Perturbation theory in chemistry: Stark effect and the helium atom first-order correction

Intuition Beginner

Many quantum-mechanical problems cannot be solved exactly. The hydrogen atom has an exact solution, but helium — with two electrons repelling each other — does not. Perturbation theory is a systematic strategy: start from a problem you can solve, then add a small correction for the part you cannot.

Think of the true Hamiltonian $\hat{H}$ as a solved part ${\hat{H}}^{(0)}$ plus a small perturbation ${\hat{H}}^{\prime }$. The label $(0)$ means "unperturbed" — the part with known solutions. The prime on ${\hat{H}}^{\prime }$ marks the perturbation. If the perturbation is small, the true energy and wave function are close to the unperturbed ones. Perturbation theory constructs a series of increasingly precise corrections.

The first-order energy correction is simple: it is the expectation value of the perturbation computed with the unperturbed wave function. You evaluate the perturbation operator using the wave function you already know. No new differential equation needs to be solved for this first step.

The second-order correction involves a sum over all other states of the unperturbed system. It accounts for how the perturbation mixes the ground state with excited states. Each excited state contributes an amount proportional to the coupling squared, divided by the energy gap. States close in energy contribute more than distant ones.

Perturbation theory and the variational method are the two main approximation strategies in quantum chemistry. The variational method always gives an upper bound to the true energy but requires a guessed trial function. Perturbation theory does not guarantee a bound, but it is systematic: each order corrects the previous one.

Visual Beginner

The Stark effect demonstrates perturbation theory in action. An external electric field $\mathcal{E}$ applied to an atom shifts its energy levels. For the ground state of hydrogen, the shift is quadratic in the field strength: $\Delta E\propto -{\mathcal{E}}^2$. The electron cloud polarises — it distorts slightly toward the field — and the induced dipole lowers the energy.

The helium atom provides a chemical application. Helium has two electrons. If you ignore their mutual repulsion, each electron occupies a hydrogen-like $1s$ orbital with $Z=2$. The unperturbed energy is $-108.8$ eV. The true ground-state energy is $-79.0$ eV. The electron-electron repulsion — the perturbation — raises the energy by about $29.8$ eV. First-order perturbation theory captures most of this shift.

Worked example Beginner

Compute the first-order energy correction for the ground state of helium using the electron-electron repulsion as the perturbation.

Setup. The helium Hamiltonian is

$$\hat{H}={\hat{H}}^{(0)}+{\hat{H}}^{\prime },$$

where ${\hat{H}}^{(0)}$ treats the two electrons as independent particles in the field of the $Z=2$ nucleus, and ${\hat{H}}^{\prime }=e^2/(4\pi {\epsilon }_0{r}_{12})$ is the electron-electron repulsion.

Unperturbed ground state. Each electron occupies a hydrogen-like $1s$ orbital with $Z=2$:

$${\psi }^{(0)}={\varphi }_{1s}({\mathbf{r}}_1){\varphi }_{1s}({\mathbf{r}}_2),$$

where ${\varphi }_{1s}(\mathbf{r})=(Z^3/\pi a_0^3{)}^{1/2}{e}^{-Zr/a_0}$.

First-order correction. The first-order energy correction is

$$E^{(1)}=⟨{\psi }^{(0)}|{\hat{H}}^{\prime }|{\psi }^{(0)}⟩=\frac{e^2}{4\pi {\epsilon }_0}⟨{\psi }^{(0)}\left|\frac{1}{r_{12}}\right|{\psi }^{(0)}⟩.$$

This integral evaluates to $\frac{5}{8}Z\cdot R_H$ for the $1s^2$ configuration. With $Z=2$ and $R_H=13.6$ eV:

$$E^{(1)}=\frac{5}{8}(2)(13.6)=\frac{5}{4}(13.6)=34.0\text{ eV}.$$

Result. The unperturbed energy is $E^{(0)}=2\times (-54.4)=-108.8$ eV (two independent electrons, each at $-Z^2{R}_H=-54.4$ eV). The first-order corrected energy is

$$E^{(0)}+E^{(1)}=-108.8+34.0=-74.8\text{ eV}.$$

The experimental value is $-79.0$ eV. First-order perturbation theory captures the dominant effect but overestimates the energy by about $4.2$ eV because the unperturbed orbitals are too compact — they do not account for how the electrons push each other apart.

