The MP2 family

The largest method family here. Every variant is density-fitted (RI): each needs an orbital basis and a matching RI auxiliary basis ([mp2] auxbasis, or the auxbasis argument in Python). Formal cost is O(N⁵) for the RI-MP2 transformation. Grades per method.kind are on Capabilities and validation. The [mp2] keys are in Input file. The working equations of every variant on this page (spin factors, frozen core, operator and unit conventions), with the code path behind each, are in crates/ferric-mp2/theory.md; the identities it relies on are checked by scripts/mp2_variant_identities.py.

RI-MP2

What it is. Second-order Møller–Plesset theory on an RHF reference (UHF for an open shell, below) with 3-centre/2-centre density fitting. Canonical (non-RI) MP2 is also implemented, for cross-validation, not production.

Open shells. rimp2 and oo-rimp2 accept an open-shell molecule (multiplicity > 1) from the CLI, for task = "energy" only: they solve the same plain UHF that kind = "uhf" runs, then take unrestricted RI-MP2 (UMP2, as PySCF mp.MP2(uhf)) or unrestricted OO-RI-MP2. There is no unrestricted MP2 nuclear gradient, and [mp2] kappa is refused on an open shell. From Python, run_rimp2 does the same (UHF + UMP2 when multiplicity > 1; result.reference says which, and kappa raises); run_oo_rimp2 is closed shell only. Both unrestricted methods are validated on the CH3 doublet at cc-pVDZ: U-RI-MP2 against ORCA RI-MP2 NoRI, U-OO-RI-MP2 against an independent numpy OO-RI-MP2 (see the anchors).

The other MP2-family kinds (mp3, att-rimp2, scs-mp2, scs-mp2-2terfc, laplace-mp2, laplace-sos-mp2, rs-mp2-rpa) and their Python drivers are closed shell only: set multiplicity = 1. mp2-v switches to a UHF reference when multiplicity > 1 (CLI, energy only); Python run_mp2_v is closed shell only.

The reference SCF: exact, RI-JK or RIJCOSX. The auxbasis above fits the correlation step only. The reference SCF is a separate choice, scf_jk on every MP2-family Python driver ([scf] jk in the CLI, task energy): "exact" (four-centre J and K, the default), "rijk" (RI-J + RI-K) or "rijcosx" (RI-J + COSX K). The fitted modes use jk_auxbasis ([scf] jk_aux), a JK-fit basis with default def2-universal-jkfit. It is never the MP2 auxbasis, because an MP2-fit set used for K costs mHa. rs-mp2-rpa is the exception: its default is "rijk". Every result echoes scf_jk and jk_auxbasis. An unknown mode, jk_auxbasis with "exact", or a k_builder the mode would override is an error (so is any k_builder on rs-mp2-rpa's RI-JK default, in the CLI as in Python). A jk_auxbasis whose name lacks "jkfit" draws a warning. oo-rimp2 accepts only "exact": orbital optimisation rebuilds exact four-centre J/K at every orbital step, so a fitted reference would change only the starting orbitals; threading RI-JK through the OO Fock builds is not implemented. In the CLI a fitted reference cannot be combined with [scf] df_increments (an exact-J/K scheme) or an explicit df_guess = true, and it skips the default DF-guess pre-stage. At aug-cc-pVTZ the four-centre SCF is most of an MP2 single point, so a fitted reference is the usual choice for interaction-energy benchmarks. The fitting error is small and largely cancels in a counterpoise-corrected interaction energy. Measured with PySCF DF-MP2 (frozen core, aug-cc-pvXz-ri for MP2) on the S66 water dimer:

Run it. method.kind = "rimp2" (examples/water-rimp2.toml, examples/water-rimp2-frozen-core.toml); Python ferric.run_rimp2(mol, bs, aux).

Accuracy. Proven. RI-MP2 is size-extensive to 2e-12 Ha for a well-separated dimer against twice the monomer (ferric-mp2/tests/rimp2_size_extensivity.rs, which asserts 1e-7).

Knobs. frozen_core defaults to 0 (all electrons correlated). Several published parameterizations below assume frozen core, so set it when you use them.

Attenuated MP2

What it is. MP2 with the Coulomb operator in the correlation energy replaced by a short-range one. Two forms are implemented:

• erfc: \( \mathrm{erfc}(\omega r)/r \), which is \( 1/r \) at short range and decays to zero beyond roughly \( 1/\omega \).
• terfc: a short-range operator that is Coulombic inside a cutoff radius \( r_0 \) and attenuated outside it, with a controlled curvature (Dutoi & Head-Gordon 2008). It needs precomputed interpolation tables (FERRIC_TERF_TABLE_DIR, generated under terf-tables/); without them the run errors rather than substituting another operator.

