Difference between revisions of "How to analyse excitons - ICTP 2022 school"

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This is demonstrated for 3D hBN, for which you should have obtained the needed data from previous calculations.  
This is demonstrated for 3D hBN, for which you should have obtained the needed data from previous calculations.  


'''Warning''': for this demonstration, we did not use converged parameters (in particular they are not converged for the k-grid). Note that one needs to converge carefully all parameters to obtain scientific relevant results. This will be the topic of [[How to choose the input parameters | one of the next tutorials]].


   
   
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!-->
!-->
==List the excitonic energies==
==List the excitonic energies==
First, we will obtain the list of all the exciton energies (that is the eigenvalues of the two-particle Hamiltonian) sorted both by energies and oscillator strength.


To read this information from the Yambo databases produced in [[Bethe-Salpeter solver: diagonalization| this previous step]], we use the Yampo pre- and post-processing utility <code>ypp</code>.
First, we will obtain the list of all the exciton energies (that is the eigenvalues of the two-particle Hamiltonian) sorted both by energy and strength. The exciton strengths or intensities are defined from the expression of the optical absorption spectrum as
 
[[File:strengh.png|none|x150px|]]
 
That is, they are given by the linear combination of the square of the dipole transition matrix elements between electron-hole pairs, where A<sup>&lambda;</sup><sub>''eh''</sub>  is the exciton composition in terms of electron-hole pairs.
 
To read this information from the Yambo databases produced in [[Bethe-Salpeter solver: diagonalization| this previous step]], we use the Yambo pre- and post-processing utility <code>ypp</code>.


Type:
Type:
Line 40: Line 46:
  $ ypp -J 3D_BSE -e s 1
  $ ypp -J 3D_BSE -e s 1


This instructs the code to list the excitons for the q-index = 1 (optical limit q=0).  
This instructs the code to list the excitons (<code>-e s</code>) in the database ''3D_BSE'' (<code>-J 3D_BSE</code>) for the q-index = 1 (optical limit q=0).  


The results are output in the files ''o-3D_BSE.exc_qpt1_E_sorted'' and ''o-3D_BSE.exc_qpt1_I_sorted''.  
The results are output in the files ''o-3D_BSE.exc_qpt1_E_sorted'' and ''o-3D_BSE.exc_qpt1_I_sorted''.  
They report the energies of the excitons and their oscillator strengths. The strengths are normalised to the largest strength. Thus in the list, the brightest exciton has strength 1.  
They report the energies of the excitons and their strengths. The strengths are normalised to the largest strength. Thus in the list, the brightest exciton has strength 1.  
In ''o-3D_BSE.exc_qpt1_E_sorted'' the exciton energies are sorted by energy  (lower to higher), in ''o-3D_BSE.exc_qpt1_I_sorted'' by the oscillator strength (larger to smaller).
In ''o-3D_BSE.exc_qpt1_E_sorted'' the exciton energies are sorted by energy  (lower to higher), in ''o-3D_BSE.exc_qpt1_I_sorted'' by the oscillator strength (larger to smaller).
[[File:strengh.png|none|x120px|]]


Inspect 'o-3D_BSE.exc_qpt1_E_sorted'':
The exciton with the lowest energy (3.54 eV) is doubly degenerate. The second-lowest is also doubly degenerate and is the brightest (oscillator strength of 1). The latter exciton is the one responsible for the largest peak in the optical absorption plots of the [[Bethe-Salpeter solver: diagonalization|previous tutorials]].


To visualize these results, we plot them:
Inspect ''o-3D_BSE.exc_qpt1_E_sorted'':
 
The exciton with the lowest energy (3.54 eV) is doubly degenerate. The strength is ~0, thus this exciton is dark, does not contribute to the absorption spectrum.
 
The second-lowest exciton is also doubly degenerate and it is the brightest (one of the components has strength 1, the other ~0.5). The latter exciton is the one responsible for the largest peak in the optical absorption plots of the [[Bethe-Salpeter solver: diagonalization|previous tutorials]].
 
To better visualize these results, we plot the exciton strength (normalized to 1, arbitrary units) versus the energy (in eV):
  $ gnuplot
  $ gnuplot
  gnuplot> set style line 2 lc rgb 'black' pt 7  # circle
  gnuplot> set style line 2 lc rgb 'black' pt 7  # circle
Line 57: Line 67:
[[File:ExcWeights3DhBN.png|none|600px]]
[[File:ExcWeights3DhBN.png|none|600px]]


<---!'''Warning''': the convergence of these results with different k-points grids is mandatory! --->
== Calculate the exciton oscillator strength and amplitude ==


== Calculate the exciton oscillator strength and amplitude ==
Once inspected the list of excitons sorted by energies and strengths, we will look at the composition of some of these excitons in terms of single-particle states (electron-hole pairs).