Check your understanding Beginner

Formal definition Intermediate+

Time-independent nondegenerate perturbation theory applies when the unperturbed Hamiltonian ${\hat{H}}^{(0)}$ has a discrete, nondegenerate spectrum. Write the full Hamiltonian as

$$\hat{H}={\hat{H}}^{(0)}+\lambda {\hat{H}}^{\prime },$$

where $\lambda$ is a dimensionless bookkeeping parameter (set to 1 at the end). Expand the energy and wave function as power series in $\lambda$:

$$E=E^{(0)}+\lambda E^{(1)}+{\lambda }^2{E}^{(2)}+\cdots ,$$ $$\psi ={\psi }^{(0)}+\lambda {\psi }^{(1)}+{\lambda }^2{\psi }^{(2)}+\cdots .$$

Substitute into $\hat{H}\psi =E\psi$, collect terms by order in $\lambda$, and solve recursively.

First-order energy. Equating coefficients of ${\lambda }^1$:

$$E^{(1)}=⟨{\psi }^{(0)}|{\hat{H}}^{\prime }|{\psi }^{(0)}⟩.$$

This is the expectation value of the perturbation in the unperturbed state.

First-order wave function. The correction ${\psi }^{(1)}$ is expanded in the complete set of unperturbed eigenfunctions $\{{\psi }_n^{(0)}\}$:

$${\psi }^{(1)}=\sum _{n\ne 0}\frac{⟨{\psi }_n^{(0)}|{\hat{H}}^{\prime }|{\psi }^{(0)}⟩}{E_0^{(0)}-E_n^{(0)}}{\psi }_n^{(0)}.$$

Each excited state contributes proportionally to its coupling with the ground state and inversely to the energy gap. States close in energy with large coupling matrix elements dominate the correction.

Second-order energy. Equating coefficients of ${\lambda }^2$:

$$E^{(2)}=\sum _{n\ne 0}\frac{|⟨{\psi }_n^{(0)}|{\hat{H}}^{\prime }|{\psi }^{(0)}⟩{|}^2}{E_0^{(0)}-E_n^{(0)}}.$$

The numerator is always non-negative. For the ground state ($E_0^{(0)}<E_n^{(0)}$ for all $n$), every denominator is negative, so $E^{(2)}\le 0$. Second-order perturbation theory always lowers the ground-state energy.

Degenerate perturbation theory

When the unperturbed level is $g$-fold degenerate, the expressions above diverge (zero denominators). The correct procedure is to diagonalise the perturbation within the degenerate subspace. Construct the $g\times g$ matrix $W_{ij}=⟨{\psi }_i^{(0)}|{\hat{H}}^{\prime }|{\psi }_j^{(0)}⟩$ and solve the secular equation $\det (\mathbf{W}-E^{(1)}\mathbf{I})=0$. The eigenvalues of $\mathbf{W}$ are the first-order energy corrections, and the eigenvectors specify the "correct" linear combinations of degenerate states that the perturbation selects.

Application: helium atom

The helium atom Hamiltonian separates as

$${\hat{H}}^{(0)}=\left(-\frac{{\hslash }^2}{2m_e}{\nabla }_1^2-\frac{2e^2}{4\pi {\epsilon }_0{r}_1}\right)+\left(-\frac{{\hslash }^2}{2m_e}{\nabla }_2^2-\frac{2e^2}{4\pi {\epsilon }_0{r}_2}\right),$$ $${\hat{H}}^{\prime }=\frac{e^2}{4\pi {\epsilon }_0{r}_{12}}.$$

Each electron is an independent hydrogen-like ion with $Z=2$. The unperturbed energy is $E^{(0)}=2\times (-Z^2{R}_H)=-8R_H=-108.8$ eV.