Why. MP2's error for non-covalent interactions has two parts that partly cancel in small basis sets: basis-set superposition error, and an error in the long-range part of the correlation energy. Attenuation removes the long-range part, so attenuated MP2 has no long-range dispersion: its asymptotic \( C_6 \) is zero. The method works in the basis and on the systems it was fitted for; MP2-V (below) adds long-range dispersion back through VV10.

Parameterization. The attenuation parameters are fitted, and each fit belongs to one basis. The aug-cc-pVTZ sets that ferric ships as defaults were fitted on S66 without counterpoise and with frozen core (stated in crates/ferric-mp2/src/scs.rs):

Running a fitted variant in another basis, with counterpoise, or with frozen_core = 0 (the default) is extrapolation outside the fit.

Run it.

ω is in Å⁻¹ in the CLI and Python and in Bohr⁻¹ in the Rust API. The SCS-MP2 defaults are Grimme's c_OS = 1.2, c_SS = 0.333.

Accuracy. att-rimp2, scs-mp2 and scs-mp2-2terfc are Proven (the erfc energy is pinned against PySCF/libcint in testdata/reference/h2o_cc-pvdz_attenuated-rimp2-erfc0p420.json). mp2-v's implementation is Proven (narrow): the damped VV10 half, the terfc-attenuated MP2 half and the erfc control each match an independent PySCF/numpy reference. Its parameterization is unbenchmarked: there is no published MP2-V total energy to compare with, only an aug-cc-pVTZ S66 subset run (n=5, H-bonded/mixed, S66x8 geometry, no CP-anchored reference), and ferric evaluates the VV10 term on the converged HF density (the post-HF variant) on its own 50×50 grid rather than SG-1. Read the header of examples/water-mp2v.toml before quoting a number.

Fitting metric. Each attenuated energy fits its integrals in the metric of the same operator, \( (P|\hat w|Q) \), so the fit is variational for that operator. Robust (Dunlap) fitting is required only when the metric operator differs from the integral operator.

The range split is not additive. MP2 is quadratic in the integrals, so the erfc and erf energies at the same ω do not sum to the Coulomb MP2 energy; the difference is the short-range × long-range cross term. RS-MP2 + LR-RPA (below) is built on it.

RS-MP2 + LR-RPA

What it is. Short-range MP2 plus long-range direct RPA: the attenuated-MP2 idea with the missing long-range correlation supplied by response rather than by a fitted dispersion term. Two formulations:

• delta-lr (default): \( E_{MP2}[\text{Coulomb}] + (E_{dRPA}[\text{erf}] - 2E_{OS}[\text{erf}]) \), one dRPA call.
• coupled-rings: \( E_{MP2}[\text{Coulomb}] + \Delta dRPA[\text{Coulomb}] - \Delta dRPA[\text{erfc}] \), two dRPA calls; includes the mixed short/long-range ring diagrams.

Run it. method.kind = "rs-mp2-rpa" (examples/water-rs-mp2-rpa.toml, [mp2] formulation); Python run_rs_mp2_rpa(..., omega=0.420, formulation="delta-lr").

Accuracy. Smoke. The ω→0 and ω→∞ limits reduce exactly to MP2 and to MP2 + dRPA; numbers at production ω are not established on new systems.

Orbital-optimized MP2 and MP3

OO-RI-MP2 (oo-rimp2, examples/water-oo-rimp2.toml, run_oo_rimp2): orbitals optimized for the MP2 Lagrangian, with a level-shifted Newton step, orbital DIIS, Cayley rotations and backtracking. Proven (narrow): the energy matches an independent numpy OO-RI-MP2 to 7.5e-13 Ha (H2O, NH3, UHF CH3 at cc-pVDZ), and the closed-shell analytic gradient matches a finite difference of its own energy to 8e-9 Ha/Bohr. ORCA 6.1.1's OO-RI-MP2 stops 3.7e-8 to 7.0e-8 Ha above the same minimum, so it is only a loose cross-check.

MP3 (mp3, examples/water-mp3.toml, run_mp3): spin-orbital third-order Møller–Plesset through the einsum! framework. Proven.

