Brillouin–Wigner Perturbation Theory

Brillouin–Wigner perturbation theory is an energy-dependent formulation of stationary perturbation theory. Instead of expanding every energy denominator around an unperturbed value from the start, it keeps the exact energy $E$ inside a resolvent and solves a self-consistency problem.

This makes it a natural bridge between ordinary Rayleigh–Schrödinger perturbation theory, projection methods, and effective Hamiltonians. It is powerful, but the energy dependence introduces normalization and root-selection caveats.

Required background. Use projectors for block elimination and nondegenerate perturbation theory to compare expansions.

Eliminating the complementary subspace

Let

$$H=H_0+V$$

and choose a model-space projector $P$. Its complement is

$$Q=I-P.$$

For an exact eigenstate

$$H\lvert\psi\rangle=E\lvert\psi\rangle,$$

split the state into retained and eliminated components:

$$\lvert\psi\rangle = P\lvert\psi\rangle + Q\lvert\psi\rangle.$$

Projecting the eigenvalue equation gives

$$PHP\lvert\psi_P\rangle + PHQ\lvert\psi_Q\rangle = E\lvert\psi_P\rangle,$$

and

$$QHP\lvert\psi_P\rangle + QHQ\lvert\psi_Q\rangle = E\lvert\psi_Q\rangle,$$

where

$$\lvert\psi_P\rangle=P\lvert\psi\rangle, \qquad \lvert\psi_Q\rangle=Q\lvert\psi\rangle.$$

Energy-Dependent Effective Hamiltonian

Take $H$ self-adjoint and $P$ orthogonal. For real $E$ outside the spectrum of $QHQ$, the inverse on $Q$ exists and the $Q$-space equation gives

$$\lvert\psi_Q\rangle = \frac{1}{E-QHQ} QHP\lvert\psi_P\rangle.$$

Substituting into the $P$-space equation gives

$$H_{\mathrm{eff}}(E) \lvert\psi_P\rangle = E\lvert\psi_P\rangle,$$

with

$$H_{\mathrm{eff}}(E) = PHP + PHQ \frac{1}{E-QHQ} QHP.$$

This is the central Brillouin–Wigner structure. The effective Hamiltonian depends on the eigenvalue being solved for.

One-State Model Space

For a nondegenerate state $\lvert n^{(0)}\rangle$, take

$$P=\lvert n^{(0)}\rangle\langle n^{(0)}\rvert.$$

If $QHQ$ is approximated by $QH_0Q$ in the resolvent, the energy equation becomes

$$E = E_n^{(0)} + V_{nn} + \sum_{m\ne n} \frac{|V_{mn}|^2}{E-E_m^{(0)}} + \cdots.$$

The energy $E$ appears on both sides. Solving this equation self-consistently resums some denominator effects that ordinary order-by-order perturbation theory would expand.

If one expands

$$\frac{1}{E-E_m^{(0)}}$$

around $E_n^{(0)}$, the usual Rayleigh–Schrödinger denominators reappear order by order.

Two-Level Check

Consider

$$H = \begin{pmatrix} E_1&v\\ v^*&E_2 \end{pmatrix}, \qquad E_1\lt E_2.$$

Choose $P=\lvert1\rangle\langle1\rvert$. The Brillouin–Wigner equation is

$$E = E_1 + \frac{|v|^2}{E-E_2}.$$

Multiplying through gives

$$(E-E_1)(E-E_2)-|v|^2=0,$$

which is the exact characteristic equation for the two-level Hamiltonian when $v\ne0$. For $v=0$, the pure $Q$ eigenstate at $E_2$ is invisible to this scalar projected equation: multiplying by a vanishing denominator must not be used to claim it was a solution of the original equation. The energy dependence has captured the exact answer because the eliminated space is one-dimensional and no further approximation was made.

For $|v|\ll E_2-E_1$, the lower root expands to

$$E = E_1 - \frac{|v|^2}{E_2-E_1} + O(|v|^4),$$

matching ordinary perturbation theory at second order.

