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Molecular orbitals describe how electrons can be spread across an entire molecule rather than belonging to a single atom. In quantum terms, they are wave-like solutions used to model bonding, antibonding, electron density, and molecular energy levels.

In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region.

The terms atomic orbital and molecular orbital^{[lower-alpha 1]} were introduced by Robert S. Mulliken in 1932 to mean one-electron orbital wave functions.^[2] At an elementary level, they are used to describe the region of space in which a function has significant amplitude.

In an isolated atom, electron locations are described by atomic orbitals. When multiple atoms combine chemically into a molecule by forming a chemical bond, the electrons' locations are determined by the molecule as a whole, so the atomic orbitals combine to form molecular orbitals. The electrons from the constituent atoms occupy these molecular orbitals.

Mathematically, molecular orbitals are approximate solutions to the Schrödinger equation for electrons in the field of the molecule's atomic nuclei. They are usually constructed by combining atomic orbitals or hybrid orbitals from each atom of the molecule, or other molecular orbitals from groups of atoms. They can be quantitatively calculated using the Hartree–Fock or self-consistent field (SCF) methods.

Molecular orbitals are commonly divided into three types: bonding orbitals, which have an energy lower than the atomic orbitals that formed them and promote chemical bonding; antibonding orbitals, which have higher energy and oppose bonding; and non-bonding orbitals, which have approximately the same energy as their constituent atomic orbitals and have little effect on bonding.

Overview

A molecular orbital (MO) can be used to represent the regions in a molecule where an electron occupying that orbital is likely to be found. Molecular orbitals are approximate solutions to the Schrödinger equation for electrons in the electric field of the molecule's nuclei. Direct calculation from this equation is usually too difficult, so molecular orbitals are often obtained from combinations of atomic orbitals.

A molecular orbital can specify the electron configuration of a molecule: the spatial distribution and energy of one electron, or one pair of electrons. Most commonly, a MO is represented as a linear combination of atomic orbitals using the LCAO-MO method. This gives a simple model of bonding in molecules and is central to molecular orbital theory.

Most present-day methods in computational chemistry begin by calculating the molecular orbitals of the system. A molecular orbital describes the behavior of one electron in the electric field generated by the nuclei and by an average distribution of the other electrons. If two electrons occupy the same orbital, the Pauli principle requires that they have opposite spin. This is an approximation: highly accurate descriptions of molecular electronic wave functions do not rely only on orbitals, as in configuration interaction.

Molecular orbitals are usually delocalized over the whole molecule. If a molecule has symmetry elements, its nondegenerate molecular orbitals are either symmetric or antisymmetric with respect to those symmetries. For example, applying a symmetry operation S to a molecular orbital ψ can leave it unchanged or reverse its sign: Sψ = ±ψ. In planar molecules, orbitals may be symmetric (sigma) or antisymmetric (pi) with respect to reflection in the molecular plane. More generally, molecular orbitals form bases for the irreducible representations of the molecule's symmetry group.^[4]

The delocalized nature of molecular orbitals distinguishes molecular orbital theory from valence bond theory, where bonds are often treated as localized electron pairs. The two descriptions are complementary. Localized molecular orbitals can also be formed by mathematical transformations of canonical orbitals; these localized orbitals resemble bonds in a Lewis structure, but their individual energy levels no longer have the same direct physical meaning.

Formation of molecular orbitals

Molecular orbitals arise from allowed interactions between atomic orbitals. Such interactions are allowed when the symmetries of the atomic orbitals are compatible. The strength of the interaction depends on orbital overlap and on how close the atomic orbitals are in energy.

The number of molecular orbitals formed must equal the number of atomic orbitals combined to form the molecule.

Qualitative discussion

For a qualitative description of molecular structure, molecular orbitals can be obtained from the Linear combination of atomic orbitals molecular orbital method ansatz. In this approach, molecular orbitals are expressed as linear combinations of atomic orbitals.^[5]

Linear combinations of atomic orbitals (LCAO)

Molecular orbitals were first introduced by Friedrich Hund^[6]^[7] and Robert S. Mulliken^[8]^[9] in 1927 and 1928.^[10]^[11] The linear combination of atomic orbitals or LCAO approximation for molecular orbitals was introduced in 1929 by Sir John Lennard-Jones.^[12]

Linear combinations of atomic orbitals can be used to estimate the molecular orbitals formed when atoms bond. For simple diatomic molecules, the bonding and antibonding wavefunctions may be represented as:

