NUMERICAL SOLUTION OF KOHN–SHAM EQUATION FOR ATOM

Zbigniew Romanowski 

Interdisciplinary Centre for Materials Modelling
Pawinskiego 5a, 02-106 Warsaw, Poland

(Received April 7, 2007; revised version received May 20, 2007)

Abstract:

The numerical algorithm solving Kohn–Sham (KS) equation for the free or confined atom on uniform mesh is presented. Using the spherical symmetry of system, the three-dimensional KS equation is reduced to the one-dimensional functional eigenproblem. The functional eigenproblem is transformed to the algebraic eigenproblem with symmetric, tridiagonal matrix where the smallest eigenvalues are searched by the bisection. The Poisson equation is solved using Numerov algorithm. The atom electron density is calculated by the self-consistent-field procedure. The final solution is obtained by the Richardson extrapolation. The RAtom program is presented, where all described algorithms are implemented. The RAtom is used to analyze N, Al, Ga and In atom applying Slater and VWN approximation for the correlation and exchange energy. For these atoms the eigenvalues, the components of total energy and the electron distribution are calculated. The obtained eigenvalues agree with the results reported for the shooting method on the logarithmic mesh within 10−5 Ha.


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