Calculation Of GGA potential and kernel

In DFT and TD-DFT in order to calcualte ground state properties and transition intesities we need exchange and correlation energy ($E_{xc}$), potential ($v_{xc}$) and kernel ($f_{xc}$). $E_{xc}$, $v_{xc}$ and $f_{xc}$ are functional of the real space density $n({\bf r})$. The easiest exchange-correlation approximation for the functional in the so-called Local Density Approximation (LDA) which depend only on the density. A more complex functional can be calculated in the so-called General Gradient Approximation (GGA) in which exchange-correlation energy depends upon the density and the gradient of the density $\nabla n({\bf r})$. In this section we will focus on the derivation of GGA exchange-correlation kernel.

Energy, potential and kernel are connected by means of functional derivatives

(1)
\begin{eqnarray} v_{xc}[n] &= \frac{\delta E_{xc}[n]}{\delta n} \\ f_{xc}[n] &= \frac{\delta v_{xc}[n]}{\delta n}, \end{eqnarray}

where the exchange and correlation energy is computed as

(2)
\begin{align} E[n] = \int \epsilon(n,\nabla n) \mathrm{d}{\bf r}, \end{align}

where $\epsilon$ is the energy per unit volume and dropping the modifier xc for the sake of brevity.

Potential

The functional derivative is defined by the relation

(3)
\begin{align} \delta E = E[n+\delta n]-E[n] = \int \frac{\delta E[n]}{\delta n({\bf r})} \delta n({\bf r}) \mathrm{d}{\bf r}. \end{align}

Therefore, adding a small variation of density and it's gradient independently, and using a Taylor expansion of the energy density around $n, \nabla n$ we will get

(4)
\begin{align} \delta E = \int \, \bigg[\frac{\partial \epsilon(n,\nabla n)}{\partial n({\bf r})} \delta n({\bf r}) + \frac{\partial \epsilon(n,\nabla n)}{\partial \nabla n({\bf r})} \delta \nabla n({\bf r})\bigg] \mathrm{d}{\bf r}. \end{align}

In the last equation we can "convert" $\delta \nabla n({\bf r})$ into $\delta n({\bf r})$ by doing an integration by parts

(5)
\begin{align} \delta E = \int \, \bigg[\frac{\partial \epsilon(n,\nabla n)}{\partial n({\bf r})} - \nabla \frac{\partial \epsilon(n,\nabla n)}{\partial \nabla n({\bf r})} \bigg] \delta n({\bf r}) \mathrm{d}{\bf r}. \end{align}

Thus, comparing the equations (4) and (3) we get

(6)
\begin{align} v({\bf r}) = \frac{\delta E[n]}{\delta n} = \frac{\partial \epsilon(n,\nabla n)}{\partial n({\bf r})} - \nabla \frac{\partial \epsilon(n,\nabla n)}{\partial \nabla n({\bf r})}. \end{align}

If we want to express the functional derivative in function of $(\nabla n) ^2=\sigma$,

(7)
\begin{align} \color{blue}{v_{xc}[n] = \frac{\partial \epsilon_{xc}(n,\nabla n)}{\partial n({\bf r})} - 2 \nabla \cdot \bigg[ \frac{\partial \epsilon_{xc}(n,\nabla n)}{\partial \sigma} \nabla n({\bf r}) \bigg]} \end{align}

Kernel

From the potential then we can calculate the kernel [1]. Let's start with the variation of potential

(8)
\begin{align} \ \delta v({\bf r}) = \frac{\partial^2 \epsilon(n,\nabla n)}{\partial n^2} \delta n + \frac{\partial^2 \epsilon(n,\nabla n)}{\partial n\partial \nabla_i n} \nabla_i \delta n - \nabla_i \bigg [ \delta n \frac{\partial^2 \epsilon(n,\nabla n)}{\partial n \partial \nabla_i n} + \delta \nabla_k n \,\frac{\partial^2 \epsilon(n,\nabla n)}{\partial \nabla_k n \partial \nabla_i n} \bigg]. \end{align}

To get rid of $\nabla \delta n$ lets rewrite the last equation in form of an integral with a $\delta$-function

(9)
\begin{align} \ \delta v({\bf r}) = \int \delta({\bf r} - {\bf r}') \frac{\partial^2 \epsilon(n',\nabla' n')}{\partial n'\partial n'} \delta n' d{\bf r}' + \\ \int \delta({\bf r} - {\bf r}') \frac{\partial^2 \epsilon(n',\nabla' n')}{\partial n'\partial \nabla'_i n'} \nabla'_i \delta n' d{\bf r}' - \int \delta({\bf r} - {\bf r}') \nabla'_i \bigg[ \delta n' \frac{\partial^2 \epsilon(n',\nabla' n')}{\partial n' \partial \nabla_i n'}\bigg] d{\bf r}' - \\ \int \delta({\bf r} - {\bf r}') \nabla'_i \bigg[ \delta \nabla'_k n' \,\frac{\partial^2 \epsilon(n',\nabla' n')}{\partial \nabla'_k n' \partial \nabla'_i n'}\bigg] d{\bf r}'. \end{align}

