[allthingschm]

Usually singlet carbenes require an acid catalyst for dimerization however there are cases where the catalyst  is not needed when the LUMO level is low enough (especially in cases when HOMO level is high as well). My hypothesized explanation for this is that the lower the LUMO, the easier it is to donate the electrons into it as less energy is required to excite the electrons. I'm not sure if this explanation is correct and if so, it seems a bit simplistic. Any suggestions to improve this?

Also, how does protonation of the singlet carbenes aid dimerization. I'm guessing that it helps to lower the LUMO but I'm not sure on how to explain this.

[Irlanur]

You should go back to the principles of Frontier Molecular Orbital Theory.

First of all, the HOMO-LUMO interaction has to be the Most-most-most important one, this is not always the case, but let's assume it here.
Second, we look at the HOMO-LUMO interaction in the transition state and estimate/compare if the reaction is "allowed".

Now specifically to your question: how does the Orbital diagramm look like if you just draw the HOMO and the LUMO? how does it look like if they are closer in Energy?