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Highly accurate numerical solution of Hartree-Fock equation with pseudospectral method for closed-shell atoms. (English) Zbl 1448.81472
Summary: The Hartree-Fock (HF) equation for atoms with closed (sub)shells is transformed with the pseudospectral (PS) method into a discrete eigenvalue equation for scaled orbitals on a finite radial grid. The Fock exchange operator and the Hartree potential are obtained from the respective Poisson equations also discretized using the PS representation. The numerical solution of the discrete HF equation for closed-(sub)shell atoms from He to No is robust, fast and gives extremely accurate results, with the accuracy superior to that of the previous HF calculations. A very moderate number of 33 to 71 radial grid points is sufficient to obtain total energies with 14 significant digits and occupied orbital energies with 12 to 14 digits in numerical calculations using the double precision (64-bit) of the floating-point format. The electron density at the nucleus is then determined with 13 significant digits and the Kato condition for the density and s orbitals is satisfied with the accuracy of 11 to 13 digits. The node structure of the exact HF orbitals is obtained and their asymptotic dependence, including the common exponential decay, is reproduced very accurately. The accuracy of the investigated quantities is further improved by performing the PS calculations in the quadruple precision (128-bit) floating-point arithmetic which provides the total energies with 25 significant digits while using only 80 to 130 grid points.
##### MSC:
 81V45 Atomic physics 34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
##### Software:
HF86; LAPACK; MKL; splib; Wolfram Functions Site
Full Text:
##### References:
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