File: lesson_tddft.html

package info (click to toggle)
abinit 7.8.2-2
links: PTS, VCS
area: main
in suites: jessie, jessie-kfreebsd
size: 278,292 kB
ctags: 19,095
sloc: f90: 463,759; python: 50,419; xml: 32,095; perl: 6,968; sh: 6,209; ansic: 4,705; fortran: 951; objc: 323; makefile: 43; csh: 42; pascal: 31
file content (528 lines) | stat: -rw-r--r-- 26,899 bytes
<html>
<head>
<title>Tutorial "TDDFT"</title>
</head>
<body bgcolor="#ffffff">
<hr>
<h1>ABINIT, lesson "TDDFT":</h1>
<h2>Time-Dependent Density Functional Theory (Casida's approach)</h2>
<hr>

<p>This lesson aims at showing how to get the following physical properties, for finite systems :
  <ul>
    <li>Excitation energies
    <li>Associated oscillator strengths
    <li>Frequency-dependent polarizability and optical spectra
  </ul>
  in the Casida approach, within Time-Dependent Density Functional Theory.
<p>This lesson should take about 30 minutes. </p>

<h5>Copyright (C) 2005-2014 ABINIT group (XG)
<br> This file is distributed under the terms of the GNU General Public License, see
~abinit/COPYING or <a href="http://www.gnu.org/copyleft/gpl.txt">
http://www.gnu.org/copyleft/gpl.txt </a>.
<br> For the initials of contributors, see ~abinit/doc/developers/contributors.txt .
</h5>

<script type="text/javascript" src="list_internal_links.js"> </script>

<h3><b>Contents of lesson "TDDFT" :</b></h3>

<ul>
  <li><a href="lesson_tddft.html#1">1.</a> Brief theoretical introduction
  <li><a href="lesson_tddft.html#2">2.</a> A first computation of electronic excitation energies and oscillator strengths
  <li><a href="lesson_tddft.html#2">3.</a> Convergence studies
  <li><a href="lesson_tddft.html#2">4.</a> The choice of the exchange-correlation potential
</ul>
<hr>

<a name="1">&nbsp;</a>
<h3><b>
1. Brief theoretical introduction
</b></h3>

<div id="content" style="background-color:#FFFFFF;height:220px;width:200px;float:right;">
<a href="https://sites.google.com/site/markcasida">
<img src="http://dcm.ujf-grenoble.fr/DCM-SITE/Trombinoscope/ccasma.jpg" align="center" /><br>
Mark E. Casida
</a>
</div>

<p>In order to do Time-Dependent Density Functional Theory calculations (TDDFT) of electronic excitations
and oscillator strengths, in the Casida's approach, you should first have some theoretical background.
<br>
TDDFT was first developed in the eighties, but the direct calculation of electronic excitations
was introduced much later, by Casida and co-workers. A comprehensive description of the underlying
formalism is given in
<pre>
"Time-Dependent Density Functional Response Theory of Molecular Systems:
 Theory, Computational Methods, and Functionals"
 M. E. Casida
 in Recent Development and Applications of Modern Density Functional Theory,
 edited by J.M. Seminario (Elsevier, Amsterdam, 1996), p. 391.
 <a href="http://dx.doi.org/10.1016/S1380-7323(96)80093-8" >http://dx.doi.org/10.1016/S1380-7323(96)80093-8</a>
</pre>
However this reference might be hard to get, that is why we have based the tutorial
instead on the following (also early) papers :
<p>
<a name="CasidaJCP1998">CasidaJCP1998:</a>
<pre>
"Molecular excitation energies to high-lying bound states from time-dependent density-functional response theory: 
 Characterization and correction of the time-dependent local density approximation ionization threshold"
 Mark E. Casida, Christine Jamorski, Kim C. Casida, and Dennis R. Salahub 
 J. Chem. Phys. 108, 4439 (1998)
 <a href="http://dx.doi.org/10.1063/1.475855">http://dx.doi.org/10.1063/1.475855</a>
</pre>
<p>
<a name="CasidaIJQC1998">CasidaIJQC1998:</a>
<pre>
"Excited-state potential energy curves from time-dependent density-functional theory: 
 A cross section of formaldehyde's 1A1 manifold"
 Mark E. Casida, Kim C. Casida, Dennis R. Salahub
 International Journal of Quantum Chemistry Volume 70, Issue 4-5, pages 933-941, 1998
 <a href="http://dx.doi.org/10.1002/(SICI)1097-461X(1998)70:4/5<933::AID-QUA39>3.0.CO;2-Z">http://dx.doi.org/10.1002/(SICI)1097-461X(1998)70:4/5<933::AID-QUA39>3.0.CO;2-Z</a>
</pre>
<p>
<a name="Vasiliev1999">Vasiliev1999:</a>
<pre>
"Ab initio Excitation Spectra and Collective Electronic Response in Atoms and Clusters"
 I. Vasiliev, S. Ogut, and J. R. Chelikowsky
 Phys. Rev. Lett. 82, 1919-1922 (1999) 
 <a href="http://dx.doi.org/10.1103/PhysRevLett.82.1919">http://dx.doi.org/10.1103/PhysRevLett.82.1919</a>
</pre>
The first of these papers, <a href="lesson_tddft.html#CasidaJCP1998">CasidaJCP1998</a>,
will be used as main reference for our tutorial.

