Density Functional Theory with Generalized Gradient Approxmations (GGA) - What Happens if the Electron Density Shows a "Kink" at the Ion Position?
My question may be stupid, so please correct me if you find anything which is obviously erroneous.
In the following I will place a question mark (?) besides points/steps I consider doubtful.
My observations:
1. In density functional theory (DFT) (at zero temperature), the energy of the system $E$ is broken up as $E = \int_{V} d v \left[ e_{K}(n(\vec{r})) + e_{H}(n(\vec{r})) + e_{XC}(n(\vec{r})) \right]$, where $V$ is the system size, $e$ the energy density per unit volume, $n(\vec{r})$ the electronic density, subscript "$K$" for "kinetic", "$H$" for "<PERSON>" and "$XC$" for "exchange-correlation".
2. In the generalized gradient approximation (gga), $e$ also depends on $\nabla n(\vec{r})$. Thus write $e(n(\vec{r}), \nabla n(\vec{r}))$.
Assumptions: 1. spherical symmetry (closed-shell)
By spherical symmetry $e_{XC}(n(\vec{r}), \nabla n(\vec{r})) = e(n(r), \nabla n(r))$. Of course in writing this we are restricting the variation in density $\delta n$ in the radial direction only. I'm doubtful about the validity of this step.
Anyway, proceed as usual. The exchange-correlation potential is defined as the functional derivative of $e_{XC}$ w.r.t. 649
$\delta n$. Now please point out if there are any mistakes in the following steps:
$$\begin{eqnarray} \delta \int dv e_{XC}(n(r), n^{\prime}(r)) &=& \int dv \left[ \frac{\partial e_{XC}}{\partial n(r)} \delta n(r) + \frac{\partial e_{XC}}{\partial n^{\prime}(r)} \delta n^{\prime}(r) \right] \end{eqnarray}$$,
where $n^{\prime}(r) = \partial_r n(r)$. The first term is done; consider the second term. $\delta n^{\prime} (r) = \partial_r \delta n(r)$. Thus we integrate the second term on the right hand side by parts, and obtain
$4 \pi \int_0^{r_0} dr \partial_r \left[ r^2 \frac{\partial e_{XC}}{\partial n^{\prime}} \delta n(r) \right] - 4 \pi \int_0^{r_0} dr \left[ \partial_r \left( r^2 \frac{e_{XC}}{n^{\prime}}\right) \delta n(r) \right]$. The first term is the boundary term; ignore it for the time being. The second term seems to give the correction to the exchange correlation potential due to the introduction of gradient-dependence. Note that the second term is equivalent to $\int_0^{r_0} dr 4 \pi r^2 \left[ - \frac{1}{r^2} \partial_r \left( r^2 \frac{\partial e_{XC}}{\partial n^{\prime}} \right)\right]$, the square-bracketed term seems to give the exchange-correlation potential (?).
Now $- \frac{1}{r^2} \partial_r \left( r^2 \frac{\partial e_{XC}}{\partial n^{\prime}} \right) = - \frac{2}{r} \frac{\partial e_{XC}}{\partial n^{\prime}} - \partial_r \frac{\partial e_{XC}}{\partial n^{\prime}}$. What I worry about is the first term, which seems to diverge at $r=0$ (but it shouldn't; or is it cancelled by a piece in the second term?)
A expression for $e_{XC}$ may be the becke88 functional (exchange only) (PRA, 38, 3098). 374