Laplace formulations

RI-Laplace MP2 (laplace-mp2, examples/water-laplace-rimp2.toml, run_laplace_mp2): MP2 through a Laplace transform of the energy denominator, using pseudo-density matrices in the AO basis. The implementation is dense. It is the correctness reference for that formulation; no reduced scaling has been measured.

Laplace SOS-MP2 (laplace-sos-mp2, examples/water-laplace-sos-mp2.toml, run_laplace_sos_mp2): opposite-spin-only MP2 (Jung et al. 2004, c_OS = 1.3 by default) with a minimax Laplace quadrature (n_quad must be 3, 5 or 7; anything else is an error). There is no c_SS: dropping the same-spin term is what lets the denominator factorize. With c_OS = 1.0 it reproduces the opposite-spin component of RI-MP2 to quadrature error, which is the test anchor. Three formulations (sos_formulation):

• mo (default) and ao are both exact and agree to round-off.
• ao-sparse restricts each Boys-localized orbital's pseudo-density to an AO domain of radius domain_cutoff_bohr (required, no default). It is the one approximate variant.

What is measured for ao-sparse:

• Against the exact AO path on n-alkanes (cc-pVDZ, c_OS = 1, n_quad = 7, 2026-07-28), chemical accuracy (absolute error below 1.6 mHa) needs a domain radius of 3, 3, 3, 4, 5 and 5 Bohr for C2, C4, C6, C8, C10 and C12. The diameter grows fivefold over that series, so radius/diameter falls from 0.52 to 0.17.
• In the STO-3G tests, a 12 Bohr domain that is exact for butane (10.5 Bohr across) is also exact for octane (19.9 Bohr across), and a 4 Bohr domain on butane is already within 0.1%.
• A 71-atom drug molecule (danuglipron, 31.3 Bohr across, STO-3G) is within 0.05% at 4 Bohr.

The test sos_ao_sparse_truncation_radius_is_transferable_across_sizes in crates/ferric-mp2/src/laplace.rs pins the STO-3G butane/octane comparison: at 12 Bohr butane is exact (relative error below 1e-9) and octane is within 1e-6, and octane at 3 Bohr is worse than at 12 Bohr. The 4 Bohr butane figure and the danuglipron run (recorded in the test's doc comment) are measurements, not assertions. The alkane sweep is kept in the project's working notes.

How to read it (provisional): the radius needed grows, but far more slowly than the molecule. That points to a finite decay length rather than strict saturation: no single radius is shown to suffice at every size.

What is not claimed: any speedup. The domains discard contributions but the tensor algebra is still dense, so there are no timings to report.

Exact and local MP2

method.kind = "rimp2" is the method; it is computed exactly unless a [local] section asks for a local approximation (see Exact and local correlation). The local approximation is closed shell and energy only.

Amplitude-threshold local MP2 (examples/water-rimp2-local.toml; [local] scheme = "amplitude-threshold", eps = 1e-4; Python run_rimp2(..., local="amplitude-threshold", eps=1e-4)): the single-threshold local MP2 of Wang, Aldossary, Shi, Liu, Li & Head-Gordon (2023), with localized virtuals and per-pair domain-local RI fits. eps has no default: it is part of the model. eps = 0 reproduces RI-MP2 exactly; a finite eps carries a one-sided truncation error that grows faster than linearly in eps. Every printout and run-log record of a local run states eps and the fraction of amplitudes kept.

The amplitude equations are solved by preconditioned conjugate gradients to a relative residual of 1e-6 by default. The energy is the stationary Hylleraas form, so its solver error is second order in that residual. Using that form with a loose stop comes from the method's authors: Wang et al. (JCTC 19, 7577, 2023; Eq. 10 and Sec. 3.1) stop at an absolute ‖r‖ < 2.5·√eps, and the 2025 follow-up (ChemRxiv preprint of JCTC 21, 10910) at 5·√eps. ferric uses a fixed relative tolerance instead and gates it on a per-run bound, 3‖PJ‖²·relres²/λ_min with λ_min ≥ 2(ε_LUMO − ε_HOMO). The bound holds for the true residual, up to rounding, so it is evaluated on ‖J + At‖ from the energy's own matvec, not on the solver's recursive residual. If the bound at the solver's stop exceeds 1e-10 Ha, the solve continues from where it stopped to a tighter tolerance, never past 1e-11; with no positive orbital gap the bound is undefined and the solve runs at 1e-11 from the start. Running out of iterations before the tolerance being sought, first or tighter, stops the run with an error; cg_max_iter and an explicit cg_rtol are fields of the Rust AmplitudeLmp2Config, not CLI or Python options. The certificate (result.local in Python, "local" in the CLI JSON) records cg_rtol, cg_rtol_source, cg_tightened, cg_relres, cg_true_relres, cg_energy_error_bound, cg_energy_error_target and cg_energy_target_met, and solver_warnings says if a converged solve did not meet the target, because the target needs more than the 1e-11 floor or the true residual drifted from the recursive one. The bound limits solver error only, not the truncation error of eps.

