Solving the Kohn-Sham equation

Question

Consider the Kohn-Sham equation \begin{align} \left( - \frac{\hbar^2}{2m} \nabla^2 + \nu_\mathrm{eff}(\mathbf{r}) \right) \varphi_j(\mathbf{r}) &= \varepsilon_j\varphi_j(\mathbf{r}) \end{align} The external potential will be considered the interaction potential of electrons and nuclei.

Then \begin{align} \nu_{eff}=-\sum_{j}^{K} \frac{e^2 \, Z_{j}}{|\mathbf{r_1}-\mathbf{R_j}|}+\int\frac{n(\mathbf{r_2})e^2}{r_{12}}\mathbf{dr_2} +\nu_{xc} \end{align}

This equation is solved by the method of a self-consistent field: $\varphi$ are first set, then $n$ is calculated from them, then the effective functional $\nu_{eff}$ is calculated.

However, I do not understand something.

At each step of calculating the effective potential, the term denoting the interaction of nuclei with electrons is constant? $r_{12}$ is the distance between electrons. In terms of this model, no changes in the coordinates of electrons occur at each step?

Answer 1

To start it is very important to be very careful when writing the relevant coordinates. In your Kohn-Sham equation you are writing the coordinate of the non-interacting electron that you are solving for as $\mathbf{r}$. To be consistent with this, your second equation should read:

$$ v_{\mathrm{eff}}(\mathbf{r})=v_{\mathrm{ext}}(\mathbf{r})+\int\frac{n(\mathbf{r}_2)e^2}{|\mathbf{r}-\mathbf{r}_2|}d\mathbf{r}_2+v_{\mathrm{xc}}(\mathbf{r}), $$

where I replaced your expression for the external potential with $v_{\mathrm{ext}}(\mathbf{r})$, as to answer your question it is enough to look at the Hartree potential (the second term). In the second term you have two electron coordinates, $\mathbf{r}$ and $\mathbf{r}_2$. The first is the electron coordinate at which you are calculating the effective potential $v_{\mathrm{eff}}(\mathbf{r})$, and to build the full potential you need to know its value for every $\mathbf{r}$. The second coordinate is the coordinate at which you calculate the electron charge density $n(\mathbf{r}_2)$ to then build the Hartree potential. As the integral is over all space, you consider all possible values of $\mathbf{r}_2$. Therefore, both coordinates $\mathbf{r}$ and $\mathbf{r}_2$ are changing: you are considering all possible values for both.