<p>
From these papers, you will learn that a TDDFT calculation of electronic excitation
energies start first from a usual ground-state calculation, with a chosen XC functional.
Such a calculation produces a spectrum of Kohn-Sham electronic energies.
It is widely known that
differences between occupied and unoccupied Kohn-Sham electronic energies resemble
excitation energies (the difference in energy between an excited state and the ground state),
although there is no real theoretical justification for this
similarity.
<p>These differences between Kohn-Sham electronic energies are the starting point of Casida's approach :
in the framework of TDDFT,
their square give the main contribution to the diagonal part of a matrix, whose eigenvalues
will be the square of the seeked excitation energies. One has to add to the diagonal matrix made from the
squares of Kohn-Sham energy differences, a coupling matrix, whose elements are
four-wavefunction integrals of the Coulomb and Exchange-Correlation kernel.
The exchange-correlation kernel contribution will differ in the
spin-singlet and in the spin-triplet states, this being the only difference between
spin-singlet and spin-triplet states.
See Eqs.(1.3) and (1.4) of <a href="lesson_tddft.html#CasidaIJQC1998">CasidaIJQC1998</a>,
and Eqs.(1-2) of <a href="lesson_tddft.html#Vasiliev1999">Vasiliev1999</a>
<p>
The construction of the coupling matrix can be done on the basis of an exchange-correlation
kernel that is derived from the exchange-correlation functional used for the ground-state,
but this is not a requirement of the theory, since such a correspondance only holds
for the exact functional. In practice, the approximation to the XC potential and the one
to the XC kernel are often different.
See section III of <a href="lesson_tddft.html#CasidaJCP1998">CasidaJCP1998</a>.
<p>
A big drawback of the currently known XC potentials and XC kernels is observed when the
system is infinite in at least one direction (e.g. polymers, slabs, or solids).
In this case, the addition of the coupling matrix is unable to shift the
edges of the Kohn-Sham band structure (each four-wavefunction integral becomes too small).
There is only a redistribution of the oscillator strengths.
In particular, the DFT band gap problem is NOT solved by TDDFT. Also, the Casida's approach
relies on the discreteness of the Kohn-Sham spectrum.
<p>
Thus, the TDDFT approach to electronic excitation energies in ABINIT is ONLY
valid for finite systems (atoms, molecules, clusters). Actually, only one k-point can be
used, and a "box center" must be defined, close to the center of gravity of the system.
<p>
The Casida formalism also gives access to the oscillator strengths, needed to obtain
the frequency-dependent polarizability, and corresponding optical spectrum.
In the ABINIT implementation, the oscillators strengths are given as a second-rank tensor,
in cartesian coordinates, as well as the average over all directions usually
used for molecules and clusters. It is left to the user to generate the
polarisability spectrum, according to e.g. Eq.(1.2) of
<a href="lesson_tddft.html#CasidaIJQC1998">CasidaIJQC1998</a>.
<p>
One can also combine the ground state total energy with the electronic excitation energies
to obtain Born-Oppenheimer potential energy curves for excited states. This is illustrated
for formaldehyde in <a href="lesson_tddft.html#CasidaIJQC1998">CasidaIJQC1998</a>.
<p>
Given its simplicity, and the relatively modest CPU cost of this type of calculation,
Casida's approach enjoys a wide popularity. There has been hundreds of papers published
on the basis of methodology. Still, its accuracy might be below the expectations,
as you will see. As often, test this method to see if it suits your needs,
and read the recent litterature ...

