GW and dielectric matrix

The GW routine also determines the frequency dependent dielectric matrix ${\bf {\epsilon }}$ in the Random Phase Approximation (RPA) via the polarizability ${\bf {\chi }}$ and the Coulomb potential $V$ using $\epsilon (\omega )=1+V\cdot \chi (\omega )$.

Note, low-scaling GW algorithms determine the dielectric matrix on the imaginary frequency axis. These algorithms cannot be used to calculate ${\bf {\epsilon }}$ on the real frequency axis. Thus one, has to rely on the quartic-scaling GW implementation, which usability is limited to relatively small unit cells containing a few dozen atoms at maximum.

For instance, if one studies only the frequency dependent dielectric matrix, ALGO=CHI' can be used and the calculation of GW quasi-particle energies is skipped. To this end, VASP calculates the polarizability ${\bf {\chi }}_{{\bf {GG}}'}({\bf {q}},\omega )$ in two steps. First the spectral density of ${\bf {\chi }}$ is calculated with Fermi's golden rule Failed to parse (syntax error): {\displaystyle \sum_{ij}\delta(\epsilon_i - \epsilon_j -\omega) \left\langle i\right\mid {\bf r} \left\mid i\right\rangle }

The latter algorithm is of particular interest if one studies local field effects in the dielectric matrix (LRPA=.TRUE.). To this end, the polarizability is determined

Note, the calculated microscopic frequency dependent dielectric function without local field effects corresponds to the same function obtained using LOPTICS=.TRUE..

and the density functional perturbation routines (LEPSILON=.TRUE.).

In fact, it is guaranteed that the results are identical to those determined using a summation over conduction band states (LOPTICS). Differences for LSPECTRAL=.FALSE. must be negligible, and can be solely related to a different complex shift CSHIFT (defaults for CSHIFT are different in both routines). Setting CSHIFT manually in the INCAR file will remedy this issue. If differences prevail, it might be required to increase NEDOS (in this case the LOPTICS routine was suffering from an inaccurate frequency sampling, and the GW routine was most likely performing perfectly well). For LSPECTRAL=.TRUE. differences can arise, because (i) the GW routine uses less frequency points and different frequency grids than the optics routine or again (ii) from a different complex shift. Increasing NOMEGA should remove all discrepancies. Finally, the GW routine is the only routine capable to include local field effects for the frequency dependent dielectric function.

The imaginary and real part of frequency dependent dielectric function is always determined by the GW routine. It can be conveniently grepped from the file using the command (note two blanks between the two words)

grep " dielectric  constant" OUTCAR

The first value is the frequency (in eV) and the other two are the real and imaginary part of the trace of the dielectric matrix. Note that two sets can be found on the OUTCAR file. The first one corresponds to the head of the microscopic dielectric matrix (and therefore does not include local field effects), whereas the second one is the inverse of the dielectric matrix with local field effects included in the random phase approximation or density functional approximation (depending on LRPA).

If full GW calculations are not required, it is possible to greatly accelerate the calculations, by calculating the response functions only at the $\Gamma$-point. This can be achieved by setting the following values in the INCAR file:

 NKREDX = number of k-points in direction of first lattice vector
 NKREDY = number of k-points in direction of second lattice vector
 NKREDZ = number of k-points in direction of third lattice vector

The calculation of the QP shifts can be bypassed by setting ALGO=CHI. Furthermore, if only the static response function is required the number of frequency points should be set to NOMEGA=1 and LSPECTRAL=.FALSE.

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