We can now analyze the excitons in terms of single-particle states: which electronic transitions are the most relevant? In order to do that, create the appropriate input with
Type:  
  $ ypp -F ypp_AMPL.in -J 3D_BSE -e a 1  
  $ ypp -F ypp_AMPL.in -J 3D_BSE -e a 1  


Suppose you wish to analyze the first 4 excitons (i.e., the first 2 doubly-degenerate excitons). Then, change this line as:
This instructs the code to create the the input file ''ypp_AMPL.in'' (<code>-F ypp_AMPL.in</code>) to analyze the composition of the excitons (<code>-e a</code>) in the database ''3D_BSE'' (<code>-J 3D_BSE</code>) for the q-index = 1 (optical limit q=0).
 
You can now edit the input file 'ypp_AMPL.in''. From the list of excitons, we should have got an idea of which excitons we wish to analyze. The ordering from the list of exciton sorted by energies in ''o-3D_BSE.exc_qpt1_E_sorted'' is used to identify the excitons to analyse.
 
For example, to analyse the 4 lowest-energy excitons (i.e., the first 2 doubly-degenerate excitons of which the first is dark and the second is bright) change this line as:
  States= "1 - 4"              # Index of the BS state(s)
  States= "1 - 4"              # Index of the BS state(s)


Close the input and run ypp
Close the input file and run <code>ypp</code> again by typing


  $ ypp -F ypp_AMPL.in -J 3D_BSE
  $ ypp -F ypp_AMPL.in -J 3D_BSE
This carries out the analysis for the specified excitons. The results are output in ''o-3D_BSE.exc_qpt1_amplitude_at_*'' and  ''o-3D_BSE.exc_qpt1_weights_at_*'', where ''*'' identifies the exciton according to the order in ''o-3D_BSE.exc_qpt1_E_sorted''.


  $ ls  o*exc*at*
  $ ls  o*exc*at*
  o-3D_BSE.exc_qpt1_amplitude_at_1 o-3D_BSE.exc_qpt1_weights_at_1 ...
  o-3D_BSE.exc_qpt1_amplitude_at_1 o-3D_BSE.exc_qpt1_weights_at_1 ...
   
   
For an exciton  <math>|\lambda></math> , ''o-3D_BSE.exc_qpt1_weights_at_*'' report the Weights
For an exciton  <math>|\lambda></math> , ''o-3D_BSE.exc_qpt1_weights_at_*'' reports the weights defined as
[[File:Weights.png|none|x60px|]]
[[File:Weights.png|none|x60px|]]
and ''o-3D_BSE.exc_qpt1_amplitude_**'' report the amplitudes  
and ''o-3D_BSE.exc_qpt1_amplitude_* '' reports the amplitudes defined as
[[File:Ampl.png|none|x70px|]]
[[File:Ampl.png|none|x70px|]]


Open the file ''o-3D_BSE.exc_weights_at_3''
 
Inspect, for example, the file ''o-3D_BSE.exc_weights_at_3'' (which we learned from ''o-3D_BSE.exc_qpt1_E_sorted'' has the largest strength)


  # Band_V          Band_C          Kv-q ibz        Symm_kv        Kc ibz          Symm_kc        Weight          Energy
  # Band_V          Band_C          Kv-q ibz        Symm_kv        Kc ibz          Symm_kc        Weight          Energy
Line 92: Line 109:
   8.00000000      9.00000000      13.0000000      1.00000000      13.0000000      1.00000000    0.738964081E-1  4.81087065   
   8.00000000      9.00000000      13.0000000      1.00000000      13.0000000      1.00000000    0.738964081E-1  4.81087065   


The third exciton is mostly composed of single-particle transitions from VBM to CBM at point H (last k-point of the grid, number 14) of the 3D hexagonal Brillouin zone, with contributions also coming from point K (number 13). All the contributions weighing less than 5% are not shown by default. Recall from the previous analysis that exciton states 3 and 4 (degenerate) are the optically active ones.
The third exciton is mostly composed of single-particle transitions from VBM to CBM at point H (last k-point of the grid, number 14) of the 3D hexagonal Brillouin zone, with contributions also coming from point K (number 13). All the contributions weighing less than 5% are not shown by default.  


