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The first order variation of the wave function $\Delta \psi_n$ is obtained by standard perturbation theory (Eq. 25 of ref 1):

\begin{equation} (H_{SCF}-\epsilon_n)|\Delta \psi_n \rangle = -(\Delta V_{SCF} - \Delta \epsilon)|\psi_n\rangle \tag{1} \end{equation}

However, in practical implementations, the following equation (Eq. 30 of the same ref) is adopted and an iterative algorithm like conjugate-gradient approach is employed to find the solution:

\begin{equation} (H_{SCF}-\epsilon_n-\alpha P_v)|\Delta \psi_n \rangle = -P_c\Delta V_{SCF}|\psi_n\rangle \tag{2} \end{equation} where $P$ is the projection operator with $v$ and $c$ for occupied and empty states, respectively. The $\alpha P_v$ term is added to eliminate the singularity of the linear operator $H_{SCF}-\epsilon_n$.

As Eq. (2) is used to evaluate the occupied states due to the projection operators, how can we calculate the derivative of conduction states?

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    $\begingroup$ This is a really interesting set of questions, but it may be a bit too broad as is. I think if you narrowed it down to one or two points from your list, with the other points possibly moved to separate questions, it will be easier for someone to provide an answer. $\endgroup$
    – Tyberius
    Dec 28 '20 at 3:40
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    $\begingroup$ I agree. This question has gone almost 2 months without a comment or answer, and it's probably because answering five questions can be a daunting task. If you can ask separate questions that would be great, and people who the answer to one of them can individually answer each question that they know how to answer. We need more questions anyway: area51.stackexchange.com/proposals/122958?phase=beta ! $\endgroup$ Dec 28 '20 at 3:43
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    $\begingroup$ @Tyberius NikeDattani Thanks for your comments. Now the original question is edited to show my most central concern. $\endgroup$ Dec 28 '20 at 14:14
  • $\begingroup$ @XiaomingWang Excellent! $\endgroup$ Dec 28 '20 at 17:27

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