Exercise 2.5 We want to refine our calculation of the ground state of the He atom in Exercise 2.2 by including the Coulo

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Exercise 2 5 We Want To Refine Our Calculation Of The Ground State Of The He Atom In Exercise 2 2 By Including The Coulo 1 (97.63 KiB) Viewed 16 times

Exercise 2.5 We want to refine our calculation of the ground state of the He atom in Exercise 2.2 by including the Coulomb repulsion between the electrons within the mean-field approximation. We start by evaluating the Hartree potential of eqn 2.46. By using the density in eqn 2.45 as a first approximation to the exact electron density of the He atom. show that the Hartree potential is given by: ² yer) -2 nur) -ur V (r) (1 - (1 + 2r) exp(-4r). (2.51) 2 nina For this exercise it is useful to remember that the radial part of the Laplace operator in spherical coordinates is: 1 a a v2 r2 ar (2.52) ar and that in the limit r → the Hartree potential should reduce to the electrostatic potential of a point charge corresponding to two electrons. Determine the total potential, V. + Vh, felt by the electrons within the mean-field approximation. The total potential calculated in this way decays exponentially fast as one moves away from the He nucleus. This trend is actually incorrect, as very general considerations indicate that in the case of He the total potential should decay as -1/r at large distance (Umrigar and Gonze, 1994). The wrong trend stems from the fact that we are neglecting an important ingredient, the exchange interaction, which will be introduced in the next section.