The first-order correction is the Coulomb integral:

$$J=\frac{5}{4}Z{R}_H=34.0\text{ eV}(Z=2).$$

To second order, the sum-over-states formula includes contributions from virtual excitations $1s\to nl$. Each excitation lowers the energy (for the ground state, $E^{(2)}<0$). The second-order correction for helium is approximately $-4.1$ eV, bringing the total to $E^{(0)}+E^{(1)}+E^{(2)}\approx -78.9$ eV — close to the exact $-79.01$ eV.

Application: Stark effect

The Stark effect Hamiltonian for a hydrogen atom in a uniform electric field $\mathcal{E}$ along $z$ is

$${\hat{H}}^{\prime }=e\mathcal{E}z=e\mathcal{E}r\cos \theta .$$

Linear Stark effect ($n=2$). The $n=2$ level is four-fold degenerate ($2s$, $2p_0$, $2p_{+1}$, $2p_{-1}$). The perturbation matrix within this subspace has nonzero off-diagonal elements connecting $2s$ and $2p_0$:

$$⟨2s|e\mathcal{E}z|2p_0⟩=-3e\mathcal{E}{a}_0.$$

Diagonalising the $2\times 2$ block gives eigenvalues $\pm 3e\mathcal{E}{a}_0$. The four-fold level splits into three: $+3e\mathcal{E}{a}_0$, $0$, $0$, $-3e\mathcal{E}{a}_0$. The splitting is linear in $\mathcal{E}$.

Quadratic Stark effect (ground state). For the $1s$ ground state, $⟨1s|z|1s⟩=0$ by parity. The first-order shift vanishes. The second-order shift is

$$E^{(2)}=-\frac{1}{2}{\alpha }_d{\mathcal{E}}^2,$$

where ${\alpha }_d=\frac{9}{2}{a}_0^3$ is the static dipole polarisability of hydrogen. The ground state is lowered and the shift is quadratic in the field.

Key theorem with proof Intermediate+

Theorem (Rayleigh-Schrodinger perturbation series). Let ${\hat{H}}^{(0)}$ be a self-adjoint operator on a Hilbert space $\mathcal{H}$ with discrete spectrum $E_n^{(0)}$ and corresponding normalised eigenfunctions ${\psi }_n^{(0)}$. Let $E_0^{(0)}$ be nondegenerate and let ${\hat{H}}^{\prime }$ be a self-adjoint perturbation. Then for sufficiently small ${\hat{H}}^{\prime }$, the perturbed ground-state energy and wave function have convergent expansions

$$E_0=E_0^{(0)}+⟨{\psi }_0^{(0)}|{\hat{H}}^{\prime }|{\psi }_0^{(0)}⟩+\sum _{n\ne 0}\frac{|⟨{\psi }_n^{(0)}|{\hat{H}}^{\prime }|{\psi }_0^{(0)}⟩{|}^2}{E_0^{(0)}-E_n^{(0)}}+\cdots$$

and

$${\psi }_0={\psi }_0^{(0)}+\sum _{n\ne 0}\frac{⟨{\psi }_n^{(0)}|{\hat{H}}^{\prime }|{\psi }_0^{(0)}⟩}{E_0^{(0)}-E_n^{(0)}}{\psi }_n^{(0)}+\cdots .$$

Proof. Write $\hat{H}(\lambda )={\hat{H}}^{(0)}+\lambda {\hat{H}}^{\prime }$. By the analytic perturbation theory of Kato, if $E_0^{(0)}$ is an isolated eigenvalue of finite multiplicity (here multiplicity 1), then for $|\lambda |$ sufficiently small, $E_0(\lambda )$ and ${\psi }_0(\lambda )$ are analytic functions of $\lambda$.

Write $E_0(\lambda )={\sum }_k{\lambda }^k{E}_0^{(k)}$ and ${\psi }_0(\lambda )={\sum }_k{\lambda }^k{\psi }_0^{(k)}$, substitute into $\hat{H}(\lambda ){\psi }_0(\lambda )=E_0(\lambda ){\psi }_0(\lambda )$, and collect powers of $\lambda$.

