ much
|
|
from the reference one. Let us see the other cases:
|
|
\begin{verbatim}
|
|
3 2 3D 1( 2.00) -0.83550 -0.83256 -0.00295
|
|
1 0 4S 1( 1.00) -0.76075 -0.76163 0.00088
|
|
2 1 4P 1( 0.00) -0.48549 -0.48617 0.00068
|
|
dEtot_ae = 0.539968 Ry
|
|
dEtot_ps = 0.540344 Ry, Delta E= -0.000376 Ry
|
|
|
|
3 2 3D 1( 2.00) -1.44648 -1.44538 -0.00110
|
|
1 0 4S 1( 0.00) -1.24186 -1.24652 0.00465
|
|
2 1 4P 1( 0.00) -0.91224 -0.91599 0.00375
|
|
dEtot_ae = 1.537516 Ry
|
|
dEtot_ps = 1.540285 Ry, Delta E= -0.002769 Ry
|
|
|
|
3 2 3D 1( 1.00) -0.68514 -0.74236 0.05722
|
|
1 0 4S 1( 2.00) -0.45729 -0.45802 0.00073
|
|
2 1 4P 1( 1.00) -0.18855 -0.18471 -0.00383
|
|
dEtot_ae = 0.343391 Ry
|
|
dEtot_ps = 0.371650 Ry, Delta E= -0.028259 Ry
|
|
|
|
3 2 3D 1( 1.00) -1.16621 -1.21438 0.04817
|
|
1 0 4S 1( 2.00) -0.87720 -0.87620 -0.00100
|
|
2 1 4P 1( 0.00) -0.56807 -0.56137 -0.00670
|
|
dEtot_ae = 0.716203 Ry
|
|
dEtot_ps = 0.739110 Ry, Delta E= -0.022907 Ry
|
|
|
|
3 2 3D 1( 1.00) -1.82248 -1.87471 0.05223
|
|
1 0 4S 1( 1.00) -1.39447 -1.39936 0.00489
|
|
2 1 4P 1( 0.00) -1.03942 -1.03465 -0.00476
|
|
dEtot_ae = 1.848995 Ry
|
|
dEtot_ps = 1.873240 Ry, Delta E= -0.024245 Ry
|
|
|
|
3 2 3D 1( 1.00) -2.54976 -2.61959 0.06983
|
|
1 0 4S 1( 0.00) -1.94361 -1.96745 0.02383
|
|
2 1 4P 1( 0.00) -1.53584 -1.54419 0.00835
|
|
dEtot_ae = 3.518170 Ry
|
|
dEtot_ps = 3.554733 Ry, Delta E= -0.036564 Ry
|
|
|
|
3 2 3D 1( 0.00) -3.84145 -3.95251 0.11106
|
|
1 0 4S 1( 0.00) -2.73793 -2.81405 0.07612
|
|
2 1 4P 1( 0.00) -2.25938 -2.28768 0.02831
|
|
dEtot_ae = 6.699594 Ry
|
|
dEtot_ps = 6.831938 Ry, Delta E= -0.132344 Ry
|
|
\end{verbatim}
|
|
It is evident that configurations with $3d^2$ occupancy are well
|
|
reproduced, with errors on total energy differences $<3$ mRy and
|
|
on eigenvalues$<5$ mRy. Configurations with different $3d$ occupancy,
|
|
however, have errors one order of magnitude higher. For the extreme
|
|
case of Ti$^{4+}$, the error is $\sim$ 0.1 Ry.
|
|
|
|
In order to better understand what is going on, let us have a
|
|
look at the AE vs PS orbitals and logarithmic derivatives for
|
|
configuration 9 (i.e. for the bare PP). Let us add a line
|
|
like this:
|
|
\begin{verbatim}
|
|
rlderiv=2.90, eminld=-4.0, emaxld=0.0, deld=0.01, nld=3,
|
|
\end{verbatim}
|
|
and plot files \texttt{ld19ps.wfc}, \texttt{ld19.dlog},
|
|
\texttt{ld19ps.dlog} using \texttt{gnuplot} as above :
|
|
|
|
\includegraphics[width=7.5cm]{pseudo-gen-fig3.pdf}
|
|
\includegraphics[width=7.5cm]{pseudo-gen-fig4.pdf}
|
|
|
|
Both the orbitals and the logarithmic derivatives (note the
|
|
different energy range) start to exhibit some visible
|
|
discrepancy now.
