Rayleigh–Schrödinger Perturbation Theory

Rayleigh–Schrödinger perturbation theory is the standard energy-independent, order-by-order construction for following an eigenvalue and its eigenvector away from a solvable reference Hamiltonian. It converts one exact eigenvalue equation into a hierarchy of linear equations whose coefficients depend only on information from earlier orders.

The name is sometimes used loosely for all stationary perturbation theory. Here it has a narrower meaning: expand the energy and state in a bookkeeping parameter, equate equal powers, and remove the unknown exact energy from every denominator. The method is local to a chosen spectral branch and to a chosen interpolation between a reference problem and the physical Hamiltonian.

This page owns the named formalism, its normalization convention, its recursive organization, and its relation to Brillouin–Wigner theory. The canonical derivations and physical interpretation of individual low-order coefficients remain on First-Order Energy Corrections, First-Order State Corrections, Second-Order Energy Corrections, and Higher-Order Structure.

Historical and Conceptual Setting

The compound name reflects two strands of spectral approximation. Rayleigh developed perturbative methods for classical vibration and wave eigenproblems; Schrödinger’s 1926 wave mechanics turned the quantum stationary-state problem into an operator eigenvalue equation. Modern presentations use “Rayleigh–Schrödinger” for the power-series method in which both the eigenvalue and eigenvector are expanded about a reference problem.

Historical naming should not be mistaken for a unique modern algorithm. Diagrammatic many-body perturbation theory, Møller–Plesset theory, effective Hamiltonians, and analytic operator perturbation theory all refine the same broad idea under different assumptions and organizational choices.

Three conceptual points matter from the start:

• the series follows a particular eigenvalue branch, not merely an integer label in a sorted spectrum;
• the coefficients depend on the chosen split into reference and perturbation, although the exact endpoint Hamiltonian does not;
• a formal coefficient hierarchy does not by itself establish convergence at the physical coupling.

Problem and Assumptions

Choose an operator family

$$H(\lambda)=H_0+\lambda V$$

and a normalized reference eigenstate

$$H_0\lvert n^{(0)}\rangle = E_n^{(0)}\lvert n^{(0)}\rangle.$$

The elementary nondegenerate construction assumes that $E_n^{(0)}$ is an isolated simple eigenvalue. Define

$$P_n = \lvert n^{(0)}\rangle \langle n^{(0)}\rvert, \qquad Q_n=I-P_n.$$

The complementary inverse required below exists only if $E_n^{(0)}-H_0$ is invertible on the relevant part of $Q_n\mathcal H$. In a discrete basis this is the familiar nonzero-denominator condition. For unbounded operators, a rigorous statement additionally requires control of domains or quadratic forms and of the spectral isolation of the target eigenvalue.

The physical coupling is often $\lambda=1$. Introducing $\lambda$ does not prove that the perturbation is small there. Control comes from dimensionless ratios such as

$$\begin{aligned} \epsilon_{mn} &= \frac{ \lvert\lambda V_{mn}\rvert }{ \lvert E_n^{(0)}-E_m^{(0)}\rvert }, \\ V_{mn} &= \langle m^{(0)}\rvert V \lvert n^{(0)}\rangle. \end{aligned}$$

together with convergence of sums over intermediate states and stability against nearby spectral clusters. Small Parameters and Error Estimates develops that control language.

Formal Expansions

Seek a branch in the form

$$\begin{aligned} E_n(\lambda) &= \sum_{r=0}^{\infty} \lambda^r E_n^{(r)}, \\ \lvert n(\lambda)\rangle &= \sum_{r=0}^{\infty} \lambda^r \lvert n^{(r)}\rangle. \end{aligned}$$

These expressions may be convergent Taylor series, asymptotic series, or formal series used only to generate coefficients. The notation alone does not decide which interpretation is valid.

