B3 - method for determining the correct linear combination of zeroth order eigenfunctions

recognise that there is a $s$ -fold degeneracy
write down the complete set of degenerate eigenfunctions $| u_{n α}^{0} ⟩$ with the same unperturbed energy $E_{n}^{0}$ for $D_{s}$
calculate the elements in the $s \times s$ matrix, $V_{α β} = ⟨ u_{n β}^{0} | \hat{V} | u_{n α}^{0} ⟩$

⟨ u_{n β}^{0} | \hat{V} | ϕ_{n α}^{0} ⟩ = E_{n i}^{1} ⟨ u_{n β}^{0} | ϕ_{n i}^{0} ⟩

\sum_{α = 1}^{s} ({⟨ u |}_{n β}^{0} \hat{V} | ϕ_{n α}^{0} ⟩ - E_{n i}^{1} ⟨ u_{n β}^{0} | ϕ_{n i}^{0} ⟩) c_{i β} = 0, i = 1, \dots, s

these $s$ equations give the required values for $c_{i α}$
setting the determinant of the term in the brackets to zero gives the $s$ roots $E_{n i}^{1}$ , the first-order energy shifts

solve the characteristic determinant to find the $s$ eigenvalues, the first-order corrections
use these eigenvalues to calculate $c_{i α}$