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Some more stuff about pseudopotential generation
git-svn-id: http://qeforge.qe-forge.org/svn/q-e/trunk/espresso@6876 c92efa57-630b-4861-b058-cf58834340f0
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giannozz
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@ -742,9 +742,9 @@ calculations respectively (file names can be changed using variable
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\texttt{prefix}). They can be plotted using for instance
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\texttt{gnuplot} and the following commands:
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\begin{verbatim}
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plot [-2:2][-20:20] 'ld1.dlog' u 1:2 w l lt 1, 'ld1.dlog' u 1:3 w l lt 2, \
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'ld1.dlog' u 1:4 w l lt 3, 'ld1ps.dlog' u 1:2 lt 1, \
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'ld1ps.dlog' u 1:3 lt 2, 'ld1ps.dlog' u 1:4 lt 3
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plot [-2:2][-20:20] 'ld1.dlog' u 1:2 w l lt 1, 'ld1.dlog' u 1:3 w l lt 2,\
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'ld1.dlog' u 1:4 w l lt 3, 'ld1ps.dlog' u 1:2 lt 1, \
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'ld1ps.dlog' u 1:3 lt 2, 'ld1ps.dlog' u 1:4 lt 3
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\end{verbatim}
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PS orbitals and the corresponding AE ones are written to file
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\texttt{ld1ps.wfc} (PS on the left, AE on the right). They can be
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@ -762,20 +762,229 @@ logarithmic derivatives as $s$, $p$, $d$).
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\includegraphics[width=7.5cm]{pseudo-gen-fig1.pdf}
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\includegraphics[width=7.5cm]{pseudo-gen-fig2.pdf}
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We observe that our first PP seems to reproduced fairly well
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We observe that our PP seems to reproduce fairly well
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the logarithmic derivatives, with deviations appearing only at
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relatively high ($> 1$ Ry) energies. AE and PS orbitals match
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very well beyond the pseudization radii; the $3d$ orbital is
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slightly deformed, because we have chosen a relatively large
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$r_c^{(l=2)}$.
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$r_c^{(l=2)}=1.3$ a.u.. It is easy to verify that a smaller
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$r_c^{(l=2)}$ yields a better $3d$ PS orbital, but also a harder
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$d$ potential: e.g., for $r_c^{(l=2)}=1.0$ a.u., you get
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\begin{verbatim}
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Wfc 3D rcut= 1.009 Estimated cut-off energy= 225.64 Ry
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\end{verbatim}
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Before proceding, it is wise to verify whether our PP has ``ghosts''.
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Let us prepare the following input for the testing phase
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(note the variable \texttt{iswitch=2} and the \texttt{\&test}
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namelist):
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\begin{verbatim}
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&input
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atom='Ti', dft='PBE', config='[Ar] 3d2 4s2 4p0',
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iswitch=2
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/
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&test
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file_pseudo='Ti.pbe-rrkj.UPF',
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nconf=1, configts(1)='3d2 4s2 4p0',
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ecutmin=50, ecutmax=200, decut=50
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/
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\end{verbatim}
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This will solve the Kohn-Sham equation for the PP read from
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\texttt{file\_pseudo}, for a single valence configuration
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(\texttt{nconf=1}) listed in \texttt{configts(1)} (the ground state
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in this case), using a base of spherical waves whose cutoff
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(in Ry) ranges from \texttt{ecutmin} to \texttt{ecutmax} in steps of
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\texttt{decut}. The initial part of the output looks good, but let us
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look at the test with spherical waves, towards the end:
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\begin{verbatim}
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Cutoff (Ry) : 200.0
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N = 1 N = 2 N = 3
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E(L=0) = -0.7483 Ry -0.3282 Ry -0.0042 Ry
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E(L=1) = -0.1077 Ry 0.0192 Ry 0.0630 Ry
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E(L=2) = -0.2961 Ry 0.0304 Ry 0.0654 Ry
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\end{verbatim}
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The lowest levels found in this way should be the same\footnote{actually
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there are numerical differences, especially large for localized states
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like $3d$, whose origin is under investigation}
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as those calculated from radial integration (see above).
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This is true for the $4p$ state (-0.1077 Ry),
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for the $3d$ state (-0.2961 Ry vs -0.31302 Ry, see footnote),
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for the $4s$ state (-0.3282 Ry)....but note the spurious $4s$
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level at -0.7483 Ry! Our PP has a ghost and is unusable.
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What should be do now? we may try to change the definition of the
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local potential. We had chosen $l=1$, let us try $l=2$ and $l=0$.
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The former has the same pathology, the latter has no ghosts.
