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.

ADF 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 ADF 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 with scalar relativistic effects within the ZORA, X2C, or RA-X2C formalism. GW should not be used in combination with solvent models, like COSMO, or other environments. In ADF2022 and later MP2 and double-hybrid functionals, GW and RPA can be used with spin-orbit coupling. Note that in case of spin-orbit coupling approximate SS and OS contributions are calculated. In ADF2022 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, ADF 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.