Order ${\lambda }^0$: ${\hat{H}}^{(0)}{\psi }_0^{(0)}=E_0^{(0)}{\psi }_0^{(0)}$. This is the unperturbed eigenvalue equation.

Order ${\lambda }^1$: $({\hat{H}}^{(0)}-E_0^{(0)}){\psi }_0^{(1)}=(E_0^{(1)}-{\hat{H}}^{\prime }){\psi }_0^{(0)}$. Taking the inner product with ${\psi }_0^{(0)}$ and using $⟨{\psi }_0^{(0)}|({\hat{H}}^{(0)}-E_0^{(0)})|{\psi }_0^{(1)}⟩=0$ (since ${\hat{H}}^{(0)}$ is self-adjoint and $E_0^{(0)}$ is the eigenvalue), the left side vanishes. The right side gives

$$0=E_0^{(1)}-⟨{\psi }_0^{(0)}|{\hat{H}}^{\prime }|{\psi }_0^{(0)}⟩,$$

yielding $E_0^{(1)}=⟨{\psi }_0^{(0)}|{\hat{H}}^{\prime }|{\psi }_0^{(0)}⟩$.

To find ${\psi }_0^{(1)}$, expand it in the unperturbed basis: ${\psi }_0^{(1)}={\sum }_{n\ne 0}{c}_n^{(1)}{\psi }_n^{(0)}$ (the $n=0$ component is set to zero by the intermediate normalisation convention $⟨{\psi }_0^{(0)}|{\psi }_0⟩=1$). Substitute back into the first-order equation and project onto ${\psi }_n^{(0)}$ for $n\ne 0$:

$$(E_n^{(0)}-E_0^{(0)})c_n^{(1)}=-⟨{\psi }_n^{(0)}|{\hat{H}}^{\prime }|{\psi }_0^{(0)}⟩,$$

giving $c_n^{(1)}=⟨{\psi }_n^{(0)}|{\hat{H}}^{\prime }|{\psi }_0^{(0)}⟩/(E_0^{(0)}-E_n^{(0)})$.

Order ${\lambda }^2$: A similar procedure applied to the second-order equation, projected onto ${\psi }_0^{(0)}$, yields

$$E_0^{(2)}=⟨{\psi }_0^{(0)}|{\hat{H}}^{\prime }|{\psi }_0^{(1)}⟩=\sum _{n\ne 0}\frac{|⟨{\psi }_n^{(0)}|{\hat{H}}^{\prime }|{\psi }_0^{(0)}⟩{|}^2}{E_0^{(0)}-E_n^{(0)}}.$$

Higher orders follow by the same recursive procedure. Convergence of the series is guaranteed for $|\lambda |$ below the radius determined by the distance from $E_0^{(0)}$ to the rest of the spectrum of ${\hat{H}}^{(0)}$ (Kato bound). $\square$

Intermediate worked example: second-order correction for a two-level system

A system has unperturbed states $|a⟩$ and $|b⟩$ with energies $E_a^{(0)}=-5.0$ eV and $E_b^{(0)}=-3.0$ eV. The perturbation has matrix element $V=⟨b|{\hat{H}}^{\prime }|a⟩=0.4$ eV.

First-order energy corrections: $E_a^{(1)}=⟨a|{\hat{H}}^{\prime }|a⟩$ and $E_b^{(1)}=⟨b|{\hat{H}}^{\prime }|b⟩$. Suppose the diagonal elements are zero (off-diagonal-only perturbation).

Second-order correction to state $|a⟩$:

$$E_a^{(2)}=\frac{|V{|}^2}{E_a^{(0)}-E_b^{(0)}}=\frac{(0.4{)}^2}{-5.0-(-3.0)}=\frac{0.16}{-2.0}=-0.08\text{ eV}.$$

The ground state is lowered by $0.08$ eV. For state $|b⟩$:

$$E_b^{(2)}=\frac{|V{|}^2}{E_b^{(0)}-E_a^{(0)}}=\frac{0.16}{2.0}=+0.08\text{ eV}.$$

The excited state is raised by the same amount. The perturbation pushes the two levels further apart — a general feature of second-order corrections.

