A closer look: plots
The electronic density
The easiest quantity we can compare for the two chains (2.05 and 2.5 intratomic
distance) is the electronic density.
To this end we can use the yambo post/pre processor (YPP). YPP is capable of
performing some basic analyses using the pre-calculated informations
stored in the yambo databases.
YPP works like yambo. It uses a series of options to create an input file
containing only the variables that are relevant to that type of calculation.
localhost>cd 2.05/ localhost>ypp -H ___ __ _____ __ __ _____ _____ | Y || _ || Y || _ \ | _ | | | ||. | ||. ||. | / |. | | \ _/ |. _ ||.\ / ||. _ \ |. | | |: | |: | ||: | ||: | \|: | | |::| |:.|:.||:.|:.||::. /|::. | `--" `-- --"`-- --"`-----" `-----" Tool: ypp 3.2.1 rev.506 Description: Y(ambo) P(ost) P(rocessor) -h :Short Help -H :Long Help -J:Job string identifier -V :Input file verbosity [opt=gen] -F :Input file -I :Core I/O directory -O :Additional I/O directory -C :Communications I/O directory -N :Skip MPI initialization -S :DataBases fragmentation -k :BZ Grid generator [(k)pt,(q)pt,(l)ongitudinal] -e :Excitons [(s)ort,(a)mplitude,(w)ave] -s :Electrons [(w)ave,(d)ensity,do(s)] -f :Free hole position [excitons plot] -r :BZ energy RIM analyzer By yambo developers group http://www.yambo-code.org
To plot the density we need to type
localhost> ypp -p dThe input file ypp.in will be
density # [R] Density electrons # [R] Electrons (and holes) Format= "g" # Output format [(g)nuplot/(x)crysden] Direction= "1" # [rlu] [1/2/3] for 1d or [12/13/23] for 2d [123] for 3D FFTGvecs= 7659 RL # [FFT] Plane-waves
As our system is one-dimensional and it is lying parallel to the z axis
we need to change the Direction="12" so to perform a contour plot
on the "XY" plane. To use Xcrysden switch
to "x" the value of the Format variable.
Now you can launch ypp.
At the end of the quick calculation you will find in your directory
the file o.density_2d.xsf that you can plot using Xcrysden. We have
provided a script to automatically view this file in the ../bin folder.
To use it type
localhost>../bin/launch_xcrysden.sh o.density_2d.xsf
Follow the same procedure for the other distance and compare the two densities. You should see something like
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We see that the smaller the gap (@ 2.05 a.u.) the more delocalized the electronic density. The consequence is that electrons can move more freely in the 2.05 case. At the same time, however, the polarization will be more "metallic-like" where the ALDA is expected to work better.
The TDLDA excitations wavefunction
Using YPP we can do something more. We can plot the wavefunction corresponding
to the most intense peak in the dielectric function.
The excitation wavefunction is a two-coordinate (r1, and r2) function and
represents the probability amplitude of finding the electron at position r1
when the hole is in r2. The larger this probability the more delocalized is the
perturbation induced by the external field.
A mean large distance between the electron and hole also reflects the poor correlation
felt by the two bodies.
YPP can plot either the case where the electron and the hole are forced to be in the same
space point, or the case where the hole's position is fixed somewhere and the
electron is left free to move.
This second case is more appropriate here.
We need first to identify the index of the corresponding
eigenstate.
Running
localhost>ypp -e s
YPP will create a file named o.exc_I_sorted that contains the list of peaks in the dielectric function ordered with increasing peak intensity.
# # E [ev] Strength Index # 3.783688 1.000000 1.000000 4.38457 0.03179 3.00000 5.171 0.3731E-2 5.000
We see that the peak with energy 3.78 dominates the spectra and it
corresponds to the index 1.
If we type now
localhost>ypp -e wto get the input file
exc_wf # [R] Excitonic Wavefunction excp # [R] Excitonic Properties plot # [R] Plot Format= "x" # Output format [(g)nuplot/(x)crysden] Direction= "12" # [rlu] [1/2/3] for 1d or [12/13/23] for 2d [123] for 3D FFTGvecs= 7659 RL # [FFT] Plane-waves States= "1 - 1" # Index of the BS state(s) Degen_Step= 0.0100 eV # Maximum energy separation of two degenerate states % Cells 1 | 1 | 1 | # Number of cell repetitions (even or 1) % % Hole 0.000 | 0.000 | 0.000 | # [cc] Hole position in unit cell %
It is interesting to do a 3D plot. To this end we set Direction="123"
and replace 1 with 16 in Cells (this value will expand the unit cell
along the X direction). The States variable is already set to the peak number 1.
We put the hole in the middle of the chain by setting
% Hole 1.02500 | 12.50000 | 12.50000 | # [cc] Hole position in unit ce %
remember that the first field must be set to the interatomic
distance divided by two. In the case of 2.5 a.u. the first field will be 1.25.
If we run now ypp it will create the file o.exc_3d_1.xsf that you can plot
by using
localhost>../bin/launch_xcrysden.sh o.density_2d.xsfIf we compare the plot with the one obtained by solving the BS equation we should see
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We immediately see that, while the ALDA excitation is always spread all over the chain,
in the 2.5 distance case the "true" wavefunction acquires a tail that decreases the
probability of finding the electron and the hole very far from each other.
This saturation of the excitation wavefunction is the physical reason for the
poor performance of the ALDA. As it is a local approximation it cannot take into account
a long-range correlation between the electron and the hole.