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First order variation of the wave function of conduction states

Let's start by unpacking Eq. (1), which we rewrite here: $$\begin{equation} (H_{SCF} - \varepsilon_n)\lvert \Delta \psi_n \rangle = -(\Delta V_{SCF} + \Delta \varepsilon_n) \lvert \psi_n \rangle. \tag{...
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