after ice SO stuff

diff --git a/physics_compact/math.tex b/physics_compact/math.tex
index b649597..bd2a888 100644 (file)
--- a/physics_compact/math.tex
+++ b/physics_compact/math.tex
@@ -21,7 +21,7 @@ which satisfies the properties of an inner product (see \ref{math_app:product})
 \begin{equation}
 ||\vec{a}||=\sqrt{(\vec{a},\vec{a})}
 \end{equation}
-that just corresponds to the length of vector \vec{a}.
+that just corresponds to the length of vector $\vec{a}$.
 Evaluating the scalar product $(\vec{a},\vec{b})$ by the sum representation of \eqref{eq:vec_sum} leads to
 \begin{equation}
 (\vec{a},\vec{b})=(\sum_i\vec{e}_ia_i,\sum_j\vec{e}_jb_j)=

diff --git a/physics_compact/phys_comp.tex b/physics_compact/phys_comp.tex
index b4d5e66..5974619 100644 (file)
--- a/physics_compact/phys_comp.tex
+++ b/physics_compact/phys_comp.tex
@@ -9,6 +9,7 @@
 \usepackage[T1]{fontenc}
 \usepackage{amsmath}
 \usepackage{amssymb}
+\usepackage{bm}
 \usepackage{ae}
 \usepackage{aecompl}
 \usepackage[dvips]{graphicx}
@@ -108,7 +109,7 @@
 \newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}
 
 % vectors are simply represented by bold font characters
-\renewcommand{\vec}[1]{{\bf #1{}}}
+\renewcommand{\vec}[1]{{\bm #1{}}}
 
 %
 % theorem environment

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index 01f545c..ee0cc2a 100644 (file)
--- a/physics_compact/solid.tex
+++ b/physics_compact/solid.tex
@@ -203,7 +203,7 @@ V(\vec{r})=\sum_l
 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}
+\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.
@@ -272,7 +272,8 @@ 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{ .}
+&=&\int_{r'}{\delta V_l^{\text{SO}}}^2(r') u_l^2(r') dr' \text{ .}\\
+&=&\braket{\delta V_l^{\text{SO}}u_l}{u_l\delta V_l^{\text{SO}}}
 \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}
@@ -298,25 +299,25 @@ P_l\left(\frac{\vec{r}'\vec{r}''}{r'r''}\right)=
 \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}'})
+&&-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}''}=\nonumber\\
+&=&-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
+P_l\left(\frac{\vec{r}'\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} \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{ ,}
+\frac{2l+1}{4\pi}\text{ ,}\nonumber\\
 \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}.
+since derivatives of functions that depend on the absolute value of $\vec{r}'$ do not contribute due to the cross product as is illustrated below (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}
@@ -335,13 +336,28 @@ r_if'(r)\frac{1}{2r}2r_j-r_jf'(r)\frac{1}{2r}2r_i=0
 \label{eq:solid:rxp2}
 \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
+If these projectors are considered to be centered around atom positions $\vec{\tau}_{\alpha n}$ of atoms $n$ of species $\alpha$, the variable $\vec{r}'$ in the previous equations is changed to $\vec{r}'_{\alpha n}=\vec{r}'-\vec{\tau}_{\alpha n}$, which implies
+\begin{eqnarray}
+r'&\rightarrow&r_{\alpha n}=|\vec{r}'-\vec{\tau}_{\alpha n}|\\
+\Omega_{\vec{r}'}&\rightarrow&\Omega_{\vec{r'}-\vec{\tau}_{\alpha n}}\\
+\delta V_l(r')&\rightarrow&\delta V_l(|\vec{r}'-\vec{\tau}_{\alpha n}|)\\
+u_l(r')&\rightarrow&u_l(|\vec{r}'-\vec{\tau}_{\alpha n}|)\\
+Y_{lm}(\Omega_{\vec{r}'})&\rightarrow&
+Y_{lm}(\Omega_{\vec{r}'-\vec{\tau}_{\alpha n}})
+\text{ .}
+\end{eqnarray}
+Within an iterative treatment on a real space grid consisting of $n_{\text{g}}$ grid points, the sum
 \begin{equation}
+\sum_{\vec{r}''_{\alpha n}}
+\sum_{lm}-i\hbar(\vec{r}'_{\alpha n}\times \nabla_{\vec{r}'_{\alpha n}})
+\braket{\vec{r}'_{\alpha n}}{\chi^{\text{SO}}_{lm}}
+E^{\text{SO,KB}}_l\braket{\chi^{\text{SO}}_{lm}}{\vec{r}''_{\alpha n}}
+\braket{\vec{r}''_{\alpha n}}{\Psi}
+\qquad\forall\,\bra{\vec{r}'_{\alpha n}}
 \end{equation}
+to obtain all elements $\bra{\vec{r}'_{\alpha n}}$, involves $n_{\text{g}}^2$ evaluations of equation~\eqref{eq:solid:so_fin} for eeach atom, if the projectors are short-ranged, i.e.\  $\delta V_l=0$ outside a certain cut-off radius.
+Thus, this method scales linearly with the number of atoms.
+
 The $E_l^{\text{SO,KB}}$ are given by
 \begin{equation}
 E_l^{\text{SO,KB}}=
@@ -351,7 +367,4 @@ E_l^{\text{SO,KB}}=
      {\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}