X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=physics_compact%2Fsolid.tex;h=e8c61e1daf2f6742304af5fd348820d2a42bebe3;hp=ed9bddbcc907b4f82addad8cd899305a2801193a;hb=62d768c49f54dc3c03f0e3df3d08c37af1815aeb;hpb=4069f849b1973e2eb9020fa2e190f93b56537c60

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index ed9bddb..e8c61e1 100644
--- a/physics_compact/solid.tex
+++ b/physics_compact/solid.tex
@@ -2,6 +2,10 @@
 
 \chapter{Atomic structure}
 
+\chapter{Reciprocal lattice}
+
+Example of primitive cell ...
+
 \chapter{Electronic structure}
 
 \section{Noninteracting electrons}
@@ -105,3 +109,239 @@ E_1 + E_2 < E_2 + E_1 +
 is revealed, which proofs the Hohenberg Kohn theorem.% \qed
 \end{proof}
 
+\section{Electron-ion interaction}
+
+\subsection{Pseudopotential theory}
+
+The basic idea of pseudopotential theory is to only describe the valence electrons, which are responsible for the chemical bonding as well as the electronic properties for the most part.
+
+\subsubsection{Orthogonalized planewave method}
+
+Due to the orthogonality of the core and valence wavefunctions, the latter exhibit strong oscillations within the core region of the atom.
+This requires a large amount of planewaves $\ket{k}$ to adequatley model the valence electrons.
+
+In a very general approach, the orthogonalized planewave (OPW) method introduces a new basis set
+\begin{equation}
+\ket{k}_{\text{OPW}} = \ket{k} - \sum_t \ket{t}\bra{t}k\rangle \text{ ,}
+\end{equation} 
+with $\ket{t}$ being the eigenstates of the core electrons.
+The new basis is orthogonal to the core states $\ket{t}$.
+\begin{equation}
+\braket{t}{k}_{\text{OPW}} =
+\braket{t}{k} - \sum_{t'} \braket{t}{t'}\braket{t'}{k} =
+\braket{t}{k} - \braket{t}{k}=0
+\end{equation}
+The number of planewaves required for reasonably converged electronic structure calculations is tremendously reduced by utilizing the OPW basis set.
+
+\subsubsection{Pseudopotential method}
+
+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(\vec{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}}(\vec{r})+\ldots
+\end{equation}
+
+\subsubsection{Norm conserving pseudopotentials}
+
+HSC potential \ldots
+
+\subsubsection{Fully separable form of the pseudopotential}
+
+KB transformation \ldots
+
+\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(\vec{r})=\sum_{l,m}\left[
+\ket{l+\frac{1}{2},m+{\frac{1}{2}}}V_{l,l+\frac{1}{2}}(\vec{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}}(\vec{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}}(\vec{r})+(l+1)V_{l,l+\frac{1}{2}}(\vec{r})\right)
+\end{equation}
+and a potential describing the difference in the potential with respect to the spin
+\begin{equation}
+V^{\text{SO}}_l(\vec{r})=\frac{2}{2l+1}\left(
+V_{l,l+\frac{1}{2}}(\vec{r})-V_{l,l-\frac{1}{2}}(\vec{r})\right)
+\end{equation}
+the total potential can be expressed as
+\begin{equation}
+V(\vec{r})=\sum_l
+\ket{l}\left[\bar{V}_l(\vec{r})+V^{\text{SO}}_l(\vec{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.
+
+
+\subsubsection{Excursus: real space representation within an iterative treatment}
+
+In the following, the spin-orbit part is evaluated in real space.
+Since spin is treated in another subspace, it can be treated separately.
+The matrix elements of the orbital angular momentum part of the potential in KB form for orbital angular momentum $l$ read
+\begin{equation}
+\bra{\vec{r'}}(r\times p)\ket{\chi_{lm}}E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\vec{r''}}
+\text{ .}
+\end{equation}
+With
+\begin{eqnarray}
+\bra{\vec{r'}}p\ket{\chi_{lm}} & = & -i\hbar\nabla_{\vec{r'}} \braket{\vec{r'}}{\chi_{lm}}
+=-i\hbar\nabla_{\vec{r'}}\,\chi_{lm}(\vec{r'}) \\
+r\ket{\vec{r'}} & = & r'\ket{\vec{r'}}
+\end{eqnarray}
+we get
+\begin{equation}
+-i\hbar(\vec{r'}\times \nabla_{\vec{r'}})\braket{\vec{r'}}{\chi_{lm}}
+E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\vec{r''}}
+\text{ .}
+\label{eq:solid:so_me}
+\end{equation}
+To further evaluate this expression, the KB projectors
+\begin{equation}
+\chi_{lm}=\frac{\ket{\delta V_l^{\text{SO}}\Phi_{lm}}}
+{\braket{\delta V_l^{\text{SO}}\Phi_{lm}}
+        {\Phi_{lm}\delta V_l^{\text{SO}}}^{1/2}}
+\end{equation}
+must be known in real space (with respect to $\vec{r'}$).
+\begin{equation}
+\braket{\vec{r'}}{\chi_{lm}}=
+\frac{\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}}{
+\braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\Phi_{lm}\delta V_l^{\text{SO}}}
+^{1/2}}
+\end{equation}
+and
+\begin{equation}
+\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}=
