# The HOMO-LUMO Gap in Open Shell Calculations. Meaningful or meaningless?

The HOMO – LUMO orbitals are central to the Frontier Molecular Orbital (FMO) Theory devised by Kenichi Fukui back in the fifties. The central tenet of the FMO theory resides on the idea that most of chemical reactivity is dominated by the interaction between these orbitals in an electron donor-acceptor pair, in which the most readily available electrons of the former arise from the HOMO and will land at the LUMO in the latter. The energy difference between the HOMO and LUMO of any chemical species, known as the HOMO-LUMO gap, is a very useful quantity for describing and understanding the photochemistry and photophysics of organic molecules since most of the electronic transitions in the UV-Vis region are dominated by the electron transfer between these two frontier orbitals.

But when we talk about Frontier Orbitals we’re usually referring to their doubly occupied version; in the case of open shell calculations the electron density with *α* spin is separate from the one with *β* spin, therefore giving rise to two separate sets of singly occupied orbitals and those in turn have a *α-*HOMO/LUMO and *β-*HOMO/LUMO, although SOMO (Singly Occupied Molecular Orbital) is the preferred nomenclature. Most people will then dismiss the HOMO/LUMO question for open shell systems as meaningless because ultimately we are dealing with two different sets of molecular orbitals. Usually the approach is to work backwards when investigating the optical transitions of a, say, organic radical, e.g. by calculating the transitions with such methods like TD-DFT (Time Dependent DFT) and look to the main orbital components of each within the set of *α* and *β* densities.

To the people who have asked me this question I strongly suggest to first try Restricted Open calculations, RODFT, which pair all electrons and treat them with identical orbitals and treat the unpaired ones independently. As a consequence, RO calculations and Unrestricted calculations vary due to variational freedom. RO calculations could yield wavefunctions with small to large values of spin contamination, so beware. Or just go straight to TDDFT calculations with hybrid orbitals which include a somewhat large percentage of HF exchange and polarized basis sets, but to always compare results to experimental values, if available, since DFT based calculations are Kohn-Sham orbitals which are defined for non-interacting electrons so the energy can be biased. Performing CI or CASSCF calculations is almost always prohibitive for systems of chemical interest but of course they would be the way to go.

Posted on September 27, 2018, in Computational Chemistry, DFT, Fukui, TD-DFT, Theoretical Chemistry, White papers and tagged #CompChem, Computational Chemistry, Frontier Orbitals, HOMO, HOMO/LUMO, LUMO, molecular orbitals, Molecular Orbitals Theory, restricted calculations, Restricted Open Shell, RODFT, spin, spin contamination, Theoretical Chemistry, unrestricted calculations. Bookmark the permalink. 12 Comments.

Great post!

Just a question: what about spintronics? Open shell systems can not be used to control spin guide reaction/transitions/transport?

Regards,

Camps

Thanks! Spintronic materials take advantage of open shell systems so as to have spin densities available that can be manipulated via magnetic fields. So, yes, you can use open shell calculations (restricted or unrestricted) to study spintronic materials or at least their basic building blocks.

Thanks for reading

Thank you for your valuable information about homo-lumo.

I hve one question .

I have read lot of articles about homo-lumo plot which was computed using DFT. How to identify the location of Homo-Lumo in molecule how? No one can give clear information. I am also doing research in DFT.

You need to plot the frontier orbitals with any visualization tool but the procedure varies from one software to another. You can try Chimera if you only have access to freeware. Perhaps I should write a post about it.

What if highest energy orbital from a-homo and b-homo consider as homo? Similarly, What if lower energy orbital from a-lumo and b-lumo consider as lumo? (open shell calculation using dft)..

Sir please see my queries also as above.

That can work, of course, but it’s tricky and dependent on how you did your calculation. As a first step I’d agree.

Thanks for this full information..

I have question concerning the positives HOMO?

That would mean your highest energy electrons are not bound and therefore not a stable part of your molecule/species. Can you provide more information about it?

Thank you for your interesting, well in my investigation i’m trying to calculate the HOMO and the LUMO energies for the anion [VMo5O19]3- using b3lyp functional. At least i found that the HOMO energy was positive. My question why it was positive?!

My purpese study is to calculate the gap energies for an anion of oxo-cluster..well for this dft calculation performed using b3lyp functional.. in the results i found the HOMO energy positive while it’s not the case.my problem is why this result! Thanks a lot for you interesting for my question.

Excuse my weighing in, but I found this a really enjoyable thought problem. I’d say the answer depends on how you define meaningful.

Does it shed some light on the properties of the system? It probably crudely approximates the properties of the 1st/2nd excited states, so then it has a meaning, albeit a semi-quantitative one.

Does it have a formal meaning? Not that I’m aware of in our usual HF, KS or GKS frameworks. In ensemble KS theory it can.

Also, the idea that the (S/H)OMO/LUMO gap is always spin-dependent reflects biases in the way we usually do calculations, rather than being a consequence of anything physical. The natural orbitals of a broken symmetry state will be different per spin, but one can also take a symmetric superposition over spin symmetries to yield identical gaps on each channel. The exact KS or GKS orbitals likewise by adopting an equi-ensemble formalism.