Past exam of the chemistry A course of the University of Cambridge / 2022 / Paper 1 / A1 / b / v / Solution

Project the first-order equation onto ${\Psi }_j^{(0)}$, with $j\ne i$. By orthonormality and the Hermitian operator property,ConsequentlyandThe numerator order is important: the question defines $H_{ij}$ with $i$ on the bra, whereas projecting the correction onto state $j$ gives $H_{ji}$. The two coincide only when the relevant matrix element is real.

Excluding an $i$ component chooses $⟨{\Psi }_i^{(0)}|{\Psi }_i^{(1)}⟩=0$, a permitted first-order phase and wavefunction normalization convention. A component parallel to ${\Psi }_i^{(0)}$ is not determined by the projected equation. Nondegeneracy ensures nonzero denominators; a small energy gap can still make the perturbative expansion inaccurate unless $|\lambda H_{ji}|\ll |E_i^{(0)}-E_j^{(0)}|$.