Exercises Intermediate+

Moller-Plesset perturbation theory Master

The helium atom perturbation theory treats electron-electron repulsion as the perturbation. For molecular systems, the standard formulation is Moller-Plesset (MP) perturbation theory, which uses a different partitioning. The unperturbed Hamiltonian is taken to be the sum of Fock operators from the Hartree-Fock solution:

$${\hat{H}}^{(0)}=\sum _{i=1}^N\hat{F}(i)=\sum _{i=1}^N\left[\hat{h}(i)+{\hat{v}}^{\text{HF}}(i)\right],$$

where $\hat{h}(i)$ is the one-electron kinetic energy plus nuclear attraction, and ${\hat{v}}^{\text{HF}}(i)$ is the Hartree-Fock average electron-electron potential. The perturbation — called the fluctuation potential — is the difference between the full electronic Hamiltonian and the Fock-operator sum:

$${\hat{H}}^{\prime }=\hat{H}-{\hat{H}}^{(0)}=\sum _{i<j}\frac{e^2}{4\pi {\epsilon }_0{r}_{ij}}-\sum _{i=1}^N{\hat{v}}^{\text{HF}}(i).$$

This choice is motivated by the Hartree-Fock starting point: the HF energy is $E_{\text{HF}}=E^{(0)}+E^{(1)}$, and the MP correction begins at second order. The HF Slater determinant is the unperturbed ground state, and the HF occupied and virtual orbitals form the basis for the sum-over-states.

MP2 (second-order Moller-Plesset). The MP2 energy correction involves double excitations from occupied to virtual orbitals:

$$E_{\text{MP2}}=\frac{1}{4}\sum _{ij}^{\text{occ}}\sum _{ab}^{\text{virt}}\frac{|⟨ij||ab⟩{|}^2}{{\epsilon }_i+{\epsilon }_j-{\epsilon }_a-{\epsilon }_b},$$

where ${\epsilon }_i$ are HF orbital energies and $⟨ij||ab⟩$ are antisymmetrised two-electron integrals in Dirac notation. The sum runs over all pairs of occupied orbitals $(i,j)$ and all pairs of virtual orbitals $(a,b)$.

MP2 recovers $80$–$90\%$ of the correlation energy for small molecules at a cost that scales as $O(N^5)$ in the number of basis functions. It is the cheapest correlated method available. However, MP2 has known failures: it overbinds van der Waals complexes, it is not variational (the energy can fall below the exact energy), and it can diverge for systems with small HOMO-LUMO gaps where the denominator becomes small.

MP3 and MP4. Higher orders add progressively more terms at increasing computational cost ($O(N^6)$ for MP3, $O(N^7)$ for MP4). MP4 includes single, double, triple, and quadruple excitations. Convergence of the MP series is not guaranteed and can be oscillatory. For some systems (notably stretched bonds and strongly correlated materials), the MP series diverges.

Comparison with variational methods. The variational method (including Hartree-Fock and configuration interaction) guarantees $E\ge E_{\text{exact}}$. MP perturbation theory has no such guarantee. The tradeoff: variational methods require an explicit choice of trial wave function and are sensitive to that choice, while MP is uniquely determined once the HF reference is chosen. In practice, MP2 is used as a quick post-HF correction, CCSD(T) (coupled cluster with perturbative triples) is the gold standard for single-reference systems, and multireference methods handle strong correlation where both MP and single-reference CC fail.

The convergence behaviour of the MP series connects to deep questions in quantum chemistry. Kohn, Meir, and Makarov (1998) showed that the MP series can be asymptotic rather than convergent, with an initial decrease followed by divergence after a certain order. The radius of convergence depends on the gap between occupied and virtual orbital energies. Systems with near-degeneracies (transition states, stretched bonds, diradicals) can have zero radius of convergence, rendering MP theory meaningless. This failure mode motivated the development of alternative approaches such as the similarity renormalisation group and driven similarity renormalisation group methods, which circumvent the small-denominator problem.

