MBPT scheme


This page describes technical aspects of the MBPT (Many-Body Perturbation Theory) module which is used in double-hybrid and MP2, RPA, GW and GW-BSE calculations. In order to use double-hybrids, MP2 or RPA in your calculation you should request it in the XC input block. In order to perform a GW calculation, you should request it in the GW input block.

BAND implements RPA, GW, and SOS-MP2 (spin-opposite-scaled) using a newly designed algorithm which in all cases scales quadratically with system size 1 3. Full MP2 is at the moment only implemented using the canonical RI-algorithm which scales to the fifth power with system size. Thus, we strongly discourage using full MP2 or double-hybrids employing full MP2 for system larger than 1000-1500 basis functions. At the moment BAND features a large number of double-hybrids using SOS-MP2 only (For a list of implemented functionals see XC input block) which are significantly faster than conventional double-hybrids while offering the same level of accuracy 2.

GW, MP2, RPA and double-hybrid functionals can be used icw scalar relativistic effects within the ZORA, X2C, or RA-X2C formalism. In BAND2022 in case of ZORA by default the so called scaled ZORA orbital energies are used in the MBPT expressions.

The Formalism used in the double-hybrid calculation can be changed using the Formalism key. By default, BAND selects the most appropriate algorithm for your system and functional.

The calculation of the independent-particle polarizability or Kohn-Sham density response function in imaginary time is the key step in SOS-MP2, RPA and GW. The equations are solved in the atomic orbital basis exploiting sparsity via advanced density fitting techniques (so-called pair-atomic resolution of the identity or pair-atomic density fitting) 1. In case of a SOS-MP2 or RPA calculation, the polarizability is than contracted with the Coulomb potential. For SOS-MP2, the correlation energy is then immediately evaluated in imaginary time while in a RPA calculation the product of Coulomb potential and polarizability is Fourier transformed to the imaginary frequency axis where the correlation energy is evaluated using a matrix logarithm. In a GW calculation, the polarizability is Fourier transformed to the imaginary frequency axis as well where the so-called screened interaction is calculated. The QP states are then evaluated along the real-frequency axis using analytical continuation techniques.