Substitute both expansions into

$$H(\lambda)\lvert n(\lambda)\rangle = E_n(\lambda)\lvert n(\lambda)\rangle.$$

The coefficient of $\lambda^r$ is

$$H_0\lvert n^{(r)}\rangle + V\lvert n^{(r-1)}\rangle = \sum_{s=0}^{r} E_n^{(s)} \lvert n^{(r-s)}\rangle,$$

where $\lvert n^{(-1)}\rangle\equiv0$. For $r\ge1$, isolate the unknown state correction:

$$\begin{aligned} \bigl(H_0-E_n^{(0)}\bigr) \lvert n^{(r)}\rangle ={}& -V\lvert n^{(r-1)}\rangle \\ &+ \sum_{s=1}^{r} E_n^{(s)} \lvert n^{(r-s)}\rangle. \end{aligned}$$

This is the master coefficient equation. At each order it is linear in the new unknowns $E_n^{(r)}$ and $\lvert n^{(r)}\rangle$ because every lower-order quantity is already known.

If the Hamiltonian itself contains higher powers,

$$H(\lambda) = H_0+ \sum_{j=1}^{\infty} \lambda^j V^{(j)},$$

then the driving term becomes a convolution:

$$\sum_{j=1}^{r} V^{(j)} \lvert n^{(r-j)}\rangle.$$

Using formulas derived for $H_0+\lambda V$ without adding these terms is a common source of missing contributions.

Intermediate Normalization

An eigenvector may be multiplied by a nonzero scalar, so the projected equation cannot determine the component of $\lvert n^{(r)}\rangle$ parallel to $\lvert n^{(0)}\rangle$. Rayleigh–Schrödinger calculations commonly fix this freedom by imposing

$$\langle n^{(0)}\vert n(\lambda)\rangle=1.$$

Therefore

$$\begin{aligned} \langle n^{(0)}\vert n^{(0)}\rangle &=1, \\ \langle n^{(0)}\vert n^{(r)}\rangle &=0 \qquad (r\ge1). \end{aligned}$$

This intermediate normalization is available locally while the exact branch has nonzero overlap with the reference state. It fixes the scale and phase relative to that reference, but it does not make the perturbative state unit normalized. Writing $N_n(\lambda)=\langle n(\lambda)\vert n(\lambda)\rangle$,

$$\begin{aligned} N_n(\lambda) ={}& 1 \\ &+ \lambda^2 \langle n^{(1)}\vert n^{(1)}\rangle \\ &+ 2\lambda^2\operatorname{Re} \\ &\qquad{}\times \langle n^{(0)}\vert n^{(2)}\rangle +O(\lambda^3). \end{aligned}$$

Under intermediate normalization the second line’s inner product vanishes, leaving a norm correction at order $\lambda^2$. For an expectation value, either divide by the norm,

$$\langle A\rangle = \frac{ \langle n(\lambda)\rvert A\lvert n(\lambda)\rangle }{ \langle n(\lambda)\vert n(\lambda)\rangle },$$

or first convert the state to unit normalization. Mixing these conventions partway through a calculation changes individual state coefficients and produces false observable terms.

Projector and Resolvent Recursion

Project the master equation with $\langle n^{(0)}\rvert$. Intermediate normalization removes every overlap except the one multiplying $E_n^{(r)}$, giving

$$E_n^{(r)} = \langle n^{(0)}\rvert V \lvert n^{(r-1)}\rangle.$$

Define the reduced resolvent

$$R_n' \equiv Q_n \frac{1}{E_n^{(0)}-H_0} Q_n.$$

Applying $Q_n$ to the coefficient equation and inverting on the complementary subspace gives

$$\begin{aligned} \lvert n^{(r)}\rangle = R_n' \Bigg[ &V\lvert n^{(r-1)}\rangle \\ &- \sum_{s=1}^{r-1} E_n^{(s)} \lvert n^{(r-s)}\rangle \Bigg]. \end{aligned}$$

The pair of recursions has a simple rhythm:

1. use the previous state correction to compute the new energy coefficient;
2. subtract lower-order energy feedback from the new driving vector;
3. apply the reduced resolvent to obtain the complementary state correction.

