--- title: "When a small correlation correction becomes too large" description: "Derive a second-order energy correction and connect shrinking orbital gaps to the failure of restricted MP2 for a stretched bond." canonical_url: "https://sandboxphysics.com/stories/molecular-electronic/perturbation-correlation/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/molecular-electronic/perturbation-correlation/" last_modified: "2026-10-28" --- [Molecular electrons and the breaking of a bond](https://sandboxphysics.com/stories/molecular-electronic/) L3 · Chapter 6 # When a small correlation correction becomes too large Sandbox Physics · 2026-10-28 Perturbation theory is useful when the configurations missing from a reference state contribute small amplitudes. A stretched bond can close the energy gap that makes those amplitudes small, so a correction that works near equilibrium loses its organizing principle. ## Before you begin The two-configuration Hamiltonian from chapter five and RHF orbital energies from chapter three. An expansion approximates a function by terms ordered in a small parameter. [Review the preceding chapter](https://sandboxphysics.com/stories/molecular-electronic/configuration-interaction/) - Derive the sign and denominator of a second-order correction. - Explain the gap diagnostic in restricted MP2. - Compare a perturbative energy with a variational result without assuming the same bound. ## An exact two-state answer contains a small-parameter expansion Start with an illustrative reference of energy zero and another configuration at energy $\Delta>0$, coupled by $V$. The lower eigenvalue can be expanded when $|V|/\Delta$ is small. The first nonzero term lowers the energy by $V^2/\Delta$. The next term is smaller only when the ratio really is small. For $\Delta=1$ and $V=0.1$ hartree, second order gives $-0.01$ hartree; the exact lower eigenvalue is about $-0.00990195$. Keep the coupling but shrink the gap to $0.02$. Second order now predicts $-0.5$, whereas the exact value is about $-0.0904988$. The large negative correction signals failure of the expansion. $$ H=\begin{pmatrix}0&V\\V&\Delta\end{pmatrix},\qquad E_-=\frac{\Delta-\sqrt{\Delta^2+4V^2}}2 $$ This pedagogical matrix isolates coupling and an excitation gap. Its diagonal gap is not automatically an HF orbital-energy denominator. $$ E_-=-\frac{V^2}{\Delta}+\frac{V^4}{\Delta^3}+\cdots $$ The expansion assumes coupling small compared with the gap. A small gap can invalidate the ordering of terms. Reading: Sandbox Physics · C. Møller and M. S. Plesset ## MP2 organizes corrections around the Fock operator Møller–Plesset theory chooses the sum of HF one-electron Fock operators as its unperturbed operator. The remaining part of the electronic Hamiltonian supplies a perturbation. In canonical HF orbitals, meaning eigenorbitals of the converged Fock operator, the first-order wavefunction includes double substitutions, and their second-order energy denominators are sums of occupied orbital energies minus sums of virtual, or unoccupied, orbital energies. For this two-electron, two-spatial-orbital restricted reference, there is only one pair excitation. The conventional MP2 correlation correction takes the compact form below. The numerator is the square of the pair-coupling integral $(gu|gu)$. The denominator is negative when the occupied bonding orbital lies below the unoccupied antibonding orbital. As the restricted bond stretches, the orbital-energy gap narrows. The same pair excitation no longer represents a small amendment to a dominant reference. FCI adjusts both configuration amplitudes together; second-order theory keeps the reference as the starting point and can overreact to a small denominator. $$ E_{\mathrm{MP2}}=E_{\mathrm{RHF}}+\frac{|(gu|gu)|^2}{2\varepsilon_g-2\varepsilon_u} $$ This one-pair expression uses conventional integrals and a restricted canonical HF reference. A many-orbital system requires the full excitation sum. Reading: Sandbox Physics · C. Møller and M. S. Plesset · Psi4 manual ## Compare the whole curve before choosing a correction At $R=1.4$ bohr, STO-3G MP2 gives approximately $-1.129872195$ hartree, between RHF at $-1.116714325$ and FCI at $-1.137275944$. At four bohr, MP2 is about $-0.862709593$, still far above FCI at $-0.943778472$. A useful improvement at one geometry has not solved the stretched-bond problem. MP2 is not obtained by minimizing the full Hamiltonian expectation over a normalized trial wavefunction. It therefore has no general FCI upper bound. An MP2 energy that falls below same-basis FCI is possible and does not indicate a better ground state. In a practical comparison, inspect the excitation gap, occupations and force curve along with the correlation correction. A rapidly growing correction and substantial occupation of another natural orbital are reasons to consider a multiconfigurational description. Reading: Sandbox Physics · C. Møller and M. S. Plesset · Psi4 manual ## Try the prediction ### [Watch a correction change along a bond](https://sandboxphysics.com/labs/electronic-method-error-map/) 1. Select MP2 and compare its error near equilibrium with its error at stretched geometries. 2. Compare the energy trend with the growing second natural occupation in the FCI reference. 3. Compare energy and force errors at the same point, and explain which feature of the curve each tests. ## Exercises and explanations 1. In the illustrative matrix, keep $\Delta=1$ and halve $V$. How does the leading correction change? Read the explanation It becomes one quarter as large because $E^{(2)}=-V^2/\Delta$. The ratio of the next term to the first also becomes smaller, so the expansion is better controlled. 2. An MP2 result lies below same-basis FCI. Which result supplies the finite-basis ground-state energy? Read the explanation FCI diagonalizes the full finite-basis Hamiltonian. MP2 is a truncated perturbative expression and can pass below that ground-state energy. Here the lower numerical value is an error of the truncated expansion. 3. Does adding basis functions always repair static correlation in a single-reference MP2 calculation? Read the explanation No. A larger basis improves spatial flexibility, but a near-degenerate reference can still have large configuration amplitudes and small perturbation denominators. Basis convergence and reference adequacy are separate questions. [Previous: Mixing configurations to keep the electrons apart](https://sandboxphysics.com/stories/molecular-electronic/configuration-interaction/) [Next: What density functional theory changes](https://sandboxphysics.com/stories/molecular-electronic/density-functional-theory/) ## Sources and calculation scope 1. [Sandbox Physics, FR13 reference inputs, raw numerical records and reproduction guide (2026)](https://github.com/yangzichao/sandboxphysics/blob/main/docs/frontier/fr13/reference-reproduction.md) Original generated data: H2/STO-3G method energies, forces and occupations; independent Psi4 checks; basis and DFT-grid checks; fixed-orbital fractional-spin data. The guide links scripts, raw outputs, runtime versions and data licenses. 2. [Psi4 manual, HF: Hartree–Fock Theory](https://psicode.org/psi4manual/master/scf.html) Theory and numerical methods sections: restricted and unrestricted references, generalized eigenproblem, total energy and convergence. The reproducible reference manifest records the installed version and settings. 3. [Psi4 manual, CI: Configuration Interaction](https://psicode.org/psi4manual/master/detci.html) CI expansion and full configuration interaction within a chosen orbital space. Here FCI is exact only for the stated finite-basis Hamiltonian. 4. [C. Møller and M. S. Plesset, Note on an Approximation Treatment for Many-Electron Systems (1934)](https://doi.org/10.1103/PhysRev.46.618) Original perturbation partition. The two-orbital worked example uses ordinary restricted second-order perturbation theory with conventional integrals.