Disadvantages and Cautions

The price of energy dependence is that the eigenvalue problem becomes nonlinear:

$$H_{\mathrm{eff}}(E)\lvert\psi_P\rangle=E\lvert\psi_P\rangle.$$

This creates several caveats:

• multiple roots may appear, and not all belong to the desired state;
• normalization of the projected state is not the same as normalization of the full state;
• derivatives of $H_{\mathrm{eff}}(E)$ can enter normalization and response formulas;
• naive truncations may behave differently from Rayleigh–Schrödinger truncations at the same nominal order;
• size-extensivity issues can arise in many-body applications.

The method reorganizes the approximation; self-consistency alone does not control neglected terms.

For a real bound-state root, the normalization follows directly from reconstructing $|\psi_Q\rangle$:

$$\|\psi\|^2 =\langle\psi_P|[I_P-\partial_EH_{\mathrm{eff}}(E)]|\psi_P\rangle.$$

The sign is a useful check: differentiating $(E-QHQ)^{-1}$ gives $-(E-QHQ)^{-2}$. Thus $-\partial_EH_{\mathrm{eff}}$ is positive semidefinite and adds the omitted probability. For the two-level example, a projected amplitude $c$ has total norm $|c|^2[1+|v|^2/(E-E_2)^2]$. Resonances require a different, non-Hermitian normalization treatment.

Relation to Resolvents

The operator

$$\frac{1}{E-QHQ}$$

is a resolvent in the excluded subspace. Its poles mark energies where the eliminated sector can strongly mix with the model space.

This is why Brillouin–Wigner theory naturally warns about near-degeneracy: if $E$ approaches an eigenvalue of $QHQ$, the effective Hamiltonian changes rapidly and the chosen model space may be too small.

Relation to Effective Hamiltonians

Schrieffer–Wolff methods construct energy-independent effective Hamiltonians by a perturbative unitary block diagonalization. Brillouin–Wigner instead gives an energy-dependent effective Hamiltonian by eliminating $Q$ through a resolvent.

Both methods encode virtual excursions into excluded states. The difference is organizational:

• Schrieffer–Wolff expands in powers of coupling over gaps and keeps unitarity explicit;
• Brillouin–Wigner keeps an energy-dependent resolvent and solves self-consistently.

Folded Effective Hamiltonians reorganizes derivatives of the same energy-dependent Q-box into one energy-independent model-space operator. Projection Methods gives the exact block reduction from which both organizations begin.

For a controlled calculation, the chosen convention should be stated before formulas are compared.

Exercises

1. Starting from

$$E = E_1 + \frac{|v|^2}{E-E_2},$$

derive the exact two-level eigenvalue equation.

Solution

Multiply by $E-E_2$:

$$E(E-E_2) = E_1(E-E_2)+|v|^2.$$

Rearranging gives

$$(E-E_1)(E-E_2)-|v|^2=0.$$

This is the characteristic equation of the $2\times2$ Hamiltonian.

2. Why does a pole of $(E-QHQ)^{-1}$ indicate that the model space may be too small?

Solution

A pole occurs when $E$ approaches an eigenvalue of the excluded-space Hamiltonian $QHQ$. If its residue couples to $P$, the eliminated sector can mix strongly with the retained sector. A symmetry-decoupled $Q$ level can instead have zero residue and remain absent from the projected equation. The perturbative separation between $P$ and $Q$ is no longer controlled, so the near-resonant excluded state should usually be included in the model space.

References

• P.-O. Löwdin, “Studies in perturbation theory. IV. Solution of eigenvalue problem by projection operator formalism,” Journal of Mathematical Physics 3, 969–982, 1962, doi:10.1063/1.1724312.
• S. Bravyi, D. P. DiVincenzo, and D. Loss, “Schrieffer–Wolff transformation for quantum many-body systems,” Annals of Physics 326, 2793–2826, 2011.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• I. Lindgren and J. Morrison, Atomic Many-Body Theory, 2nd ed., Springer, 1986.
• A. Messiah, Quantum Mechanics, Dover, 1999.