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So our data for PP generation are as follows:
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\begin{verbatim}
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&input
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atom='Ti', dft='PBE', config='[Ar] 3d2 4s2 4p0',
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rlderiv=2.90, eminld=-2.0, emaxld=2.0, deld=0.01, nld=3,
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iswitch=3
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/
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&inputp
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pseudotype=1, nlcc=.true., lloc=0,
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file_pseudopw='Ti.pbe-rrkj.UPF',
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/
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3
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4P 2 1 0.00 0.00 2.9 2.9
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3D 3 2 2.00 0.00 1.3 1.3
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4S 1 0 2.00 0.00 2.9 2.9
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\end{verbatim}
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(note \texttt{lloc=0} and the $4s$ state at the end of the list).
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Let us plot again logarithmic derivatives and orbitals (they look
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quite the same as before) and run again the test with spherical
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waves. We get (see the last section in the output):
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\begin{verbatim}
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Cutoff (Ry) : 50.0
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N = 1 N = 2 N = 3
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E(L=0) = -0.3282 Ry -0.0049 Ry 0.0361 Ry
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E(L=1) = -0.1077 Ry 0.0192 Ry 0.0630 Ry
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E(L=2) = -0.1469 Ry 0.0311 Ry 0.0682 Ry
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Cutoff (Ry) : 100.0
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N = 1 N = 2 N = 3
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E(L=0) = -0.3282 Ry -0.0049 Ry 0.0361 Ry
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E(L=1) = -0.1077 Ry 0.0192 Ry 0.0630 Ry
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E(L=2) = -0.2959 Ry 0.0303 Ry 0.0652 Ry
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Cutoff (Ry) : 150.0
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N = 1 N = 2 N = 3
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E(L=0) = -0.3282 Ry -0.0049 Ry 0.0361 Ry
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E(L=1) = -0.1077 Ry 0.0192 Ry 0.0630 Ry
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E(L=2) = -0.2961 Ry 0.0303 Ry 0.0652 Ry
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\end{verbatim}
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This time the first column yields (with a small discrepancy for $3d$)
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the expected levels, and only those levels. It is wise to inspect
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the second column as well for absence of suspiciously low levels:
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ghosts may appear also as spurious excited states close to occupied
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states. Note how bad the energy for the $3d$ level is at 50 Ry.
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At 100 Ry however we are close to convergence and at 150 Ry
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well converged, in agreement with the estimate given during
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the PP generation (138 Ry).
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We have now our first candidate (i.e. not surely wrong) PP.
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In order to 1) verify if it really does the job, 2) quantify
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its transferability, 3) quantify its hardness, and 4) improve
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it if possible, we need to perform some testing.
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it, if possible, we need to perform some more testing.
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\subsubsection{Testing}
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As a first idea of how good our PP is, let us verify how it
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behaves on differente electronic configuration. The code
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allows to test several configurations in the following way:
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\begin{verbatim}
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&input
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atom='Ti', dft='PBE', config='[Ar] 3d2 4s2 4p0',
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iswitch=2
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/
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&test
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file_pseudo='Ti.pbe-rrkj.UPF',
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nconf=10
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configts(1)='3d2 4s2 4p0'
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configts(2)='3d2 4s1 4p1'
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configts(3)='3d2 4s1 4p0'
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configts(4)='3d2 4s0 4p0'
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configts(5)='3d3 4s1 4p0'
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configts(6)='3d1 4s2 4p1'
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configts(7)='3d1 4s2 4p0'
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configts(8)='3d1 4s1 4p0'
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configts(9)='3d1 4s0 4p0'
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configts(10)='3d0 4s0 4p0'
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/
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\end{verbatim}
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here we have chosen 10 different valence configurations
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(the corresponding AE configurations are obtained by
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superimposing \texttt{configts} to core states in \texttt{config}).
