HOMO-LUMO Gap Formula
The HOMO-LUMO gap is a simple subtraction, but it is easy to get the sign, the units or the meaning wrong. This guide works through the formula, converts the result between units and explains why the orbital gap is not the optical gap.
Key takeaways
- The HOMO-LUMO gap is ELUMO minus EHOMO, both from the same calculation and in the same unit.
- The orbital gap, the fundamental gap (IE minus EA) and the optical gap are three different quantities.
- A smaller gap usually signals a more polarizable, more reactive or more easily excited molecule, as a trend only.
The formula
The HOMO-LUMO gap is the energy difference between the lowest unoccupied and the highest occupied molecular orbital:
Because the LUMO lies above the HOMO, the gap is positive for any closed-shell ground state. Both energies must come from the same calculation: the same method, basis set and geometry. Subtracting a HOMO from one source and a LUMO from another produces a number with no clear meaning.
Worked examples
Example 1: from orbital energies in eV
This site's calculator reports benzene (Extended Hückel, standard formula) with EHOMO = −12.804 eV and ELUMO = −8.307 eV.
Watch the double negative. Subtracting a negative HOMO energy adds its magnitude, so the gap is the distance between the two levels.
Example 2: from orbital energies in Hartree
Many quantum chemistry programs print orbital energies in Hartree (Eh). Suppose an output lists EHOMO = −0.2500 Eh and ELUMO = −0.0100 Eh (hypothetical values).
Convert at the end, after subtracting, or convert both energies first; the result is the same as long as both are in one unit.
Orbital, fundamental and optical gaps
Three related quantities are often called "the gap", and mixing them up is the most common error in using HOMO-LUMO data.
| Gap | Definition | How it is obtained |
|---|---|---|
| Orbital (HOMO-LUMO) gap | ELUMO − EHOMO | One ground-state calculation; depends strongly on method |
| Fundamental gap | IE − EA | Energies of the cation, neutral and anion, or photoelectron data |
| Optical gap | Lowest allowed excitation energy | UV-visible absorption, or an excited-state calculation such as TD-DFT |
The optical gap is smaller than the fundamental gap because the excited electron and the hole it leaves behind still attract each other (the exciton binding energy). Hartree-Fock orbital gaps are usually too large compared with the fundamental gap, while common density functionals give orbital gaps that are too small. Extended Hückel gaps are of a similar order of magnitude to optical gaps for conjugated molecules but are not designed to match either one.
What the gap tells you
- Kinetic stability: a large gap means adding or removing electrons, or mixing in excited configurations, costs a lot of energy. Such molecules tend to be less reactive.
- Chemical hardness: in conceptual density functional theory the hardness is often approximated as η ≈ (ELUMO − EHOMO) / 2. Hard molecules have large gaps; soft, polarizable molecules have small ones.
- Conjugation length: in a homologous series of conjugated molecules, the gap shrinks as conjugation grows. Ethylene, benzene and naphthalene give EHT gaps of 5.019, 4.497 and 2.738 eV on this site.
- Materials screening: the gap of a candidate semiconductor or dye is a first, cheap filter before more expensive calculations.
Common mistakes
- Reporting the gap with the wrong sign after subtracting in the wrong order.
- Mixing units, for example a HOMO in Hartree and a LUMO in eV.
- Comparing a gap from one method with a gap from another, or with an experimental optical gap, without saying so.
- Using a gap from an open-shell calculation without specifying which spin orbitals were used.
References and further reading
- Parr, R. G.; Pearson, R. G. Absolute Hardness: Companion Parameter to Absolute Electronegativity. J. Am. Chem. Soc. 1983, 105, 7512-7516.
- Bredas, J.-L. Mind the gap! Mater. Horiz. 2014, 1, 17-19.
- Zhan, C.-G.; Nichols, J. A.; Dixon, D. A. J. Phys. Chem. A 2003, 107, 4184-4195.
- Tiesinga, E. et al. CODATA Recommended Values of the Fundamental Physical Constants: 2018. Rev. Mod. Phys. 2021, 93, 025010.