In this equation we have to do integration by parts several times. The second term in the last equation will read

(10)
\begin{align} \ \ \delta v_2({\bf r}) = - \int \delta n' \nabla'_i \bigg[ \delta({\bf r} - {\bf r}') \frac{\partial^2 \epsilon(n',\nabla' n')}{\partial n'\partial \nabla'_i n'} \bigg] d{\bf r}' = \\ - \int \delta n' \frac{\partial^2 \epsilon(n',\nabla' n')}{\partial n'\partial \nabla'_i n'} \nabla'_i \delta({\bf r} - {\bf r}') d{\bf r}' - \int \delta n' \delta({\bf r} - {\bf r}') \nabla'_i \frac{\partial^2 \epsilon(n',\nabla' n')}{\partial n'\partial \nabla'_i n'} d{\bf r}'. \end{align}

The third term in equation (9) reads

(11)
\begin{align} \ \ \delta v_3({\bf r}) = - \int \delta({\bf r} - {\bf r}') \nabla'_i \delta n' \frac{\partial^2 \epsilon(n',\nabla' n')}{\partial n' \partial \nabla_i n'} d{\bf r}' = + \int \delta n' \frac{\partial^2 \epsilon(n',\nabla' n')}{\partial n' \partial \nabla_i n'} [\nabla'_i \delta({\bf r} - {\bf r}')] d{\bf r}'. \end{align}

Adding up second and third terms we will have terms with $\nabla$ of delta function canceled

(12)
\begin{align} \ \ \delta v_2({\bf r}) + \delta v_3({\bf r}) = - \int \delta n' \delta({\bf r} - {\bf r}') \nabla'_i \frac{\partial^2 \epsilon(n',\nabla' n')}{\partial n'\partial \nabla'_i n'} d{\bf r}'. \end{align}

The fourth term in equation (9) needs two partial integration procedures

(13)
\begin{align} \delta v_4({\bf r}) = -\int \delta({\bf r} - {\bf r}') \nabla'_i \bigg[ \delta \nabla'_k n' \,\frac{\partial^2 \epsilon(n',\nabla' n')}{\partial \nabla'_k n' \partial \nabla'_i n'}\bigg] d{\bf r}' = \int \nabla'_i[\delta({\bf r} - {\bf r}')] \delta \nabla'_k n' \,\frac{\partial^2 \epsilon(n',\nabla' n')}{\partial \nabla'_k n' \partial \nabla'_i n'} d{\bf r}'=\\ -\int \delta n' \nabla'_k \bigg [ \nabla'_i[\delta({\bf r} - {\bf r}')] \frac{\partial^2 \epsilon(n',\nabla' n')}{\partial \nabla'_k n' \partial \nabla'_i n'}\bigg] d{\bf r}'. \end{align}

Let's now gather all terms and state the result

(14)
\begin{align} \delta v({\bf r}) = \int \delta n' \bigg \{ \delta({\bf r} - {\bf r}') \frac{\partial^2 \epsilon(n',\nabla' n')}{\partial n'\partial n'} - \delta({\bf r} - {\bf r}') \nabla'_i \frac{\partial^2 \epsilon(n',\nabla' n')}{\partial n\partial \nabla'_i n} - \\ \nabla'_k \bigg [ \nabla'_i[\delta({\bf r} - {\bf r}')] \frac{\partial^2 \epsilon(n',\nabla' n')}{\partial \nabla'_k n' \partial \nabla'_i n'}\bigg] \bigg \} d{\bf r}'. \end{align}

The result agrees, it's GGA part with the Nazarov-Vignale PRL [2].

(15)
\begin{align} \ \ f({\bf r}, {\bf r}') = \delta({\bf r} - {\bf r}') \bigg[ \frac{\partial^2 \epsilon(n,\nabla n)}{\partial n\partial n} - \nabla_i \frac{\partial^2 \epsilon(n,\nabla n)}{\partial n\partial \nabla_i n} \bigg] - \nabla_k \bigg [ \nabla_i[\delta({\bf r} - {\bf r}')] \frac{\partial^2 \epsilon(n,\nabla n)}{\partial \nabla_k n \partial \nabla_i n}\bigg]. \end{align}
Bibliography
1. V. Nazarov, private communication.
2. V. Nazarov, G. Vignale, Phys. Rev. Lett. 107 216402 (2011).
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