From dc409c933867747e200594c1a0eab7512060e2c4 Mon Sep 17 00:00:00 2001
From: hackbard <hackbard@hackdaworld.org>
Date: Mon, 18 Jun 2012 09:52:21 +0200
Subject: [PATCH] more pp

---
 physics_compact/solid.tex | 32 ++++++++++++++++++++++++++++++--
 1 file changed, 30 insertions(+), 2 deletions(-)

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index e19b7ce..e4c2362 100644
--- a/physics_compact/solid.tex
+++ b/physics_compact/solid.tex
@@ -135,14 +135,42 @@ The number of planewaves required for reasonably converged electronic structure
 
 \subsubsection{Pseudopotential method}
 
-\subsubsection{Norm conserving pseudopotentials}
+Following the idea of orthogonalized planewaves leads to the pseudopotential idea, which --- in describing only the valence electrons --- effectively removes an undesriable subspace from the investigated problem.
+
+Let $\ket{\Psi_\text{V}}$ be the wavefunction of a valence electron with the Schr\"odinger equation 
+\begin{equation}
+H \ket{\Psi_\text{V}} = \left(\frac{1}{2m}p^2+V\right)\ket{\Psi_\text{V}}=
+E\ket{\Psi_\text{V}} \text{ .}
+\end{equation}
+\ldots projection operatore $P_\text{C}$ \ldots
+
+\subsubsection{Semilocal form of the pseudopotential}
+
+Ionic potentials, which are spherically symmteric, suggest to treat each angular momentum $l,m$ separately leading to $l$-dependent non-local (NL) model potentials $V_l(r)$ and a total potential
+\begin{equation}
+V=\sum_{l,m}\ket{lm}V_l(r)\bra{lm} \text{ .}
+\end{equation}
+In fact, applied to a function, the potential turns out to be non-local in the angular coordinates but local in the radial variable, which suggests to call it asemilocal (SL) potential.
 
+Problem of semilocal potantials become valid once matrix elements need to be computed.
+Integral with respect to the radial component needs to be evaluated for each planewave combination, i.e.\  $N(N-1)/2$ integrals.
 \begin{equation}
-V=\ket{lm}V_l(r)\bra{lm}
+\bra{k+G}V\ket{k+G'} = \ldots
 \end{equation}
 
+A local potential can always be separated from the potential \ldots
+\begin{equation}
+V=\ldots=V_{\text{local}}(r)+\ldots
+\end{equation}
+
+\subsubsection{Norm conserving pseudopotentials}
+
+HSC potential \ldots
+
 \subsubsection{Fully separable form of the pseudopotential}
 
+
+
 \subsection{Spin orbit interaction}
 
 
-- 
2.39.2