X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=physics_compact%2Fsolid.tex;h=c4cf86947ad49fa982159d62d439a116024bc4ee;hp=e19b7cef4823afd4f95daf2d2fdc34b7f7df73a3;hb=fa167b99a2c520549296e61b92d56bbdd44d3849;hpb=eac23ae428984d20c62851681234b206ec1e3dc7

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index e19b7ce..c4cf869 100644
--- a/physics_compact/solid.tex
+++ b/physics_compact/solid.tex
@@ -135,18 +135,68 @@ 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}
-V=\ket{lm}V_l(r)\bra{lm}
+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{Fully separable form of the pseudopotential}
+\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}
+\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}
 
-\subsection{Spin orbit interaction}
+\subsubsection{Norm conserving pseudopotentials}
+
+HSC potential \ldots
 
+\subsubsection{Fully separable form of the pseudopotential}
 
-\subsubsection{Perturbative treatment}
+KB transformation \ldots
 
-\subsubsection{Non-perturbative method}
+\subsection{Spin-orbit interaction}
 
+Relativistic effects can be incorporated in the normconserving pseudopotential method up to but not including order $\alpha^2$ with $\alpha$ being the fine structure constant.
+This is advantageous since \ldots
+With the solutions of the all-electron Dirac equations, the new pseudopotential reads
+\begin{equation}
+V(r)=\sum_{l,m}\left[
+\ket{l+\frac{1}{2},m+{\frac{1}{2}}}V_{l,l+\frac{1}{2}}(r)
+\bra{l+\frac{1}{2},m+{\frac{1}{2}}} +
+\ket{l-\frac{1}{2},m-{\frac{1}{2}}}V_{l,l-\frac{1}{2}}(r)
+\bra{l-\frac{1}{2},m-{\frac{1}{2}}}
+\right] \text{ .}
+\end{equation}
+By defining an averaged potential weighted by the different $j$ degeneracies of the $\ket{l\pm\frac{1}{2}}$ states
+\begin{equation}
+\bar{V}_l(r)=\frac{1}{2l+1}\left(
+l V_{l,l-\frac{1}{2}}(r)+(l+1)V_{l,l+\frac{1}{2}}(r)\right)
+\end{equation}
+and a potential describing the difference in the potential with respect to the spin
+\begin{equation}
+V^{\text{SO}}_l(r)=\frac{2}{2l+1}\left(
+V_{l,l+\frac{1}{2}}(r)-V_{l,l-\frac{1}{2}}(r)\right)
+\end{equation}
+the total potential can be expressed as
+\begin{equation}
+V(r)=\sum_l \ket{l}\left[\bar{V}_l(r)+V^{\text{SO}}_l(r)LS\right]\bra{l}
+\text{ ,}
+\end{equation}
+where the first term correpsonds to the mass velocity and Darwin relativistic corrections and the latter is associated with the spin-orbit (SO) coupling.