The exact RI-MP2 reference that measures that error is opt-in. It is a full canonical RI-MP2 and forms the global (naux, nocc·nvir) tensor, so a run with it switched on is not reduced-cost. In the CLI, [local] reference = true computes it and prints the local error against it; without it the output reads E_corr(canonical RI) = not computed (opt-in: set [local] reference = true) and the run log's e_corr_canonical_ri is null. In Python, compute_reference=True computes it; by default result.local["e_corr_canonical_ri"] is None.

Integral-direct local MP2 ([local] integral_direct = true, examples/alkane8-rimp2-local-direct.toml; Python integral_direct=True) is the reduced-cost path. Its correlation assembly never forms the global 3-index tensor; only the opt-in reference does. Locality comes from an integral-free pair gate (gate_cal), per-occupied auxiliary-fit and virtual domains (aux_radius, virt_radius), and truncation of each orbital's AO support (ao_tail). With every map at its trivial setting it reproduces the global 3-index path and canonical RI-MP2 (tests/lmp2_direct.rs).

What is measured for the integral-direct local MP2 (n-alkanes C20 → C48, 6-31G with cc-pVDZ-RI, a quiet machine, 2026-09-07; benchmark bench_direct_alkane_series in crates/ferric-mp2/tests/lmp2_direct.rs). The run froze the carbon cores (frozen_core = number of carbons) and calibrated the pair gate (0.7 Coulomb, 0.02 erfc with ω = 1.0) at eps = 1e-4; the defaults are all-electron with no pair gate. Timings are the correlation stage (assembly + solve) and exclude the canonical reference:

• Correlation-stage cost grows as about N^1.24 with the erfc kernel and N^1.4 with Coulomb, fitted to the last three sizes.
• It is at about parity with canonical RI-MP2 at C20 (about 1.1× slower) and about 6× faster at C32 (5.7–6.3×).

How to read it (provisional): sub-quadratic on this series, but the fit has three points on one family of molecules in one basis. It is not shown to be linear, and it is not measured on 3-D or diffuse systems.

Schwarz triple cut (schwarz_skip, default 1e-5) applies to the Coulomb, erfc and terfc kernels alike. A shell triple is skipped when sqrt(max (P|P)) * sqrt(max (ab|ab)) < schwarz_skip, both factors from the run's own operator; schwarz_skip = 0 evaluates every triple. For terfc the factors come from the table engine's own diagonals and carry a 1e-5 relative inflation for table error, and the cut is only available inside the checked box: orbital shells up to g, r0 between 0.3 and 2.5 Å and r0·ω ≤ 8.9 (outside it the run errors and names schwarz_skip = 0.0). Measured for terfc (n-alkanes, 6-31G with cc-pVDZ-RI, eps = 5e-5, default maps, plain Schwarz, 2026-10-11, a loaded machine so ratios only, order alternated, two repeats):

At 1e-5 the cut skips 55% of the shell triples at C12 and 67% at C16 (51% and 63% at 1e-6). The pair gate (gate_cal = 0.7) leaves the C12 result unchanged. At 1e-6 the energy changes by 1e-10 (C12 linked) to 1.8e-9 (C16 NCP).

Local MP2 without integral_direct makes no scaling claim. Its assembly is pair-local (no dense J is formed), but it still builds the global 3-index tensor. Use it as the reference implementation and for small systems.

Cite

RI-MP2 auxiliary sets: Weigend et al. 1998. SCS-MP2: Grimme 2003. SOS-MP2: Jung et al. 2004. OO-MP2: Lochan & Head-Gordon 2007; Bozkaya et al. 2011. Laplace MP2: Häser & Almlöf 1992. Attenuated MP2: Goldey & Head-Gordon 2012; terfc: Dutoi & Head-Gordon 2008, Goldey, Dutoi & Head-Gordon 2013; SCS-MP2(2terfc): Goldey & Head-Gordon 2014; MP2-V: Goldey, Belzunces & Head-Gordon 2015. LMP2: Wang et al. 2023. Robust fitting: Dunlap 2000. Full entries in References.