The MP2 family

The largest method family here, and the one most directly tied to the response framing.

Why attenuation

MP2 builds dispersion from an uncoupled polarizability — the density fluctuation does not feel the field it creates. That over-polarizes, producing:

• \( C_6 \) coefficients that are too large
• overestimated π-stacking
• an error partly masked by BSSE in small basis sets, which is why the problem is easy to miss

Attenuating the correlation operator damps the long-range part where the uncoupled approximation is worst. One parameter, and the error it targets is identifiable rather than empirical.

Variants

RI-MP2 — density-fitted via 3-center/2-center integrals. Canonical MP2 is also implemented, for cross-validation rather than production.

Attenuated RI-MP2 — \( \mathrm{erfc}(\omega r)/r \) and terfc operators (Goldey & Head-Gordon, JPCL 2012).

SCS-MP2 — Grimme spin-component scaling (JCP 2003), and SCS-MP2(2terfc), dual-attenuated (Goldey, Dutoi & Head-Gordon, PCCP 2013).

OO-RI-MP2 — orbital-optimized, with level-shifted Newton, orbital DIIS, Cayley rotations and backtracking.

MP3 — spin-orbital third-order Møller–Plesset via the einsum! framework.

MP2-V — attenuated MP2 combined with VV10 nonlocal correlation.

Laplace formulations

RI-Laplace MP2 — AO-Laplace via pseudo-density matrices. The implementation is dense; it is the correctness reference for that formulation, not a reduced-scaling path. No O(N) has been measured.

Laplace SOS-MP2 — opposite-spin-only with a Laplace-factorized denominator (c_os scaling, minimax quadrature). The MO and AO formulations agree to machine precision.

The AO-sparse variant is measured negative: its truncation radius tracks the molecular diameter instead of saturating, so it does not deliver reduced scaling. That is reported rather than omitted.

Local MP2

Amplitude-threshold LMP2 — WSHG23 single-threshold, with localized virtuals and per-pair domain-local RI fits.

Counters only — no scaling claim is made. The J build is still dense-from-RI, so while the amplitude machinery is in place and anchored, the end-to-end cost is not reduced. The assembly step, not the solve, is the measured wall.

Size-extensivity

RI-MP2's total energy is size-extensive to 2e-12 Ha for a well-separated dimer versus twice the monomer — pinned by a test, not asserted. That is five orders inside the test's tolerance, so it passes on physics rather than on a loose bound.

Robust fitting

When the RI metric differs from the physical kernel — as it does for attenuated operators — robust (Dunlap) density fitting is required, not optional. Domain-local \( V^{-1} \) plus robust fitting gives size-extensive µHa-level MP2 error; the non-robust form collapses in a way driven by the metric, not by the domain size.