We can now plot the amplitude of the bright exciton:
Recall from the previous analysis that of the first 4 exciton states, exciton states 3 and 4 (degenerate) are the optically active ones. We plot then the amplitude of the lowest energy bright exciton:
  $ paste o-3D_BSE.exc_qpt1_amplitude_at_3 o-3D_BSE.exc_qpt1_amplitude_at_4 > o-3D_BSE.exc_qpt1_amplitude_at_3_4
  $ paste o-3D_BSE.exc_qpt1_amplitude_at_3 o-3D_BSE.exc_qpt1_amplitude_at_4 > o-3D_BSE.exc_qpt1_amplitude_at_3_4
  $ gnuplot
  $ gnuplot
Line 103: Line 120:
[[File:ExcAmp3DhBN.png|none|600px]]
[[File:ExcAmp3DhBN.png|none|600px]]


Recall that while this plot is related to the peaks in the optical absorption spectrum, it lacks the coupling with the external field, which is encoded in the dipole matrix elements.
Note that while this plot is related to the peaks in the optical absorption spectrum, it lacks the coupling with the external field, which is encoded in the dipole matrix elements.


== Plot the exciton spatial distribution ==
== Plot the exciton spatial distribution ==


To see the spatial character of the exciton, YPP writes the exciton spatial distribution, e.g., the probability to find the electron when the hole is fixed in a given position.
The final analysis we carry out is the plot of the spatial distribution of the exciton, that is of the probability of finding the electron at a certain position r when the hole is fixed in a position r'.
Different output formats can be selected and to obtain 1D, 2D, and 3D plots.


Create the input and change the various parameters as shown below.
Create the input file ``ypp_WF.in`` using the command below and change the various parameters as shown below:


  $ ypp -F ypp_WF.in -J 3D_BSE -e w
  $ ypp -F ypp_WF.in -J 3D_BSE -e w
Line 128: Line 144:
  %
  %


One of the parameters, <code>Cells</code>, is the size of the cell where the exciton will be visualised.
<code>Cells</code>, is the size of the cell where the exciton will be visualised.
Note that if the k-grid of the BSE simulation is XxYxZ, then the exciton has an induced fictitious periodicity for every XxYxZ cells of the simulation.
If the k-grid used for the BSE calculations is XxYxZ, then the exciton has an induced fictitious periodicity for every XxYxZ cells of the simulation.
For hBN, this is not a problem because the exciton is strongly localized, but in other systems, where excitons are more delocalized, it may b necessary to use  
For hBN, this is not a problem because the first bright exciton is strongly localized, but in other systems, where excitons are more delocalized, one must use very large k-grids for the BSE calculation.
very large k-grids for the BSE calculation.


Also notice that, because of the value of <code>Degen_Step</code>, the code will automatically recognize and merge the degenerate excitons 3 and 4: you will see this in the log file.
<code>Degen_Step</code>, tells the maximum energy separation of two degenerate states. Since this value is small enough, the code will automatically recognize excitons 3 and 4 are degenerate and merge them: you can check this in the log file.


Finally: how did we know at which coordinate to place the hole? It's best placed in the position where the valence electrons contributing to the exciton are localised. In the case of hBN, they are on the nitrogen atoms. We can obtain the Cartesian coordinates of these atoms from our previously generated ''o-3D_BSE.exc_qpt1_E_sorted'' file:
Plots can be in 1, 2 or 3D (<code>Direction</code>) and the output format is chosen with <code>Format</code>. 
 
A non-trivial input parameter is the hole position. The best position for the hole is where the valence electrons contributing to the exciton are localised. In the case of hBN, they are on the nitrogen atoms. We can obtain the Cartesian coordinates of these atoms from our previously generated ''o-3D_BSE.exc_qpt1_E_sorted'' file:
  # Atom 1 with Z 5 [cc]:  2.35800028      1.36139178      0.00000000   
  # Atom 1 with Z 5 [cc]:  2.35800028      1.36139178      0.00000000   
  # Atom 2 with Z 5 [cc]: -2.35800028    -1.36139178      6.08835602   
  # Atom 2 with Z 5 [cc]: -2.35800028    -1.36139178      6.08835602   
  # Atom 1 with Z 7 [cc]: -2.35800028    -1.36139178      0.00000000   
  # Atom 1 with Z 7 [cc]: -2.35800028    -1.36139178      0.00000000   
  # Atom 2 with Z 7 [cc]:  2.35800028      1.36139178      6.08835602  
  # Atom 2 with Z 7 [cc]:  2.35800028      1.36139178      6.08835602  
We take the positive values of the nitrogen(Z=7, last line) and shift the vertical position of the hole from 6.088 to 7.088 so that the hole does not end up in the center of the atom (and therefore, possibly in a node of the wave function).   
We take the positive values of the nitrogen (Z=7, last line) and shift the vertical position of the hole from 6.088 to 7.088 so that the hole does not end up in the center of the atom (and therefore, possibly in a node of the wave function).   


