View source for Optical properties at finite temperature

[[File:High low.png|thumb | 200px|Yambo tutorial image]]

In this tutorial, we will show you how to calculate optical properties including thermal effects due to the electron-phonon coupling.<br>
This tutorial assumes that you have completed all the steps from the previous tutorial on [[Electron Phonon Coupling]].<br>
The tutorial is dived into different steps first we will calculate absorption at independent particle approximation, <br>then we will include excitonic effects,
and finally, we will show how to analyze the data.<br>

''' <span style="color:red">WARNING</span>''': Calculations in this tutorial are performed without double-grid, please check if the file <code>SAVE/ndb.PH_DoubleGrid</code>  is present in your calculations, then remove it and rerun electron-phonon coupling before proceed with optics.

'''Notice that there are two ways to include finite temperature effects in the BSE''':
* The first is to use a full Bethe-Salpeter equation with all the coupling terms in such a way to treat complex quasi-particles that cannot be handled in standard (Hermitian) BSE, this is explained in section [[https://www.yambo-code.org/wiki/index.php?title=Optical_properties_at_finite_temperature#Bethe-Salpeter_at_finite_temperature_.28diagonalization_solver.29 Bethe-Salpeter at finite temperature (diagonalization solver)]]. This approach has the advantage that one can obtain excitonic eigenvalues and eigenfunctions at finite temperature and not only the optical spectrum.
 
* The second method is use of a standard Hermitinan (resonant) BSE plus the inversion solver <ref name='dbgrid'>D. Kammerlander et al. [https://arxiv.org/abs/1209.1509 Phys. Rev. B '''86''', 125203] </ref>. In this case, a Lo two-particle Green's function that contains the complex quasi-particles is used in the solver to get a finite temperature spectrum. This approach has the advantage that can be used with the double-grid but it does not give access to eigenvalues and eigenfunctions of the BSE. This approach is described in the section: [[https://www.yambo-code.org/wiki/index.php?title=Optical_properties_at_finite_temperature#Bethe-Salpeter_at_finite_temperature_.28inversion_solver.29 Bethe-Salpeter at finite temperature (inversion solver)]]


== Absorption at finite temperature ==

Now you repeat the previous calculation but including all k-points, the last 3 valence and the first 3 conduction bands:

 .....
 %QPkrange                        # [GW] QP generalized Kpoint/Band indices
 1|8|2|7|
 %
 ....
and save the results of the 0K and 300K temperature in two separate folder with the -J option. 
Now you can use the correction to the energy levels and the induced width to calculate the optical absorption at finite temperature. Generate the input with the command  <code>yambo_ph -o c -V qp -F yambo.in_300K</code>:

 optics                           # [R] Linear Response optical properties
 chi                              # [R][CHI] Dyson equation for Chi.
 dipoles                          # [R] Oscillator strenghts (or dipoles)
 Chimod= "IP"                     # [X] IP/Hartree/ALDA/LRC/PF/BSfxc
 % QpntsRXd
 <span style="color:red"> 1 | 1 |   </span>                          # [Xd] Transferred momenta
 %
 % BndsRnXd
  <span style="color:red"> 2 |  7 |  </span>                         # [Xd] Polarization function bands
 %
 % EnRngeXd
 <span style="color:red"> 0.000000 | 5.000000 | </span>        eV    # [Xd] Energy range
 %
 % DmRngeXd
  <span style="color:red">0.0100000 | 0.0100000 | </span>        eV    # [Xd] Damping range
 %
 <span style="color:red">ETStpsXd=  1500  </span>                  # [Xd] Total Energy steps
 % LongDrXd
  1.000000 | 0.000000 | 0.000000 |        # [Xd] [cc] Electric Field
 %
 <span style="color:red">XfnQPdb= "E W < T300/ndb.QP" </span>     # [EXTQP Xd] Database action

set the path of the ''ndb.QP'' you want to read and perform the calculations. Notice that from the QP database we read two quantities the correction to the energy levels  <span style="color:blue">E</span> and the width  <span style="color:blue">W</span>. In this calculation we also included a small smearing 0.01eV to mimic the electronic smearing. Run your calculations with the command: <code>yambo_ph -F yambo.in_300K -J T300</code> and do the same for the 0K case.<br>
<br>You can plot the results with gnuplot using the command: <code> plot 'o-T300.eps_q1_ip' w l t '300 K', 'o-T0.eps_q1_ip' w l t '0 K'</code>.<br> Hereafter the result without and with electron-phonon coupling at two different temperatures:
[[File:Si absorption finite t.png|800px|center |Absorption of bulk silicon at finite temperature]]

The temperature effect is clearly visible in the figure.