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{r'})
+\text{ .}
+\label{eq:solid:so_r1}
+\end{equation}
+In this expression, only the spherical harmonics are complex functions.
+Thus, the complex conjugate with respect to $\vec{r''}$ is given by
+\begin{equation}
+\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}=
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{r''})
+\text{ .}
+\label{eq:solid:so_r2}
+\end{equation}
+Using the orthonormality property 
+\begin{equation}
+\int Y^*_{l'm'}(\Omega_r)Y_{lm}(\Omega_r) d\Omega_r = \delta_{ll'}\delta_{mm'}
+\label{eq:solid:y_ortho}
+\end{equation}
+of the spherical harmonics, the squared norm of the $\chi_{lm}$ reduces to
+\begin{eqnarray}
+\braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\Phi_{lm}\delta V_l^{\text{SO}}} &=&
+\int \braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\vec{r'}}
+\braket{\vec{r'}}{\Phi_{lm}\delta V_l^{\text{SO}}} d\vec{r'}\\
+&=&\int 
+{\delta V_l^{\text{SO}}}^2(r')\frac{u_l^2(r')}{r'^2}Y^*_{lm}(\Omega_{r'})
+Y_{lm}(\Omega_{r'})
+r'^2 dr' d\Omega_{r'} \\
+&=&\int_{r'}
+{\delta V_l^{\text{SO}}}^2(r') u_l^2(r') dr'
+\int_{\Omega_{r'}}Y^*_{lm}(\Omega_{r'})Y_{lm}(\Omega_{r'}) d\Omega_{r'}\\
+&=&\int_{r'}{\delta V_l^{\text{SO}}}^2(r') u_l^2(r') dr' \text{ .}
+\end{eqnarray}
+To obtain a final expression for the matrix elements \eqref{eq:solid:so_me}, the product of \eqref{eq:solid:so_r1} and \eqref{eq:solid:so_r2} must be further evaluated.
+\begin{eqnarray}
+\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}
+\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}&=&
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{r'})
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{r''})\nonumber\\
+&=&
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}
+Y^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'})
+\end{eqnarray}
+and if all states with magnetic quantum numbers $m=-l,-l+1,\ldots,l-1,l$ that contribute to the potential for angular momentum $l$ are considered
+\begin{equation}
+\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}
+\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}=
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}
+\sum_mY^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'}) \text{ ,}
+\end{equation}
+which can be rewritten as
+\begin{equation}
+\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}
+\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}=
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}
+\frac{2l+1}{4\pi}P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)
+\end{equation}
+using the vector addition theorem
+\begin{equation}
+P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)=
+\frac{4\pi}{2l+1}\sum_mY^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'})
+\end{equation}
+In total, the matrix elements of the potential for angular momentum $l$ can be calculated as
+\begin{eqnarray}
+\bra{\vec{r'}}V^{\text{KB,SO}}\ket{\vec{r''}}&=&
+\bra{\vec{r'}}(r\times p)\ket{\chi_{lm}}E^{\text{SO,KB}}_l
+\braket{\chi_{lm}}{\vec{r''}}\\
+&=&
+-i\hbar(\vec{r'}\times \nabla_{\vec{r'}})
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
+P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)\times\nonumber\\
+&&\times\frac{E^{\text{SO,KB}}_l\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}}
+             {\int_{r}{\delta V_l^{\text{SO}}}^2(r) u_l^2(r) dr} \,
+\frac{2l+1}{4\pi}\\
+&=&
+-i\hbar
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
+P'_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)\times\nonumber\\
+&&\left(\frac{\vec{r'}\times\vec{r''}}{r'r''}\right)
+\frac{E^{\text{SO,KB}}_l\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}}
+        {\int_{r}{\delta V_l^{\text{SO}}}^2(r) u_l^2(r) dr} \,
+\frac{2l+1}{4\pi}
+\end{eqnarray}
+If the potential at position $\vec{r}$ is considered a sum of atomic potentials $v_{\alpha}(\vec{r}-\vec{\tau}_{\alpha n})$ (atom $n$ of species $\alpha$)
+\begin{equation}
+V(\vec{r})=\sum_{\alpha}\sum_n v_{\alpha}(\vec{r}-\vec{\tau}_{\alpha n})
+\end{equation}
+and the SO projectors are likewise centered on atoms, the SO potential contribution reads
+\begin{equation}
+\end{equation}
+The $E_l^{\text{SO,KB}}$ are given by
+\begin{equation}
+E_l^{\text{SO,KB}}=
+\frac{\braket{\delta V_lu_l}{u_l\delta V_l}}
+     {\bra{u_l}\delta V_l\ket{u_l}}=
+\frac{\int_{r}\delta V^2_l(r)u^2_l(r)}r^2dr
+     {\int_{r'}\int_{r''}\braket{u_l}{r'}\bra{r'}\delta V_l
+\ket{r''}\braket{r''}{u_l}}=
+\end{equation}
+Finally, to evaluate $V^{\text{SO}}_l\ket{\Psi}$, the integral \ldots
+\begin{equation}
+\end{equation}
+