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The theory behind the trick is based on the Hellmann-Feynman (HF) theorem $$\frac{dE_{\lambda}}{d\lambda}~=~\langle \psi_{\lambda} | \frac{d\hat{H}_{\lambda}}{d\lambda}| \psi_{\lambda} \rangle,\tag{A}$$ which works with a single derivative, but not with a square of a derivative, cf. OP's failed calculation (5) for the expectation value $\langle\frac{1}{r^2}... 3 That the trick works once is lucky coincidence, it is actually non-sense as presented. Consider, as an extreme counterexample, an operator$O$independent from$e$, and a state$\lvert \psi\rangle$whose eigenvalue w.r.t.$O$is$e$. Then, by the same logic we have$\$ 0 = \langle \psi \vert 0 \vert \psi \rangle = \langle \psi \vert \frac{\partial}{\partial ...