Approximations to HF Theory for Calculation

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The Herman-Skillman program is not a true Hartree-Fock method. Although one of the best self-consistent theories for single-electron wave functions describing the motion of electrons in the field of atomic nuclei, implementing a Hartree-Fock algorithm for many-electron systems with large numbers of electrons, such as heavy atoms, is very time consuming. This is primarily due to the Hartree-Fock exchange term which can become very complicated to calculate. The program originally conceived by Herman and Skillman has four main areas of approximation introduced to improve calculation efficiency. The variational principle of quantum mechanics provides justification for this schema. It is true the accuracy of the radial wave functions (and hence, the electron charge density or potential energy) generated by iteratively solving of the Hartree-Fock equations in an approximative manner will be diminished. However, the variational principle assures us the values for the spin-orbital energies determined by these wave functions will still be quite good and so further calculations made using the wave functions will still be highly reliable.

I. Slater's Approximations for HF
As stated in the outline of the Hartree-Fock theory, the basis for formulating the HF equations is the assumption that the many-electron wave function for a free atom or an ion with only closed shells can be represented by a single determinantal wave function composed of single-electron spin-orbitals.

  1. Calculation of the exchange term for each occupied orbital is an exhaustive use of computer resources. Slater noted that all orbital exchange integrals have the same general character. Thus, we can use an average exchange potential for all of the exchange terms.
  2. The essential physics will be retained if instead of calculating the average exchange potential for the actual system, the value is obtained from an analytic expression derived from the theory of a free-electron gas. This is a local density approximation; claiming non-local contributions to the exchange potential term are negligible. The new net potential, the exact Coulomb term plus the approximated average exchange term, is the Hartree-Fock-Slater potential.

II. Shell Approximations
To make a full Hartree-Fock calculation, the many-electron wave function should be represented by a linear combination of Slater determinantal wave functions. Closed shell systems have a well defined electronic configuration that can be represented with a single determinantal wave function. However, open shell systems will have a multiplet structure associated with all possible combinations of placing the open shell electrons and aligning their spins. Ideally, each determinantal term would have a weighting representing the probability of that particular electronic configuration.

  1. To minimize complications in energy determinations, the energy splitting of orbital subshells are ignored. Contributing factors to orbital splitting, such as L.S interactions, are considered negligible energy corrections.
  2. To keep the problem direct, the multiplet of possible electronic configuration within open shell systems are ignored. Thus, a configuration with one or more open shells is treated the same as a closed shell configuration. This is another form of ignoring L.S interactions. The effective result of these two approximations to L.S effects is a simplified average system. The single determinantal wave function behaves like an average of the full multiplet of configurations and yields an average energy for the subshell given the average number of electrons filling that subshell.
  3. To simplify the mathematics of the theory, a spherical average of the electron density is taken to reduce the electronic structure problem to that of a central field problem. This allows the orbital component of the wave function to be treated by separation of variable into radial and angular parts. The Hartree-Fock equation then becomes a purely radial wave equation.

III. Latter Compensation for the Free-electron gas Approximation
The free-electron gas exchange term introduced by Slater's approximations to the Hartree-Fock theory is an average potential term used to replace a complicated integral. It greatly increases the speed of calculations and is quite good at the small r, but it fails to treat the self-coulomb potential properly at large r. Generally the average potential term reduces the effect of the self-exchange potential at large r. Instead of cancelling out the self-coulomb term as the radial distance approaches large values, the average free-electron exchange goes to zero.

  1. A simple solution to this problem is to use the ideal asymptotic potential for large r. The "boundary" distance, ro, must be located at which the value of the Hartree-Fock-Slater potential matches the value of the ideal asymptotic potential. From that distance on, the iterative potential is then given the ideal asymptotic value. This corrects the potential value, but introduces a poorly defined intermediate region at r=ro which will have a discontinuous radial derivative.

IV. Non-Relativistic
In keeping with creating a fast, simple program for generating functional radial wave functions, relativistic effects were not included in the formulation of the program. From a theoretical perspective, the non-relativistic equations are simpler to work with. From a working perspective, a majority of elements studied with the intent for application in industry come from the lighter elements of the periodic table. If a more accurate determination of the energy for a subshell is desired, a perturbative correction to the energy can be calculated separately.

The question that arises is how accurate are the results? The simple answer is "pretty good", with two primary exceptions that are expected from the above approximations. The average shell system is an increasingly poor approximation for atoms with an open shell configuration which have a prominent multiplet structure. Also, the error introduced in ignoring relativistic effects increases with the Z value of the element. Heavy atoms need a first order perturbation treatment to correct for this and the error introduced by ignoring subshell splitting due to magnetic L.S.

About the Herman-Skillman program.

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Created: November 12, 1998 --- Last Updated: November 28, 2001
By Mark D. Pauli