Real time approach to linear response

In this example, we will consider a single layer of hexagonal boron nitride (hBN). If you didn't before you can download input files and Yambo databases for this tutorial here: hBN-2D-RT.tar.gz.

The first input file is a self-consistent(SCF) calculation that is used to generate the density of the system. The second input file is a non-self consistent(NSCF) calculation to diagonalize the KS Hamiltonian, which depends on the density of the first run, on for a given number of bands and k-points. Notice that parameters in the NSCF calculation determine the number of k-points and the maximum number of bands that can be used in Lumen. Run this calculation with the command:

pw.x -inp hBN_2D_scf.in  > output_scf
pw.x -inp hBN_2D_nscf.nscf.in > output_nscf

Notice that in the NSCF file of QuantumEspresso we use the flag force_symmorphic=.true. to exclude the non-symmorphic symmetries that are not supported by Yambo.

Real-time dynamics

In order to calculate linear-response in real-time, we will perturb the system with a delta function in time external field. Use the command yambo_nl -u -F input_lr.in to generate the input:

nlinear                      # [R NL] Non-linear optics
NL_Threads= 1                # [OPENMP/NL] Number of threads for nl-optics
% NLBands
  3 |  6 |                   # [NL] Bands
%
NLstep=   0.0100       fs    # [NL] Real Time step length
NLtime=55.000000      fs    # [NL] Simulation Time
NLverbosity= "high"           # [NL] Verbosity level (low | high)
NLintegrator= "INVINT"       # [NL] Integrator ("EULEREXP/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "IPA"         # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/JGM/SEX/HF")
NLLrcAlpha= 0.000000         # [NL] Long Range Correction
% NLEnRange
 0.200000 | 8.000000 | eV    # [NL] Energy range 
%
NLEnSteps= 1                 # [NL] Energy steps
NLDamping= 0.000000    eV    # [NL] Damping
% ExtF_Dir
 0.000000 | 1.000000  | 0.000000 |        # [NL ExtF] Versor
%
ExtF_kind=  "DELTA"          # [NL ExtF] Kind(SIN|SOFTSIN|RES|ANTIRES|GAUSS|DELTA|QSSIN)

The standard input of Lumen is thought for the non-linear response so we have to change some parameters in order to calculate the linear response. Set the field direction along y, the field type to DELTA, the length of the simulation to 55 fs, number of bands from 3 to 6 dephasing to zero and the number of energy steps to one, as shown above in red. We set the verbosity to "high" in such a way to print real-time output files. We set the differential equation integrator to INVINT that is faster but less accurate than the default (see Ref. ) . This integrator is ok in case of independent partcicles but I advise you to use CRANKNIC integrator when correlation effects are present. Now run yambo_nl -F input_lr.in The code will produce different files: o.polarization_F1 that contains the polarization, o.external_potential_F1 the external field we used, and finally r_optics_nloptics a report with all information about the simulation. If you plot the third column of o.polarization_F1 versus the first one (time-variable) you will get the time-dependent polarization along the y-direction:

Results Analysis

Now we can use ypp_nl -u to analyze the results:

nonlinear                    # [R] NonLinear Optics Post-Processing
Xorder= 1                    # Max order of the response functions
% TimeRange
-1.000000 |-1.000000 | fs    # Time-window where processing is done
%
ETStpsRt= 200                # Total Energy steps
% EnRngeRt
  0.00000 | 10.00000 | eV    # Energy range
%
DampMode= "LORENTZIAN"             # Damping type ( NONE | LORENTZIAN | GAUSSIAN )
DampFactor=  0.10000   eV    # Damping parameter

where we set a Lorentzian smearing corresponding to 0.1 eV. Notice that due to the finite time of our simulation smearing is always necessary to Fourier transform the result. Then we run ypp_nl and obtain the following files: the dielectric constant along with the field direction o.YPP-eps_along_E, the EELS along with the same direction o.YPP-eels_along_E, and the damped polarization o.YPP-damped_polarization. Now we can plot the dielectric constant and compare it with the linear response:

The input for the linear response can be downloaded here. Notice that in a real-time simulation we obtain directly the

[math]\displaystyle{ \chi(\omega) = \frac{P(\omega)}{E(\omega)} }[/math]

that is realted to the dielectric constant throught the relation [math]\displaystyle{ \epsilon(\omega) = 1 + 4 \pi \chi(\omega) }[/math]