10. Perturbation theory and response
Position in the course: Lesson 10 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
Prerequisites: Eigenstates and matrix elements
1. Identify a small parameter
Write \(H=H_0+\lambda V\), where \(H_0|n\rangle=E_n^{(0)}|n\rangle\). Assume a nondegenerate discrete level separated from nearby levels and matrix elements small enough relative to the gaps. Expand both energy and state, using intermediate normalization \(\langle n|\psi_n\rangle=1\). Comparing powers of \(\lambda\) gives
Projection onto \(n\) yields \(E_n^{(1)}=V_{nn}\). Projection onto \(m\ne n\) yields \(\langle m|\psi_n^{(1)}\rangle=V_{mn}/(E_n^{(0)}-E_m^{(0)})\). Projecting the second-order equation back onto \(n\) then gives
Small denominators reveal the approximation's danger. The ground-state second-order correction is nonpositive when every coupled state lies above it. Excited states have contributions of both signs. These are coefficients in a power series, not independent variational bounds.
2. Check against a solvable two-state problem
Take \(H=\begin{pmatrix}0&v\\v&\Delta\end{pmatrix}\) with \(\Delta>0\). The lower exact root is \((\Delta-\sqrt{\Delta^2+4v^2})/2\). Expanding the square root for \(|v|/\Delta\ll1\) yields
At \(v/\Delta=0.1\), second order is close. At a vanishing gap it diverges while the exact energy remains finite at \(-|v|\). One must diagonalize the coupled low-energy subspace rather than keep shrinking a denominator in a nondegenerate formula. This is the conceptual bridge to multireference electron correlation.
3. Degenerate perturbation theory
If several unperturbed states share an energy, choose their projector \(P\) and diagonalize \(PVP\) first. Its eigenvectors define the combinations that split at first order. Only then treat coupling to the complementary space. An arbitrary original basis inside the degenerate space does not have separately meaningful first-order diagonal shifts. Symmetry often makes the required small matrix easier to construct.
4. Worked electric response
For an electric field \(F\) along \(z\), the interaction is \(V=-F\hat\mu_z\). A parity-even nonpolar ground state has zero first-order shift. The second-order energy is \(-\alpha F^2/2\), with
For a two-level teaching model with gap \(\Delta\) and transition dipole \(d\), \(\alpha=2d^2/\Delta\). Halving the gap doubles the weak-field polarizability, but also halves the field scale at which mixing \(Fd/\Delta\) becomes large. A huge response may therefore signal proximity to the perturbative boundary rather than a robust linear behavior.
5. Exercises and limitations
Why does a diagonal perturbation have no second-order mixing correction in this eigenbasis? All off-diagonal \(V_{mn}\) vanish, so the sum is zero. Its energies can shift at first order without its eigenvectors changing.
What changes if the state has a permanent dipole? A linear term \(-F\langle\mu_z\rangle\) appears before the quadratic response. Separate permanent polarization from induced polarization.
Continuum states may require integrals, resonances need special treatment and strong fields require nonperturbative methods. In practical response calculations basis incompleteness and missing correlation affect transition matrix elements as well as energy gaps; converging total energy alone does not establish an accurate polarizability.
Analytical teaching schematic, not simulation data.
Further conceptual check
For a harmonic oscillator perturbed by a constant force, only the first excited state couples to the ground state through \(x\). Substituting its matrix element and gap gives \(\Delta E=-F^2/(2m\omega^2)\). Completing the square in \(m\omega^2x^2/2-Fx\) gives the same shift exactly. This is an independent algebraic verification of the second-order sign and factor.
6. References and study connections
The derivations and toy arithmetic are original teaching constructions. No molecular simulation is reported here.
7. Related theory and practice
Molecular methods · Quantum Monte Carlo · Gaussian · VASP