I have a mean-field Hamiltonian for N electrons. The mean-field potential felt by electron $i$ at position ${\bf r}_i$ is given by

$V^{(i)}_{int}({\bf r}_i)=\sum_{j\ne i}|\psi_j({\bf r}_i)|^2$

I can understand why this is the case. However, I need to clear up my understanding of the term $\psi_j({\bf r}_i)$. Is it simply the wavefunction of the $j^{th}$ electron at position ${\bf r}_i$?


The key idea in the mean field approach is taking into account the contribution, to the potential(the cause; see below) at each point ${\bf r}$, of the complete electronic configuration. By electronic configuration, we mean the probability density distribution of each of the one electron states $|\psi_j({\bf r}^')|^2 $. So the cause of the $j^{th}$ electron(at ${\bf r}^'$) "on" `the' electron at ${\bf r}$ is given by

$u({\bf r}) = -e \frac{|\psi_j({\bf r}^')|^2}{|{\bf r} - {\bf r}^'|}$.

Hence the total cause would be integrating over all the possible positions ${\bf r}^'$ of the $j^{th}$ electron and summing over all the states $j$ including 'the' electron which feels the effect. This total cause is used then to deduce the wavefunction of `the' electron using Schr\:odinger equation (the effect).

This may be confusing as we are including the wavefunction of the interested electron (the effect) in calculating what could be the possible cause "on" it to have that effect. This is the crux of the mean field approach where effect feedbacks the cause until we reach at a 'self-consistent' description of the cause-effect process! (self-consistent description is the physics lexicon used for this process.)

Some caveats : While integrating over all possible ${\bf r}^'$ we encounter a singularity when ${\bf r}^' = {\bf r}$. This is dealt with what is called "self-interaction" term and correction. I dont have good understanding of this.


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