In an eigenbasis of $H_0$, write

$$\lvert n^{(r)}\rangle = \sum_m C_m^{(r)} \lvert m^{(0)}\rangle,$$

with $C_n^{(0)}=1$ and $C_n^{(r)}=0$ for $r\ge1$. For $m\ne n$,

$$\begin{aligned} C_m^{(r)} = \frac{1}{E_n^{(0)}-E_m^{(0)}} \Bigg[ &\sum_k V_{mk}C_k^{(r-1)} \\ &- \sum_{s=1}^{r-1} E_n^{(s)}C_m^{(r-s)} \Bigg]. \end{aligned}$$

This component form is convenient for symbolic or numerical generation, but the projector form shows more clearly where spectral isolation enters. Higher-Order Structure gives the detailed all-order derivation, third-order formula, normalization bookkeeping, and large-order diagnostics.

Low-Order Landmarks

The first coefficients are landmarks rather than separate methods:

Order	Coefficient	Canonical treatment
First-order energy	$E_n^{(1)}=V_{nn}$	First-Order Energy Corrections
First-order state	$\lvert n^{(1)}\rangle=R_n'V\lvert n^{(0)}\rangle$	First-Order State Corrections
Second-order energy	$E_n^{(2)}=\langle n^{(0)}\rvert VR_n'V\lvert n^{(0)}\rangle$	Second-Order Energy Corrections
Third order and beyond	nested resolvents plus subtraction terms	Higher-Order Structure

The subtraction terms are not cosmetic. They remove repeated lower-order energy shifts from the state recursion, enforce the chosen normalization, and participate in the cancellation of disconnected contributions in many-body energies.

Worked Audit: A General Two-Level Matrix

Consider

$$\begin{aligned} H_0 &= \begin{pmatrix} 0&0\\ 0&\Delta \end{pmatrix}, \\ V &= \begin{pmatrix} a&v\\ v^*&b \end{pmatrix}, \qquad \Delta\gt0. \end{aligned}$$

Follow the branch that begins at $E_0^{(0)}=0$. The first three corrections are

$$E_0^{(1)}=a,$$ $$E_0^{(2)} = -\frac{\lvert v\rvert^2}{\Delta},$$

and

$$E_0^{(3)} = \frac{(b-a)\lvert v\rvert^2}{\Delta^2}.$$

Define

$$\begin{aligned} D(\lambda)^2 ={}& \bigl(\Delta+\lambda(b-a)\bigr)^2 \\ &+ 4\lambda^2\lvert v\rvert^2. \end{aligned}$$

The exact lower eigenvalue is

$$\begin{aligned} E_-(\lambda) = \frac{1}{2} \Bigg[ &\Delta+\lambda(a+b) \\ &-D(\lambda) \Bigg]. \end{aligned}$$

Expanding the square root gives

$$\begin{aligned} E_-(\lambda) ={}& \lambda a - \lambda^2\frac{\lvert v\rvert^2}{\Delta} \\ &+ \lambda^3 \frac{(b-a)\lvert v\rvert^2}{\Delta^2} +O(\lambda^4), \end{aligned}$$

which checks the recursion. The example also reveals its limitation: when $\lvert\lambda v\rvert$ is comparable to $\Delta$, the expansion is a poor substitute for diagonalizing the two-state sector.

Degenerate and Nearly Degenerate Branches

For a degenerate reference eigenvalue, $E_n^{(0)}-H_0$ is not invertible on all directions orthogonal to one arbitrarily chosen basis vector. Ordinary one-state recursion therefore fails before any question of “small coupling” is asked.

The cure is structural:

• for an exact finite degeneracy, diagonalize $PVP$ inside the complete degenerate eigenspace;
• for levels whose internal splitting competes with their coupling, retain the whole cluster in a model space and construct an effective Hamiltonian;
• for a level approaching a continuum threshold, move to resolvent or scattering methods with the correct boundary conditions.