== Bethe-Salpeter at finite temperature (diagonalization solver) ==

In this section we will calculate the Bethe-Salpeter at finite temperature using the full-BSE.

You can generate the input using the command <code>yambo_ph -X s -o b -k sex -y d -V qp -F yambo.in_BSE</code>

 em1s                             # [R][Xs] Statically Screened Interaction
 optics                           # [R] Linear Response optical properties
 bss                              # [R] BSE solver
 bse                              # [R][BSE] Bethe Salpeter Equation.
 dipoles                          # [R] Oscillator strenghts (or dipoles)
 DIP_Threads=0                    # [OPENMP/X] Number of threads for dipoles
 X_Threads=0                      # [OPENMP/X] Number of threads for response functions
 K_Threads=0                      # [OPENMP/BSK] Number of threads for response functions
 Chimod= "HARTREE"                # [X] IP/Hartree/ALDA/LRC/PF/BSfxc
 <span style="color:red">BSEmod= "coupling"      </span>         # [BSE] resonant/retarded/coupling
 BSKmod= "SEX"                    # [BSE] IP/Hartree/HF/ALDA/SEX/BSfxc
 BSSmod= "d"                      # [BSS] (h)aydock/(d)iagonalization/(s)lepc/(i)nversion/(t)ddft`
 BSENGexx=  9257            RL    # [BSK] Exchange components
 BSENGBlk=-1                RL    # [BSK] Screened interaction block size [if -1 uses all the G-vectors of W(q,G,Gp)]
 #WehCpl                        # [BSK] eh interaction included also in coupling
 <span style="color:red">KfnQPdb= "E W < T0/ndb.QP"  </span>     # [EXTQP BSK BSS] Database action
 % KfnQP_E
 <span style="color:red">0.500000 | 1.000000 | 1.000000 |  </span>      # [EXTQP BSK BSS] E parameters  (c/v) eV|adim|adim
 %
 BSEprop= "abs"                   # [BSS] Can be any among abs/jdos/kerr/magn/dich/photolum/esrt
 BSEdips= "none"                  # [BSS] Can be "trace/none" or "xy/xz/yz" to define off-diagonal rotation plane
 % BSEQptR
  1 | 1 |                             # [BSK] Transferred momenta range
 %
 % BSEBands
  <span style="color:red"> 2 |  7 |</span>                           # [BSK] Bands range
 %
 % BEnRange
  <span style="color:red">0.000000 | 5.000000 |</span>         eV    # [BSS] Energy range
 %
 % BDmRange
 <span style="color:red">0.010000 | 0.010000 | </span>        eV    # [BSS] Damping range
 %
 <span style="color:red">BEnSteps= 1500    </span>               # [BSS] Energy steps
 % BLongDir
  1.000000 | 0.000000 | 0.000000 |        # [BSS] [cc] Electric Field
 %
 #WRbsWF                        # [BSS] Write to disk excitonic the WFs
 % BndsRnXs
   1 | 12 |                           # [Xs] Polarization function bands
 %
 <span style="color:red">NGsBlkXs= 113                RL </span>   # [Xs] Response block size
 % LongDrXs
  1.000000 | 0.000000 | 0.000000 |        # [Xs] [cc] Electric Field
 %
 XTermKind= "none"                # [X] X terminator ("none","BG" Bruneval-Gonze)  