<hr>

<a name="2">&nbsp;</a>
<h3><b>
2. A first computation of electronic excitation energies and oscillator strengths
</b></h3>

<p>
We will now compute and analyse the excitation energies of the diatomic molecule N2.
This is
a rather simple system, with cylindrical symmetry, allowing interesting
understanding. Although we will suppose that you are familiarized with quantum numbers
for diatomic molecules, this should not play an important role in the
understanding of the way to use Abinit's implementation of Casida's formalism.

<p><i>Before beginning, you might consider to work in a different
subdirectory as for the other lessons. Why not "Work_tddft" ?
</i>

<p>Now, you are ready to run ABINIT and prepare the needed file.
<p>Copy the files ttddft_x.files and ttddft_1.in in "Work_tddft" :
<li><i>cp ../ttddft_x.files . </i></li>
<li><i>cp ../ttddft_1.in . </i></li>

<p>
So, issue now :
<li><i> ../../abinit < ttddft_x.files >& log </i></li>
<p>

<br>
The computation is quite fast : about 15 secs on a 2.8 GHz PC.
<br>
Let's examine the input file ttddft_1.in.
<br>
There are two datasets : the first one corresponds to a typical ground-state
calculation, with only occupied bands. The density and wavefunctions
are written, for use in the second data set. The second dataset is the one
where the TDDFT calculation is done. Moreover, the non-self-consistent
calculation of the occupied eigenfunctions and corresponding eigenenergies
is also accomplished. This is obtained by setting
<a href="../input_variables/varbas.html#iscf" target="kwimg">iscf</a> to -1.
Please, take now some time to read the information about this value of
<a href="../input_variables/varbas.html#iscf" target="kwimg">iscf</a>, and the
few input variables that acquire some meaning in this context
(namely,
<a href="../input_variables/vargs.html#boxcenter" target="kwimg">boxcenter</a>,
<a href="../input_variables/varrf.html#td_mexcit" target="kwimg">td_mexcit</a>,
and <a href="../input_variables/varrf.html#td_maxene" target="kwimg">td_maxene</a>).
Actually,
this is most of the information that should be known
to use the TDDFT in ABINIT !
<br>
You will note that we have 5 occupied bands (defined for dataset 1), and that
we add 7 unoccupied bands in the dataset 2, to obtain a total of 12 bands.
The box is not very large
(6x5x5 Angstrom), the cutoff is quite reasonable, 25 Hartree), and as requested
for the Casida's formalism, only one k point is used.
We have chosen the Perdew-Wang 92 LDA functional for both the self-consistent
and non-self-consistent calculations
(<a href="../input_variables/varbas.html#ixc" target="kwimg">ixc</a>=7).