|
|
|
|
One can try to fiddle with all generation parameters,
|
|
better if one at the time, to see whether things improve.
|
|
Curiously enough, the pseudization radius for the core
|
|
correction, which in principle should be as small as
|
|
possible, seems to improve things if pushed slightly
|
|
outwards (try \texttt{rcore=2.0}). Also surprisingly,
|
|
a smaller pseudization radius for the $3d$ state, 0.9
|
|
or 1.0 a.u., doesn't bring any visible improvement
|
|
to transferability
|
|
(but it increases a lot the required cutoff!).
|
|
Changing the pseudization radii for $4s$ and $4p$ states
|
|
doesn't affect much the results.
|
|
|
|
A different local potential -- a pseudized version
|
|
of the total self-consistent potential -- can be chosen
|
|
by setting \texttt{lloc=-1} and setting \texttt{rcloc}
|
|
to the desired pseudization radius (a.u.). For small
|
|
\texttt{rcloc} ghosts re-appear; \texttt{rcloc=2.9}
|
|
yields slighty better total energy differences but slightly worse
|
|
eigenvalues. Note that the PP so generated will also have a
|
|
$s$ projector, while the previous
|
|
ones had only $p$ and $d$ projectors.
|
|
|
|
One could also generate the PP from a different electronic
|
|
configuration. Since Ti tends to lose rather than to attract
|
|
electrons, it will be more easily found in a ionized state than
|
|
in the neutral one. One might for instance use the electronic
|
|
configuration of the Bachelet-Hamann-Schl\"uter paper\cite{BHS}:
|
|
$3d^2 4s^{0.75} 4p^{0.25}$. This however doesn't seem to improve
|
|
much.
|
|
|
|
Finally we end up with these generation data:
|
|
\begin{verbatim}
|
|
&input
|
|
atom='Ti', dft='PBE', config='[Ar] 3d2 4s2 4p0',
|
|
iswitch=3
|
|
/
|
|
&inputp
|
|
pseudotype=1, nlcc=.true., rcore=2.0, lloc=0,
|
|
file_pseudopw='Ti.pbe-n-rrkj.UPF'
|
|
/
|
|
3
|
|
4P 2 1 0.00 0.00 2.9 2.9
|
|
3D 3 2 2.00 0.00 1.3 1.3
|
|
4S 1 0 2.00 0.00 2.9 2.9
|
|
\end{verbatim}
|
|
|
|
\subsection {Single-projector, norm-conserving, with semicore states}
|
|
|
|
The results of transferability tests suggest that a Ti PP with only
|
|
$3d$, $4s$, $4p$ states have limited transferability to cases with
|
|
different $3d$ configurations. In order to improve it, a possible
|
|
way is to put semicore $3s$ and $3p$ states in valence. The maximum
|
|
for those states (0.87 a.u. and 0.90 a.u. respectively) is in the
|
|
same range as for $3d$ (0.98 a.u.). Let us try thus the following:
|
|
\begin{verbatim}
|
|
&input
|
|
atom='Ti', dft='PBE', config='[Ar] 3d2 4s2 4p0',
|
|
rlderiv=2.90, eminld=-4.0, emaxld=2.0, deld=0.01, nld=3,
|
|
iswitch=3
|
|
/
|
|
&inputp
|
|
pseudotype=1, rho0=0.001, ...
|
|
file_pseudopw='Ti.pbe-sp-rrkj.UPF'
|
|
/
|
|
3
|
|
3S 1 0 2.00 0.00 1.1 1.1
|
|
3P 2 1 6.00 0.00 1.2 1.2
|
|
3D 3 2 2.00 0.00 1.3 1.3
|
|
&test
|
|
configts(1)='3s2 3p6 3d2 4s2 4p0',
|
|
/
|
|
\end{verbatim}
|
|
Note the presence of the \texttt{\&test} namelist: it is used in this
|
|
context to supply the electronic valence configuration, to be used
|
|
for unscreening. As a first step, we do not include the core correction.
|
|
In place of the dots we should specify the local reference potential.
|
|
If we use \texttt{lloc=-1} with large values of \texttt{rcloc},
|
|
(comparable to pseudization radii for the previous case)
|
|
we get all kinds of mysterious errors:
|
|
\begin{verbatim}
|
|
from compute_chi : error # 1
|
|
n is too large
|
|
\end{verbatim}
|
|
for \texttt{rcloc=2.5}, while \texttt{rcloc=2.7} produces an equally
|
|
mysterious
|
|
\begin{verbatim}
|
|
from run_pseudo : error # 1
|
|
Errors in PS-KS equation
|
|
\end{verbatim}
|
|
while smaller values (e.g. 1.5) lead to other errors:
|
|
\begin{verbatim}
|
|
WARNING! Expected number of nodes: 0 = 2-1-1, number of nodes found: 1.