Ψ = c_{a} ψ_{a} + c_{b} ψ_{b}

Ψ^{*} = c_{a} ψ_{a} - c_{b} ψ_{b}

where $Ψ$ and $Ψ^{*}$ are the wavefunctions for bonding and antibonding molecular orbitals, $ψ_{a}$ and $ψ_{b}$ are atomic wavefunctions from atoms a and b, and $c_{a}$ and $c_{b}$ are adjustable coefficients. As atoms approach each other, their atomic orbitals overlap, creating regions of high electron density. The atoms are held together by electrostatic attraction between positively charged nuclei and negatively charged electrons occupying bonding molecular orbitals.^[13]

Bonding, antibonding, and nonbonding MOs

When atomic orbitals interact, the resulting molecular orbital can be bonding, antibonding, or nonbonding.

Bonding molecular orbitals

arise from constructive, in-phase interactions between atomic orbitals;
are lower in energy than the atomic orbitals that combine to produce them.

Antibonding molecular orbitals

arise from destructive, out-of-phase interactions;
contain a nodal plane between the interacting atoms;
are higher in energy than the atomic orbitals that combine to produce them.

Nonbonding molecular orbitals

result when atomic orbitals do not interact effectively because of incompatible symmetry or poor energy matching;
have approximately the same energy as the atomic orbital of one atom in the molecule.

Sigma and pi labels for MOs

Molecular orbitals are often categorized by symmetry labels such as σ (sigma), π (pi), δ (delta), and φ (phi). These labels describe how the orbital behaves with respect to the internuclear axis. The number of nodal planes containing that axis is zero for σ orbitals, one for π orbitals, two for δ orbitals, and three for φ orbitals.

σ symmetry

A molecular orbital with σ symmetry can result from interaction between two s orbitals or between two p_z orbitals. It is symmetric with respect to rotation around the internuclear axis. A σ* antibonding orbital also has σ symmetry, but contains a nodal plane between the nuclei and perpendicular to the internuclear axis.^[14]

π symmetry

A molecular orbital with π symmetry results from side-by-side interaction of p orbitals, such as p_x or p_y orbitals. A π orbital changes phase under rotation about the internuclear axis and has one nodal plane containing that axis. A π* antibonding orbital has an additional nodal plane between the nuclei.^[14]^[15]^[16]^[17]

δ symmetry

A molecular orbital with δ symmetry results from interaction between suitable d orbitals. Such orbitals are important in some transition-metal complexes. A δ bonding orbital has two nodal planes containing the internuclear axis, while a δ* antibonding orbital also has a nodal plane between the nuclei.

φ symmetry

Theoretical chemists have proposed higher-order bonds such as φ bonds, which would correspond to overlap of f atomic orbitals. No established molecule is known to contain a φ bond.

Gerade and ungerade symmetry

For molecules with a center of inversion, additional labels can be applied to molecular orbitals. If inversion through the center of symmetry leaves the orbital phase unchanged, the orbital is said to have gerade (g) symmetry. If inversion changes the phase, the orbital has ungerade (u) symmetry.

Centrosymmetric molecules include homonuclear diatomics, octahedral molecules, and square-planar molecules. Non-centrosymmetric molecules include heteronuclear diatomics and tetrahedral molecules.

For a bonding MO with σ symmetry, the orbital may be σ_g, while an antibonding σ orbital may be σ_u. For π orbitals, the bonding and antibonding gerade/ungerade assignments depend on the phase relationship of the interacting p orbitals.^[14]

MO diagrams

Molecular orbital diagrams visualize bonding interactions by arranging molecular orbitals according to energy. Orbitals are represented by horizontal lines, with higher lines indicating higher energy. Electrons are then placed into these orbitals according to the Pauli exclusion principle and Hund's rule.

Some useful rules are:

The number of molecular orbitals equals the number of atomic orbitals included in the basis set.
Orbitals of compatible symmetry and similar energy mix most strongly.
Symmetry-adapted linear combinations (SALCs) can be used to combine atomic orbitals systematically.
The relative energies of the molecular orbitals depend on overlap and parent-orbital energies.