Comparison: perturbation theory versus variational methods Master

Perturbation theory and the variational principle attack the same problem from opposite directions. Understanding their relative strengths is essential for practical quantum chemistry.

The variational principle states that for any normalised trial function $\Phi$, $⟨\Phi |\hat{H}|\Phi ⟩\ge E_0$ where $E_0$ is the exact ground-state energy. This provides a guaranteed upper bound. Hartree-Fock is the best single-determinant variational ansatz. Full configuration interaction (FCI) in a complete basis is exact but intractable for all but the smallest systems. Truncated CI (CISD, CISDT) is variational but not size-consistent: the energy of two noninteracting He atoms computed separately differs from the energy computed at infinite separation in a single calculation.

Perturbation theory does not provide a bound. The second-order correction can overshoot, giving an energy below the exact result. However, perturbation theory is size-consistent at every order: MP2 gives the same energy for $N$ noninteracting helium atoms whether computed separately or together. Size-consistency is essential for computing reaction energies, dissociation curves, and intermolecular interactions.

Property	Variational (CI)	Perturbation (MP)
Energy bound	Yes ($E\ge E_0$)	No
Size-consistent	Only full CI	Yes, all orders
Systematic improvability	Yes (add excitations)	Yes (higher orders)
Cost scaling	CISD: $O(N^6)$; FCI: factorial	MP2: $O(N^5)$; MP4: $O(N^7)$
Convergence	Monotone from above	Not guaranteed
Strong correlation	Multireference CI needed	Often diverges

For the helium atom specifically, a variational calculation with a single parameter (varying the effective nuclear charge $Z_{\text{eff}}$) gives $E=-77.5$ eV, already better than first-order perturbation theory ($-74.8$ eV). Adding more variational parameters rapidly approaches the exact energy. The Hylleraas variational calculation (using basis functions containing $r_{12}$ explicitly) converges to $-79.01$ eV with a few hundred terms. Perturbation theory converges more slowly for helium because the perturbation (electron-electron repulsion) is not small — it accounts for about $27\%$ of the unperturbed energy.

Connections Master

• Hydrogen atom quantum chemistry 14.04.01 provides the hydrogenic orbitals used as the unperturbed basis for the helium atom calculation. The $1s$ orbital with $Z=2$ is the starting point for both the perturbation theory and variational treatments of helium.

• Many-electron atoms: Hartree-Fock 14.01.03 develops the self-consistent field method whose Fock-operator partitioning underpins Moller-Plesset perturbation theory. The Hartree-Fock energy is the sum of zeroth- and first-order MP energies, and the MP2 correction begins where HF leaves off.

• Stark effect and spectroscopy. The Stark effect connects directly to atomic spectroscopy 14.12.01: the splitting of energy levels in an electric field produces observable spectral shifts. The linear Stark effect in hydrogen and the quadratic Stark effect in atoms with nondegenerate ground states are both measured by high-resolution laser spectroscopy.

• Molecular orbital theory 14.05.02 builds MOs from atomic orbitals. When the molecular Hamiltonian is treated by MP theory, the atomic orbital basis enters through the two-electron integrals. The quality of the atomic orbital basis set directly affects the convergence of the MP series.

• Variational methods and post-HF methods. This unit's perturbation-theoretic approach complements the variational methods discussed in 14.01.03. Coupled-cluster theory, which uses an exponential ansatz $\Psi =e^{\hat{T}}{\Phi }_0$ where $\hat{T}$ is the cluster operator, can be viewed as summing certain classes of perturbation theory diagrams to infinite order. CCSD(T) combines variational and perturbative elements.

Historical notes Master

Rayleigh's original insight (1877). Lord Rayleigh developed perturbation theory for vibrating strings with small inhomogeneities. He showed that the frequency shifts could be expressed as a series of corrections, with the first-order term being an integral of the perturbation weighted by the unperturbed mode shape. Rayleigh's work applied classical mechanics, but the mathematical structure carries over directly to quantum mechanics. Schrodinger himself cited Rayleigh in his 1926 papers.