<p>
We can now examine the output file ttddft_1.out.
<br>
One can jump to the second dataset section, and skip a few
non-interesting information, in order to reach the following information :
<pre>
 *** TDDFT : computation of excited states ***
 Splitting of  12 bands in   5 occupied bands, and   7 unoccupied bands,
 giving    35 excitations.
</pre>
The matrix that is diagonalized, in the Casida's formalism, is thus a 35x35 matrix.
It will give 35 excitation energies.
<br>
Then, follows the list of excitation energies, obtained from the difference of Kohn-Sham
eigenvalues (occupied and unoccupied), for further reference.
They are ordered by increasing energy. In order to analyze the TDDFT as well as experimental
data in the next section, let us mention that the Kohn-Sham eigenfunctions
in this simulation have the following characteristics :
<ul>
 <li>The first and fifth states are (non-degenerate) occupied sigma states (m=0), with even parity
 <li>The second state is a (non-degenerate) occupied sigma state (m=0), with odd parity
 <li>The third and fourth states are degenerate occupied pi states (m=+1,-1), with odd parity
 <li>The sixth and seventh states are degenerate unoccupied pi states (m=+1,-1), with even parity
 <li>The state 8 is a (non-degenerate) unoccupied sigma state (m=0), with even parity
</ul>
Combining states 3,4 and 5 with 6, 7 and 8, give the first nine
Kohn-Sham energy differences :
<pre>
  Transition  (Ha)  and   (eV)   Tot. Ene. (Ha)  Aver     XX       YY       ZZ
   5->  6 3.10888E-01 8.45969E+00 -1.92741E+01 0.0000E+00 0.00E+00 0.00E+00 0.00E+00
   5->  7 3.10888E-01 8.45969E+00 -1.92741E+01 0.0000E+00 0.00E+00 0.00E+00 0.00E+00
   5->  8 3.44036E-01 9.36171E+00 -1.92409E+01 0.0000E+00 0.00E+00 0.00E+00 0.00E+00
   4->  6 3.64203E-01 9.91046E+00 -1.92207E+01 1.4460E-01 4.34E-01 0.00E+00 0.00E+00
   3->  6 3.64203E-01 9.91046E+00 -1.92207E+01 4.2302E-01 1.27E+00 0.00E+00 0.00E+00
   4->  7 3.64203E-01 9.91046E+00 -1.92207E+01 4.2302E-01 1.27E+00 0.00E+00 0.00E+00
   3->  7 3.64203E-01 9.91046E+00 -1.92207E+01 1.4460E-01 4.34E-01 0.00E+00 0.00E+00
   4->  8 3.97351E-01 1.08125E+01 -1.91876E+01 4.0028E-02 0.00E+00 1.20E-01 0.00E+00
   3->  8 3.97351E-01 1.08125E+01 -1.91876E+01 4.0028E-02 0.00E+00 0.00E+00 1.20E-01
</pre>
Without the coupling matrix, these would be the excitation energies, for both
the spin-singlet and spin-triplet states. The coupling matrix modifies the eigenenergies,
by mixing different electronic excitations, and also lift some degeneracies,
e.g. the quadruplet formed by the combination of the degenerate states 3-4 and 6-7
that gives the excitation energies with 3.64203E-01 Ha in the above table.
<p>
Indeed, concerning the spin-singlet, the following excitation energies are obtained
(see the next section of the output file):
<pre>
  TDDFT singlet excitation energies (at most 20 of them are printed),
  and corresponding total energies.
  Excit#   (Ha)    and    (eV)    total energy (Ha)    major contributions
   1    3.47952E-01   9.46826E+00   -1.923699E+01    0.83(  5->  6)  0.17(  5->  7)
   2    3.48006E-01   9.46971E+00   -1.923693E+01    0.83(  5->  7)  0.17(  5->  6)
   3    3.62425E-01   9.86208E+00   -1.922251E+01    0.99(  5->  8)  0.00(  2-> 10)
   4    3.64202E-01   9.91044E+00   -1.922074E+01    0.37(  3->  7)  0.37(  4->  6)
   5    3.84223E-01   1.04553E+01   -1.920072E+01    0.37(  4->  6)  0.37(  3->  7)
   6    3.84236E-01   1.04556E+01   -1.920070E+01    0.37(  4->  7)  0.37(  3->  6)
   7    3.96699E-01   1.07947E+01   -1.918824E+01    0.99(  3->  8)  0.01(  4->  8)
   8    3.96723E-01   1.07954E+01   -1.918822E+01    0.99(  4->  8)  0.01(  3->  8)
   9    4.54145E-01   1.23579E+01   -1.913079E+01    1.00(  5->  9)  0.00(  3-> 12)
   ...
</pre>
The excitation energies are numbered according to increasing energies,
in Ha as well as in eV. The total energy is also given (adding excitation
energy to the the ground-state energy), and finally,
the two major contributions to each of these excitations are mentioned
(size of the contribution then identification).
<p>
It is seen that the first and second excitations are degenerate
(numerical inaccuracies accounts for the meV difference), and mainly comes from
the first and second Kohn-Sham energy differences (between occupied state 5
and inoccupied states 6 and 7). This is also true for the third
excitation, that comes from the third Kohn-Sham energy difference (between occupied
state 5 and unoccupied state 8).
The quadruplet of Kohn-Sham energy differences, that was observed at 3.64203E-01 Ha,
has been split into one doublet and two singlets, with numbers 4 (the lowest singlet), 5-6
(the doublet) while the last singlet is not present in the 20 lowest excitations.
<br>
The list of oscillator strength is then provided.
<pre>
  Oscillator strengths :  (elements smaller than 1.e-6 are set to zero)
  Excit#   (Ha)   Average    XX        YY        ZZ         XY        XZ        YZ
   1 3.47952E-01 0.000E+00 0.000E+00 0.000E+00 0.000E+00  0.00E+00  0.00E+00  0.00E+00
   2 3.48006E-01 0.000E+00 0.000E+00 0.000E+00 0.000E+00  0.00E+00  0.00E+00  0.00E+00
   3 3.62425E-01 0.000E+00 0.000E+00 0.000E+00 0.000E+00  0.00E+00  0.00E+00  0.00E+00
   4 3.64202E-01 0.000E+00 0.000E+00 0.000E+00 0.000E+00  0.00E+00  0.00E+00  0.00E+00
   5 3.84223E-01 0.000E+00 0.000E+00 0.000E+00 0.000E+00  0.00E+00  0.00E+00  0.00E+00
   6 3.84236E-01 0.000E+00 0.000E+00 0.000E+00 0.000E+00  0.00E+00  0.00E+00  0.00E+00
   7 3.96699E-01 5.759E-02 0.000E+00 1.928E-03 1.709E-01  0.00E+00  0.00E+00 -1.81E-02
   8 3.96723E-01 5.544E-02 0.000E+00 1.645E-01 1.855E-03  0.00E+00  0.00E+00  1.75E-02
   9 4.54145E-01 0.000E+00 0.000E+00 0.000E+00 0.000E+00  0.00E+00  0.00E+00  0.00E+00
  10 4.60223E-01 9.496E-02 2.849E-01 0.000E+00 0.000E+00  0.00E+00  0.00E+00  0.00E+00
   ...
</pre>
The first six transitions are forbidden, with zero oscillator strength. The
seventh and eighth transitions are allowed, with sizeable YY, YZ and ZZ components.
<p>
Next, one finds the excitation energies for the spin-triplet states :
<pre>
  TDDFT triplet excitation energies (at most 20 of them are printed),
  and corresponding total energies.
  Excit#   (Ha)    and    (eV)    total energy (Ha)    major contributions
   1    2.88423E-01   7.84838E+00   -1.929652E+01    0.82(  5->  6)  0.18(  5->  7)
   2    2.88424E-01   7.84842E+00   -1.929652E+01    0.82(  5->  7)  0.18(  5->  6)
   3    2.99762E-01   8.15693E+00   -1.928518E+01    0.37(  3->  6)  0.37(  4->  7)
   4    3.33749E-01   9.08177E+00   -1.925119E+01    0.37(  4->  6)  0.37(  3->  7)
   5    3.33809E-01   9.08339E+00   -1.925113E+01    0.37(  4->  7)  0.37(  3->  6)
   6    3.36922E-01   9.16812E+00   -1.924802E+01    1.00(  5->  8)  0.00(  2-> 10)
   7    3.64202E-01   9.91045E+00   -1.922074E+01    0.37(  3->  7)  0.37(  4->  6)
   8    3.90779E-01   1.06336E+01   -1.919416E+01    0.67(  3->  8)  0.27(  2->  6)
   9    3.90834E-01   1.06351E+01   -1.919411E+01    0.67(  4->  8)  0.27(  2->  7)
   ...
</pre>
Spin-triplet energies are markedly lower than the corresponding spin-singlet energies.
Also, the highest singlet derived from the Kohn-Sham quadruplet is now the excitation
number 7. The oscillator strengths also follow. At this stage, we are in position to
compare with experimental data, and try to improve the quality of our calculation.