A general procedure for constructing a qualitative molecular orbital diagram is:

Assign a point group to the molecule.
Identify symmetry-adapted combinations of atomic orbitals.
Arrange fragment orbitals in approximate energy order.
Combine orbitals of the same symmetry.
Estimate molecular orbital energies from overlap and energy matching.
Refine the diagram using molecular orbital calculations when needed.^[18]

Bonding in molecular orbitals

Orbital degeneracy

Molecular orbitals are degenerate when they have the same energy. For example, in homonuclear diatomic molecules, molecular orbitals derived from p_x and p_y atomic orbitals may form two degenerate bonding orbitals and two degenerate antibonding orbitals.^[13]

Ionic bonds

In an ionic bond, oppositely charged ions are held together by electrostatic attraction.^[19] Molecular orbital theory can describe ionic bonding as an extreme form of polar bonding. In such cases, the bonding orbitals are close in energy and character to the atomic orbitals of the anion, so the electron density is shifted strongly toward the anion.^[20]

Bond order

The bond order of a molecule can be determined by comparing the number of electrons in bonding and antibonding molecular orbitals. A pair of electrons in a bonding orbital contributes to bonding, while a pair in an antibonding orbital cancels a bond contribution.

For example, N₂, with eight electrons in bonding orbitals and two electrons in antibonding orbitals, has bond order three, corresponding to a triple bond. Bond strength generally increases with bond order, while bond length generally decreases.

There are exceptions. Although Be₂ has bond order zero in simple MO analysis, experimental evidence indicates a weakly bound Be₂ molecule with bond length 245 pm and bond energy 10 kJ/mol.^[14]^[21]

HOMO and LUMO

The highest occupied molecular orbital is called the HOMO, and the lowest unoccupied molecular orbital is called the LUMO. The energy difference between them is the HOMO–LUMO gap. It is important in chemical reactivity, optical absorption, and electronic transport.

For Hartree–Fock calculations where Koopmans' theorem applies, the HOMO–LUMO gap can be interpreted as an approximation to the difference between vertical ionization potential and vertical electron affinity. This interpretation must be used carefully, because the HOMO–LUMO gap is not always the same as the optical gap, fundamental gap, or bulk-material band gap.^[22]^: 17–18^[22]^: 18

Examples

Homonuclear diatomics

Homonuclear diatomic molecular orbitals contain equal contributions from each atomic orbital in the basis set. This is seen in simple molecules such as H₂, He₂, and Li₂.^[14]

H₂

Bonding and antibonding molecular orbitals of H₂ formed from hydrogen 1s atomic orbitals.

In the hydrogen molecule H₂, two hydrogen 1s atomic orbitals combine into one bonding orbital and one antibonding orbital. The bonding combination is lower in energy, while the antibonding combination is higher. Since H₂ has two electrons, both occupy the bonding orbital, making the molecule more stable than two separated hydrogen atoms. The bond order is one.

He₂

For hypothetical He₂, two electrons would occupy the bonding orbital and two would occupy the antibonding orbital. These effects cancel, giving bond order zero. Thus ordinary He₂ is not expected to form a stable covalent molecule, although weak van der Waals interactions can occur.

Li₂

Dilithium Li₂ forms from overlap of atomic orbitals on two lithium atoms. Each lithium atom contributes three electrons, and the molecular orbital filling gives a bond order of one.^[23]

Heteronuclear diatomics

Molecular orbitals in heteronuclear diatomic molecules contain unequal contributions from the two atoms. Orbital interactions occur if there is sufficient overlap and if the interacting atomic orbitals have compatible symmetry and similar energies.^[24]

HF

In hydrogen fluoride, HF, overlap between the hydrogen 1s orbital and the fluorine 2p_z orbital is symmetry-allowed and energetically favorable. This produces bonding and antibonding σ orbitals and gives HF a bond order of one. Since HF is not centrosymmetric, gerade and ungerade labels do not apply.^[25]

Quantitative approach

Quantitative molecular orbital calculations require orbitals that allow the electronic wavefunction to converge toward the full configuration-interaction limit. The most common starting point is the Hartree–Fock method, which expresses molecular orbitals as eigenfunctions of the Fock operator.

In practical calculations, molecular orbitals are expanded as linear combinations of basis functions, often Gaussian functions centered on the atomic nuclei. The resulting equations for the coefficients form a generalized eigenvalue equation known as the Roothaan equations. Quantum chemical programs such as Spartan can perform such molecular orbital calculations.^[26]

Experimental techniques such as ultraviolet photoelectron spectroscopy and X-ray photoelectron spectroscopy measure ionization energies rather than orbital energies directly. Ionization energies can sometimes be related approximately to orbital energies through Koopmans' theorem, but the agreement can be poor in some molecules.^[27]

Table of contents (84 articles)

