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Perturbation Theory in Quantum Chemistry
Rayleigh-Schrödinger perturbation theory expresses exact energies and wavefunctions as power series in a perturbation parameter $\lambda$. In quantum chemistry, Møller-Plesset (MP) theory partitions the full Hamiltonian into a Hartree-Fock reference and a correlation perturbation, yielding systematic corrections to the HF energy. The MP2 correction captures roughly 80–90% of the dynamic correlation energy at $O(N^5)$ computational cost.
Rayleigh-Schrödinger Perturbation Theory
| Partition the Hamiltonian as $\hat{H} = \hat{H}^{(0)} + \lambda\hat{H}’$ where $\hat{H}^{(0)} | n\rangle = E_n^{(0)} | n\rangle$ is exactly solvable. Expanding $E = E^{(0)} + \lambda E^{(1)} + \lambda^2 E^{(2)} + \cdots$ and matching powers of $\lambda$: |
The second-order correction is always negative (stabilising) for the ground state, since all energy denominators $E_0^{(0)}-E_n^{(0)} < 0$.
Møller-Plesset Theory
In MP theory, $\hat{H}^{(0)} = \sum_i \hat{F}i$ is the sum of Fock operators and $\hat{H}’ = \hat{H}{elec} - \hat{H}^{(0)}$ is the fluctuation potential. The MP2 energy correction, involving double excitations from occupied ${i,j}$ to virtual ${a,b}$ orbitals, is
\[E_{MP2} = \sum_{i<j}\sum_{a<b}\frac{|\langle ij||ab\rangle|^2}{\varepsilon_i+\varepsilon_j-\varepsilon_a-\varepsilon_b}\]| where $\langle ij | ab\rangle$ are antisymmetrised two-electron integrals. |
Convergence and Limitations
| Method | Scaling | Captures |
|---|---|---|
| HF | $O(N^4)$ | Mean-field exchange |
| MP2 | $O(N^5)$ | ~80–90% of $E_{corr}$ |
| MP3 | $O(N^6)$ | Diminishing returns |
| MP4 | $O(N^7)$ | Near CCSD quality |
MP series can diverge for systems with small HOMO-LUMO gaps or significant multi-reference character, where the HF reference is a poor starting point. Spin-component-scaled MP2 (SCS-MP2) empirically corrects for the imbalanced treatment of same-spin and opposite-spin correlation.