Now close the input and run ypp
Close the input file and run <code>ypp</code> by typing
  $ ypp -F ypp_WF.in -J 3D_BSE
  $ ypp -F ypp_WF.in -J 3D_BSE


After it's finished, we can visualise with xcrysden or with VESTA:
After the calculation is completed, we can visualise the output with xcrysden or with VESTA:
  $ xcrysden --xsf o-3D_BSE.exc_qpt1_3d_3.xsf
  $ xcrysden --xsf o-3D_BSE.exc_qpt1_3d_3.xsf


Line 151: Line 168:


[[File:ExcWF3DhBN.jpg|none|600px]]
[[File:ExcWF3DhBN.jpg|none|600px]]
We notice that the electron is completely confined on the boron atoms and on the same layer of the hole. For comparison, see for example ref. <ref>Huge Excitonic Effects in Layered Hexagonal Boron Nitride, B. Arnaud et al., [https://arxiv.org/abs/cond-mat/0503390 preprint ArXiv]</ref>.
We notice that the electron is completely confined on the boron atoms and on the same layer of the hole. For comparison, see for example ref. <ref>Huge Excitonic Effects in Layered Hexagonal Boron Nitride, B. Arnaud et al., [https://arxiv.org/abs/cond-mat/0503390 preprint ArXiv]</ref>.


Revision as of 19:31, 28 March 2021

In this tutorial you will learn how to:

  • analyze an optical spectrum obtained from BSE in terms of excitonic energies and composition
  • look at the spatial distribution of the exciton

This is demonstrated for 3D hBN, for which you should have obtained the needed data from previous calculations.

Warning: for this demonstration, we did not use converged parameters (in particular they are not converged for the k-grid). Note that one needs to converge carefully all parameters to obtain scientific relevant results. This will be the topic of one of the next tutorials.


Prerequisites

Previous modules

You will need:

  • The SAVE databases for 3D hBN
  • The 3D_BSE directory containing the ndb.BS_diago* databases for 3D hBN
  • ypp executable
  • xcrysden or VESTA executables
  • gnuplot executable

All the databases required for this tutorial should be in the YAMBO_TUTORIALS/hBN/ directory.

List the excitonic energies

First, we will obtain the list of all the exciton energies (that is the eigenvalues of the two-particle Hamiltonian) sorted both by energy and strength. The exciton strengths or intensities are defined from the expression of the optical absorption spectrum as

Strengh.png

That is, they are given by the linear combination of the square of the dipole transition matrix elements between electron-hole pairs, where Aλeh is the exciton composition in terms of electron-hole pairs.

To read this information from the Yambo databases produced in this previous step, we use the Yambo pre- and post-processing utility ypp.

Type: 

$ ypp -J 3D_BSE -e s 1


This instructs the code to list the excitons (-e s) in the database 3D_BSE (-J 3D_BSE) for the q-index = 1 (optical limit q=0).

The results are output in the files o-3D_BSE.exc_qpt1_E_sorted and o-3D_BSE.exc_qpt1_I_sorted. They report the energies of the excitons and their strengths. The strengths are normalised to the largest strength. Thus in the list, the brightest exciton has strength 1. In o-3D_BSE.exc_qpt1_E_sorted the exciton energies are sorted by energy (lower to higher), in o-3D_BSE.exc_qpt1_I_sorted by the oscillator strength (larger to smaller).


Inspect o-3D_BSE.exc_qpt1_E_sorted:

The exciton with the lowest energy (3.54 eV) is doubly degenerate. The strength is ~0, thus this exciton is dark, does not contribute to the absorption spectrum.

The second-lowest exciton is also doubly degenerate and it is the brightest (one of the components has strength 1, the other ~0.5). The latter exciton is the one responsible for the largest peak in the optical absorption plots of the previous tutorials.

