Documentation
Basic formulation of ESM:
Expression for Green's function:
We use the Green's function to solve the following Poisson equations:
\begin{align}
\nabla\cdot\left[\epsilon(\boldsymbol{r})\nabla\right]V(\boldsymbol{r})&=-4\pi\rho(\boldsymbol{r}),\\
\nabla\cdot\left[\epsilon(\boldsymbol{r})\nabla\right]G(\boldsymbol{r},\boldsymbol{r}')&=-4\pi\delta(\boldsymbol{r}-\boldsymbol{r}').
\end{align}
Adopting the Laue expression, the above equations become:
\begin{align}
\left\{\partial_z\left[\epsilon(z)\partial_z\epsilon(z)-g_{\|}^2\right]\right\}V(\boldsymbol{g}_{\|},z)&=-4\pi\rho(z), \\
\left\{\partial_z\left[\epsilon(z)\partial_z\epsilon(z)-g_{\|}^2\right]\right\}G(\boldsymbol{g}_{\|},z,z')&=-4\pi\delta(z-z').
\end{align}
Hartree term:
The classical electrostatic interaction energy (Hartree energy) in terms of density is given by
\begin{align}
E_\mathrm{H}=\frac{1}{2}\iint\mathrm{d}\boldsymbol{r}\mathrm{d}\boldsymbol{r}'\rho_\mathrm{e}(\boldsymbol{r})G(\boldsymbol{r},\boldsymbol{r}
')\rho_\mathrm{e}(\boldsymbol{r}')
\end{align}
Local potential term:
The local potential in real space and reciprocal space are defined as follows:
\begin{align}
\label{v_loc}
V_\mathrm{loc}(\boldsymbol{r})&=\sum_{m\nu}v_{\mathrm{loc},\nu}(\boldsymbol{r}-\boldsymbol{R}_{m\nu}),\\
\label{vloc_xtalg}
V_\mathrm{loc}(\boldsymbol{G})&=\frac{1}{\Omega_0}\int_{\Omega_0}\mathrm{d}\boldsymbol{r} V_\mathrm{loc}(\boldsymbol{r})\mathrm{e}^{-\mathrm{i}\boldsymbol{G}\cdot\boldsymbol{r}} \notag\\
&=\frac{1}{\Omega_0}\sum_\nu\mathrm{e}^{-\mathrm{i}\boldsymbol{G}\cdot\boldsymbol{\tau}_\nu}v_{\mathrm{loc},\nu}(\boldsymbol{G}),
\end{align}
Ewald term:
The ion-ion interaction energy:
\begin{align}
\label{ewald-energy}
E_\mathrm{ion}=&\frac{1}{2}\sum_{\ell\mu}\sum_{m\nu}Z_\mu Z_\nu\iint\mathrm{d}\boldsymbol{r}\mathrm{d}\boldsymbol{r}'\delta(\boldsymbol{r}-\boldsymbol{R}_{\ell\mu})G^0(\boldsymbol{r}-\boldsymbol{r}')\delta(\boldsymbol{r}'-\boldsymbol{R}_{m\nu}) \notag \\
&-\frac{1}{2}\sum_{\ell}\sum_{\mu\rightarrow\nu}Z_\mu Z_\nu\iint\mathrm{d}\boldsymbol{r}\mathrm{d}\boldsymbol{r}'\delta(\boldsymbol{r}-\boldsymbol{R}_{\ell\mu})G^0(\boldsymbol{r}-\boldsymbol{r}')\delta(\boldsymbol{r}'-\boldsymbol{R}_{\ell\nu}).
\end{align}
is computed by Ewald method, where \(G^0\) is the bare Coulomb interaction eq.(\ref{g0_real}). The second term will delete the self interaction in the first term.
Modified source list:
Following files are modified or added in Quantum ESPRESSO package.
Added
- PW: esm.f90
Modified
- PW: init_run.f90