|
|
\end{verbatim}
|
|
Even if the code doesn't stop, the presence of such messages
|
|
is a signal of something going wrong in the generation algorithm.
|
|
With some more experiments, though, one finds that \texttt{rcloc=1.3}
|
|
yields a good potential. We still have other choices. In this case,
|
|
$d$ as reference potential: \texttt{lloc=2}, seems to work as well
|
|
(and produces a PP with less projectors: only $s$ and $p$).
|
|
The generation algorithm in the latter case yields these
|
|
results for Kohn-Sham energies:
|
|
\begin{verbatim}
|
|
n l nl e AE (Ry) e PS (Ry) De AE-PS (Ry)
|
|
1 0 3S 1( 2.00) -4.60347 -4.60348 0.00001
|
|
2 1 3P 1( 6.00) -2.85621 -2.85623 0.00002
|
|
3 2 3D 1( 2.00) -0.31302 -0.31301 -0.00001
|
|
2 0 4S 1( 2.00) -0.32830 -0.32892 0.00062
|
|
3 1 4P 1( 0.00) -0.10777 -0.10732 -0.00045
|
|
\end{verbatim}
|
|
Note that the $3s$, $3p$, $3d$ levels should be the same by construction
|
|
(the difference is numerical noise); the $4s$ and $4p$ levels are not
|
|
guaranteed to be the same. The fact that they are, to a very good degree,
|
|
is very reassuring. A look at the orbitals will reveal that $3s, 3p, 3d$
|
|
are nodeless, $4s$ and $4p$ have one node. The spherical wave basis
|
|
set confirms the absence of ghosts:
|
|
\begin{verbatim}
|
|
Cutoff (Ry) : 50.0
|
|
N = 1 N = 2 N = 3
|
|
E(L=0) = -4.5385 Ry -0.3263 Ry -0.0047 Ry
|
|
E(L=1) = -2.8427 Ry -0.1071 Ry 0.0193 Ry
|
|
E(L=2) = -0.1511 Ry 0.0311 Ry 0.0685 Ry
|
|
|
|
Cutoff (Ry) : 100.0
|
|
N = 1 N = 2 N = 3
|
|
E(L=0) = -4.5883 Ry -0.3279 Ry -0.0048 Ry
|
|
E(L=1) = -2.8547 Ry -0.1073 Ry 0.0193 Ry
|
|
E(L=2) = -0.2918 Ry 0.0303 Ry 0.0649 Ry
|
|
|
|
Cutoff (Ry) : 150.0
|
|
N = 1 N = 2 N = 3
|
|
E(L=0) = -4.5899 Ry -0.3280 Ry -0.0048 Ry
|
|
E(L=1) = -2.8549 Ry -0.1073 Ry 0.0193 Ry
|
|
E(L=2) = -0.2936 Ry 0.0303 Ry 0.0649 Ry
|
|
\end{verbatim}
|
|
Note that for $l=0$ the first ($N=1$) level is the $3s$ level,
|
|
the second ($N=2$) level is the $4s$ level, and the like for
|
|
$l=1$. Let us now repeat the testing on the nine
|
|
selected configurations as for the 4-electron PP. You will
|
|
have to add \texttt{3s2 3p6} to all test configurations
|
|
\texttt{configts}. Let us see check the errors on total energy
|
|
differences:
|
|
\begin{verbatim}
|
|
$ grep Delta ld1.test
|
|
dEtot_ps = 0.227291 Ry, Delta E= -0.001230 Ry
|
|
dEtot_ps = 0.540886 Ry, Delta E= -0.000918 Ry
|
|
dEtot_ps = 1.540155 Ry, Delta E= -0.002640 Ry
|
|
dEtot_ps = 0.343314 Ry, Delta E= 0.000077 Ry
|
|
dEtot_ps = 0.715061 Ry, Delta E= 0.001142 Ry
|
|
dEtot_ps = 1.849816 Ry, Delta E= -0.000820 Ry
|
|
dEtot_ps = 3.522904 Ry, Delta E= -0.004735 Ry
|
|
dEtot_ps = 6.702626 Ry, Delta E= -0.003032 Ry
|
|
\end{verbatim}
|
|
Energy differences are reproduced with an
|
|
error that does not exceed a few mRy (see column at the rhs).
