The CASE Approximation

  

Figure: Graphs of 1/r, $\erfc (r)/r$ and $\erf (r)/r$

LegA1/r LegB $\erfc (r)/r$ LegC$\erf (r)/r$

Examining the KWIK $\erfc(\omega r)/r$ separator in figure 6.1 shows two distinctly contrasting curves. The first is singular at the origin and decays very rapidly. The second, accounting for the long-range behaviour of 1/r, is exceptionally smooth (that is, lacking in high-frequency components) and finite at the origin. If the long range operator was a flat line it would have no effect on the wavefunction, only altering the total energy. Given the flatness of the long-range operator it is reasonable to assume that it is of less importance to the wavefunction than the short-range function.

The CASE (Coulomb-Attenuated-Schrödinger-Equation) approximation is to take this idea to its logical conclusion and completely neglect the long-range component. This seems a drastic step to take, but it is not as preposterous as it may first appear. At worst it could be viewed as a test for the short-sightedness of electrons [192]. The CASE results may be strictly only applicable to a universe where the Coulomb operator decays rapidly, yet it is possible that they could be a useful guide to chemistry. It is well-known to chemists that molecules are essentially non-polar over large distance scales. Therefore it is not unreasonable to expect the attractive (nuclear-electron) and repulsive (nuclear-nuclear and electron-electron) Coulomb interactions between widely separated regions of a molecule to approximately cancel. What is required, therefore, is a way of smoothly cutting off the long-range interaction -- something that the $\erfc(\omega r)/r$ function provides.

If the approximation proves to be a disaster all is not lost, as CASE can be employed as a zeroth-order approximation upon which higher-order corrections can be constructed. The value of CASE in its own right should be examined first, however.