8. MP2: correlation as controlled perturbation
Position in the course: Lesson 8 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
Prerequisites: Canonical HF and perturbation theory
1. Choose the reference partition
Møller-Plesset theory expands around canonical Hartree-Fock orbitals. Use an orthonormal spin-orbital basis with occupied indices \(i,j\) and virtual indices \(a,b\). The zeroth-order operator is the sum of Fock operators; its determinant eigenvalues are sums of orbital energies. The perturbation restores the difference between that mean field and the true electronic Hamiltonian. The first-order corrected total is the HF energy; second order introduces correlation through virtual substitutions.
The quality of this partition depends on the reference. A stable single determinant with sizable excitation gaps is a more suitable starting point than a stretched bond with nearly degenerate occupations. Increasing the basis does not cure a poor partition.
2. Derive doubles amplitudes and energy
Project the first-order perturbation equation onto an excited determinant \(I\). It gives \(c_I^{(1)}=\langle I|V|0\rangle/(E_0^{(0)}-E_I^{(0)})\). Brillouin's theorem eliminates singles for a stationary canonical HF reference; the two-body Hamiltonian couples directly to doubles. Therefore define
The unrestricted index sum counts each pair interchange twice, yielding the one-quarter factor. Different spatial-orbital formulas regroup opposite-spin and same-spin terms; do not copy this factor into them blindly. For a normal Aufbau reference with all virtual energies above occupied energies, denominators are negative, so the second-order correction is nonpositive.
3. Worked small-gap warning
Consider a two-determinant model \(H=\begin{pmatrix}0&v\\v&\Delta\end{pmatrix}\). The exact low energy is \([\Delta-\sqrt{\Delta^2+4v^2}]/2\), whereas second order is \(-v^2/\Delta\). With \(\Delta=1,v=0.1\), they are approximately \(-0.009902\) and \(-0.010000\). With \(\Delta=0.02,v=0.1\), they are \(-0.090499\) and \(-0.500000\). The second-order value is lower but substantially worse. These are deterministic teaching values, not executed molecular MP2 calculations.
A denominator alone is not a complete diagnostic: amplitudes combine coupling strength and gap. Inspect unusually large amplitudes together with spin symmetry, occupations and reference stability. When near-degeneracy is intrinsic, expand the reference space instead of assuming a higher perturbation order will fix it.
4. Cost, size consistency and practical approximations
Conventional canonical MP2 has formal fifth-power computational scaling and substantial integral storage. Density fitting or local approximations can reduce practical cost but introduce additional thresholds and auxiliary-basis choices. For noninteracting fragments with consistent references and bases, MP2 correlation contributions add, giving size consistency. This does not guarantee uniform accuracy for strong correlation, metallic small-gap limits or all noncovalent interactions.
Frozen-core calculations exclude selected occupied orbitals from the correlation sums. State that choice and test it when core-valence correlation matters. Correlation energy converges more slowly with orbital basis size than HF energy, so a convincing HF basis test does not automatically validate MP2 results.
5. Exercises and explained answers
If every gap doubles while integrals remain fixed, what happens? Each energy contribution halves in magnitude and each amplitude halves. In a real basis change the integrals also change, so this scaling is only a controlled model.
Why is MP2 not variational? The truncated perturbation energy is not the exact Rayleigh quotient of its truncated wavefunction. It can lie below the exact ground energy without contradicting the variational theorem.
Does removing a core orbital mean removing its electrons from HF? No. Frozen-core correlation normally keeps the occupied core density in the reference and omits its excitations from the correlation treatment.
Further conceptual check
Opposite-spin and same-spin correlation channels use different exchange structure. Empirical spin-component scaling changes their weights and defines a different approximation; it is not the unmodified perturbation theorem. State such modifications explicitly and test them for the intended observable.
6. References and study connections
- DePrince group academic HF tutorial
- DePrince group academic tutorial collection
- Psi4 official coupled-cluster documentation
- Helgaker, Jørgensen and Olsen: Molecular Electronic-Structure Theory
The derivations and toy arithmetic are original teaching constructions. No molecular simulation is reported here.
7. Related theory and practice
Quantum mechanics · Density functional theory · Quantum Monte Carlo · Gaussian