|
|
Eigenvalues are also well reproduced, e.g.:
|
|
\begin{verbatim}
|
|
1 0 3S 1( 2.00) -8.37382 -8.37230 -0.00152
|
|
2 1 3P 1( 6.00) -6.57173 -6.57195 0.00021
|
|
3 2 3D 1( 0.00) -3.84145 -3.83518 -0.00627
|
|
2 0 4S 1( 0.00) -2.73793 -2.74985 0.01192
|
|
3 1 4P 1( 0.00) -2.25938 -2.25525 -0.00412
|
|
\end{verbatim}
|
|
although errors may reach 0.01 Ry (still one order of magnitude
|
|
better than what we get with the previous 4-electron PP). The price
|
|
to pay is the presence of more electrons in the valence.
|
|
\newpage
|
|
\subsection{Testing in molecules and solids}
|
|
|
|
Even if our PP looks good
|
|
(or not too bad) on paper based on the results of atomic calculations,
|
|
it is always a good idea to test it in simple molecular or solid-state
|
|
systems, for which all-electron data (i.e. calculations performed with
|
|
the same XC functional but without PP's, such as e.g. FLAPW, LMTO,
|
|
Quantum Chemstry calculations) is available. The comparison with
|
|
experiments is of course interesting, but the goal of PP's is to
|
|
reproduce AE data, not to improve DFT.
|
|
|
|
|
|
\appendix
|
|
|
|
\section{Atomic Calculations}
|
|
|
|
\subsection{Nonrelativistic case}
|
|
|
|
Let us assume that the charge density $n(r)$ and the potential $V(r)$
|
|
are spherically symmetric. The Kohn-Sham (KS) equation:
|
|
\begin{equation}
|
|
\left(-{\hbar^2\over 2m}\nabla^2 + V(r)-\epsilon\right) \psi(\r)=0
|
|
\end{equation}
|
|
can be written in spherical coordinates. We write the wavefunctions as
|
|
\begin{equation}
|
|
\psi(\r)=\left({R_{nl}(r)\over r}\right)Y_{lm}(\hat\r),
|
|
\end{equation}
|
|
where $n$ is the main quantum
|
|
number $l=n-1,n-2,\dots,0$ is angular momentum, $m=l,l-1,\dots,-l+1,-l$
|
|
is the projection of the angular momentum on some axis.
|
|
The radial KS equation becomes:
|
|
\begin{eqnarray}
|
|
\left(-{\hbar^2\over 2m} {1\over r} {d^2 R_{nl}(r)\over dr^2}
|
|
+(V(r)-\epsilon) {1\over r} R_{nl}(r)
|
|
\right) Y_{lm}(\hat\r) \hskip 4cm \nonumber \\ \mbox{} -
|
|
{\hbar^2\over 2m} \left({1\over\mbox{sin}\theta}{\partial\over\partial\theta}
|
|
\left(\mbox{sin}\theta{\partial Y_{lm}(\hat\r)
|
|
\over\partial\theta}\right)
|
|
+ {1\over\mbox{sin}^2\theta}
|
|
{\partial^2Y_{lm}(\hat\r)\over\partial\phi^2}
|
|
\right) {1\over r^3} R_{nl}(r) = 0.
|
|
\end{eqnarray}
|
|
This yields an angular equation for the spherical harmonics $Y_{lm}(\hat\r)$:
|
|
\begin{equation}
|
|
-\left({1\over\mbox{sin}\theta}{\partial\over\partial\theta}
|
|
\left(\mbox{sin}\theta{\partial Y_{lm}(\hat\r)\over\partial\theta}
|
|
\right)
|
|
+ {1\over\mbox{sin}^2\theta}
|
|
{\partial^2Y_{lm}(\hat\r)\over\partial\phi^2}
|
|
\right) = l(l+1)Y_{lm}(\hat\r)
|
|
\end{equation}
|
|
and a radial equation for the radial part $R_{nl}(r)$:
|
|
\begin{equation}
|
|
-{\hbar^2\over 2m} {d^2 R_{nl}(r)\over dr^2}
|
|
+\left( {\hbar^2\over 2m} {l(l+1)\over r^2} + V(r)-\epsilon
|
|
\right) R_{nl}(r) = 0.
