# Molecular orbital
As defined by IUPAC, a **molecular orbital** is a wavefunction for a single electron that describes how that electron moves within the effective field generated by the nuclei together with every other electron in a molecular entity containing more than one atom[1](https://goldbook.iupac.org/terms/view/M03996). Molecular orbitals belong to the molecule as a whole rather than to any single atom, and they are most often built mathematically as a linear combination of atomic orbitals (LCAO)[1](https://goldbook.iupac.org/terms/view/M03996) • [2](https://openstax.org/books/organic-chemistry/pages/1-11-describing-chemical-bonds-molecular-orbital-theory). The theory built on them, molecular orbital (MO) theory, explains chemical bonding, electron energies, and magnetic properties that localized-bond models cannot account for, such as the paramagnetism of the oxygen molecule[3](https://openstax.org/books/chemistry-atoms-first-2e/pages/5-4-molecular-orbital-theory?query=intensive+).
| Key fact | Detail |
|---|---|
| Definition | A one-electron wavefunction for an electron in the effective field of the nuclei and all other electrons of a multi-atom molecular entity[1](https://goldbook.iupac.org/terms/view/M03996) |
| Construction | Formed by the linear combination of atomic orbitals (LCAO); the number of molecular orbitals formed equals the number of atomic orbitals combined[3](https://openstax.org/books/chemistry-atoms-first-2e/pages/5-4-molecular-orbital-theory?query=intensive+) • [4](https://ndl.iitkgp.ac.in/he_document/aklectures/aklectures/1_3_2_103) |
| Bonding vs antibonding | Each pair of combining atomic orbitals gives one lower-energy bonding orbital and one higher-energy antibonding orbital, the latter marked with an asterisk and containing a node between the nuclei[3](https://openstax.org/books/chemistry-atoms-first-2e/pages/5-4-molecular-orbital-theory?query=intensive+) • [5](https://chem.libretexts.org/Bookshelves/General_Chemistry/Map%3A_General_Chemistry_%28Petrucci_et_al.%29/11%3A_Chemical_Bonding_II%3A_Additional_Aspects/11.6%3A_Delocalized_Electrons%3A_Bonding_in_the_Benzene_Molecule) |
| Bond order | \( \text{bond order} = (N_{\mathrm{bonding}} - N_{\mathrm{antibonding}})/2 \); H2 has bond order 1, He2 has 0 and does not form[6](https://chemistry.illinois.edu/system/files/2023-07/Molecular%20Orbital%20Theory.pdf) • [7](https://mitocw.ups.edu.ec/courses/chemistry/5-112-principles-of-chemical-science-fall-2005/video-lectures/lecture-27-molecular-orbital-theory-for-diatomic-molecules/) |
| Magnetic test case | The MO diagram of O2 places two unpaired electrons in degenerate orbitals, predicting the paramagnetism observed for liquid oxygen[3](https://openstax.org/books/chemistry-atoms-first-2e/pages/5-4-molecular-orbital-theory?query=intensive+) • [8](https://chemed.chem.purdue.edu/genchem/topicreview/bp/ch8/mo.html) |
| Level ordering | The relative order of the 1p\(_{u}\) and 3s\(_{g}\) levels is reversed for homonuclear diatomic molecules after C\(_{2}\)[9](https://www.chemistry.mcmaster.ca/esam/Chapter_8/section_3.html) |
| Delocalized systems | In benzene, six 2p orbitals combine into six delocalized π MOs; the three bonding MOs exactly accommodate the six π electrons[10](https://research.cm.utexas.edu/nbauld/teach/ch610bnotes/ch20.htm) |
## Formation from atomic orbitals
When two atomic orbitals on different atoms combine, the wavefunctions can interfere constructively or destructively. The in-phase (additive) combination produces a bonding orbital, lower in energy than the parent atomic orbitals, with most of its electron density between the two nuclei; electrons there are attracted by both nuclei at once and hold the atoms together[3](https://openstax.org/books/chemistry-atoms-first-2e/pages/5-4-molecular-orbital-theory?query=intensive+) • [6](https://chemistry.illinois.edu/system/files/2023-07/Molecular%20Orbital%20Theory.pdf). The out-of-phase (subtractive) combination produces an [antibonding orbital](https://www.edgepedia.org/antibonding-orbital), always higher in energy than the atomic orbitals, with a node, a region of zero electron density, between the nuclei; electrons in it pull the nuclei apart[3](https://openstax.org/books/chemistry-atoms-first-2e/pages/5-4-molecular-orbital-theory?query=intensive+) • [6](https://chemistry.illinois.edu/system/files/2023-07/Molecular%20Orbital%20Theory.pdf) • [4](https://ndl.iitkgp.ac.in/he_document/aklectures/aklectures/1_3_2_103).
The count is fixed by the mathematics: combining n atomic orbitals yields exactly n molecular orbitals[4](https://ndl.iitkgp.ac.in/he_document/aklectures/aklectures/1_3_2_103). Electrons then fill the orbitals by the [Aufbau principle](https://www.edgepedia.org/aufbau-principle), lower-energy orbitals first, spreading among degenerate orbitals before pairing, with at most two opposite-spin electrons per orbital[3](https://openstax.org/books/chemistry-atoms-first-2e/pages/5-4-molecular-orbital-theory?query=intensive+).
