From: hackbard Date: Mon, 18 Jun 2012 07:52:21 +0000 (+0200) Subject: more pp X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=dc409c933867747e200594c1a0eab7512060e2c4;p=lectures%2Flatex.git more pp --- 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}