Wolf (Wolf, Keblinksi, Phillpot, and Eggebrecht, J. Chem. Phys. 110 (1999) 8254) came up with a very clever way to approximate the Madelung potential in infinite solids that is much less expensive to calculate than the traditional reciprocal-space-based Ewald methods, and in his seminal paper, Wolf showed that his approach works wonderfully for both crystalline and amorphous solids like NaCl and MgO.
Implementing this so-called Wolf sum isn’t hard; there are two parameters to devise and picking the right one is quite straightforward:
Wolf-approximated Madelung potential for halite as a function of the two empirical parameters rc and beta (=1/alpha)
In the case of halite, if you want a cutoff of 10 Å, it looks like β = 3.46 Å is a good choice; the Madelung potential appears fully converged, and unlike the β=2.46Å case, there is no systematic error due to overdamping. Life is good, right?
As it turns out, applying the Wolf summation method to slightly more complicated crystals isn’t as nice. Take, for example, alumina (Al2O3):
Wolf-approximated Madelung potential for alumina.
Suddenly the Madelung potential doesn’t oscillate nicely around the true value; rather, the converged value decreases monotonically with increasing damping. This offers no indication of what the true converged Madelung energy for this crystal is. What about other relevant materials that aren’t a simple 1:1 stoichiometry?
Madelung potential for water assuming partial charges
Water looks a lot like alumina. The Wolf sum isn’t working very well here.
Madelung potential for water assuming partial charges with diffuse character
Using a more realistic (but still empirical) treatment of the Coulombic nature of water makes things worse.
Madelung potential for amorphous silica assuming partial charges with diffuse character
…and amorphous silica also doesn’t work.
So what’s going on here? This method seems scientifically sound, and I’m reasonably sure my implementation of it is correct since I can match the results of other codes, but the only systems with which it seems to work reliably are electronically very simple.
What frustrates me about this is that my work for the last five years has been using this Wolf method for both water and silica. The parameters were published before I started, and I had assumed that they were derived using some sort of sensible procedure. Unfortunately, I can’t figure out what that method was because re-parameterizing the Wolf method myself has revealed the process to be nothing but fragile and murky.
I can’t say that this is unexpected given how my research has almost invariably panned out for the last five years, but it is frustrating nonetheless. Consequentially, I will probably spend the next few days fighting code and methods rather than doing real science that can contribute to my dissertation. But such is the nature of graduate work.
