formulas ...

[lectures/latex.git] / posic / talks / mpi_app.tex

diff --git a/posic/talks/mpi_app.tex b/posic/talks/mpi_app.tex
index 167579f..67a4d40 100644 (file)
--- a/posic/talks/mpi_app.tex
+++ b/posic/talks/mpi_app.tex
@@ -1001,61 +1001,81 @@ r = \unit[2--4]{nm}
  Utilized computational methods
 }
 
- \vspace{0.1cm}
+\vspace{0.2cm}
 
- \small
+\small
 
-{\bf Molecular dynamics (MD):}\\
+{\bf Molecular dynamics (MD)}\\
 \scriptsize
-\begin{tabular}{l r}
-\hline
+\begin{tabular}{p{4.5cm} p{7.5cm}}
 Basics & Details\\
 \hline
-Microscopic description of N particle system & \\
-Analytical interaction potential & Tersoff-like bond order potential (Erhart/Albe) \\
-Numerical integration using Newtons equation of motion as a propagation rule in 6N-dimensional phase space & Velocity Verlet | timestep: \unit[1]{fs} \\
-Observables obtained by time and/or ensemble averages & NpT (isothermal-isobaric)\\
-%\begin{itemize}
-%\item Berendsen thermostat:
-%      $\tau_{\text{T}}=100\text{ fs}$
-%\item Berendsen barostat:\\
-%      $\tau_{\text{P}}=100\text{ fs}$,
-%      $\beta^{-1}=100\text{ GPa}$
-%\end{itemize}\\
+System of $N$ particles &
+$N=5832\pm 1$ (Defects), $N=238328+6000$ (Precipitation)\\
+\hline
+Phase space propagation &
+Velocity Verlet | timestep: \unit[1]{fs} \\
+\hline
+Analytical interaction potential &
+Tersoff-like {\color{red}short-range}, {\color{blue}bond order} potential
+(Erhart/Albe)
+$\displaystyle
+E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
+    \pot_{ij} = {\color{red}f_C(r_{ij})}
+    \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
+$\\
+\hline
+Observables: time/ensemble averages &
+NpT (isothermal-isobaric) | Berendsen thermostat/barostat\\
 \hline
 \end{tabular}
 
- \begin{itemize}
-  \item Microscopic description of N particle system
-  \item Analytical interaction potential
-  \item Numerical integration using Newtons equation of motion\\
-        as a propagation rule in 6N-dimensional phase space
-  \item Observables obtained by time and/or ensemble averages
- \end{itemize}
- {\bf Details of the simulation:}
- \begin{itemize}
-  \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
-  \item Ensemble: NpT (isothermal-isobaric)
-        \begin{itemize}
-         \item Berendsen thermostat:
-               $\tau_{\text{T}}=100\text{ fs}$
-         \item Berendsen barostat:\\
-               $\tau_{\text{P}}=100\text{ fs}$,
-               $\beta^{-1}=100\text{ GPa}$
-        \end{itemize}
-  \item Erhart/Albe potential: Tersoff-like bond order potential
-  \vspace*{12pt}
-        \[
-        E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
-        \pot_{ij} = {\color{red}f_C(r_{ij})}
-        \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
-        \]
- \end{itemize}
+\small
+
+\vspace{0.1cm}
+
+{\bf Density functional theory (DFT)}
+
+\scriptsize
+
+\begin{minipage}[t]{6cm}
+\underline{Basics}
+\begin{itemize}
+ \item Born-Oppenheimer approximation:\\
+       Decouple electronic \& ionic motion
+ \item Hohenberg-Kohn theorem:\\
+       $n_0(r) \stackrel{\text{uniquely}}{\rightarrow}$
+       $V_0$ / $H$ / $\Phi_i$ / \underline{$E_0$}
+\end{itemize}
+\underline{Details}
+\begin{itemize}
+\item Code: \textsc{vasp}
+\item Plane wave basis set $\{\phi_j\}$\\[0.1cm]
+$\displaystyle
+\Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_j^i \phi_j(r)
+$\\
+$\displaystyle
+E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}=\unit[300]{eV}
+$
+\item Ultrasoft pseudopotential
+\item Brillouin zone sampling: $\Gamma$-point
+\end{itemize}
+\end{minipage}
+\begin{minipage}[t]{6cm}
+
+\[
+\left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) - \epsilon_i \right] \Phi_i(r) = 0
+\]
+\[
+n(r)=\sum_i^N|\Phi_i(r)|^2
+\]
+\[
+V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
+                 +V_{\text{XC}}[n(r)]
+\]
+
+\end{minipage}
 
- \begin{picture}(0,0)(-230,-30)
-  \includegraphics[width=5cm]{tersoff_angle.eps} 
- \end{picture}
- 
 \end{slide}
 
 \end{document}
@@ -1063,12 +1083,11 @@ Observables obtained by time and/or ensemble averages & NpT (isothermal-isobaric
 
 \begin{slide}
 
+ \small
  {\large\bf
   Density functional theory (DFT) calculations
  }
 
- \small
-
  Basic ingredients necessary for DFT
 
  \begin{itemize}
@@ -1131,11 +1150,6 @@ which in turn depends on $n(r)$
         \end{itemize}
   \item \underline{Plane wave basis set}
         - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
-\[
-\rightarrow
-\text{Fourier series: } \Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_j^i \phi_j(r), \quad E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}
-\qquad ({\color{blue}300\text{ eV}})
-\]
   \item \underline{Brillouin zone sampling} -
         {\color{blue}$\Gamma$-point only} calculations
   \item \underline{Pseudo potential}