To better visualize these results, we plot the exciton strength (normalized to 1, arbitrary units) versus the energy (in eV): 

$ gnuplot
gnuplot> set style line 2 lc rgb 'black' pt 7   # circle
gnuplot> plot 'o-3D_BSE.exc_qpt1_E_sorted' with points ls 2 title 'Strengths'
ExcWeights3DhBN.png

Calculate the exciton oscillator strength and amplitude

Once inspected the list of excitons sorted by energies and strengths, we will look at the composition of some of these excitons in terms of single-particle states (electron-hole pairs).

Type: 

$ ypp -F ypp_AMPL.in -J 3D_BSE -e a 1

This instructs the code to create the the input file ypp_AMPL.in (-F ypp_AMPL.in) to analyze the composition of the excitons (-e a) in the database 3D_BSE (-J 3D_BSE) for the q-index = 1 (optical limit q=0).

You can now edit the input file 'ypp_AMPL.in. From the list of excitons, we should have got an idea of which excitons we wish to analyze. The ordering from the list of exciton sorted by energies in o-3D_BSE.exc_qpt1_E_sorted is used to identify the excitons to analyse.

For example, to analyse the 4 lowest-energy excitons (i.e., the first 2 doubly-degenerate excitons of which the first is dark and the second is bright) change this line as: 

States= "1 - 4"              # Index of the BS state(s)

Close the input file and run ypp again by typing 

$ ypp -F ypp_AMPL.in -J 3D_BSE

This carries out the analysis for the specified excitons. The results are output in o-3D_BSE.exc_qpt1_amplitude_at_* and o-3D_BSE.exc_qpt1_weights_at_*, where * identifies the exciton according to the order in o-3D_BSE.exc_qpt1_E_sorted. 

$ ls  o*exc*at*
o-3D_BSE.exc_qpt1_amplitude_at_1 o-3D_BSE.exc_qpt1_weights_at_1 ...

For an exciton [math]\displaystyle{ |\lambda\gt }[/math] , o-3D_BSE.exc_qpt1_weights_at_* reports the weights defined as

Weights.png

and o-3D_BSE.exc_qpt1_amplitude_* reports the amplitudes defined as

Ampl.png


Inspect, for example, the file o-3D_BSE.exc_weights_at_3 (which we learned from o-3D_BSE.exc_qpt1_E_sorted has the largest strength) 

# Band_V          Band_C          Kv-q ibz        Symm_kv         Kc ibz          Symm_kc         Weight          Energy
#    
  7.00000000      10.0000000      14.0000000      2.00000000      14.0000000      2.00000000     0.395135850      4.35248947   
  7.00000000      10.0000000      14.0000000      1.00000000      14.0000000      1.00000000     0.394993663      4.35248947   
  8.00000000      9.00000000      14.0000000      2.00000000      14.0000000      2.00000000     0.391943455      4.35241365   
  8.00000000      9.00000000      14.0000000      1.00000000      14.0000000      1.00000000     0.391800284      4.35241365   
  7.00000000      10.0000000      13.0000000      2.00000000      13.0000000      2.00000000     0.745555162E-1   4.81094742   
  7.00000000      10.0000000      13.0000000      1.00000000      13.0000000      1.00000000     0.745274872E-1   4.81094742   
  8.00000000      9.00000000      13.0000000      2.00000000      13.0000000      2.00000000     0.739243180E-1   4.81087065   
  8.00000000      9.00000000      13.0000000      1.00000000      13.0000000      1.00000000     0.738964081E-1   4.81087065

The third exciton is mostly composed of single-particle transitions from VBM to CBM at point H (last k-point of the grid, number 14) of the 3D hexagonal Brillouin zone, with contributions also coming from point K (number 13). All the contributions weighing less than 5% are not shown by default.

Recall from the previous analysis that of the first 4 exciton states, exciton states 3 and 4 (degenerate) are the optically active ones. We plot then the amplitude of the lowest energy bright exciton: 

$ paste o-3D_BSE.exc_qpt1_amplitude_at_3 o-3D_BSE.exc_qpt1_amplitude_at_4 > o-3D_BSE.exc_qpt1_amplitude_at_3_4
$ gnuplot
gnuplot> set xlabel 'Energy (eV)' 
gnuplot> set ylabel 'Amplitude' 
gnuplot> p 'o-3D_BSE.exc_qpt1_amplitude_at_3_4' u 1:($2+$4)/2 w l t 'Bright exciton'
ExcAmp3DhBN.png

Note that while this plot is related to the peaks in the optical absorption spectrum, it lacks the coupling with the external field, which is encoded in the dipole matrix elements.

