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 ) 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.

- Ruiz et al. used

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!