<p>
To summarize our results, we obtain the following five
lowest-lying spin-singlet excitation energies,
with corresponding quantum numbers (that we derive from the knowledge of the Kohn-Sham
states quantum numbers):
<pre>
9.47 eV   m=+1,-1  even parity (Pi_g state)
9.86 eV   m=0      even parity (Sigma_g state)
9.91 eV   m=0      odd parity  (Sigma_u state)
10.46 eV  m=+2,-2  odd parity  (Delta_u state)
10.79 eV  m=+1,-1  odd parity  (Pi_u state)
</pre>
and the following five lowest-lying spin-triplet excitations energies,
with corresponding quantum numbers :
<pre>
7.85 eV   m=+1,-1  even parity (Pi_g state)
8.16 eV   m=0      odd parity  (Sigma_u state)
9.08 eV   m=+2,-2  odd parity  (Delta_u state)
9.16 eV   m=0      even parity (Sigma_g state)
9.91 eV   m=0      odd parity  (Sigma_u state)
</pre>
The quantum number related to the effect of a mirror plane, needed for Sigma states,
could not be attributed on the sole basis of the knowledge of Kohn-Sham orbitals
quantum numbers.
<p>
The lowest-lying experimental spin-singlet excitation energies,
see table III of <a href="lesson_tddft.html#CasidaJCP1998">CasidaJCP1998</a>, are as follows :
<pre>
9.31 eV   m=+1,-1  even parity (Pi_g state)
9.92 eV   m=0      odd parity  (Sigma_u- state)
10.27 eV  m=+2,-2  odd parity  (Delta_u state)
</pre>
and the lowest-lying experimental spin-triplet excitations energies are :
<pre>
7.75 eV   m=0      odd parity  (Sigma_u+ state)
8.04 eV   m=+1,-1  even parity (Pi_g state)
8.88 eV   m=+2,-2  odd parity  (Delta_u state)
9.67 eV   m=0      odd parity  (Sigma_u- state)
</pre>
In several cases, the agreement is quite satisfactory, on the order of 0.1-0.2 eV. However, there
are also noticeable discrepancies. Indeed, we have to understand, in our simulation :
<ul>
 <li>The appearance of the spin-singlet Sigma_g state at 9.86 eV (Spin-singlet state 2)</li>
 <li>The inversion between the spin-triplet Pi_g and Sigma_u states (Spin-triplet states 1 and 2)</li>
 <li>The appearance of the spin-triplet Sigma_g state at 9.16 eV (Spin-triplet state 4)</li>
</ul>