Plot the exciton spatial distribution

The final analysis we carry out is the plot of the spatial distribution of the exciton, that is of the probability of finding the electron at a certain position r when the hole is fixed in a position r'.

Create the input file ``ypp_WF.in`` using the command below and change the various parameters as shown below: 

$ ypp -F ypp_WF.in -J 3D_BSE -e w

excitons                     # [R] Excitons
wavefunction                 # [R] Wavefunction
Format= "x"                  # Output format [(c)ube/(g)nuplot/(x)crysden]
Direction= "123"               # [rlu] [1/2/3] for 1d or [12/13/23] for 2d [123] for 3D
FFTGvecs=  30        Ry    # [FFT] Plane-waves
States= "3 - 3"              # Index of the BS state(s)
Degen_Step=   0.0100   eV    # Maximum energy separation of two degenerate states
% Cells
 5 | 5 | 1 |                             # Number of cell repetitions in each direction (odd or 1)
%
% Hole
2.35800028 | 1.36139178 | 7.08835602 # [cc] Hole position in unit cell (positive)
%

Cells, is the size of the cell where the exciton will be visualised. If the k-grid used for the BSE calculations is XxYxZ, then the exciton has an induced fictitious periodicity for every XxYxZ cells of the simulation. For hBN, this is not a problem because the first bright exciton is strongly localized, but in other systems, where excitons are more delocalized, one must use very large k-grids for the BSE calculation.

Degen_Step, tells the maximum energy separation of two degenerate states. Since this value is small enough, the code will automatically recognize excitons 3 and 4 are degenerate and merge them: you can check this in the log file.

Plots can be in 1, 2 or 3D (Direction) and the output format is chosen with Format.

A non-trivial input parameter is the hole position. The best position for the hole is where the valence electrons contributing to the exciton are localised. In the case of hBN, they are on the nitrogen atoms. We can obtain the Cartesian coordinates of these atoms from our previously generated o-3D_BSE.exc_qpt1_E_sorted file: 

# Atom 1 with Z 5 [cc]:  2.35800028      1.36139178      0.00000000   
# Atom 2 with Z 5 [cc]: -2.35800028     -1.36139178      6.08835602   
# Atom 1 with Z 7 [cc]: -2.35800028     -1.36139178      0.00000000   
# Atom 2 with Z 7 [cc]:  2.35800028      1.36139178      6.08835602

We take the positive values of the nitrogen (Z=7, last line) and shift the vertical position of the hole from 6.088 to 7.088 so that the hole does not end up in the center of the atom (and therefore, possibly in a node of the wave function).

Close the input file and run ypp by typing 

$ ypp -F ypp_WF.in -J 3D_BSE

After the calculation is completed, we can visualise the output with xcrysden or with VESTA: 

$ xcrysden --xsf o-3D_BSE.exc_qpt1_3d_3.xsf

$ VESTA o-3D_BSE.exc_qpt1_3d_3.xsf
ExcWF3DhBN.jpg

We notice that the electron is completely confined on the boron atoms and on the same layer of the hole. For comparison, see for example ref. [1].

Plot electron/hole average density (only in Yambo 5.x)

Another way to analyze excitons, it is the possibility to plot the average electron/hole densities defined as:

Electron hole density.png

to generate the corresponding input just type 

ypp -F ypp_WF.in -e w -avehole

and choose the exciton you want to plot. The electron/hole average densities correspond to generalized valence/conduction orbitals for a given exciton. They are interesting in particular for molecular crystals because they allow distinguishing charge-transfer versus Frenkel excitons, from the relative position of the electron/hole densities.

If you want to see an example of hole/electron density of excitons please have a look to ref. [2].

Summary

From this tutorial you've learned:

  • How sort excitonic states by energy and intensity
  • Analyse their composition in reciprocal space and in terms of single-particle transitions
  • Visualize the exciton wave function in real space

Navigate

  1. ↑ Huge Excitonic Effects in Layered Hexagonal Boron Nitride, B. Arnaud et al., preprint ArXiv
  2. ↑ Strongly Bound Excitons in Metal-Organic Framework MOF-5: A Many-Body Perturbation Theory Study, A. R. Kshirsagar et al., preprint ChemRxiv
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