Developers Manual:Dielectric function

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This is based on G. F. Bertsch et al. Phys. Rev. B 62, 7998 (2000).

For the moment we will only consider the terms of the Lagrangian with A^\alpha\!.

Standard formulation


L=\int d^3r\sum_k\frac12\left(i\partial^\alpha\phi_k^*(r)-\frac1c A^\alpha\phi_k^*(r)\right)\left(-i\partial^\alpha\phi_k(r)-\frac1c A^\alpha\phi_k(r)\right)


+\int d^3r\,d^3r'\,\sum_k e^{i/cA^\alpha(r^\alpha-{r'}^\alpha)}\phi_k^*(r)V(r,r')\phi_k(r')

-\frac{\Omega}{8\pi c^2}\dot{A}^\alpha \dot{A}^\alpha

-i\int d^3r\,\sum_k\phi_k^*(r)\partial_t\phi_k(r)

Equation of motion for A^\alpha\!


\frac{d}{dt}\frac{\partial L}{\partial \dot{A}^\beta}=-\frac{\Omega}{4\pi c^2}\ddot{A}^\beta


\frac{\partial L}{\partial A^\beta}=\int d^3r\sum_k\frac12\left[
\left(-\frac1c\delta^{\alpha\beta}\phi_k^*(r)\right)\left(-i\partial^\alpha\phi_k(r)-\frac1c{}A^\alpha\phi_k(r)\right)
+
\left(i\partial^\alpha\phi_k^*(r)-\frac1c A^\alpha\phi_k^*(r)\right)\left(-\frac1c\delta^{\alpha\beta}\phi_k(r)\right)
\right]

+\int d^3r\,d^3r'\,\sum_k \frac{i}c\left(r^\beta-{r'}^\beta\right)e^{i/cA^\alpha(r^\alpha-{r'}^\alpha)}\phi_k^*(r)V(r,r')\phi_k(r')


\frac{\partial L}{\partial A^\beta}=\int d^3r\sum_k\frac12\left[
\frac{i}c\phi_k^*(r)\partial^\beta\phi_k(r)+\frac1{c^2}\phi_k^*(r)\phi_k(r)A^\beta
-\frac{i}c\phi_k(r)\partial^\beta\phi_k^*(r)+\frac1{c^2}A^\beta\phi_k^*(r)\phi_k(r)
\right]

+\frac{i}{c}\int d^3r\,d^3r'\,\sum_ke^{i/cA^\alpha(r^\alpha-{r'}^\alpha)}
 \phi_k^*(r)\left[r^\beta{}V(r,r')-V(r,r'){r'}^\beta\right]\phi_k(r')


\frac{\partial L}{\partial A^\beta}=\frac{i}c\int d^3r\sum_k\phi_k^*(r)\partial^\beta\phi_k(r)
+\frac1{c^2}A^\beta\int d^3r\sum_k\phi_k^*(r)\phi_k(r)

+\frac{i}{c}\int d^3r\,d^3r'\,\sum_ke^{i/cA^\alpha{r^\alpha}}
 \phi_k^*(r)\left[r^\beta{}V(r,r')-V(r,r'){r'}^\beta\right]e^{-i/cA^\alpha{r'}^\alpha}\phi_k(r')


\ddot{A}^\beta=-i\frac{4\pi{c}}\Omega\int d^3r\sum_k\phi_k^*(r)\partial^\beta\phi_k(r)
-\frac{4\pi{N}}{\Omega}A^\beta-i\frac{4\pi{c}}{\Omega}\int d^3r\,d^3r'\,\sum_ke^{i/cA^\alpha{r^\alpha}}
 \phi_k^*(r)\left[\hat{r}^\beta{},\hat{V}\right]e^{-i/cA^\alpha{r'}^\alpha}\phi_k(r')

Finally:


\ddot{A}^\beta=\frac{4\pi{c}}{i\Omega}\int d^3r\sum_k\phi_k^*(r)\left\{\partial^\beta\phi_k(r)
+e^{i/cA^\alpha{r^\alpha}}\int d^3r'\,\left[\hat{r}^\beta{},\hat{V}\right]e^{-i/cA^\alpha{r'}^\alpha}\phi_k(r')\right\}
-\frac{4\pi{N}}{\Omega}A^\beta

The coupling with the TDKS equation


\frac{\delta L}{\delta \phi^*_m(r')} =\int d^3r\sum_k\frac12\left(-i\delta(r-r')\delta_{km}\partial^\alpha-\frac1c A^\alpha\delta(r-r')\delta_{km}\right)\left(-i\partial^\alpha\phi_k(r)-\frac1c A^\alpha\phi_k(r)\right)

+\int d^3r\,d^3r''\,\sum_k e^{i/cA^\alpha(r^\alpha-{r''}^\alpha)}\delta(r-r')\delta_{km}V(r,r'')\phi_k(r'')


\frac{\delta L}{\delta \phi^*_k(r)} =\frac12\left(-i\partial^\alpha-\frac1c A^\alpha\right)\left(-i\partial^\alpha\phi_k(r)-\frac1c A^\alpha\phi_k(r)\right)

+\int\,d^3r'\,e^{i/cA^\alpha(r^\alpha-{r'}^\alpha)}V(r,r')\phi_k(r')


\frac{\delta L}{\delta \phi^*_k(r)}=-\frac12\partial^\alpha\partial^\alpha\phi_k(r)+\frac1{2c^2}A^\alpha A^\alpha \phi_k(r) + \frac{i}c{A}^\alpha\partial^\alpha\phi_k(r)

+e^{i/cA^\alpha r^\alpha}\int\,d^3r'\,V(r,r')e^{-i/cA^\alpha{r'}^\alpha}\phi_k(r')

Notation

  • Greek superscript indexes: spatial coordinates with implicit sum.
  • \Omega\!: cell volume
  • N\!: number of electrons