<p>
Still, the agreement between these TDDFT values and the experimental
values is much better than anything that can be done on the sole basis
of Kohn-Sham energy differences, that are (for spin-singlet and -triplet) :
<pre>
8.46 eV   m=+1,-1   even parity (Pi_g state)
9.36 eV   m=0       odd parity  (Sigma_u state)
9.91 eV   m=0(twice),+2,-2 odd parity  (Sigma_u and Delta_u states)
10.81 eV  m=+1,-1   odd parity  (Pi_u state)
</pre>


<hr>

<a name="3">&nbsp;</a>
<h3><b>
3. Convergence studies.
</b></h3>

There are several parameters subjet to convergence studies in this context :
the energy cut-off, the box size, and the number of unoccupied bands.
<p>
We will start with the number of unoccupied states. The only input parameter
to be changed in the input file is the value of nband2. The following results
are obtained, for nband2 = 12, 30, 60, 100 and 150 (Energies given in eV):
<pre>
Singlet 1 :  9.47   9.44   9.39   9.36   9.35
Singlet 2 :  9.86   9.74   9.68   9.66   9.66
Singlet 3 :  9.91   9.91   9.91   9.91   9.91
Singlet 4 : 10.46  10.45  10.44  10.44  10.43
Singlet 5 : 10.79  10.79  10.79  10.79  10.79
Triplet 1 :  7.85   7.84   7.83   7.82   7.82
Triplet 2 :  8.16   8.08   8.03   8.00   8.00
Triplet 3 :  9.08   9.07   9.05   9.05   9.04
Triplet 4 :  9.16   9.16   9.15   9.15   9.15
Triplet 5 :  9.91   9.91   9.91   9.91   9.91
</pre>
You might try to obtain one of these ... The computation with nband2=100
takes about 7 minutes on a 2.8 GHz PC, and gives a result likely converged
within 0.01 eV. Let's have a look at these data.
Unfortunately, none of the above-mentioned discrepancies with experimental
data is resolved, although the difference between the first and second spin-triplet
states decreases significantly. Although we see that at least 60 bands are needed to obtain
results converged within 0.05 eV, we will continue to rely on 12 bands to try to understand the
most important discrepancies, while keeping the CPU time to a low level.
<p>
We next try to increase the cut-off energy. Again, this is fairly easy. One can e.g.
set up a double dataset loop. The following results
are obtained, for ecut = 25, 35, 45, 55, 65, and 75 Ha :
<pre>
Singlet 1 :  9.47   9.41   9.39   9.36   9.36
Singlet 2 :  9.86   9.83   9.78   9.76   9.76
Singlet 3 :  9.91   9.97  10.01  10.02  10.03
Singlet 4 : 10.46  10.37  10.32  10.30  10.29
Singlet 5 : 10.79  10.87  10.90  10.91  10.92
Triplet 1 :  7.85   7.79   7.76   7.74   7.73
Triplet 2 :  8.16   8.02   7.94   7.92   7.91
Triplet 3 :  9.08   8.98   8.92   8.90   8.89
Triplet 4 :  9.16   9.28   9.33   9.34   9.34
Triplet 5 :  9.91   9.83   9.78   9.77   9.76
</pre>
You might try to obtain one of these ... The computation with ecut=75
takes about 90 secs on a 2.8 GHz PC, and gives a result likely converged
within 0.01 eV. Let us have a look at these data.
Concerning the discrepancies with the experimental results,
we see that the position of the second spin-singlet state has even worsened,
the difference between the first and second spin-triplet states decreases, so that,
together with an increase of nband, their order might become the correct one, and the
fourth spin-triplet state energy has increased, but not enough.
<p>
We finally examine the effect of the cell size. Again, this is fairly easy. One can e.g.
set up a double dataset loop. The following results
are obtained, for acell = (6 5 5), (7 6 6), (8 7 7), (9 8 8), (10 9 9) and (12 11 11) :
<pre>
Singlet 1 :  9.47   9.37   9.33   9.33   9.33   9.33
Singlet 2 :  9.86   9.78   9.84   9.91   9.96  10.03
Singlet 3 :  9.91   9.88   9.85   9.85   9.85   9.85
Singlet 4 : 10.46  10.41  10.38  10.37  10.37  10.37
Singlet 5 : 10.79  10.98  11.14  11.27  11.19  11.04
Triplet 1 :  7.85   7.75   7.72   7.71   7.72   7.72
Triplet 2 :  8.16   8.18   8.18   8.18   8.19   8.20
Triplet 3 :  9.08   9.07   9.06   9.06   9.06   9.06
Triplet 4 :  9.16   9.36   9.55   9.68   9.78   9.90
Triplet 5 :  9.91   9.88   9.85   9.85   9.85   9.85
</pre>
Obviously, the cell size plays an important role in the spurious appearance
of the states, that was remarked when comparing against experimental data.
Indeed,
the singlet 2 and triplet 4 states energy increases strongly with the cell size,
while all other states quickly stabilize (except the still higher singlet 5 state).