See Degenerate Perturbation Theory and Quasi-Degenerate Perturbation Theory. Adding more Rayleigh–Schrödinger orders does not repair an incorrectly chosen one-dimensional model space.

Comparison with Brillouin–Wigner Theory

For a one-state projector, define the excluded-space resolvent

$$G_Q(E,\lambda) = Q_n \frac{1}{E-Q_nH(\lambda)Q_n} Q_n.$$

Eliminating the $Q_n$ component of the exact eigenvector then leads schematically to

$$\begin{aligned} E ={}& E_n^{(0)} +\lambda V_{nn} \\ &+ \lambda^2 \langle n^{(0)}\rvert VG_Q(E,\lambda)V \lvert n^{(0)}\rangle. \end{aligned}$$

Brillouin–Wigner theory retains $E$ in this denominator and solves a self-consistent equation. Rayleigh–Schrödinger theory expands both the resolvent and $E$ about $E_n^{(0)}$ and collects a definite coefficient of every power of $\lambda$.

Feature	Rayleigh–Schrödinger	Brillouin–Wigner
Denominators	Reference energies after expansion	Unknown or partially resummed energy
Equation at a fixed order	Linear recursion	Nonlinear self-consistency
Order counting	Explicit	Can mix nominal orders
Root tracking	Attached to a reference branch	Requires selecting among roots
Natural normalization	Intermediate normalization	Projected-state normalization with resolvent derivatives
Typical strength	Transparent coefficients and diagrams	Compact model-space elimination and partial resummation

When both constructions are expanded consistently and apply to the same analytic branch, they reproduce the same Taylor coefficients. Their finite truncations can nevertheless differ because a self-consistent denominator resums selected higher-order terms. Neither organization is uniformly superior; each approximation needs its own error and root-selection checks.

The canonical energy-dependent treatment is Brillouin–Wigner Perturbation Theory.

Repartitioning and Scheme Dependence

A physical Hamiltonian $H_{\mathrm{phys}}$ can be split in many ways:

$$H_{\mathrm{phys}} = H_0+V = \widetilde H_0+\widetilde V.$$

The corresponding interpolations

$$H(\lambda)=H_0+\lambda V, \qquad \widetilde H(\lambda) = \widetilde H_0+\lambda\widetilde V$$

agree at $\lambda=1$ but generally trace different paths through operator space. Their exact endpoint spectra are the same, while their perturbative coefficients, convergence radii, and useful truncation orders can differ.

A good reference Hamiltonian should absorb the dominant solvable structure, preserve useful symmetries, and leave denominators separated from strongly coupled omitted states. Counterterms, mean fields, screened interactions, and optimized frequencies are all forms of repartitioning. They can improve a series, but their definition and order counting must be stated; otherwise two nominally “second-order” results may not be comparable.

Atomic Physics

Atomic calculations expose the strengths of the Rayleigh–Schrödinger organization:

• angular momentum and parity eliminate forbidden matrix elements before radial sums are attempted;
• weak static electric and magnetic fields generate level shifts, induced moments, and polarizabilities;
• relativistic, recoil, and finite-size operators can be inserted as controlled corrections after a reference Hamiltonian is chosen;
• model spaces handle fine-structure multiplets or accidental near degeneracies that cannot be treated one state at a time.

For example, a parity-odd electric-dipole perturbation has zero diagonal matrix element in a parity eigenstate, so the linear Stark shift vanishes for an isolated nondegenerate level. The leading energy response is then second order and involves a sum over opposite-parity states. Degenerate hydrogenic manifolds require degenerate theory instead; applying the isolated-state formula there produces zero denominators rather than a physical answer.

Atomic many-body implementations replace explicit sums by resolvents, inhomogeneous equations, diagrammatic expansions, or coupled effective Hamiltonians. The Rayleigh–Schrödinger label identifies the energy-independent order expansion, not one particular computational representation.

Molecular and Electronic-Structure Uses

Molecular perturbation theory must decide which degrees of freedom and correlations belong in the reference problem. Common uses include weak-field shifts, vibrational anharmonic corrections, rotational couplings, and electronic correlation around an independent-particle reference.