|
|
\end{equation}
|
|
The charge density is given by
|
|
\begin{equation}
|
|
n(r) = \sum_{nlm} \Theta_{nl}\left |{R_{nl}(r)\over r}Y_{lm}(\hat r)\right|^2
|
|
= \sum_{nl} \Theta_{nl}{R^2_{nl}(r)\over 4\pi r^2}
|
|
\end{equation}
|
|
where $\Theta_{nl}$ are the occupancies ($\Theta_{nl}\le 2l+1$)
|
|
and it is assumed that the occupancies of $m$ are such as to yield
|
|
a spherically symmetric charge density (which is true only for closed
|
|
shell atoms).
|
|
\subsubsection{Useful formulae}
|
|
|
|
Gradient in spherical coordinates $(r,\theta,\phi)$:
|
|
\begin{equation}
|
|
\nabla\psi = \left({\partial\psi\over\partial r},
|
|
{1\over r}{\partial\psi\over\partial\theta},
|
|
{1\over r \mbox{sin}\theta}
|
|
{\partial\psi\over\partial\phi}
|
|
\right)
|
|
\end{equation}
|
|
Laplacian in spherical coordinates:
|
|
\begin{equation}
|
|
\nabla^2\psi = {1\over r} {\partial^2\over\partial r^2}(r\psi)
|
|
+ {1\over r^2\mbox{sin}\theta} {\partial\over\partial\theta}
|
|
\left(\mbox{sin}\theta{\partial\psi\over\partial\theta}\right)
|
|
+ {1\over r^2\mbox{sin}^2\theta}
|
|
{\partial^2\psi\over\partial\phi^2}
|
|
\end{equation}
|
|
|
|
\subsection{Fully relativistic case}
|
|
|
|
The relativistic KS equations are
|
|
Dirac-like equations for a spinor with a ``large'' $R_{nlj}(r)$ and
|
|
a ``small'' $S_{nlj}(r)$ component:
|
|
\begin{eqnarray}
|
|
c\left({d \over dr} + {\kappa\over r}\right)R_{nlj}(r) & = &
|
|
\left(2mc^2 - V(r) + \epsilon \right)S_{nlj}(r)\\
|
|
c\left({d \over dr} - {\kappa\over r}\right)S_{nlj}(r) & = &
|
|
\left( V(r) + \epsilon \right) R_{nlj}(r)
|
|
\end{eqnarray}
|
|
where $j$ is the total angular momentum ($j=1/2$ if $l=0$,
|
|
$j=l+1/2,l-1/2$ otherwise); $\kappa=-2(j-l)(j+1/2)$ is the Dirac
|
|
quantum number ($\kappa=-1$ is $l=0$, $\kappa=-l-1,l$ otherwise);
|
|
and the charge density is given by
|
|
\begin{equation}
|
|
n(r) = \sum_{nlj} \Theta_{nlj}{R^2_{nlj}(r)+S^2_{nlj}(r)\over 4\pi r^2}.
|
|
\end{equation}
|
|
|
|
|
|
\subsection{Scalar-relativistic case}
|
|
|
|
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: $\hbar=1, m=1/2, e^2=2$):
|
|
\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}
|
|
where $\alpha=1/137.036$ is the fine-structure constant,
|
|
$\langle\kappa\rangle=-1$ is the degeneracy-weighted average value
|
|
of the Dirac's $\kappa$ for the two spin-orbit-split levels, $M(r)$ is
|
|
defined as
|
|
\begin{equation}
|
|
M(r)= 1 - {\alpha^2\over 4} \left(V(r)-\epsilon\right).
|
|
\end{equation}
|
|
The charge density is defined as in the nonrelativistic case:
|
|
\begin{equation}
|
|
n(r) = \sum_{nl} \Theta_{nl}{R^2_{nl}(r)\over 4\pi r^2}.
|
|
\end{equation}
|
|
|
|
\subsection{Numerical solution}
|
|
|
|
The radial (scalar-relativistic) KS equation is integrated
|
|
on a radial grid. It is convenient to
|
|
have a denser grid close to the nucleus and a coarser one far
|
|
away. Traditionally a logarithmic grid is used:
|
|
$r_i=r_0\mbox{exp}(i\Delta x)$. With this grid, one has
|
|
\begin{equation}
|
|
\int_0^\infty f(r) d r = \int_0^\infty f(x) r(x) dx
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
{d f(r)\over d r}={1\over r} {d f(x)\over d x},\qquad
|
|
{d^2 f(r)\over d r^2}=-{1\over r^2} {d f(x)\over d x}
|
|
+ {1\over r^2} {d^2 f(x)\over d x^2}.