## Sigma and pi orbitals
Head-on combination of two s orbitals, as in H\(_{2}\), gives σ (sigma) bonding and σ* (sigma-star) antibonding orbitals, so named because they look like s orbitals when viewed along the bond axis[8](https://chemed.chem.purdue.edu/genchem/topicreview/bp/ch8/mo.html). Sideways combination of two p orbitals, as in the C–C double bond of ethylene, gives π bonding and π* antibonding orbitals; the π bonding orbital has no node between the nuclei and results from lobes of the same algebraic sign, while π* has a node and comes from lobes of opposite sign[2](https://openstax.org/books/organic-chemistry/pages/1-11-describing-chemical-bonds-molecular-orbital-theory). Combining the six 2p atomic orbitals of a diatomic molecule yields three bonding orbitals (one σ and two degenerate π) and three antibonding orbitals (one σ* and two π*)[3](https://openstax.org/books/chemistry-atoms-first-2e/pages/5-4-molecular-orbital-theory?query=intensive+).
## Bond order and stability
In MO theory, two electrons in a bonding orbital contribute one net bond, and two electrons in an antibonding orbital cancel one bond, giving the bond order formula[8](https://chemed.chem.purdue.edu/genchem/topicreview/bp/ch8/mo.html)
\[ \text{bond order} = \frac{N_{\mathrm{bonding}} - N_{\mathrm{antibonding}}}{2} \]
For H\(_{2}\), both electrons occupy the σ bonding orbital, so the bond order is (2 − 0)/2 = 1[6](https://chemistry.illinois.edu/system/files/2023-07/Molecular%20Orbital%20Theory.pdf). A positive bond order means the species is energetically favorable to form; a bond order of zero means it will not[6](https://chemistry.illinois.edu/system/files/2023-07/Molecular%20Orbital%20Theory.pdf). The hypothetical He\(_{2}\) molecule illustrates the limit: its four electrons fill both the bonding and the antibonding orbital, the net bond order is zero, and the atoms simply bounce off each other[7](https://mitocw.ups.edu.ec/courses/chemistry/5-112-principles-of-chemical-science-fall-2005/video-lectures/lecture-27-molecular-orbital-theory-for-diatomic-molecules/). Core orbitals contribute essentially nothing to molecular stability; only combinations of valence-shell orbitals matter[8](https://chemed.chem.purdue.edu/genchem/topicreview/bp/ch8/mo.html).
## Magnetic properties and the oxygen molecule
**The paramagnetism of O\(_{2}\)** is the standard demonstration of MO theory's reach. A [Lewis structure](https://www.edgepedia.org/lewis-structure) of oxygen pairs all electrons and would predict diamagnetism, repulsion by a magnetic field. The [MO diagram](https://www.edgepedia.org/molecular-orbital-diagram) instead places the last two electrons singly in the two degenerate π* antibonding orbitals, following Hund's rule, so the molecule has two unpaired electrons and is paramagnetic, attracted into a magnetic field[8](https://chemed.chem.purdue.edu/genchem/topicreview/bp/ch8/mo.html) • [3](https://openstax.org/books/chemistry-atoms-first-2e/pages/5-4-molecular-orbital-theory?query=intensive+). [Liquid oxygen](https://www.edgepedia.org/liquid-oxygen) is attracted strongly enough to bridge the gap between the poles of a horseshoe magnet, confirming the MO prediction[8](https://chemed.chem.purdue.edu/genchem/topicreview/bp/ch8/mo.html). The same diagram gives a bond order of two for O\(_{2}\), with eight valence electrons in bonding and four in antibonding orbitals, matching the Lewis-structure bond order while correcting the magnetic prediction[8](https://chemed.chem.purdue.edu/genchem/topicreview/bp/ch8/mo.html).
## Energy-level ordering in diatomic molecules
For homonuclear diatomic molecules, the relative ordering of orbital energies is nearly universal, with one systematic exception: the relative order of the 1p\(_{u}\) and 3s\(_{g}\) levels reverses for the molecules after C\(_{2}\)[9](https://www.chemistry.mcmaster.ca/esam/Chapter_8/section_3.html). F\(_{2}\) comes out with bond order 1 and Ne\(_{2}\) with bond order 0, consistent with the existence of the former and nonexistence of the latter[11](https://www.pearson.com/content/dam/one-dot-com/one-dot-com/us/en/higher-ed/en/products-services/course-products/tro-chemistry-4e-info/pdf/chapter10.pdf).
