X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=physics_compact%2Fsolid.tex;h=01f545c984d26622bff40ebf8c614ee38efd8b18;hp=e4c23629af81992e5ca3603b66badbeb7f24daa6;hb=8d1d7674ef08669827ef927728256566d173a1b3;hpb=dc409c933867747e200594c1a0eab7512060e2c4

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index e4c2362..01f545c 100644
--- a/physics_compact/solid.tex
+++ b/physics_compact/solid.tex
@@ -148,7 +148,7 @@ E\ket{\Psi_\text{V}} \text{ .}
 
 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{ .}
+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.
 
@@ -160,7 +160,7 @@ Integral with respect to the radial component needs to be evaluated for each pla
 
 A local potential can always be separated from the potential \ldots
 \begin{equation}
-V=\ldots=V_{\text{local}}(r)+\ldots
+V=\ldots=V_{\text{local}}(\vec{r})+\ldots
 \end{equation}
 
 \subsubsection{Norm conserving pseudopotentials}
@@ -169,12 +169,189 @@ 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.
 
-\subsection{Spin orbit interaction}
 
+\subsubsection{Excursus: real space representation within an iterative treatment}
 
-\subsubsection{Perturbative 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 read
+\begin{equation}
+\sum_{lm}
+\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}'}r\ket{\chi_{lm}} & = & \vec{r}'\braket{\vec{r}'}{\chi_{lm}}\\
+\bra{\vec{r}'}p\ket{\chi_{lm}} & = & -i\hbar\nabla_{\vec{r}'}
+\braket{\vec{r}'}{\chi_{lm}}
+\end{eqnarray}
+we get
+\begin{equation}
+-i\hbar\sum_{lm}(\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}
+\ket{\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_{\vec{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_{\vec{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_{\vec{r}'})
+Y_{lm}(\Omega_{\vec{r}'})
+r'^2 dr' d\Omega_{\vec{r}'} \\
+&=&\int_{r'}
+{\delta V_l^{\text{SO}}}^2(r') u_l^2(r') dr'
+\int_{\Omega_{\vec{r}'}}Y^*_{lm}(\Omega_{\vec{r}'})Y_{lm}(\Omega_{\vec{r}'}) d\Omega_{\vec{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 sum of the products of \eqref{eq:solid:so_r1} and \eqref{eq:solid:so_r2} must be further evaluated.
+\begin{eqnarray}
+\sum_{lm}
+\braket{\vec{r}'}{\delta V_l^{\text{SO}}\Phi_{lm}}
+\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r}''}&=&\sum_{lm}
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{\vec{r}'})
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{\vec{r}''})\nonumber\\
+&=&\sum_l
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}\sum_m
+Y^*_{lm}(\Omega_{\vec{r}''})Y_{lm}(\Omega_{\vec{r}'})\nonumber\\
+&=&\sum_l
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}
+P_l\left(\frac{\vec{r}'\vec{r}''}{r'r''}\right)\frac{2l+1}{4\pi}\nonumber\\
+\end{eqnarray}
+due to 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_{\vec{r}''})Y_{lm}(\Omega_{\vec{r}'})
+\text{ .}
+\end{equation}
+In total, the matrix elements of the SO potential can be calculated by
+\begin{eqnarray}
+-i\hbar\sum_{lm}(\vec{r}'\times \nabla_{\vec{r}'})\braket{\vec{r}'}{\chi_{lm}}
+E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\vec{r}''}=\\
+=-i\hbar\sum_l(\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)\cdot\nonumber
+\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} \cdot
+\frac{2l+1}{4\pi}\nonumber\\
+=
+-i\hbar\sum_l
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
+P'_l\left(\frac{\vec{r}'\vec{r}''}{r'r''}\right)\cdot
+\left(\frac{\vec{r}'\times\vec{r}''}{r'r''}\right)\cdot
+\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}\text{ ,}
+\label{eq:solid:so_fin}
+\end{eqnarray}
+where the derivatives of functions that depend on the absolute value of $\vec{r}'$ do not contribute due to the cross product as can be seen from equations \eqref{eq:solid:rxp1} and \eqref{eq:solid:rxp2}.
+\begin{eqnarray}
+\left(\vec{r}\times\nabla_{\vec{r}}\right)f(r)&=&
+\left(\begin{array}{l}
+r_y\frac{\partial}{\partial r_z}f(r)-r_z\frac{\partial}{\partial r_y}f(r)\\
+r_z\frac{\partial}{\partial r_x}f(r)-r_x\frac{\partial}{\partial r_z}f(r)\\
+r_x\frac{\partial}{\partial r_y}f(r)-r_y\frac{\partial}{\partial r_x}f(r)
+\end{array}\right)
+\label{eq:solid:rxp1}
+\end{eqnarray}
+\begin{eqnarray}
+r_i\frac{\partial}{\partial r_j}f(r)-r_j\frac{\partial}{\partial r_i}f(r)&=&
+r_if'(r)\frac{\partial}{\partial r_j}(r_x^2+r_y^2+r_z^2)^{1/2}-
+r_jf'(r)\frac{\partial}{\partial r_i}(r_x^2+r_y^2+r_z^2)^{1/2}\nonumber\\
+&=&
+r_if'(r)\frac{1}{2r}2r_j-r_jf'(r)\frac{1}{2r}2r_i=0
+\label{eq:solid:rxp2}
+\end{eqnarray}
 
-\subsubsection{Non-perturbative method}
+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}