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**Unformatted text preview: **Steve Cronin University of Southern California Electrical Engineering - Electrophysics Semiconductor Physics EE.506 ab initio calculations The Hamiltonian for the entire electronic and ionic system without any assumptions: After the Born-Oppenheimer Approximation: The el-el interaction term is a double sum over 10 23 particles. Hohenberg, Kohn and Sham: The total energy of the 10 23 electronic system is a function of the electron density where Density functional theory (DFT): we’re actually taking into account the many-body effects of el-el interaction. (A functional is a function that takes functions as its argument.) Kohn-Sham Eigen-Equation: Local Density Approximation (LDA): simplifies the exchange correlation term The exchange interaction arises because the exchange of two identical particles is mathematically equivalent. The exchange interaction is a quantum mechanical effect which increases or decreases the energy of two or more electrons when their wave functions overlap. The Kohn-Sham equation, reduces the many-body interacting problem to a one- electron problem We weigh the exchange-correlation energy with the local electron density. Kohn-Sham Eigen-Equation: The Kohn-Sham equation (1965), reduced the many-body interacting problem to a one-electron problem, that contains a functional, that we can solve using an iterative approach: Pseudopotential Approximation We express the wavefunction as a sum of plane waves up to some cutoff energy, E cutoff The problem is that the coulomb potential diverges negatively near the ionic core. Approx: Below some critical radius, r c , we smooth out V(r) => V pseudo (r) These wiggles require high E cutoff Diverges to minus infinity! All this has done is cut out the high harmonics of the core electrons, which don’t contribute anyway. The energy of the valence electrons are calculated accurately. In order to solve equation this we need to diagonalize a 10 5 x 10 5 matrix....

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