In this input file we asked Yambo to include finite temperature quasi-particle in the BSE with the line <code>KfnQPdb= "E W < T0/ndb.QP" </code>, notice that we also introduce an additional rigid shift of 0.5 eV to mimic the GW correction with the line <code> 0.500000 | 1.000000 | 1.000000 |  \# [EXTQP BSK BSS] E parameters  (c/v) eV|adim|adim</code>. You can now run the calculation with the command <code> yambo_ph -F yambo.in_BSE -J T0_BSE</code><br>
In principle you can calculate the GW correction following the [https://www.yambo-code.org/wiki/index.php?title=How_to_obtain_the_quasi-particle_band_structure_of_a_bulk_material:_h-BN this tutorial] and then merge the corresponding ndb.QP database with the one of the electron-phonon coupling using the command <code> ypp -qpdb m</code>.
Notice that we changed the BSE type to "coupling" because you need the full Bethe-Salpeter to deal with complex quasi-particles.<br>
Now we do the same calculation at T = 300 K. Copy the previous input file <code> cp yambo.in_BSE yambo.in_BSE_300</code>. Then modify where the code reads the QP database :
 <span style="color:red">KfnQPdb= "E W < T300/ndb.QP"  </span>     # [EXTQP BSK BSS] Database action
Run the command with <code>yambo_ph -F yambo.in_BSE_300 -J T300_BSE</code><br>
Finally, you should be able to plot the absorption spectra from the outputs in gnuplot :
 <code> plot 'o-T0_BSE.eps_q1_diago_bse' w l t '0 K', 'o-T300_BSE.eps_q1_diago_bse' w l t '300 K'</code>
[[File:Si bse optics.png|800px|center |Bethe-Salpeter at finite temperature for bulk silicom]]

'''Approximations:'''<br>
In this calculation we have made different approximations:<br>
1) we did not include the renormalization of excitons due to the change of the screening potential W with temperature. Including electron-phonon coupling
in the dielectric constant by changing the line <code> XfnQPdb= "none" </code> is unfortunately not enough, for a discussion see ref. <ref>L. Adamska and P. Umari, [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.103.075201 Phys. Rev. B '''103''', 075201 (2021)] </ref><br>
2) The electron-phonon coupling should enter in the BSE through the exciton-phonon matrix elements and not from the single-particle self-energy, as we have done in this tutorial. The approximation used in this tutorial is valid for not too strong bound excitons, and in general it generated a finite life-time also for the lowest exciton in direct materials that is not correct. For a discussion see refs.<ref>G. Antonius, S. G. Louie [https://arxiv.org/abs/1705.04245 Phys. Rev. B '''105''', 085111 (2022)]</ref><ref name='claudio'> see Supp. Mat. of [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.99.081109 Phys. Rev. B '''99''', 081109(R) (2019)]</ref><ref name='fulvio'>F. Paleari, [https://orbilu.uni.lu/handle/10993/41058 Phd Thesis (2019)]</ref> and <ref>H. Chen, D. Sangalli, and M. Bernardi [https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.125.107401 Phys. Rev. Lett. '''125''', 107401 (2020)]</ref>.<br>
3) we did not include scattering between exciton and phonons, therefore our results do not contain phonon-assisted absorption peaks, you can include these terms by finite difference displacements see refs. <ref name='fulvio'></ref>.

== Result analysis ==
Results of BSE at finite temperature can be analized in the same way of standard BSE. You can plot exciton binging energy as a function of the temperature, exciton wave-function and so, see [https://www.yambo-code.org/wiki/index.php?title=How_to_analyse_excitons Exciton Plot tutorial].
For example if you sort exciton with the command <code> ypp_ph -e s -J T0_BSE</code> for the case T=0K, in addition to the exciton energy now you will find its line-width in the files .
<code>o-T0_BSE.exc_qpt1_E_sorted</code> and <code>o-T0_BSE.exc_qpt1_I_sorted</code>.
 #
 #    Maximum Residual Value = .77136E+05
 #   
 #    E [ev]             Strength           Index              W [meV]
 #
 ...............
     2.82974172        0.388171151E-1      335.000000         11.9707289
     2.82976580        0.110338017         334.000000         11.9676380
     2.82979798        0.992550552         333.000000         11.9646358
     2.83366466        0.177820283E-2      336.000000         6.12744808
     2.83368802        0.183327938E-3      338.000000         6.12689734
     2.83369708        0.679280609E-3      337.000000         6.12836981
     2.87943292        0.146849517E-3      332.000000         2.51892495
     2.87945318        0.295515812E-3      331.000000         2.51745558
     2.89804840        0.539837289E-3      330.000000         1.42265606
     2.93557525        0.532299697         329.000000         12.5491867
     2.93559408        0.214365616         391.000000         12.5504198
     2.93561029        0.175688639         390.000000         12.5498590
     2.97320676        0.158064649E-3      388.000000         9.35168839
     2.97321057        0.300355150E-4      408.000000         9.35530090
     2.97321224        0.931323011E-4      389.000000         9.35327435
     2.97682190        0.551215744E-4      405.000000         8.87609196

'''Notice''' that in the files you will find many excitons with negative energy due the anti-resonant part of the full-BSE.