Schrodinger's perturbation theory (1926). In the fourth of his series of papers "Quantisierung als Eigenwertproblem," Schrodinger developed perturbation theory for the wave equation. He applied it to the Stark effect in hydrogen, obtaining the first-order splitting of the $n=2$ level. Epstein and Schwarzschild had previously obtained the same result using the old quantum theory (1916), but Schrodinger's derivation was cleaner and generalisable.

The helium atom problem. Heisenberg and Hylleraas independently attacked the helium atom in the late 1920s. Heisenberg used perturbation theory with the electron-electron repulsion as the perturbation and obtained the first-order result of $-74.8$ eV. Hylleraas (1929) introduced variational wave functions containing $r_{12}$ explicitly, achieving unprecedented accuracy. The competition between perturbation theory and variational methods for helium drove the development of both approaches through the 1930s.

Moller-Plesset theory (1934). Moller and Plesset showed that the Hartree-Fock energy is the sum of zeroth- and first-order energies in a perturbation expansion where the Fock operator is the unperturbed Hamiltonian. Their paper was largely ignored until the development of efficient two-electron integral algorithms in the 1970s made MP2 calculations feasible. Pople's GAUSSIAN program (1970s–1980s) popularised MP2 as a standard post-HF method.

Epstein-Nesbet partitioning (1926, 1964). An alternative to MP partitioning uses the full diagonal of the Hamiltonian in the many-electron basis as ${\hat{H}}^{(0)}$, rather than the Fock-operator sum. The Epstein-Nesbet (EN) partitioning often converges faster than MP for systems with small gaps but is less systematically connected to the Hartree-Fock reference.

Convergence questions. The convergence of the MP series remained poorly understood until the work of Cremer and He (1996), who showed empirically that MP4 is more reliable than MP3 (the series oscillates), and the theoretical analysis of Stillinger (2000), who proved that the MP series for the two-electron atom is convergent for $Z>Z_c$ where $Z_c$ is a critical nuclear charge. Below $Z_c$, the exact ground state acquires an imaginary component, and the real-valued perturbation series cannot converge. Olsen and collaborators (2000) demonstrated divergent MP series for several molecular systems, sparking renewed interest in alternative perturbative frameworks.

Bibliography Master

• Schrodinger, E., "Quantisierung als Eigenwertproblem. IV," Ann. Phys. 384 (1926), 418-428. Perturbation theory for the wave equation, applied to the Stark effect.
• Hylleraas, E. A., "Neue Berechnung der Energie des Heliums im Grundzustande," Z. Physik 54 (1929), 347-366. Variational calculation of helium with explicit $r_{12}$ terms.
• Moller, C. & Plesset, M. S., "Note on an Approximation Treatment for Many-Electron Systems," Phys. Rev. 46 (1934), 618-622. The MP partitioning.
• Levine, I. N., Quantum Chemistry, 7e (Pearson, 2014), Ch. 9. Thorough treatment of perturbation theory with chemical applications.
• Sakurai, J. J. & Napolitano, J., Modern Quantum Mechanics, 3e (Cambridge, 2021), §5.1. Rigorous operator-theoretic foundation.
• Atkins, P. & Friedman, R., Molecular Quantum Mechanics, 5e (Oxford, 2010), Ch. 6. Perturbation theory applied to atomic and molecular problems.
• Szabo, A. & Ostlund, N. S., Modern Quantum Chemistry (Dover, 1996), Ch. 6. MP theory derivation and implementation details.
• Kohn, W., Meir, Y. & Makarov, D. E., "Evaluating the MP perturbation series," Phys. Rev. Lett. 80 (1998), 4153-4156. Analysis of MP series convergence.
• Olsen, J., Jorgensen, P., Helgaker, T. & Christiansen, O., "Divergence in Moller-Plesset perturbation theory," J. Chem. Phys. 112 (2000), 9736-9748. Demonstrates divergence for molecular systems.
• Cremer, D. & He, Z., "Specification of the domain of applicability of MP theory," J. Phys. Chem. 100 (1996), 6173-6188. Systematic comparison of MP2 through MP4.