Møller–Plesset perturbation theory is a prominent many-electron specialization: a Fock operator defines the zeroth-order orbital problem and the residual interaction is treated perturbatively. Its coefficients are Rayleigh–Schrödinger coefficients for that particular partition, supplemented by fermionic antisymmetry and many-body bookkeeping. Changing orbitals or the partition changes the finite-order approximation even though the target electronic Hamiltonian is unchanged.

Several cautions follow:

• a small orbital energy denominator can signal a multireference problem rather than a need for still higher single-reference order;
• nuclear motion and electronic motion may introduce separate approximation layers, including Born–Oppenheimer and nonadiabatic expansions;
• truncation of a many-electron wavefunction and truncation of an energy series have different size-consistency properties;
• basis-set error is distinct from perturbation-order error.

When one determinant is not qualitatively dominant, degenerate or quasi-degenerate model-space methods are often the correct starting point.

Linked Many-Body Organization

Repeated insertions of a two-body interaction generate rapidly growing sums over intermediate many-particle states. Second quantization and Wick’s theorem reorganize those sums into contractions and diagrams. In a linked formulation, disconnected pieces cancel from the energy, leaving contributions that scale properly when independent subsystems are combined.

This statement needs precision: bare intermediate-normalized state corrections contain disconnected products, and an arbitrary truncation of the wavefunction need not be size consistent. The linked-cluster result concerns the properly organized energy expansion and its cancellation structure. Wick’s Theorem Preview introduces the operator machinery; Higher-Order Structure shows where subtraction terms first appear in the recursion; Perturbation Theory in Many-Body Systems develops excitation-rank, volume, cumulant, and infrared counting.

Convergence and Asymptotic Meaning

If an analytic operator family has an isolated eigenvalue, the local branch may admit a convergent Taylor series. In finite dimensions, the nearest singularity in complex $\lambda$ often occurs where eigenvalues and eigenvectors coalesce, and its distance can set the convergence radius. A spectrum that looks harmless for real positive $\lambda$ can therefore have a short series radius because of a complex exceptional point.

Many important infinite-dimensional problems instead produce divergent asymptotic series. The anharmonic oscillator is the standard laboratory: low-order coefficients can be highly useful even though high-order coefficients eventually grow too rapidly for convergence. The operational questions are then:

• at which order are terms smallest for the coupling of interest?
• how stable is the result under nearby truncation orders?
• is a Borel or other resummation justified?
• what nonperturbative effects are invisible to every power of $\lambda$?

Do not infer convergence from a few decreasing terms. Use Anharmonic Oscillator and Higher-Order Structure for the canonical examples and diagnostics.

Practical Workflow

1. Specify the interpolation. Write $H(\lambda)$, identify the physical value of $\lambda$, and state any counterterms or mean fields.
2. Identify the spectral object. Decide whether the target is an isolated level, an exactly degenerate eigenspace, or a nearly degenerate cluster.
3. Exploit symmetry. Resolve conserved sectors and selection rules before summing over intermediate states.
4. Choose normalization. State intermediate or unit normalization and use it consistently.
5. Generate coefficients. Use projectors or component recursions, retaining all terms of the requested order.
6. Check denominators and tails. Inspect small gaps, continuum contributions, and basis-cutoff dependence.
7. Validate independently. Compare with exact limits, direct diagonalization, variational bounds, sum rules, or finite-difference derivatives.
8. Report truncation honestly. Distinguish perturbative order, numerical error, basis error, and model error.

Common Mistakes

• Calling $\lambda$ small without identifying dimensionless coupling-to-gap ratios.
• Following a sorted eigenvalue index through an avoided crossing instead of tracking the branch by overlap or symmetry.
• Applying a one-state reduced resolvent to an exact or near degeneracy.
• Mixing intermediate-normalized state coefficients with unit-normalized expectation-value formulas.
• Omitting $\lambda^2V^{(2)}$ or higher terms that are already present in the Hamiltonian.
• Comparing two partitions or two perturbative organizations only by their nominal order labels.
• Treating a finite set of formal coefficients as evidence that the infinite series converges.
• Assuming that a higher perturbative order compensates for a qualitatively wrong reference state.