== Excitonic Eliashberg Functions ==
If you run the BSE and save the excitonic wave-functions (uncomment the flag <code>WRbsWF</code> in the BSE input file), tt is possible to plot the excitonic Eliashberg Functions with the command <code> ypp_ph -e e</code>.
For an interpretation of these functions see the discussion in reference <ref>Andrea Marini
 [https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.101.106405 Phys. Rev. Lett. '''101''', 106405 (2008)]</ref>

== Bethe-Salpeter at finite temperature (inversion solver) ==

It is possible to take into account complex quasi-particles in the BSE solution by means of the inversion procedure.
The idea is to construct an Lo (Eq. 5 of Ref.<ref name='dbgrid'></ref>) that includes the electron-phonon corrections including the imaginary part.
In order to generate the input you can type: <code>yambo_ph -X s -o b -k sex -y i -V qp</code>

 em1s                             # [R][Xs] Statically Screened Interaction
 optics                           # [R] Linear Response optical properties
 bss                              # [R] BSE solver
 bse                              # [R][BSE] Bethe Salpeter Equation.
 dipoles                          # [R] Oscillator strenghts (or dipoles)
 DIP_Threads=0                    # [OPENMP/X] Number of threads for dipoles
 X_Threads=0                      # [OPENMP/X] Number of threads for response functions
 K_Threads=0                      # [OPENMP/BSK] Number of threads for response functions
 Chimod= "HARTREE"                # [X] IP/Hartree/ALDA/LRC/PF/BSfxc
 BSEmod= "resonant"               # [BSE] resonant/retarded/coupling
 BSKmod= "SEX"                    # [BSE] IP/Hartree/HF/ALDA/SEX/BSfxc
 BSSmod= "i"                      # [BSS] (h)aydock/(d)iagonalization/(s)lepc/(i)nversion/(t)ddft`
 BSENGexx=  9257            RL    # [BSK] Exchange components
 BSENGBlk=-1                RL    # [BSK] Screened interaction block size [if -1 uses all the G-vectors of W(q,G,Gp)]
 #WehCpl                        # [BSK] eh interaction included also in coupling
 <span style="color:red">KfnQPdb= "E W < T0/ndb.QP"</span>       # [EXTQP BSK BSS] Database action
 KfnQP_INTERP_NN= 1               # [EXTQP BSK BSS] Interpolation neighbours (NN mode)
 KfnQP_INTERP_shells= 20.00000    # [EXTQP BSK BSS] Interpolation shells (BOLTZ mode)
 KfnQP_DbGd_INTERP_mode= "NN"     # [EXTQP BSK BSS] Interpolation DbGd mode
 % KfnQP_E
 <span style="color:red">  0.500000 | 1.000000 | 1.000000 |  </span>      # [EXTQP BSK BSS] E parameters  (c/v) eV|adim|adim
 %
 BSEprop= "abs"                   # [BSS] abs/kerr/magn/dichr trace
 % BSEQptR
  1 | 1 |                             # [BSK] Transferred momenta range
 %
 % BSEBands
 <span style="color:red">   2 |  7 |  </span>                         # [BSK] Bands range
 %
 % BEnRange
 <span style="color:red">  0.00000 | 5.00000 | </span>        eV    # [BSS] Energy range 
 %
 % BDmRange
 <span style="color:red">  0.010000 | 0.010000 |  </span>       eV    # [BSS] Damping range
 %
 <span style="color:red">BEnSteps= 1500  </span>                  # [BSS] Energy steps
 % BLongDir 
  1.000000 | 0.000000 | 0.000000 |        # [BSS] [cc] Electric Field
 %
 #WRbsWF                        # [BSS] Write to disk excitonic the WFs
 XfnQPdb= "none"                  # [EXTQP Xd] Database action
 % BndsRnXs
   1 | 12 |                           # [Xs] Polarization function bands
 %
 <span style="color:red">NGsBlkXs= 113  </span>            RL    # [Xs] Response block size
 % LongDrXs
  1.000000 | 0.000000 | 0.000000 |        # [Xs] [cc] Electric Field
 %
 XTermKind= "none"                # [X] X terminator ("none","BG" Bruneval-Gonze)

