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I found a recursive scheme to solve the Kohn-Sham equation. However, I have a misunderstanding: how to choose the electron density for the initial step? There are two ways to calculate the electron density:

\begin{equation} \begin{gathered} n (\mathbf{r_1})=N \int |\Psi(\mathbf{r_1}, \sigma_1,..., \mathbf{r_N}, \sigma_N)|^2 \, d\sigma_1 \mathbf{dr_2} d \sigma_2...\mathbf{dr_N} d \sigma_N \end{gathered} \end{equation}

\begin{equation} \begin{gathered} n(\mathbf{r_1}) =\sum_{i}^{N}|\varphi_i(\mathbf{r_1})|^2 \end{gathered} \end{equation} enter image description here

However, in the beginning we do not know the functions $\varphi_i$. And in principle, we never know the multi-electron function $\Psi$.

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You can make different choices for your initial charge density. As long as your choice is "close enough" to the answer, then in the self-consistent scheme you will eventually reach the correct charge density. You can think of this as analogous to a minimization problem where you are trying to find a local minimum: if you are within the basin of your local minimum, then any starting point within that basin will eventually take you to the minimum if you go "downhill".

In practice algorithms may behave better or worse. The Kohn-Sham scheme is used to solve the electronic structure problem for molecules or solids, and a typical starting point is to use the atomic charge densities of each atom as if these atoms were isolated from each other. With this choice, the self-consistent scheme will modify these initial atomic charge densities to take into account the fact that your atoms are sitting in a molecular or solid and interact with the other atoms around them.

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    $\begingroup$ how atomic charge densities are computed? Is it an easy calculation both mathematically and computationally, or it requires dedicated algorithms and formalism? Thank you for your time making such great answers on this website as well as your valuable videos on YouTube. @ProfM $\endgroup$
    – Sha
    Commented Sep 25, 2022 at 17:09

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