Exercises

1. Derive the master coefficient equation

Starting from $H(\lambda)=H_0+\lambda V$, insert the energy and state series and derive the equation at order $\lambda^r$.

Solution

The left-hand side is

$$\begin{aligned} H(\lambda)\lvert n(\lambda)\rangle ={}& \sum_{r=0}^{\infty} \lambda^r H_0\lvert n^{(r)}\rangle \\ &+ \sum_{r=1}^{\infty} \lambda^r V\lvert n^{(r-1)}\rangle. \end{aligned}$$

The right-hand side is the Cauchy product

$$E_n(\lambda)\lvert n(\lambda)\rangle = \sum_{r=0}^{\infty} \lambda^r \sum_{s=0}^{r} E_n^{(s)}\lvert n^{(r-s)}\rangle.$$

Equating the coefficient of $\lambda^r$ and moving the $s=0$ term to the left gives

$$\begin{aligned} \bigl(H_0-E_n^{(0)}\bigr) \lvert n^{(r)}\rangle ={}& -V\lvert n^{(r-1)}\rangle \\ &+ \sum_{s=1}^{r} E_n^{(s)}\lvert n^{(r-s)}\rangle. \end{aligned}$$

2. Convert to unit normalization

An intermediate-normalized state is known through second order. Find the component parallel to $\lvert n^{(0)}\rangle$ that appears when the state is converted to unit normalization.

Solution

Intermediate normalization gives

$$\langle n^{(0)}\vert n^{(1)}\rangle = \langle n^{(0)}\vert n^{(2)}\rangle =0.$$

Hence

$$\lVert n(\lambda)\rVert^2 = 1+ \lambda^2 \langle n^{(1)}\vert n^{(1)}\rangle +O(\lambda^3).$$

Using $(1+x)^{-1/2}=1-x/2+O(x^2)$, the unit-normalized state contains

$$-\frac{\lambda^2}{2} \langle n^{(1)}\vert n^{(1)}\rangle \lvert n^{(0)}\rangle$$

in addition to the intermediate-normalized coefficients through second order.

3. Check the two-level coefficients

For the two-level matrix in the worked audit, use the standard formulas to derive $E_0^{(1)}$, $E_0^{(2)}$, and $E_0^{(3)}$.

Solution

The diagonal matrix element gives

$$E_0^{(1)}=V_{00}=a.$$

There is one intermediate state, so

$$E_0^{(2)} = \frac{V_{01}V_{10}}{0-\Delta} = -\frac{\lvert v\rvert^2}{\Delta}.$$

At third order, the connected chain and normalization subtraction are

$$\begin{aligned} E_0^{(3)} ={}& \frac{V_{01}V_{11}V_{10}}{\Delta^2} - V_{00} \frac{V_{01}V_{10}}{\Delta^2} \\ ={}& \frac{(b-a)\lvert v\rvert^2}{\Delta^2}. \end{aligned}$$

4. Recover Rayleigh–Schrödinger order from a self-consistent denominator

Suppose a Brillouin–Wigner equation has the form

$$\begin{aligned} E &= E_0+\lambda a+ \frac{\lambda^2\lvert v\rvert^2} {E-E_1-\lambda b}, \\ \Delta &= E_1-E_0\gt0. \end{aligned}$$

Expand the root connected to $E_0$ through third order.