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Some of them are neutral, some are ionic, the first five leave
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the $3d$ states unchanged, the last one is a completely ionized
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Ti$^{4+}$. For each configuration, the code writes results
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(e.g. orbitals) into files \texttt{ld1}$N.*$ and \texttt{ld1ps}$N.*$,
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where $N$ is the index of the configuration. A summary is written to
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file \texttt{ld1.test}. For the first configuration, AE and PS
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eigenvalues and total energies are written:
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\begin{verbatim}
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3 2 3D 1( 2.00) -0.31302 -0.31302 0.00000
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1 0 4S 1( 2.00) -0.32830 -0.32830 0.00000
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2 1 4P 1( 0.00) -0.10777 -0.10777 0.00000
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Etot = -1707.131006 Ry, -853.565503 Ha, -23226.698556 eV
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Etotps = -9.748745 Ry, -4.874372 Ha, -132.638416 eV
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\end{verbatim}
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(AE and PS eigenvalues are in this case the same, since this is the
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reference configuration used to build the PP). For the following
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configurations, AE and PS eigenvalues, plus total energy
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{\em differences}\footnote{Reminder: absolute PS total energies
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depend upon the specific PP! Only energy differences are significant.}
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wrt configuration 1 are printed:
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\begin{verbatim}
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3 2 3D 1( 2.00) -0.40319 -0.40457 0.00138
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1 0 4S 1( 1.00) -0.38394 -0.38420 0.00026
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2 1 4P 1( 1.00) -0.15248 -0.15237 -0.00011
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dEtot_ae = 0.226061 Ry
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dEtot_ps = 0.226250 Ry, Delta E= -0.000189 Ry
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\end{verbatim}
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The discrepancy between AE and PS energy differences (in this case,
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wrt the ground state) as well as the discrepancies in AE and PS
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eigenvalues, are a measure of how transferrable a PP is. In this case,
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the AE-PS discrepancy on $\delta E = E(4s^14p^13d^2) - E(4s^24p^03d^2)$
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(look at \texttt{Delta E}) is quite small, $<0.2$ mRy, while the
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maximum discrepancy of the eigenvalues (rightmost column) $\sim 1$ mRy.
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These are very good results.
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Unfortunately this is also a configuration that doesn't differ much
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from the reference one. Let us see the other cases:
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\begin{verbatim}
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3 2 3D 1( 2.00) -0.83550 -0.83256 -0.00295
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1 0 4S 1( 1.00) -0.76075 -0.76163 0.00088
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2 1 4P 1( 0.00) -0.48549 -0.48617 0.00068
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dEtot_ae = 0.539968 Ry
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dEtot_ps = 0.540344 Ry, Delta E= -0.000376 Ry
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3 2 3D 1( 2.00) -1.44648 -1.44538 -0.00110
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1 0 4S 1( 0.00) -1.24186 -1.24652 0.00465
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2 1 4P 1( 0.00) -0.91224 -0.91599 0.00375
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dEtot_ae = 1.537516 Ry
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dEtot_ps = 1.540285 Ry, Delta E= -0.002769 Ry
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3 2 3D 1( 3.00) -0.14077 -0.12814 -0.01263
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1 0 4S 1( 1.00) -0.27408 -0.28302 0.00894
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2 1 4P 1( 0.00) -0.08212 -0.08867 0.00655
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dEtot_ae = 0.081274 Ry
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dEtot_ps = 0.095152 Ry, Delta E= -0.013878 Ry
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3 2 3D 1( 1.00) -0.68514 -0.74236 0.05722
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1 0 4S 1( 2.00) -0.45729 -0.45802 0.00073
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2 1 4P 1( 1.00) -0.18855 -0.18471 -0.00383
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dEtot_ae = 0.343391 Ry
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dEtot_ps = 0.371650 Ry, Delta E= -0.028259 Ry
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3 2 3D 1( 1.00) -1.16621 -1.21438 0.04817
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1 0 4S 1( 2.00) -0.87720 -0.87620 -0.00100
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2 1 4P 1( 0.00) -0.56807 -0.56137 -0.00670
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dEtot_ae = 0.716203 Ry
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dEtot_ps = 0.739110 Ry, Delta E= -0.022907 Ry
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3 2 3D 1( 1.00) -1.82248 -1.87471 0.05223
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1 0 4S 1( 1.00) -1.39447 -1.39936 0.00489
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2 1 4P 1( 0.00) -1.03942 -1.03465 -0.00476
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dEtot_ae = 1.848995 Ry
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dEtot_ps = 1.873240 Ry, Delta E= -0.024245 Ry
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3 2 3D 1( 1.00) -2.54976 -2.61959 0.06983
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1 0 4S 1( 0.00) -1.94361 -1.96745 0.02383
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2 1 4P 1( 0.00) -1.53584 -1.54419 0.00835
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dEtot_ae = 3.518170 Ry
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dEtot_ps = 3.554733 Ry, Delta E= -0.036564 Ry
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3 2 3D 1( 0.00) -3.84145 -3.95251 0.11106
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1 0 4S 1( 0.00) -2.73793 -2.81405 0.07612
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2 1 4P 1( 0.00) -2.25938 -2.28768 0.02831
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dEtot_ae = 6.699594 Ry
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dEtot_ps = 6.831938 Ry, Delta E= -0.132344 Ry
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\end{verbatim}
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It is evident that configurations with $3d^2$ occupancy are well
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reproduced, with errors on total energy differences $<3$ mRy and
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on eigenvalues$<5$ mRy. Configurations with different $3d$ occupancy,
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however, have errors one order of magnitude higher. For the extreme
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case of Ti$^{4+}$, the error is $\sim$ 0.1 Ry.
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\newpage
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