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MP2 and spin-scaled MP2: adding correlation on top of Hartree-Fock

Hartree-Fock (HF) gives you a mean-field wavefunction: every electron feels the average field of all the others, but their instantaneous correlation — the way electrons dodge each other — is missing. Second-order Moller-Plesset perturbation theory (MP2) is the cheapest, most widely used way to put that correlation back. You run it when HF energies aren't accurate enough (reaction energies, conformers, weak interactions) but a full coupled-cluster treatment is too expensive. This tutorial computes a standalone MP2 energy for water and shows how to switch on the spin-component-scaled variants (SCS-MP2, SOS-MP2) and how the choice of HF reference (RHF/UHF/ROHF) enters.

A little theory

MP2 treats electron correlation as a perturbation on the converged HF reference. Its correlation energy splits cleanly into two physically distinct pieces — the same-spin part (two electrons of the same spin, E_aa + E_bb) and the opposite-spin part (E_ab):

E(MP2) = c_ss * (E_aa + E_bb) + c_os * E_ab

Conventional MP2 sets both scale factors to 1. But MP2 systematically over-counts same-spin correlation and under-counts opposite-spin correlation, so re-weighting the two pieces improves accuracy for almost no extra cost. That is the idea behind the spin-component-scaled family:

  • SCS-MP2 (Grimme): c_ss = 1/3, c_os = 1.2 — a balanced all-round improvement.
  • SOS-MP2 (Head-Gordon): c_ss = 0, c_os = 1.3 — drops same-spin work entirely (cheaper) and is well suited to Laplace/RI accelerations.

OpenQP computes the two spin components separately, so every preset is just a choice of c_ss and c_os. In OpenQP, MP2 is a post-SCF, energy-only workflow: it first converges an HF reference, then adds the correlation energy and reports the total E(HF+MP2). The reference can be closed-shell RHF, unrestricted UHF, or restricted-open-shell ROHF (ROHF orbitals are semicanonicalized before the correlation step so the energy denominators are well defined). For the derivation and the parameterizations behind each preset, see the MP2 workflow page.

Input-file style

The runnable deck is inputs/h2o_mp2_6-31g.inp — water in the 6-31G basis, a UHF reference, conventional MP2. Annotated:

[input]
system=
 8   0.000000000   0.000000000  -0.041061554   # O   (Angstrom)
 1  -0.533194329   0.533194329  -0.614469223   # H
 1   0.533194329  -0.533194329  -0.614469223   # H

charge=0
method=mp2              # select the MP2 post-SCF workflow
basis=6-31g
runtype=energy          # MP2 is energy-only; other runtypes are rejected
functional=             # MUST be empty — MP2 needs an HF reference, not KS-DFT

[guess]
type=huckel             # extended-Huckel initial orbital guess
save_mol=false

[scf]
type=uhf                # the HF reference: rhf | uhf | rohf
multiplicity=1          # singlet ground state
maxit=50                # max SCF iterations
conv=1.0e-10            # tight SCF convergence (correlation is sensitive)
save_molden=false

[mp2]
variant=mp2             # conventional (unscaled) MP2; c_ss = c_os = 1.0

Key points:

  • method=mp2 in [input] is what turns on the correlation step; functional must be empty (MP2 on a Kohn-Sham reference is rejected) and runtype must be energy (no MP2 gradients/Hessians yet).
  • The HF reference is chosen in [scf] via type. Water is closed-shell, so type=rhf would be the natural choice and gives the identical energy here; this example ships with uhf to exercise the unrestricted path. Use rohf (or uhf) with the appropriate multiplicity for open-shell systems.
  • The [mp2] section is optional. Omit it entirely for conventional MP2, or set variant to pick a spin-scaled preset. For spin-component scaling change one line:
[mp2]
variant=scs-mp2         # Grimme SCS-MP2: c_ss = 1/3, c_os = 1.2

[mp2]
variant=sos-mp2         # scaled-opposite-spin: c_ss = 0, c_os = 1.3

Other accepted values include os-mp2, ss-mp2, scs-mi-mp2, and custom. For a literature parameterization not in the table, use custom with explicit scales:

[mp2]
variant=custom
same_spin_scale=0.50
opposite_spin_scale=1.10

Python style

The equivalent calculation with the OpenQP Python API is inputs/h2o_mp2_6-31g.py. job.theory.mp2(...) sets the [input] method=mp2, the [scf] reference, and the [mp2] section in one call; extra keywords (here conv) are forwarded to [scf].

from oqp.openqp import OpenQP

job = OpenQP("h2o_mp2", silent=1)

job.molecule(
    """
O   0.000000000   0.000000000  -0.041061554
H  -0.533194329   0.533194329  -0.614469223
H   0.533194329  -0.533194329  -0.614469223
""",
    charge=0,
    multiplicity=1,
)

# reference -> [scf] type ; variant -> [mp2] variant ; conv -> [scf] conv
job.theory.mp2(basis="6-31g", reference="uhf", variant="mp2", conv=1.0e-10)

mol = job.run()
results = mol.get_results()
print("HF reference energy:", mol.get_scf_energy())
print("MP2 total energy:   ", results["energy"])

To run spin-scaled MP2, change only the variant argument:

job.theory.mp2(basis="6-31g", reference="uhf", variant="scs-mp2")   # SCS-MP2
job.theory.mp2(basis="6-31g", reference="uhf", variant="sos-mp2")   # SOS-MP2

Custom scales go through the same helper (the helper defaults variant to custom when you pass scale factors):

job.theory.mp2(
    basis="6-31g",
    reference="uhf",
    variant="custom",
    same_spin_scale=0.50,
    opposite_spin_scale=1.10,
)

Run it

Input-file style (from the inputs/ folder):

cd mp2/inputs
openqp h2o_mp2_6-31g.inp

Python style:

cd mp2/inputs
python h2o_mp2_6-31g.py

Both need OpenQP installed (pip install openqp) and produce the same numbers.

Reading the output

MP2 reports an HF reference energy, the MP2 correlation energy it adds, and their sum, the MP2 total energy — the number you almost always want. For this water / 6-31G example the validated run gives:

Quantity Value (Ha)
E(MP2, correlation) -0.1278307451
E(MP2, total) -76.1121207760
  • In the log file (<project>.log) look for the HF energy, the printed correlation energy, and the final total.
  • From Python, mol.get_scf_energy() is the HF reference and mol.get_results()["energy"] is the MP2 total (this matches the energy field written to <project>.json).
  • A spin-scaled run reuses the same HF reference but reweights the two correlation components, so its total differs from conventional MP2 by the change in c_ss (E_aa+E_bb) + c_os E_ab — the HF energy is unchanged.

References / manual