|
|
\end{equation}
|
|
We start with a given self-consistent potential $V$ and
|
|
a trial eigenvalue $\epsilon$. The equation is integrated
|
|
from $r=0$ outwards to $r_t$, the outermost classical
|
|
(nonrelativistic for simplicity) turning point, defined
|
|
by $ {l(l+1) /r_t^2} + \left(V(r_t)-\epsilon\right)=0$.
|
|
In a logarithmic grid (see above) the equation to solve becomes:
|
|
\begin{eqnarray}
|
|
{1\over r^2} {d^2 R_{nl}(x)\over d x^2} & = &
|
|
{1\over r^2} {d R_{nl}(x)\over d x} +
|
|
\left( {l(l+1)\over r^2} +
|
|
M(r)\left(V(r)-\epsilon\right) \right) R_{nl}(r) \nonumber \\
|
|
& & \mbox{} - {\alpha^2\over 4M(r)} {dV(r)\over dr}
|
|
\left({1\over r} {dR_{nl}(x)\over dx} +
|
|
\langle\kappa\rangle {R_{nl}(r)\over r}\right).
|
|
\end{eqnarray}
|
|
This determines ${d^2 R_{nl}(x)/d x^2}$ which is used to
|
|
determine ${d R_{nl}(x)/ dx}$ which in turn is used to
|
|
determine $R_{nl}(r)$, using predictor-corrector or whatever
|
|
classical integration method. ${dV(r)/dr}$ is evaluated
|
|
numerically from any finite difference method. The series
|
|
is started using the known (?) asymptotic behavior of $R_{nl}(r)$
|
|
close to the nucleus (with ionic charge $Z$)
|
|
\begin{equation}
|
|
R_{nl}(r)\simeq r^\gamma,\qquad \gamma={l\sqrt{l^2-\alpha^2 Z^2}+
|
|
(l+1)\sqrt{(l+1)^2-\alpha^2 Z^2}\over 2l+1}.
|
|
\end{equation}
|
|
The number of nodes is counted. If there are too few (many)
|
|
nodes, the trial eigenvalue is increased (decreased) and
|
|
the procedure is restarted until the correct number $n-l-1$
|
|
of nodes is reached. Then a second integration is started
|
|
inward, starting from a suitably large $r\sim 10r_t$ down
|
|
to $r_t$, using as a starting point the asymptotic behavior
|
|
of $R_{nl}(r)$ at large $r$:
|
|
\begin{equation}
|
|
R_{nl}(r)\simeq e^{-k(r)r},\qquad
|
|
k(r)=\sqrt{{l(l+1)\over r^2} + \left(V(r)-\epsilon\right)}.
|
|
\end{equation}
|
|
The two pieces are continuously joined at $r_t$ and a correction to the trial
|
|
eigenvalue is estimated using perturbation theory (see below). The procedure
|
|
is iterated to self-consistency.
|
|
|
|
The perturbative estimate of correction to trial eigenvalues is described in
|
|
the following for the nonrelativistic case (it is not worth to make relativistic
|
|
corrections on top of a correction). The trial eigenvector $R_{nl}(r)$ will have
|
|
a cusp at $r_t$ if the trial eigenvalue is not a true eigenvalue:
|
|
\begin{equation}
|
|
A = {d R_{nl}(r_t^+)\over dr} - {d R_{nl}(r_t^-)\over dr} \ne 0.
|
|
\end{equation}
|
|
Such discontinuity in the first derivative translates into a
|
|
$\delta(r_t)$ in the second derivative:
|
|
\begin{equation}
|
|
{d^2 R_{nl}(r)\over dr^2} = {d^2 \widetilde R_{nl}(r)\over dr^2}
|
|
+ A \delta(r-r_t)
|
|
\end{equation}
|
|
where the tilde denotes the function obtained by matching the
|
|
second derivatives in the $r< r_t$ and $r> r_t$ regions.
|
|
This means that we are actually solving a different problem in which
|
|
$V(r)$ is replaced by $V(r)+\Delta V(r)$,
|
|
given by
|
|
\begin{equation}
|
|
\Delta V(r) = -{\hbar^2 \over 2m} {A\over R_{nl}(r_t)}\delta(r-r_t).