<p>
There is one lesson to be learned from that convergence study : the
convergence of different states can be quite different. Usually, converging the
lower excited states do not require too much effort, while it is quite difficult,
especially concerning the supercell size, to converge higher states.

<p>
At this stage, we will simply stop this convergence study,
and give the results of an ABINIT calculation
using ecut 45 Hartree, acell 12 11 11, and  30 bands (not fully converged, though !),
then compare the results with other LDA/TDLDA results
(from <a href="lesson_tddft.html#CasidaJCP1998">CasidaJCP1998</a>)
and experimental results :
<pre>
                    present Casida experimental
Singlet Pi_g      :  9.25    9.05     9.31
Singlet Sigma_u-  :  9.72    9.63     9.92
Singlet Delta_u   : 10.22   10.22    10.27
Triplet Sigma_u+  :  7.95    7.85     7.75
Triplet Pi_g      :  7.64    7.54     8.04
Triplet Delta_u   :  8.89    8.82     8.88
Triplet Sigma_u-  :  9.72    9.63     9.67
</pre>

Our calculation is based on pseudopotentials, while Casida's calculation
is an all-electron one. This fact might account for the 0.1-0.2 eV discrepancy
between both calculations (it is of course the user's responsability to test the
influence of different pseudopotentials on his/her calculations).
The agreement with experimental data is
on the order of 0.2 eV, with the exception of the Triplet Pi_g state (0.4 eV).
In particular,
we note that LDA/TDLDA is not able to get the correct ordering of
the lower two triplet states ... One of our problems was intrinsic
to the LDA/TDLDA approximation ...


<a name="4">&nbsp;</a>
<h3><b>
4. The choice of the exchange-correlation potential.
</b></h3>
As emphasized in <a href="lesson_tddft.html#CasidaJCP1998">CasidaJCP1998</a>,
choosing a different functional for the self-consistent part (XC potential)
and the generation of the coupling matrix (XC kernel) can give a better description
of the higher-lying states. Indeed, a potential with a -1/r tail (unlike the LDA
or GGA) like the van Leeuwen-Baerends one, can reproduce fairly well the ionisation
energy, giving a much better description of the Rydberg states. Still,
the LDA kernel works pretty well.
<p>
In order to activate this procedure,
set the value of ixc in dataset 1 to the SCF functional, and the value
of ixc in dataset 2 to the XC functional to be used for the kernel.
Use pseudopotentials that agree with the SCF functional.

<script type="text/javascript" src="list_internal_links.js"> </script>

</body>
</html>