Minnesota MN15 functional (and the others) – magnetic coupling constant – a benchmark study

Honestly, I don’t like studying magnetic systems, e.g. single molecule magnet, and I don’t know why I end up testing my three favorite functionals with this toy model H-He-H. You may laugh at this system but there has been at least 6 papers (six!, including one from Gustavo E. Scuseria) used this system as a model for magnetic studies.

The calculation is simple, the magnetic coupling constant J, defined as the different between the singlet and triplet state J = E(singlet) – E(triplet). This can be easily done by DFT, but you may end up with a wrong number.

So, what is the problem? There are two:

  • The stupid singlet state is open-shell (in the language of DFT) or multi-reference (in the language of CASSCF). You have to use broken-symmetry BS-DFT to study the singlet state.
  • Once you manage to obtain the BS solution for the singlet and the triplet states, there are three ways to calculate J:
    • Ruiz et al. used J = E(BS-DFT) – E(triplet) (duh?) without any correction for the spin-contamination. They claimed that good results can be obtain with this “faulty” approach because of error cancellation. Basically, I can call this approach “right results for wrong reasons”. The paper lead to a polite comment here and a reply by the original authors here.
    • The famous Noodleman’s projection method J = 2(E(BS-DFT) – E(triplet)). This applies when the magnetic centers weakly interact, e.g. large distance between He and H.
    • The also famous Yamaguchi’s  spin projection scheme using spin contamination, that is valid regardless whether the interaction is strong or weak.

In this post, I do a simple test with this system with d(H-He) = 2.0 Å. I use B3LYP, PBE0, MN15, SCAN0, and ωB97M-V as usual. The basis set is 6-311G** since a full-CI solution with this basis set is available (J = -50 cm-1). I also test the self-interaction correction (SIC).


What I can see

  • Without SIC, Ruiz’s approach is excellent with errors less than 40% for all functionals (rrwr). Both Noodleman’s and Yamaguchi’s give very bad results (error more than 100%) using B3LYP, while using other functionals, the results are fine, especially MN15 and ωB97M-V.
  • With SIC, Ruiz’s approach screws up its performance, the errors are more than 50%, while the results with Noodleman’s and Yamaguchi’s improve.


What can I learn here?

I still hate studying magnetic systems. This is just a small system with H and He. With larger systems containing multiple metal centers, (i) a SIC calculation is too expensive and (ii) the errors “should” be larger.

PS. I use a random number generator from 0 to 100 and obtain |J| = 53 cm-1 just the first time I press the generate button. It’s a match!



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