## Delocalization and aromatic systems
Because molecular orbitals extend over the whole molecule, MO theory handles delocalized bonding naturally, where valence-bond models need multiple resonance structures[3](https://openstax.org/books/chemistry-atoms-first-2e/pages/5-4-molecular-orbital-theory?query=intensive+). In benzene, the six unhybridized 2p orbitals of the sp² carbons overlap cyclically to form six delocalized π molecular orbitals spread over the whole ring[10](https://research.cm.utexas.edu/nbauld/teach/ch610bnotes/ch20.htm). For cyclic conjugated systems, the energy levels always include a single lowest-lying MO above which the levels come in degenerate pairs; planar systems with 4n + 2 π electrons fill only bonding levels, which is the molecular-orbital basis of Hückel's aromaticity rule, while planar 4n π-electron systems are antiaromatic because delocalization would destabilize them[12](https://openstax.org/books/organic-chemistry/pages/15-3-aromaticity-and-the-huckel-4n-2-rule). Benzene's three bonding π orbitals exactly accommodate its six π electrons[10](https://research.cm.utexas.edu/nbauld/teach/ch610bnotes/ch20.htm).
## Frontier orbitals and reactivity
The highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) govern much of chemical reactivity. [Photoelectron spectroscopy](https://www.edgepedia.org/photoelectron-spectroscopy) gives a direct experimental handle on occupied orbitals: by Koopmans' theorem, the vertical ionization energy from an occupied orbital in a closed-shell molecule approximates the negative of that orbital's energy in a Hartree–Fock calculation[13](https://eprints.soton.ac.uk/464770/1/874452.pdf).
## Computation
Quantum-chemical calculations express molecular orbitals as expansions in basis sets, the fixed functions used to represent the one-electron orbitals. Modern atom-centered, energy-optimized basis-set families for molecular applications include atomic natural orbital, correlation consistent, polarization consistent, and def2 sets[14](https://onlinelibrary.wiley.com/doi/10.1002/qua.24355). Accurate correlated treatments also require reliable descriptions of virtual (unoccupied) orbitals, which conventional schemes often describe poorly; localized correlation-converged virtual orbitals have been introduced as an efficient basis for many-body Hamiltonians in molecular systems[15](https://arxiv.org/abs/2604.16928).
## References
1. [IUPAC Compendium of Chemical Terminology, molecular orbital (M03996)](https://goldbook.iupac.org/terms/view/M03996)
2. [OpenStax Organic Chemistry, 1.11 Describing Chemical Bonds: Molecular Orbital Theory](https://openstax.org/books/organic-chemistry/pages/1-11-describing-chemical-bonds-molecular-orbital-theory)
3. [OpenStax Chemistry: Atoms First 2e, 5.4 Molecular Orbital Theory](https://openstax.org/books/chemistry-atoms-first-2e/pages/5-4-molecular-orbital-theory?query=intensive+)
4. [NDLI lectures, Bonding and Antibonding Molecular Orbitals](https://ndl.iitkgp.ac.in/he_document/aklectures/aklectures/1_3_2_103)
5. [Chemistry LibreTexts, Delocalized Electrons: Bonding in the Benzene Molecule](https://chem.libretexts.org/Bookshelves/General_Chemistry/Map%3A_General_Chemistry_%28Petrucci_et_al.%29/11%3A_Chemical_Bonding_II%3A_Additional_Aspects/11.6%3A_Delocalized_Electrons%3A_Bonding_in_the_Benzene_Molecule)
6. [University of Illinois, Molecular Orbital Theory lecture notes](https://chemistry.illinois.edu/system/files/2023-07/Molecular%20Orbital%20Theory.pdf)
7. [MIT OpenCourseWare, Lecture 27: Molecular Orbital Theory for Diatomic Molecules](https://mitocw.ups.edu.ec/courses/chemistry/5-112-principles-of-chemical-science-fall-2005/video-lectures/lecture-27-molecular-orbital-theory-for-diatomic-molecules/)
8. [Purdue University, Molecular Orbital Theory topic review](https://chemed.chem.purdue.edu/genchem/topicreview/bp/ch8/mo.html)
9. [McMaster University, Molecular Orbitals for Homonuclear Diatomics](https://www.chemistry.mcmaster.ca/esam/Chapter_8/section_3.html)
10. [University of Texas at Austin, Chapter 20: Benzene and Derivatives: Aromaticity](https://research.cm.utexas.edu/nbauld/teach/ch610bnotes/ch20.htm)
11. [Tro, Chemistry 4e, Chapter 10 sample (Pearson)](https://www.pearson.com/content/dam/one-dot-com/one-dot-com/us/en/higher-ed/en/products-services/course-products/tro-chemistry-4e-info/pdf/chapter10.pdf)
12. [OpenStax Organic Chemistry, 15.3 Aromaticity and the Hückel 4n + 2 Rule](https://openstax.org/books/organic-chemistry/pages/15-3-aromaticity-and-the-huckel-4n-2-rule)
13. [University of Southampton, Theoretical methods used in photoelectron spectroscopy](https://eprints.soton.ac.uk/464770/1/874452.pdf)
14. [Hill (2013), Gaussian basis sets for molecular applications, Int. J. Quantum Chem.](https://onlinelibrary.wiley.com/doi/10.1002/qua.24355)
15. [Correlation-Converged Virtual Orbitals for Accurate and Efficient Quantum Molecular Simulations, arXiv](https://arxiv.org/abs/2604.16928)
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*Topic: Encyclopedia › Physical world and mathematics › Chemistry › Chemical principles and methods › Chemical bonding and intermolecular forces*
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