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3.6.1 Elecron-Electron Interaction

The self-consistent HARTREE self-energy due to electron-electron interaction is given by [205]

\begin{displaymath}\begin{array}{ll} \Sigma_\mathrm{el-el}({\bf r_1},t_1) \ &\... ...e = \ -\ensuremath {\mathrm{q}}\phi({\bf r_1}) \ . \end{array}\end{displaymath} (3.46)

where $ \varrho({\bf r},t)/(-\ensuremath {\mathrm{q}}) =n({\bf r},t)=-i\hbar G({\bf r},t,{\bf r},t)$ (see Section 3.9.1). The potential $ \phi$ resulting from the HARTREE self-energy is in fact the solution of the POISSON equation with the charge density $ \varrho$. The HARTREE self-energy is instantaneous.

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M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors