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The Interacting response function: Many-Body and TDDFT

In LR we have introduced the response function in the Kubo formalism. The calculation of the response function can be done in yambo in two different theoretical frameworks: the Many Body ( BSK ) and the TDDFT ( TDDFT ) schemes. In both approaches is crucial the knowledge of the Reciprocal Space and/or Bloch representation of the response function.

Bloch representation and the Non-interacting response function

With the response function written in Reciprocal Space

the density operators can be expanded in the basis of the single-particle electronic states

one easily obtains in the non-interacting picture

Here the band summation is governed by the BndsRn , while QpntsR decides the momenta at which the response function is calculated. We have introduced the interacting electron-hole Green's function

that in the non-interacting picture becomes

in yambo you can control the "type" of Green's function (causal or T-ordered) using the GrFnTp . The energy range is decided in EnRnge , while the damping is assumed linear in energy with bounds decided in DmRnge . The energy range is divided in ETStps uniform steps. Electron-hole pairs can be selected in energy using the EhEngy We have also introduced the oscillators matrix elements

with

that are used also in the exchange and correlation self-energy, XX and GW. In Eq.(1) is in the difference vector shifted in the IBZ and

See KPT for mode details about k-points sampling and FFT for further technical details about the evaluation of real-space integrals.

The Optical limit

In the limit of vanishing momentum the above expression simplifies. In particular the oscillator strengths read

The long-wavelength limit of the oscillators is easily calculated as

The Random Phase approximation

The simplest approach to the calculation of the response function is obtained assuming that the total potential the system is approximatively given by

in this approximation the relation between the exact and the non interacting response function, in reciprocal space, is given by

(with repeated indexes summed). In yambo input files the NGsBlk controls the size of the response function in Reciprocal Space. Note that when NGsBlk = 1 no Local Fields are considered. This corresponds to neglect the charge oscillations induced by the external potential.

As mentioned before Eq.(2) can be solved either directly (a method that we will call G space ) or in the Bloch representation that rewrites Eq.(2) as an equation for the electron-hole Green's function (see BSK and TDDFT for more details).

In the general Many Body language, even if Eq.(2) is only approximated it defines the relation between the reducible and the irreducible response functions

The Static approximation

When the polarization function is used to build the BSE kernel ( BSK ) the screening process is assumed to be instantaneous (see [3] for more details)

with FT for Fourier Transform. This polarization function is much faster to calculate then the dynamical one and corresponds to a specific yambo runlevel.

References

  1. G. Strinati Rivista del nuovo cimento 11,1 (1988).
  2. H. Ehrenreich, The Optical Porperties of Solids,Academic, New York (1965).
  3. A. Marini, R. Del Sole, Phys. Rev. Lett., 91, 176402 (2003).