X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=physics_compact%2Fsolid.tex;h=1fc7a17eea47e2ac231e10db76dfc7321a9e3c2c;hp=50df24d29d03741e8a4efb3933ae5b98316dcbba;hb=f47fcf0c7b7c7e2d5adc294d72a0c914289c584c;hpb=9fac15a186e73cf587fb7d9dfa1c737c778cd08a

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index 50df24d..1fc7a17 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}
@@ -18,11 +22,58 @@
 
 \subsubsection{Hohenberg-Kohn theorem}
 
-Considering a system with a nondegenerate ground state, there is obviously only one ground-state charge density $n_0(\vec{r})$ that correpsonds to a given potential $V(\vec{r})$.
+The Hamiltonian of a many-electron problem has the form
+\begin{equation}
+H=T+V+U\text{ ,}
+\end{equation}
+where
+\begin{eqnarray}
+T & = & \langle\Psi|\sum_{i=1}^N\frac{-\hbar^2}{2m}\nabla_i^2|\Psi\rangle\\
+  & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} d\vec{r}' \,
+        \langle \Psi | \vec{r} \rangle \langle \vec{r} |
+        \nabla_i^2
+        | \vec{r}' \rangle \langle \vec{r}' | \Psi \rangle\\
+  & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} d\vec{r}' \,
+        \langle \Psi | \vec{r} \rangle \nabla_{\vec{r}_i}
+        \langle \vec{r} | \vec{r}' \rangle
+        \nabla_{\vec{r}'_i} \langle \vec{r}' | \Psi \rangle\\
+  & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} d\vec{r}' \,
+        \nabla_{\vec{r}_i} \langle \Psi | \vec{r} \rangle
+        \delta_{\vec{r}\vec{r}'}
+        \nabla_{\vec{r}'_i} \langle \vec{r}' | \Psi \rangle\\
+  & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} \,
+        \nabla_{\vec{r}_i} \Psi^*(\vec{r}) \nabla_{\vec{r}_i} \Psi(\vec{r})
+        \text{ ,} \\
+V & = & \int V(\vec{r})\Psi^*(\vec{r})\Psi(\vec{r})d\vec{r} \text{ ,} \\
+U & = & \frac{1}{2}\int\frac{1}{\left|\vec{r}-\vec{r}'\right|}
+        \Psi^*(\vec{r})\Psi^*(\vec{r}')\Psi(\vec{r}')\Psi(\vec{r})
+        d\vec{r}d\vec{r}'
+\end{eqnarray}
+represent the kinetic energy, the energy due to the external potential and the energy due to the mutual Coulomb repulsion.
+
+\begin{remark}
+As can be seen from the above, two many-electron systems can only differ in the external potential and the number of electrons.
+The number of electrons is determined by the electron density.
+\begin{equation}
+N=\int n(\vec{r})d\vec{r}
+\end{equation}
+Now, if the external potential is additionally determined by the electron density, the density completely determines the many-body problem.
+\end{remark}
+
+Considering a system with a nondegenerate ground state, there is obviously only one ground-state charge density $n_0(\vec{r})$ that corresponds to a given potential $V(\vec{r})$.
+\begin{equation}
+n_0(\vec{r})=\int \Psi_0^*(\vec{r},\vec{r}_2,\vec{r}_3,\ldots,\vec{r}_N)
+                  \Psi_0(\vec{r},\vec{r}_2,\vec{r}_3,\ldots,\vec{r}_N)
+             d\vec{r}_2d\vec{r}_3\ldots d\vec{r}_N
+\end{equation}
 In 1964, Hohenberg and Kohn showed the opposite and far less obvious result \cite{hohenberg64}.
+
+\begin{theorem}[Hohenberg / Kohn]
 For a nondegenerate ground state, the ground-state charge density uniquely determines the external potential in which the electrons reside.
-The proof presented by Hohenberg and Kohn proceeds by {\em reductio ad absurdum}.
+\end{theorem}
 
+\begin{proof}
+The proof presented by Hohenberg and Kohn proceeds by {\em reductio ad absurdum}.
 Suppose two potentials $V_1$ and $V_2$ exist, which yield the same electron density $n(\vec{r})$.
 The corresponding Hamiltonians are denoted $H_1$ and $H_2$ with the respective ground-state wavefunctions $\Psi_1$ and $\Psi_2$ and eigenvalues $E_1$ and $E_2$.
 Then, due to the variational principle (see \ref{sec:var_meth}), one can write
@@ -55,5 +106,169 @@ E_1 + E_2 < E_2 + E_1 +
 \int n(\vec{r}) \left( V_2(\vec{r})-V_1(\vec{r}) \right) d\vec{r}
 }_{=0}
 \end{equation}
-is revealed.
+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(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}
 
+\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(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.
+
+
+\subsubsection{Excursus: real space representation suitable for 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{r'}(r\times p)\ket{\chi_{lm}}E^{\text{SO,KB}}_l\braket{\chi_{lm}}{r''}
+\text{ .}
+\end{equation}
+With
+\begin{eqnarray}
+\bra{r'}p\ket{\chi_{lm}} & = & -i\hbar\nabla_{r'} \braket{r'}{\chi_{lm}}
+=-i\hbar\nabla_{r'}\,\chi_{lm}(r') \\
+r\ket{r'} & = & r'\ket{r'}
+\end{eqnarray}
+we get
+\begin{equation}
+-i\hbar(r'\times \nabla_{r'})\braket{r'}{\chi_{lm}}E^{\text{SO,KB}}_l
+\braket{\chi_{lm}}{r''}
+\text{ .}
+\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 $r$).
+\begin{equation}
+\braket{r'}{\chi_{lm}}=
+\frac{\braket{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{r'}{\delta V_l^{\text{SO}}\Phi_{lm}}=
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{r'})
+\text{ .}
+\end{equation}
+In this expression, only the spherical harmonics are complex functions.
+Thus, the complex conjugate with respect to $r''$ is given by
+\begin{equation}
+\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{r''}=
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{r''})
+\text{ .}
+\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 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}
+
+Finally, to evaluate $V^{\text{SO}}_l\ket{\Psi}$, the integral \ldots
+\begin{equation}
+\end{equation}