Solution

Write

$$E=E_0+\lambda e_1+\lambda^2e_2+\lambda^3e_3+\cdots.$$

The denominator is

$$\begin{aligned} E-E_1-\lambda b ={}& -\Delta+\lambda(e_1-b) \\ &+O(\lambda^2). \end{aligned}$$

Therefore

$$\begin{aligned} \frac{1}{E-E_1-\lambda b} ={}& -\frac{1}{\Delta} \\ &- \lambda\frac{e_1-b}{\Delta^2} +O(\lambda^2). \end{aligned}$$

Matching powers gives

$$\begin{aligned} e_1 &=a, \\ e_2 &=-\frac{\lvert v\rvert^2}{\Delta}, \\ e_3 &=\frac{(b-a)\lvert v\rvert^2}{\Delta^2}. \end{aligned}$$

The self-consistent equation reproduces the Rayleigh–Schrödinger coefficients when it is expanded consistently.

5. Show that coefficients depend on the partition

For a one-dimensional Hamiltonian with physical energy $v$, compare the interpolations

$$H(\lambda)=\lambda v$$

and

$$\widetilde H(\lambda) = a+\lambda(v-a).$$

What is invariant, and what changes?

Solution

At the physical point,

$$H(1)=\widetilde H(1)=v.$$

The first interpolation has coefficients

$$E^{(0)}=0, \qquad E^{(1)}=v,$$

whereas the second has

$$\widetilde E^{(0)}=a, \qquad \widetilde E^{(1)}=v-a.$$

Thus the exact endpoint is invariant, but reference energies and perturbative coefficients depend on the path chosen to reach it. In nontrivial problems this choice also changes denominators and convergence properties.

6. Diagnose a nearly degenerate pair

Two reference levels are separated by $\delta$, with an off-diagonal perturbation matrix element $\lambda v$. Explain why the one-state expansion fails when $\lvert\lambda v\rvert/\lvert\delta\rvert$ is not small and identify the correct replacement.

Solution

The first-order state coefficient contains

$$\frac{\lambda v}{\delta},$$

and the second-order energy contains a term of order

$$\frac{\lambda^2\lvert v\rvert^2}{\delta}.$$

If $\lvert\lambda v\rvert/\lvert\delta\rvert$ is not small, the reference state is strongly mixed and the nominal hierarchy of orders is lost. Both levels should be retained in a model space and diagonalized together, exactly for a two-state model or through degenerate/quasi-degenerate effective-Hamiltonian theory when additional states contribute perturbatively.

Cross-Links

• Time-Independent Perturbation Theory for the chapter-level decision framework.
• Nondegenerate Perturbation Theory for the canonical low-order derivation.
• Hellmann–Feynman Theorem for exact parameter derivatives, generalized forces, and variational caveats.
• Higher-Order Structure for all-order recursion, third order, diagrams, and large-order behavior.
• Brillouin–Wigner Perturbation Theory for the energy-dependent alternative.
• Quasi-Degenerate Perturbation Theory for clustered levels and model-space selection.
• Resolvent Operator for the operator analytic structure behind reduced resolvents.
• Perturbation Theory Benchmarks for numerical coefficient and truncation checks.
• Atomic Physics and Molecular Physics for symmetry-resolved applications.

References

• E. Schrödinger, “Quantisierung als Eigenwertproblem,” Annalen der Physik 385, 437–490 (1926), doi:10.1002/andp.19263851302.
• T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, 1976; reprint 1995, doi:10.1007/978-3-642-66282-9.
• A. Messiah, Quantum Mechanics, Vol. II, North-Holland, 1962, Chapter XVII.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Vol. II, Wiley, 1977, Complement B.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2021, Chapter 5.
• I. Lindgren and J. Morrison, Atomic Many-Body Theory, 2nd ed., Springer, 1986.
• C. Møller and M. S. Plesset, “Note on an Approximation Treatment for Many-Electron Systems,” Physical Review 46, 618–622 (1934), doi:10.1103/PhysRev.46.618.
• J. Goldstone, “Derivation of the Brueckner Many-Body Theory,” Proceedings of the Royal Society A 239, 267–279 (1957), doi:10.1098/rspa.1957.0037.
• I. Shavitt and R. J. Bartlett, Many-Body Methods in Chemistry and Physics, Cambridge University Press, 2009, Chapters 3–5.