MP2: correlation at perturbation prices
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The cheapest serious correlation method treats what HF missed as a small disturbance. Møller and Plesset's move (1934): let the zeroth-order Hamiltonian be the sum of Fock operators — so the mean field is already inside — and perturb with the leftover, the fluctuation between the true repulsion and its average. First order just gives back E_HF; the first new physics arrives at second order:
Read it as physics, not indices: every way of promoting two electrons from occupied (i, j) to virtual (a, b) orbitals contributes its coupling squared, divided by the orbital-energy gap you'd pay for the excitation. Cheap excitations with strong coupling dominate. Every denominator is negative, so every term lowers the energy — correlation, recovered configuration by configuration.
Møller-Plesset perturbation theory splits the Hamiltonian as H = H₀ + V. What is H₀, and what is the perturbation?
For minimal-basis H₂ the sum has exactly one term — one occupied σ, one virtual σ*. The transformed integral is (ov|ov) = 0.1813, the gap is 2(ε₀ − ε₁) = −2.497, and:
Thirteen milli-hartree closer to the exact answer, for the price of one integral transformation. That trade — a polynomial-cost slice of the correlation FCI would buy at exponential cost — is why MP2 became the workhorse. The fine print lives in the denominator.
The MP2 energy is a sum of terms |coupling|² / (εᵢ + εⱼ − εₐ − εᵦ). Which systems should make you distrust it?
For H₂/STO-3G the MP2 correction is −13.2 mHa, computed from a single excitation. What made the minimal basis collapse the whole MP2 sum to one term?
Now compute it yourself: Project 7 transforms your own SCF's integrals and lands on −0.013158.