Questions: Bonding and Antibonding Orbitals: Sigma, Pi, and the HOMO-LUMO Gap

Question 1

He₂ does not exist as a stable molecule. Which explanation best captures why, based on molecular orbital theory?

• A. Helium atoms are too large for their orbitals to overlap effectively
• B. The four electrons fill both the σ bonding and σ* antibonding orbitals; the destabilization from the antibonding pair slightly exceeds the stabilization from the bonding pair, leaving no net stabilization
• C. Helium has no valence electrons available for bonding
• D. The σ bonding orbital is empty in He₂, so there is no force holding the atoms together

Answer: (B)

Helium has two electrons per atom, giving He₂ four electrons total. These fill both the σ bonding (2 electrons) and σ* antibonding (2 electrons) orbitals. The common misconception is that these exactly cancel — but they don't. The antibonding orbital is raised in energy by slightly more than the bonding orbital is lowered (due to the asymmetry of orbital mixing plus nuclear repulsion), so the net effect is slight destabilization. He₂ has a bond order of zero and does not form. This asymmetry is why antibonding electrons destabilize more than bonding electrons stabilize.

Question 2

A series of polyene molecules has increasing numbers of conjugated double bonds. As conjugation extends, the UV-Vis absorption wavelength shifts to longer values (lower energy). What is the molecular orbital explanation?

• A. More double bonds increase the molecular weight, slowing the electrons and lowering their energy
• B. Each added double bond introduces a new σ bond, which destabilizes the molecule and reduces the HOMO-LUMO gap
• C. Extended conjugation raises the HOMO energy and lowers the LUMO energy, narrowing the gap and requiring less energy (longer wavelength) to promote an electron
• D. Longer molecules absorb at longer wavelengths simply because they have more atoms to absorb photons

Answer: (C)

In a conjugated system, pi molecular orbitals spread across the entire conjugated framework. As more double bonds are added, the HOMO (highest filled pi orbital) rises in energy and the LUMO (lowest empty pi* orbital) falls — the gap narrows. A narrower HOMO-LUMO gap means a lower-energy photon suffices to promote an electron from HOMO to LUMO, and lower energy corresponds to longer wavelength. Beta-carotene (11 conjugated double bonds, orange) vs. ethylene (1 double bond, absorbs far UV) illustrates this beautifully.

Question 3

An antibonding orbital, when occupied by electrons, destabilizes a molecule by more than the corresponding bonding orbital stabilizes it.

• T. True
• F. False

Answer: True

This asymmetry is a fundamental result of quantum mechanical mixing of atomic orbitals. The antibonding orbital is raised above the original atomic orbital energies by a greater amount than the bonding orbital is lowered below them. The reason involves both the mathematical nature of the mixing (second-order perturbation terms add to antibonding) and nuclear repulsion, which increases at the internuclear distances typical of antibonding character. This is why molecules like He₂ (equal bonding and antibonding electrons) are not merely 'no net stabilization' but are actually slightly destabilized.

Question 4

A molecule with a large HOMO-LUMO gap will appear colored because the gap corresponds to a large energy transition.

• T. True
• F. False

Answer: False

This reverses the relationship. A large HOMO-LUMO gap requires a high-energy photon for the HOMO→LUMO transition. High-energy photons are in the UV range, which is invisible to human eyes. A molecule with a large gap absorbs UV light and appears colorless (like ethylene or benzene). A molecule with a small gap absorbs visible light — a lower energy photon — and therefore appears colored. Beta-carotene's small gap (due to extended conjugation) causes it to absorb blue-violet light, making it appear orange.

Question 5

Why does a node between the nuclei in an antibonding orbital cause the molecule to be destabilized, rather than simply having no bonding effect?

Model answer: In an antibonding orbital, destructive interference between the two atomic wavefunctions creates a nodal plane of zero electron density between the nuclei. This means the electrons are actually concentrated outside the internuclear region — in the 'back lobes' on either side. In this configuration, the electrons do not screen the nuclear repulsion between the two positively charged nuclei; in fact, electrons localized behind the nuclei can even pull the nuclei apart. The result is net destabilization: the nuclei repel each other while the electron density is positioned where it cannot counteract that repulsion.

Compare with a bonding orbital, where electron density concentrates between the nuclei. Those electrons simultaneously attract both nuclei, effectively holding them together and screening their mutual repulsion. In an antibonding orbital, electrons are in the worst possible position relative to the nuclei — this is active destabilization, not merely the absence of bonding.