then if you run the job you will get the spectrum at finite temperature. Notice that with the inversion solver you can choose if you want the BSE with or without the coupling by changing the <code>BSEmod</code> variable.
This approach can be combined with double-grid as explained here: [http://www.attaccalite.com/speed-up-dielectric-constant-calculations-using-the-double-grid-method-with-yambo/ Speed up dielectric constant calculations using the double-grid method with Yambo ]. Yambo automatically will interpolate quasi-particle corrections on the fine grid. <br> Notice that BSE inversion can become unfeasible for large BSE matrices.

<span style="color:green">Nota bene:</span> if you use the inversion solver you just get the optical spectra, but you do not have access to the excitonic eigenvalues and eigenvectors and by consequence you cannot plot the excitonic wave-function or print the  excitonic Eliashberg functions.

== Using electron-phonon quasi-particles in real-time simulations ==
The electron-phonon quasi-particle correction can be used also in the real-time dynamics. In order to do so, put the <code>ndb.QP</code> in the SAVE <br>folder and then remove the symmetries with comand <code>ypp_nl -y</code>, see the tutorial [[Linear response_using Dynamical Berry Phase]].

 fixsyms                          # [R] Remove symmetries not consistent with an external perturbation
 % Efield1
  <span style="color:red"> 1.000000 | 0.000000 | 0.000000 |   </span>     # First external Electric Field
 %
 % Efield2
  0.000000 | 0.000000 | 0.000000 |        # Additional external Electric Field
 %
 BField= 0.000000           T     # [MAG] Magnetic field modulus
 Bpsi= 0.000000             deg   # [MAG] Magnetic field psi angle [degree]
 Btheta= 0.000000           deg   # [MAG] Magnetic field theta angle [degree]
 #RmAllSymm                     # Remove all symmetries
 <span style="color:red">RmTimeRev</span>                     # Remove Time Reversal
 #RmSpaceInv                    # Remove Spatial Inversion
 #KeepKGrid                     # Do not expand the k-grid

<code>Ypp_nl</code> will create a new folder <code>FixSymm</code> with a <code>SAVE</code> folder that will contains a new <code>ndb.QP</code> with removed symmetries. Now you can run a real-time simulation to get the linear response with the command <code>yambo_nl -u p</code>;
 
 nloptics                         # [R] Non-linear spectroscopy
 % NLBands
 <span style="color:red">  2 |  7 | </span>                          # [NL] Bands range
 %
 NLverbosity= "high"              # [NL] Verbosity level (low | high)
 <span style="color:red">NLtime=150.000000</span>           fs    # [NL] Simulation Time
 NLintegrator= "INVINT"           # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
 NLCorrelation= "IPA"             # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX/LSEX/LHF")
 NLLrcAlpha= 0.000000             # [NL] Long Range Correction
 <span style="color:red">NLDamping= 0.015000</span>        eV    # [NL] Damping (or dephasing)        
 RADLifeTime=-1.000000      fs    # [RT] Radiative life-time (if negative RADLifeTime=Phase_LifeTime)
 #EvalCurrent                   # [NL] Evaluate the current
 HARRLvcs=  9257            RL    # [HA] Hartree     RL components
 EXXRLvcs=  9257            RL    # [XX] Exchange    RL components
 CORRLvcs=  9257            RL    # [GW] Correlation RL components
 <span style="color:red">GfnQPdb= "E W < SAVE/ndb.QP"</span>                  # [EXTQP G] Database action
 Field1_Freq= 0.100000      eV    # [RT Field1] Frequency
 Field1_Int=  1000.00       kWLm2 # [RT Field1] Intensity
 Field1_Width= 0.000000     fs    # [RT Field1] Width
 <span style="color:red">Field1_kind= "DELTA"</span>          
 Field1_pol= "linear"             # [RT Field1] Pol(linear|circular)
 % Field1_Dir
 <span style="color:red">  1.000000 | 0.000000 | 0.000000 |  </span>      # [RT Field1] Versor
 %
In this input we read the quasi-particle life time with the line: <span style="color:red">GfnQPdb= "E W < SAVE/ndb.QP"</span>.  Notice that we introduce also a small damping factor <span style="color:red">NLDamping= 0.01 [eV]</span> to take into account the dephasing due to the electron-eletrcon interaction and the electron-hole recombination.

