Basis set name versus number of total orbitals
I would like to first address a part of the question that appears to be a misconception about the use of a 6-31+G(d,p) basis set, since you wrote:
"In my understanding of such basis sets, it is difficult to do this."
6-31+G(d,p) is not a "big" or "small" basis set, unless we're dealing with a big or small number of atoms (and aiming for a big or small amount of accuracy in our solution to the electronic Schrödinger equation). 6-31+G(d,p) is a fairly small basis set if we are dealing with a small molecule and only aiming to do TD-DFT. As currently written, the question only indicates that the paper was studying "azo" compounds, so the question alone does not tell us if the basis set is manageable or not manageable on a laptop computer. The question mentioned a journal publication, and we have to look into that paper to see what specific "azo" molecules are being discussed, before we can assess anything.
The calculation mentioned in the paper is possible on a laptop
The largest XYZ file in the articles Supplementary PDF is for a molecule with 46 atoms: bis(9H-fluoren-2-yl)diazene (which they call AzoFL). I just ran for you a CCSD(T)/6-31+G(d,p) calculation in MOLPRO, which seems to show that 6-31+G(d,p) seems manageable for a 46-atom molecule. I am aware the authors of the paper mentioned in the question, used Gaussian16, but I do not have access to Gaussian, so I used MOLPRO, which I do not think will affect whether or not the calculation is feasible on a laptop.
Since CSCD(T) is much more expensive than B3-LYP/TD-DFT (which is what is used in the paper), I conclude that B3-LYP/TD-DFT is feasible with the same or even smaller resources. The CCSD(T) calculation is still running, but the fact that it is running, is enough to know that it is possible (if the calculation is not possible due to lack of disk space or RAM, MOLPRO would have told us earlier in the output file).
What I have finished though is HF and MP2, and neither of them (nor the integrals) took more than 1000 seconds:
PROGRAMS * TOTAL RMP2 RHF INT
CPU TIMES * 1872.67 488.34 832.73 551.49
REAL TIME * 3816.76 SEC
DISK USED * 84.74 GB
The disk space needed was only 85GB, which most laptops do have available. Unfortunately MOLPRO did not print the "peak RAM usage", and as user Anyon has pointed out, the CPU you mention in your question has a maximum capacity of 32GB of RAM, so this is the main limitation. I can re-run the calculation to only use 32GB of RAM and see what happens, but when the total disk usage is 85GB, it essentially means that a machine with 32GB of RAM will be enough, even if it means some of the disk has to act as SWAP space. I would not recommend doing this very often though, because if you're reading and writing 85GB too often, the lifetime of your disk might be affected.
- It is possible to do a 46-atom TD-DFT-B3LYP/6-31+G(d,p) calculation on a laptop with an Intel Core i3-6006U CPU, but it requires 85GB of disk space, and I wouldn't recommend doing such calculations often if you care about the lifetime of your disk.
- The name of the basis set, for example: 6-31+G(d,p), is not what matters in determining whether or not TD-DFT-B3LYP is possible, as much as the number of atoms, and more importantly the total number of orbitals.