author Frank Zirkelbach Wed, 20 Jun 2012 14:08:50 +0000 (16:08 +0200) committer Frank Zirkelbach Wed, 20 Jun 2012 14:08:50 +0000 (16:08 +0200)

index e8c61e1..01f545c 100644 (file)
@@ -207,49 +207,51 @@ where the first term correpsonds to the mass velocity and Darwin relativistic co

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
+The matrix elements of the orbital angular momentum part of the potential in KB form read

-\bra{\vec{r'}}(r\times p)\ket{\chi_{lm}}E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\vec{r''}}
+\sum_{lm}
+\bra{\vec{r}'}(r\times p)\ket{\chi_{lm}}E^{\text{SO,KB}}_l
+\braket{\chi_{lm}}{\vec{r}''}
\text{ .}

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'}}
+\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

--i\hbar(\vec{r'}\times \nabla_{\vec{r'}})\braket{\vec{r'}}{\chi_{lm}}
-E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\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}''}
\text{ .}
\label{eq:solid:so_me}

To further evaluate this expression, the KB projectors

-\chi_{lm}=\frac{\ket{\delta V_l^{\text{SO}}\Phi_{lm}}}
+\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}}

-must be known in real space (with respect to $\vec{r'}$).
+must be known in real space (with respect to $\vec{r}'$).

-\braket{\vec{r'}}{\chi_{lm}}=
-\frac{\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}}{
+\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}}

and

-\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'})
+\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}

In this expression, only the spherical harmonics are complex functions.
-Thus, the complex conjugate with respect to $\vec{r''}$ is given by
+Thus, the complex conjugate with respect to $\vec{r}''$ is given by

-\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''})
+\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}

@@ -261,70 +263,78 @@ Using the orthonormality property
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 \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'} \\
+{\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_{r'}}Y^*_{lm}(\Omega_{r'})Y_{lm}(\Omega_{r'}) d\Omega_{r'}\\
+\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 product of \eqref{eq:solid:so_r1} and \eqref{eq:solid:so_r2} must be further evaluated.
+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}
-\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
-
-\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}
-\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}=
+\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_mY^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'}) \text{ ,}
-
-which can be rewritten as
-
-\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''}\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''}
-\frac{2l+1}{4\pi}P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)
-\end{equation}
+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

-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'})
+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{ .}

-In total, the matrix elements of the potential for angular momentum $l$ can be calculated as
+In total, the matrix elements of the SO potential can be calculated by
\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'}})
+-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)\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
+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)\times\nonumber\\
-&&\left(\frac{\vec{r'}\times\vec{r''}}{r'r''}\right)
+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}
+       {\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}
+
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 $\alpha$)

V(\vec{r})=\sum_{\alpha}\sum_n v_{\alpha}(\vec{r}-\vec{\tau}_{\alpha n})