where Ψ and Φ represent the first-order corrections (in ζ) to the wavefunctions Ψ and Φ. Note that we use [1] to indicate first-order with respect to the X operator. The induced property ∆X, introduced in Eq. ( ), is of first order by definition as we have obtained it from the Hellmann-Feynman theorem, so for that particular quantity, we remove the [1] superscript. The functions Ψ and Φ are the solutions of the following linear equations: