Coulomb backends: exact, density-fitted, streamed, screened

The two-electron Coulomb (and exact-exchange) term is the engine's cost center. dftax offers a ladder of backends with the same energy but different memory/compute trade-offs.

Exact 4-center ERIs (default)

from dftax import run_rks
from dftax.system import Molecule
from dftax.energy.xc import PBE

water = Molecule.from_xyz("O 0 0 0; H 0.7586 0 0.5043; H 0.7586 0 -0.5043", "sto-3g")
run_rks(water, PBE())                      # builds the (nao⁴) ERI tensor

Exact (μν|λσ) via McMurchie-Davidson, using the 8-fold permutational symmetry. O(N⁴) memory, best for small systems and as the validation reference.

Density fitting (RI)

run_rks(water, PBE(), auxbasis="weigend")  # RI-J Coulomb via a fitting basis

Resolution-of-identity reduces the 4-center integral to 2- and 3-center pieces against an auxiliary basis: O(N³) compute, O(N²·N_aux) memory for the 3-center tensor.

Streaming + screening (large systems)

For memory-light runs, stream the RI-J contraction over auxiliary chunks (never forming the 3-center tensor) and Cauchy-Schwarz-screen negligible bra pairs:

from dftax import RKS, rks_scf
from dftax.grid import becke_grid

gc, gw = becke_grid(water.symbols, water.atom_coords(), 75, 302)
ks = RKS.from_molecule(water, PBE(), gc, gw,
                       auxbasis="weigend", df_chunk=64, df_screen=1e-10)
rks_scf(ks).e_tot

df_chunk sets the auxiliary streaming chunk (O(chunk·nao) grid memory); df_screen drops bra pairs whose self-Schwarz factor is below the relative cutoff (O(N) survivors for extended systems). Hybrids stream RI-K the same way.

Hybrids and exact exchange

PBE0 / B3LYP add a fraction of exact exchange K. On the exact path it is contracted on the fly (exchange_k_4c); under density fitting it uses streamed RI-K. The L ≥ 2 exact path is compile- and memory-bounded by scanning the bra primitive pair in the ERI kernel (so cc-pVDZ compiles in seconds rather than OOMing).

Choosing a backend

System size	Backend
small / validation	exact 4-center (default)
medium	RI density fitting (auxbasis=...)
large	streamed + screened DF (df_chunk, df_screen)