|
|
\end{equation}
|
|
The energy difference between the solution to such fictitious potential
|
|
and the solution to the real potential can be estimated from
|
|
perturbation theory:
|
|
\begin{equation}
|
|
\Delta\epsilon_{nl} = - \langle\psi|\Delta V |\psi\rangle
|
|
= {\hbar^2 \over 2m} R_{nl}(r_t) A.
|
|
\end{equation}
|
|
|
|
\section{Equations for the Troullier-Martins method}
|
|
|
|
We assume a pseudowavefunction $R^{ps}$ having the following form:
|
|
\begin{eqnarray}
|
|
R^{ps}(r)&=&r^{l+1}e^{p(r)} \quad r\le r_c \\
|
|
R^{ps}(r)&=&R(r) \quad r\ge r_c
|
|
\end{eqnarray}
|
|
where
|
|
\begin{equation}
|
|
p(r)=c_0+c_2r^2+c_4r^4+c_6r^6+c_8r^8+c_{10}r^{10}+c_{12}r^{12}.
|
|
\end{equation}
|
|
On this pseudowavefunction we impose the norm conservation
|
|
condition:
|
|
\begin{equation}
|
|
\int_{r<r_c} (R^{ps}(r))^2 dr=\int_{r<r_c} (R(r))^2 dr
|
|
\end{equation}
|
|
and continuity conditions on the wavefunction and its derivatives up
|
|
to order four at the matching point:
|
|
\begin{equation}
|
|
{d^nR^{ps}(r_c)\over dr^n}={d^nR(r_c)\over dr^n}, \quad n=0,...,4
|
|
\end{equation}
|
|
|
|
\noindent$\bullet$ Continuity of the wavefunction:
|
|
\begin{equation}
|
|
R^{ps}(r_c)=r_c^{l+1}e^{p(r_c)}=R(r_c)
|
|
\end{equation}
|
|
\begin{equation}
|
|
p(r_c) = \mbox{log}{R(r_c)\over r_c^{l+1}}
|
|
\end{equation}
|
|
|
|
\noindent$\bullet$ Continuity of the first derivative of the wavefunction:
|
|
\begin{equation}
|
|
{dR^{ps}(r)\over dr} = (l+1)r^le^{p(r)} + r^{l+1}e^{p(r)}p'(r)
|
|
={l+1\over r}R^{ps}(r) + p'(r)R^{ps}(r)
|
|
\end{equation}
|
|
that is
|
|
\begin{equation}
|
|
p'(r_c) = {dR(r_c)\over dr} {1\over R^{ps}(r_c)} -
|
|
{l+1\over r_c}.
|
|
\end{equation}
|
|
|
|
\noindent$\bullet$ Continuity of the second derivative of the wavefunction:
|
|
\begin{eqnarray}
|
|
{d^2R^{ps}(r)\over d^2r}
|
|
& = &
|
|
{d\over dr}\left((l+1)r^le^{p(r)} + r^{l+1}e^{p(r)}p'(r)\right)
|
|
\nonumber \\ & = &
|
|
l(l+1)r^{l-1}e^{p(r)} + 2(l+1)r^le^{p(r)}p'(r) +
|
|
r^{l+1}e^{p(r)}\left[p'(r)\right]^2 + r^{l+1}e^{p(r)}p''(r)
|
|
\nonumber\\ & = &
|
|
\left( {l(l+1)\over r^2}+ {2(l+1)\over r}p'(r) +
|
|
\left[p'(r)\right]^2 + p''(r) \right) r^{l+1}e^{p(r)}.
|
|
\end{eqnarray}
|
|
From the radial Schr\"odinger equation:
|
|
\begin{equation}
|
|
{d^2R^{ps}(r)\over dr^2} =
|
|
\left ( {l(l+1)\over r^2} +{2m\over\hbar^2}(V(r)-\epsilon)\right)R^{ps}(r)
|
|
\end{equation}
|
|
that is
|
|
\begin{equation}
|
|
p''(r_c) = {2m\over\hbar^2}(V(r_c)-\epsilon) - 2 {l+1\over r_c}p'(r_c)
|
|
-\left[p'(r_c)\right]^2
|
|
\end{equation}
|
|
|
|
\noindent$\bullet$ Continuity of the third and fourth derivatives of the
|
|
wavefunction. This is assured if the third and fourth derivatives of
|
|
$p(r)$ are continuous. By direct derivation of the expression of
|
|
$p''(r)$:
|
|
\begin{equation}
|
|
p'''(r_c) = {2m\over\hbar^2}V'(r_c) + 2 {l+1\over r^2_c}p'(r_c)
|
|
- 2{l+1\over r_c}p''(r_c) -2 p'(r_c)p''(r_c)
|
|
\end{equation}
|
|
|
|
\begin{eqnarray}
|
|
p''''(r_c) & = & {2m\over\hbar^2}V''(r_c) - 4 {l+1\over r^3_c}p'(r_c)
|
|
+ 4{l+1\over r^2_c}p''(r) \nonumber \\ & & \mbox{}
|
|
- 2{l+1\over r_c}p'''(r_c)
|
|
- 2 \left[p''(r_c)p''(r_c)\right]^2 -2 p'(r_c)p'''(r_c)
|
|
\end{eqnarray}
|
|
|
|
The additional condition: $V''(0)=0$ is imposed.
