Approximations to HF Theory for Calculation
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
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.
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
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.
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.
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.
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
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.
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.
Go to top of page
Created: November 12, 1998 --- Last Updated: November 28, 2001
By Mark D. Pauli