Quantal Density Functional Theory (QDFT): Further Understandings

We consider electrons in the following external fields: (a) {\boldmath $\cal{E}$} $({\bf{r}}t)=-$ {\boldmath $\nabla$ } $v ({\bf{r}}t)$, ${\bf{B}}({\bf{r}}t)=\nabla\times{\bf{A}}({\bf{r}}t)$, and ${\bf{E}}({\bf{r}}t)=-\nabla\phi({\bf{r}}t)-(1/c) \partial {\bf{A}}({\bf{r}}t)/\partial t$, (b) {\boldmath $\cal{E}$ } $({\bf{r}}t)=-$ {\boldmath $\nabla$} $v({\bf{r}} t)$, (c) {\boldmath $\cal{E}$ } $({\bf{r}})=-$ {\boldmath $\nabla$ } $v ({\bf{r}})$ and ${\bf{B}}({\bf{r}}t)=\nabla\times{\bf{A}}({\bf{r}} t)$, and (d) {\boldmath $\cal{E}$ } $({\bf{r}})=-$ {\boldmath $\nabla$ } $v({\bf{r}})$. The basic variables for these systems are for (a) the density $\rho({\bf{r}}t)$ and physical current density ${\bf{j}}({\bf{r}} t)$, (b) $\rho({\bf{r}}t)$ and (paramagnetic) ${\bf{j}}({\bf{r}}t)$, (c) $\rho({\bf{r}})$ and ${\bf{j}}({\bf{r}})$, (d) $\rho ({\bf{r}})$. In QDFT, the local potential of the model fermions is the work done in a conservative effective field. In each of the above cases the effective field is representative of the \textit{same} correlations, \textit{viz}. due to the Pauli exclusion principle, Coulomb repulsion and Correlation-Kinetic effects.