A long-range kernel beyond the TDLDA
We have seen the failure of the ALDA in the case of the H2 chain
when the intra-molecular distance increases. We also have seen how the
excitation wave function differs from the one obtained by solving the
Bethe Salpeter equation in the case of 2.5 a.u. intermolecular distance.
Such discrepancy has been traced back to the the long-range correlation
between the electron and hole, that cannot be captured by the simple local
approximation.
Now, we use yambo to check whether simple approximations for the fxc kernel
can cure the ALDA drawbacks. To this end we use the
Reciprocal space Dyson equation
for the response function (-o c option).
The key point is to realize that, indeed, the fxc kernel
acts like a self-energy in the Dyson equation for &Chi. As in the case
of quasiparticles, the most optimal self-energy can be chosen by looking
at the structure of the bare propagator, &Chi0 in this case.
If you look here
you will see that in the long wavelength limit &Chi0
behaves like q2.
As a naive consequence the self-energy must cancel this power dependence to
ensure that the product fxc&Chi0 remains finite in the
optical limit.
This simple argument is enough to introduce the long-range TDDFT kernel
Now we will use this kernel with a static approximation for the parameter alpha, first for the chain of intermolecular distance 2.5 a.u..
Enter the 2.5 directory
localhost> cd 2.5
and generate the input file for a tddft calculation in reciprocal space, specifying you want the long-range kernel:
localhost> yambo -o c -t lYou are now redirected to the editing of the yambo.in input file.
optics # [R OPT] Optics chi # [R CHI] Dyson equation for Chi. lrc_fxc # [R TDDFT] The LRC TDDFT kernel XfnQPdb= "none" # [EXTQP Xd] Database XfnQP_N= 1 # [EXTQP Xd] Interpolation neighbours % XfnQP_E 0.000000 | 1.000000 | 1.000000 | # [EXTQP Xd] E parameters (c/v) % % XfnQP_W 0.000 | 0.000 | 0.000 | 0.000 | # [EXTQP Xd] W parameters (c/v) % XfnQP_Z= ( 1.000000 , 0.000000 ) # [EXTQP Xd] Z factor (c/v) % QpntsRXd 1 | 41 | # [Xd] Transferred momenta % % BndsRnXd 1 | 20 | # [Xd] Polarization function bands % NGsBlkXd=Please change the yellow values ...1 RL # [Xd] Response block size % EnRngeXd 0.00000 | 10.00000 | eV # [Xd] Energy range % % DmRngeXd 0.10000 | 0.10000 | eV # [Xd] Damping range % ETStpsXd= 100 # [Xd] Total Energy steps % LongDrXd 1.000000 | 0.000000 | 0.000000 | # [Xd] [cc] Electric Field % LRC_alpha= 0.000000 # [TDDFT] LRC alpha factor
% XfnQP_E 3.5000000 | 1.000000 | 1.000000 | # [EXTQP Xd] E parameters (c/v) % % QpntsRXd 1 | 1 | # [Xd] Transferred momenta (We want only q=0 response) % % BndsRnXd 1 | 2 | # [Xd] Polarization function bands % NGsBlkXd=100 RL # [Xd] Response block size (Put some local fields, not too much) ... ETStpsXd= 1000 # [Xd] Total Energy steps ...
... and run yambo.
Now we can perform
different calculations assigning different values to the variable
LRC_alpha. This value is a static approximation to α(&omega) in the
long-range expression for fxc.
This must be negative (the interaction between electron
and hole is attractive). Try different numbers in the range 0 to -20.
You will see that around LRC_alpha=-19 we obtain
the same excitation energy of the BSE spectrum.
Now you can repeat the same calculations for the chain with 2.05 intermolecular distance. Once you will find the optimal value of %alpha, you will realize that as expected, it is one order of magnitude smaller than the one needed for the previous, more inhomogeneous system.
To conclude this tutorial note that the value of &alpha you found is much larger then in any solid, as showed in this picture. Can you understand why ?
