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Though DFT is developed for many electron systems with an aim to lower the computational cost, a hand-on step-by-step calculation of small systems will let learners to take a close look at the science behind the black box. This notebook implements a DFT-LDA code to calculate the ground state energy of helium atom. This implementation follows the formulations of DFT Helium in Chapter 5 of Thijssen’s book (1), where the main mathematical works and approximations were described. Different from a previous work on Helium calculation(2), which employed combination of basis Gaussian functions, this work will calculate the same system by solving one radial wavefunction. In this program, the numerical calculation incorporates local density approximation (LDA) to approximate the electron density.
The simulation will mainly contain three main calculations: (a) calculation of the radial Schrödinger equation, (b) implementing the electrostatic potential (Hartree), and (c) incorporating the electron exchange potential. The eigen value and wavefunction is calculated by integrating the wavefunction with Numerov's method with fixed boundary conditions. The first few steps cover the above key components in the self-consistent function (SCF) calculation starting from a guess of the initial value of the wave function. All steps are described in detail. The last part performs the iterative SCF calculation until energy convergence is achieved.
(1) Thijssen, J., 2007. Computational physics. Cambridge university press.
(2) Baseden, K.A. and Tye, J.W., 2014. Introduction to Density Functional Theory: Calculations by Hand on the Helium Atom. Journal of Chemical Education, 91(12), pp.2116-2123.
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