Questions: Perturbation Theory in Quantum Chemistry

Question 1

A computational chemist calculates the dissociation energy of F₂ using MP2 and gets a result that is *further* from experiment than plain Hartree-Fock. What is the most likely explanation?

• A. MP2 cannot handle fluorine because of its high electronegativity
• B. The Hartree-Fock reference for F₂ near bond dissociation is qualitatively incorrect — a single determinant poorly describes the breaking bond — causing the perturbation series to diverge or oscillate
• C. MP2 is fundamentally less accurate than Hartree-Fock for all bond energies
• D. Second-order perturbation theory overcorrects for correlation, always giving energies below the true value

Answer: (B)

Perturbation theory requires a qualitatively correct zeroth-order solution. Near dissociation, F₂ requires at least two determinants to describe correctly — the bonding and antibonding configurations become nearly degenerate. Hartree-Fock, which uses a single determinant, gives a qualitatively wrong picture, and MP2 applied to this bad reference can produce results worse than HF. This is the critical failure mode: perturbation theory does not correct a fundamentally wrong starting point — it amplifies errors in it.

Question 2

What does the first-order Møller-Plesset energy correction (MP1) contribute beyond the Hartree-Fock energy?

• A. It recovers approximately 50% of the correlation energy by including singly-excited determinants
• B. It corrects for basis set superposition error in the wave function
• C. It adds nothing — the first-order correction exactly reproduces the Hartree-Fock energy
• D. It captures triple excitations, which dominate the correlation energy

Answer: (C)

A mathematically subtle but important result: in Møller-Plesset theory, the first-order energy correction (MP1) simply recovers the Hartree-Fock energy itself — it adds no new physical content. This follows from Brillouin's theorem, which ensures that singly-excited determinants don't mix with the HF ground state. The first genuinely new contribution comes at second order (MP2), which mixes in doubly-excited determinants and captures the dominant electron correlation effects. Many students assume 'first order' means 'some correction'; here it means none.

Question 3

Higher orders of Møller-Plesset perturbation theory (MP3, MP4, ...) consistently give more accurate energies than MP2.

• T. True
• F. False

Answer: False

Perturbation theory is not variational — it is not bounded below by the true energy. The series can oscillate, with some orders giving results that are farther from the true answer than lower orders. MP3 is more computationally expensive than MP2 but often *less* accurate for molecular geometries, precisely because of this non-variational oscillation. This is why MP2 is the dominant method in practice: it captures 80–90% of the correlation energy at modest cost, while higher orders offer unreliable improvements at disproportionate expense.

Question 4

Perturbation theory is applicable to any molecular system, regardless of whether the Hartree-Fock reference is a good description.

• T. True
• F. False

Answer: False

Perturbation theory assumes the perturbation V is genuinely small relative to H₀. If the Hartree-Fock reference is qualitatively wrong — as it is for strongly correlated systems like transition metal complexes, bond-breaking situations, or open-shell molecules near degeneracy — the perturbation is not small and the series can fail catastrophically. Multi-reference methods (CASSCF, MRCI) are required when a single-determinant HF reference is insufficient. Knowing when perturbation theory applies is as important as knowing how to apply it.

Question 5

Why is MP3 rarely used in practice, even though it is one perturbation order higher than the widely-used MP2?

Model answer: MP3 is more computationally expensive than MP2 (scaling as N⁶ vs. N⁵) but does not reliably give better results. Because Møller-Plesset perturbation theory is non-variational, the series can oscillate — MP3 can be farther from the true energy than MP2 for certain properties, particularly geometries. This means the extra cost of MP3 buys unpredictable accuracy rather than a guaranteed improvement. MP2 captures 80–90% of the correlation energy with N⁵ scaling and is well-characterized in its performance; moving to MP3 breaks this favorable cost-to-accuracy ratio without providing consistent benefit.

The non-variational character is the key point. Variational methods (like CCSD or FCI) always overshoot the true energy, so higher levels always improve. Perturbation series can go above *or* below, oscillating around the true value. MP2 often undershoots (overestimates correlation energy) and MP3 often overshoots back, so they bracket the true answer — but this means neither one is uniformly better, and users prefer the cheaper, better-characterized MP2.