This small dephasing is the one that sets the total length of the simulation, because the el-ph one has different values for the different transitions. If you run the code in the output the code suggests you and optical length for the linear reponse, for the non-linear one see. Running the code, it will print in output information of the life-time read from the <code>ndb.QP</code> database:

 <---> Minimum damping detected: 0.001568 [eV]
 <---> Maximum damping detected: 1.704498 [eV]
 <---> Suggest simulation length: 131.6424 [fs]

after the you can analyze results with the command <code>ypp_nl -u</code> or using YamboPy. Hereafter the input for <code>ypp_nl</code>.

 nonlinear                        # [R] Non-linear response analysis
 Xorder= 1                        # Max order of the response/exc functions
 % TimeRange
  -1. |  -1.0000 |         fs    # Time-window where processing is done
 %
 <span style="color:red">ETStpsRt= 1500  </span>                  # Total Energy steps
 % EnRngeRt
   <span style="color:red">2.00000 | 5.00000 | </span>        eV    # Energy range
  %
 DampMode= "NONE"                 # Damping type ( NONE | LORENTZIAN | GAUSSIAN )
 DampFactor= 0.000000       eV    # Damping parameter
 PumpPATH= "none"                 # Path of the simulation with the Pump only

Then you can plot optical absorption at 300 Kelvin from real-time vs the linear response one:

[[File:Rt_vs_lr_finite_T.png|center|800px]]The small wiggles in the real-time results indicate the simulation time is not enought and it should be increased.

== Phonon-assisted density of states ==
Even if exciton-phonon scattering is not included in the BSE at finite temperature, it is possible to plot the phonon-assisted density of states (DOS) for light emission, defined as:

[[File:Formula exc ph.png | 600px | center | Phonon-assisted JDOS ]]

where '''E<sub>q</sub><sup>l</sup>''' is the energy of the '''l'''-exciton at '''q'''-momentum,  '''ω<sub>q</sub><sup>λ</sup>''' is the phonon energy, '''n<sub>B</sub>''' is the Bose function,  exciton are weighted with a Boltzamn factor and '''E<sub>min</sub>''' is the lowest exciton energy in the all BZ.
In order to calculate these DOS you need to solve BSE for all the q-points for the lowest excitons, you can use SLEPC library to speed up calculations, then you need a converged phonon calculations.
Then you interpolate phonon on a dense phonon grid using <code>matdyn.x</code> utility in QE, and Yambo will interpolate exicton on the same grid.
Here we present an example for hBN that is an indirect semiconductor.
Running <code> ypp_ph -e p</code> you will get:
 
 BoseTemp= 50 K    # Bosonic Temperature
 excitons                         # [R] Excitonic properties
 ph_ass_dos                       # [R] Phonon-assisted DOS
 States= "1 - 4"                  # Index of the BS state(s)
 PHfreqF= "./bn.freq_54"            # PWscf format file containing the phonon frequencies
 % DOSERange
  5.000000 | 5.500000 |         eV    # Energy range
 %
 DOSESteps= 1000                  # Energy steps
 DOS_broad= 0.004        eV    # Broadening of the DOS 

where "bn.freq_54" is the file produced by matdyn.x with the input:

 &input
     asr='simple',
     flfrc='bn.fc',
     flfrq='bn.freq_54',
     dos=.true.,
     fldos='bn.dos',
     ndos=2
     nk1=54, nk2=54, nk3=18
 /

and the BSE was calculated on a 18x18x6 grid. The final spectra will be:

[[File:Test dos pl bulk.png| 700px | center| Phonon-assisted JDOS]]

where the DOS has been shifted to match the experimental peaks. Notice that the peak intensities are complitely off, because of the lack of the exciton-phonon matrix elements.

==References==