|
|
The screened potential is
|
|
\begin{eqnarray}
|
|
V(r)&=&{\hbar^2\over2m}\left({1\over R^{ps}(r)}{d^2R^{ps}(r)\over dr^2}
|
|
- {l(l+1)\over r^2}\right)+\epsilon \\
|
|
&=&{\hbar^2\over2m}\left( 2{l+1\over r} p'(r)+\left[p(r)\right]^2+p''(r)
|
|
\right)+\epsilon
|
|
\end{eqnarray}
|
|
|
|
Keeping only lower-order terms in $r$:
|
|
\begin{eqnarray}
|
|
V(r) &\simeq&{\hbar^2\over2m}\left( 2{l+1\over r} (2c_2r + 4c_4r^3)
|
|
+ 4c_2^2r^2 + 2c_2 + 12c_4r^2\right) + \epsilon \\
|
|
& = & {\hbar^2\over2m}\left( 2c_2(2l+3) +
|
|
\left((2l+5)c_4+c_2^2\right) r^2\right)+ \epsilon .
|
|
\end{eqnarray}
|
|
The additional constraint is:
|
|
\begin{equation}
|
|
(2l+5)c_4+c_2^2=0.
|
|
\end{equation}
|
|
|
|
|
|
|
|
\begin{thebibliography}{10}
|
|
|
|
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|
|
D.R. Hamann, M. Schl\"uter, and C. Chiang,
|
|
Phys. Rev. Lett. {\bf 43}, 1494 (1979).
|
|
|
|
\bibitem{van} D. Vanderbilt, Phys. Rev. B {\bf 47}, 10142 (1993).
|
|
|
|
\bibitem{PAW} P.~E. Bl\"ochl, Phys. Rev. B {\bf 50}, 17953 (1994).
|
|
|
|
\bibitem{BHS}
|
|
G.B. Bachelet, D.R. Hamann and M. Schl\"uter, Phys. Rev. B {\bf 26},
|
|
4199 (1982).
|
|
|
|
\bibitem{Gonze}
|
|
X. Gonze, R. Stumpf, and M. Scheffler, Phys. Rev. B {\bf 44}, 8503 (1991).
|
|
|
|
\bibitem{Goedecker}
|
|
S. Goedecker, M. Teter, and J. Hutter, Phys. Rev. B {\bf 54}, 1703 (1996).
|
|
|
|
\bibitem{TM} N. Troullier and J.L. Martins, Phys. Rev. B {\bf 43},
|
|
1993 (1991).
|
|
|
|
\bibitem{fhi98PP}
|
|
M.Fuchs and M. Scheffler, Comput. Phys. Commun. {\bf 119}, 67 (1999).
|
|
|
|
\bibitem{RRKJ} A. M. Rappe, K. M. Rabe, E. Kaxiras, and J. D. Joannopoulos,
|
|
Phys. Rev. B {\bf 41}, 1227 (1990)
|
|
(erratum: Phys. Rev. B {\bf 44}, 13175 (1991)).
|
|
|
|
\bibitem{KB} L. Kleinman and D.M. Bylander, Phys. Rev. Lett. 48, 1425 (1982).
|
|
|
|
\bibitem{CoreCorr}
|
|
S.G. Louie, S. Froyen, and M.L. Cohen, Phys. Rev. B {\bf 26}, 1738 (1982).
|
|
|
|
\bibitem{Zunger} S.H. Wei and A. Zunger, Phys. Rev. B {\bf 37}, 8958 (1987).
|
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|
|
\end{thebibliography}
|
|
|
|
\end{document}
|