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Overview of the Herman-Skillman Program

The Herman-Skillman code is one of the simplest Self-Consistent Field computational codes for determining one-electron wave functions and the associated potential for any free atom or ion. The code originates from 1960-62 out of an effort to do energy band calculations for a wide variety of crystals. The history you can read partly in the original code itself, but the program was written in the "old days" of computing and there are not many comments explaining what is happening. The original code along with a detailed description of the numerical methods used, examples, and tables of atomic structure is available in the book by the two authors: Herman, Frank and Skillman, Sherwood, Atomic Structure Calculations, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1963.

Herman and Skillman's program is an effort to solve the Hartree-Fock equations for a complicated many-electron system in an approximate manner. In general, the code solves Slater's simplified version of the Hartree-Fock equations. As per Hartree-Fock formalism, the many-electron wave function is written as a single derminantal wave function. This is standard for a free atom or ion with only closed shells where there is only one electronic configuration. For a free atom or ion with one or more open shells, using only a single determinantal term represents the first main approximation.

The other major approximation is the replacement of the Hartree-Fock exchange potential terms with an average exchange potential. While the exchange terms are different for each occupied orbital, they have the same general character which can be approximated by results taken from the theory of a free-electron gas (See input parameter, ALPHA). The Hartree-Fock-Slater equations thus derived can be shown to have the same essential physical content of the Hartree-Fock equations.

There are two further simplifications used by Herman and Skillman. At large values of r, the potential is defined to have the correct analytic asymptotic behavior (See input parameter, KUT). Additionally, the self-consistent calculations are kept non-relativistic.

Many of the approximations are made in the interest of computational efficiency; keeping the program simple and fast. The loss of accuracy compared to solution of the rigorous Hartree-Fock model is accepted since the results will still provide good working wave functions (potentials) and energies. (Herman and Skillman include a first-order perturbation theory program in their book for estimating the relativistic and magnetic spin-orbit interaction corrections to the energies.)


Running the code, in brief

Main Inputs: Output:

Compiled in the book

Abridged information useful for starting more fine detailed runs containing:

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Created: November 12, 1998 --- Last Updated: December 4, 2001
By Mark D. Pauli