MP2 calculations

Second-order Møller–Plesset perturbation theory (MP2) estimates an electron correlation correction using molecular orbitals from a converged HF reference. It does not replace HF: RustiQ first solves HF, then evaluates MP2. MP2 is available with both RHF and UHF references. An unconverged HF result cannot be used as an MP2 reference.

The checked-in H₂ input requests RHF-MP2 with no frozen orbitals:

[basis]
name = "sto-3g"

[molecule]
geometry = "../molecule.xyz"

[method.hf]
method = "Rhf"

[method.hf.guess]
type = "CoreHamiltonian"

[method.mp2]
frozen_orbitals = 0

Run it from the repository root after importing STO-3G:

cargo run --locked -- basis import tests/data/sto-3g.json
cargo run --locked -- run samples/h2/sto-3g/mp2_calculation.toml

The MP2 correlation energy is the correction added to HF electronic energy. Total MP2 energy is HF electronic energy plus MP2 correlation energy plus nuclear repulsion. The correlation energy is usually negative, but its sign and size depend on the system and reference.

Open-shell UHF-MP2 example

OH is an open-shell doublet (multiplicity 2). For this input, Auto selects UHF, and MP2 uses the converged UHF reference. The existing OH/STO-3G input is:

[basis]
name = "sto-3g"

[molecule]
geometry = "../oh.xyz"
charge = 0
multiplicity = 2

[method.hf]
method = "Auto"
convergence_threshold = 1e-10

[method.hf.guess]
type = "CoreHamiltonian"

[method.mp2]
frozen_orbitals = 0

[method.hf.diis]
enabled = true
max_history = 6

Run it from the repository root after making STO-3G available, as described in the basis-set guide:

cargo run --locked -- run samples/oh/sto-3g/mp2_calculation.toml

The calculation reports a UHF-MP2 correlation energy. Existing PySCF comparisons cover this specific example within the tolerances documented below; they do not establish general chemical accuracy.

Frozen occupied orbitals

Occupied orbitals contain electrons in the HF reference; virtual orbitals are unoccupied orbitals available as excitation destinations. The frozen_orbitals setting excludes the lowest occupied orbitals from the MP2 correlation treatment. Its default is zero, so all occupied orbitals are included. RHF requires at least one occupied orbital to remain correlated and at least one virtual orbital; the frozen count must be smaller than the occupied count. For UHF, the count cannot exceed either spin’s occupied-orbital count, though one spin sector may be fully frozen.

For RHF, the count refers to spatial orbitals; freezing one excludes one doubly occupied orbital. For UHF, the same count is applied separately to the alpha and beta occupied spaces. The excluded orbitals can have different spatial shapes. The setting does not choose chemically defined core orbitals automatically. Select it explicitly when comparing calculations.

Memory use

MP2 transforms integrals from the atomic-orbital basis into molecular-orbital blocks. RustiQ processes these blocks to avoid storing the full transformed four-index tensor at once. memory_limit configures the estimated additional workspace used for these transformations. "auto" selects a budget based on available memory; explicit byte sizes such as "500 MB" can also be set.

This workspace budget is not a limit on total process memory. It excludes already resident HF data and atomic-orbital integrals, allocator overhead, and other numerical-library scratch space. Even a single required block may exceed a very small budget. See the source for Mp2MemoryPlan and the authoritative Nickel defaults in the input contract.

For example, the existing Nickel default is auto. An explicit byte-size budget belongs under [method.mp2]:

[method.mp2]
memory_limit = "500 MB"

If the budget is below the minimum workspace needed for one occupied block, that sector cannot be transformed and the MP2 calculation errors. Automatic budget resolution follows the shared resource policy.

The H₂/STO-3G RHF-MP2 comparison checks the HF energy within \(2\times10^{-10}\) Hartree and MP2 correlation energy within \(10^{-10}\) Hartree against PySCF 2.14.0. The OH/STO-3G UHF-MP2 case checks HF energy within \(5\times10^{-8}\), correlation energy within \(5\times10^{-9}\) Hartree, and spin expectation within \(2\times10^{-5}\). These are agreement tolerances for those inputs, not a measure of general chemical accuracy. See the reference coverage and the reference test source. Reproduce both selected cases with:

nix run .#pyscf-check -- -k 'h2-sto-3g-rhf-mp2 or oh-sto-3g-uhf-mp2'

Small energy denominators can make MP2 corrections unreliable, especially in strongly correlated or near-degenerate systems. Numerical safeguards and agreement on a few examples do not establish physical validity. The method’s perturbative basis is described in Møller and Plesset (1934); see also this discussion of problematic MP2 denominators (Lee and Head-Gordon, 2018).