Reproducing low-scaling GW QP shifts

[bprobinson102]

Hello VASP team,

I am trying to reproduce Fig. 3(c) in the 2016 Liu et al. PRB for the low-scaling GW data; however, my QP corrections are both qualitatively and quantitatively different than what the figure in the paper shows. I have attached the IOs for the GW and preceding calculations, as well as a figure of my QP shifts. For clarify, my x-axis is the DFT eigenvalues offset by the Fermi energy and the QP shifts are defined as QP-e(zeros)-KS-energies. Furthermore, I tried to use as many of the same calculation parameters as I could. I would greatly appreciate it if you are able to help me figure out the source causing this discrepancy. Thank you for your time!

Brian

[merzuk.kaltak]

Dear Brian,

The ENCUT value you have chosen for the Cu PAW potential (900 eV) is relatively large, and the POTCAR file was not constructed for such a high cutoff. Peitao used the Cu_sv_GW_nc potential in his calculations and kept the default cutoff of 728 eV.

His calculations also included the long-wave limit (see Eqs. 79 and 80). This means that a pre-calculated WAVEDER file must be present in the working directory for the GW step. Such a file is usually generated by setting LOPTICS=T during the exact-diagonalization step.

I also noticed that you select Fermi occupancies via ISMEAR=-1, which suggests that you are interested in QP energies at finite electronic temperature. If that is indeed your intention, you should set LFINITE_TEMPERATURE=T consistently across all of your calculations.

Please keep in mind, however, that the LFINITE_TEMPERATURE tag did not yet exist in 2016 — it was introduced in 2020. The original calculations were therefore carried out within the zero-temperature formalism of many-body perturbation theory, using ISMEAR=0 and SIGMA=0.1. For metallic systems this approach can be problematic and may lead to difficulties with k-point convergence. In such cases we generally recommend to use LFINITE_TEMPERATURE=T.

Best regards,

[bprobinson102]

Hi Merzuk,

Thank you for the reply. Can you clarify if the long-wavelength limit is properly described for metals in VASP? It is my understanding that LOPTICS should be off when studying a metallic system with GW.

Brian

[merzuk.kaltak]

Hi Brian,

The long-wave limit is treated correctly for metals in VASP, in the sense that both intra- and interband contributions to the polarizability are properly accounted for.

The difficulties arise instead in the product of the polarizability with the Coulomb potential, and there are in fact two separate issues here.

The first concerns the electronic temperature. The intraband contribution scales with the inverse electronic temperature, so that it becomes infinite at T=0 — this is the well-known Drude term. The mechanism implemented in VASP for the zero-temperature case can therefore be problematic for some systems.

The second issue persists even at finite temperature. There the intraband term is finite and approaches a constant as q→0, but the Coulomb potential itself diverges at q=0, so the product remains ill-behaved. Handling this properly requires a truncation of the Coulomb kernel within the Wigner–Seitz cell.

In practice, the main consequence of both effects is a non-uniform convergence with respect to k-point sampling. The RPA correlation energy, for instance, tends to oscillate as the number of k-points is increased, which makes it difficult to extrapolate reliably to the infinite k-point limit. Lattice constants obtained as minima of the energy–volume curve, by contrast, usually converge smoothly with k-points when no WAVEDER file is present. The same holds more generally for energy differences, which are typically the quantities of interest.

The example in the paper does include the long-wave limit, based on the expectation that the QP shift near the Fermi energy is a smooth function and should not exhibit the discontinuities that can appear when the WAVEDER file is neglected. Peitao tested both variants and observed better convergence when the limit was included.

So the short answer to your question is that switching LOPTICS off is a sensible default for metals in many situations, but it is not a strict rule — it depends on the quantity you are converging.

This remains an active area of investigation. The Wigner–Seitz truncated Coulomb kernel mentioned above is one of the routes we are currently pursuing for the treatment of metals within GW.

Best regards,
Merzuk