Index

Composite matter

1. Materials

2. Matter

3. Plasma and fusion physics

4. Molecules

5. Nuclear matter

Sub-molecular

6. Atoms

7. Particles

8. Composite particles

9. Fields

10. Vacuum and spacetime

Full contents

1. Materials (6) Back to index

1. Physics:Quantum materials/band structure

2. Physics:Quantum materials/phonon

3. Physics:Quantum materials/superconductor

4. Physics:Quantum materials/topological phase

5. Physics:Quantum materials/crystal lattice

6. Physics:Quantum materials/solid state

2. Matter (5) Back to index

7. Physics:Quantum matter/phase

8. Physics:Quantum matter/state of matter

9. Physics:Quantum matter/thermodynamic system

10. Physics:Quantum matter/density

11. Physics:Quantum matter/temperature

3. Plasma and fusion physics (6) Back to index

12. Physics:Quantum Tokamak

13. Physics:Quantum Fusion

14. Physics:Quantum matter/plasma

15. Physics:Quantum magnetic confinement

16. Physics:Quantum Lawson criterion

17. Physics:Quantum Quark–gluon plasma

4. Molecules (6) Back to index

18. Physics:Quantum molecular structure

19. Physics:Quantum chemical bond

20. Physics:Quantum molecular orbital

21. Physics:Quantum degenerate orbitals

22. Physics:Quantum molecular spectroscopy

23. Physics:Quantum molecular vibration

5. Nuclear matter (6) Back to index

24. Physics:Quantum atomic nucleus

25. Physics:Quantum nuclear matter

26. Physics:Quantum isotope

27. Physics:Quantum deuteron

28. Physics:Quantum nuclear binding energy

29. Physics:Quantum nuclear force

6. Atoms (7) Back to index

30. Physics:Quantum atoms/electron

31. Physics:Quantum atoms/energy level

32. Physics:Quantum atoms/electron configuration

33. Physics:Quantum atoms/transition

34. Physics:Quantum atoms/ion

35. Physics:Quantum atoms/orbital

36. Physics:Quantum atoms/hydrogen

7. Particles (12) Back to index

37. Physics:Quantum particle

38. Physics:Quantum elementary particle

39. Physics:Quantum fermion

40. Physics:Quantum boson

41. Physics:Quantum lepton

42. Physics:Quantum electron

43. Physics:Quantum neutrino

44. Physics:Quantum quark

45. Physics:Quantum photon

46. Physics:Quantum gluon

47. Physics:Quantum W and Z bosons

48. Physics:Quantum Higgs boson

8. Composite particles (12) Back to index

49. Physics:Quantum hadron

50. Physics:Quantum baryon

51. Physics:Quantum meson

52. Physics:Quantum nucleon

53. Physics:Quantum proton

54. Physics:Quantum neutron

55. Physics:Quantum pion

56. Physics:Quantum kaon

57. Physics:Quantum glueball

58. Physics:Quantum tetraquark

59. Physics:Quantum pentaquark

60. Physics:Quantum exotic hadron

9. Fields (12) Back to index

61. Physics:Quantum field theory

62. Physics:Quantum electromagnetic field

63. Physics:Quantum gauge field

64. Physics:Quantum scalar field

65. Physics:Quantum vacuum field

66. Physics:Quantum wavefunction field

67. Physics:Quantum spinor field

68. Physics:Quantum matter field

69. Physics:Quantum Higgs field

70. Physics:Quantum Yang-Mills field

71. Physics:Quantum photon field

72. Physics:Quantum gluon field

10. Vacuum and spacetime (12) Back to index

73. Physics:Quantum spacetime

74. Physics:Quantum vacuum energy

75. Physics:Quantum virtual particle

76. Physics:Quantum zero-point energy

77. Physics:Quantum curved spacetime

78. Physics:Quantum quantum gravity

79. Physics:Quantum vacuum state

80. Physics:Quantum false vacuum

81. Physics:Quantum Planck scale

82. Physics:Quantum spacetime foam

83. Physics:Quantum metric field

84. Physics:Quantum field theory in curved spacetime

Notes

↑ Prior to Mulliken, the word "orbital" was used only as an adjective, for example "orbital velocity" or "orbital wave function."^[1] Mulliken used orbital as a noun, when he suggested the terms "atomic orbitals" and "molecular orbitals" to describe the electronic structures of polyatomic molecules.^[2]^[3]

References

↑ "orbital". Dictionary by Merriam-Webster: America's most-trusted online dictionary. Merriam-Webster. https://www.merriam-webster.com/dictionary/orbital. Retrieved April 18, 2021.
↑ ^2.0 ^2.1 Mulliken, Robert S. (July 1932). "Electronic Structures of Polyatomic Molecules and Valence. II. General Considerations". Physical Review 41 (1): 49–71. doi:10.1103/PhysRev.41.49. Bibcode: 1932PhRv...41...49M.
↑ Brown, Theodore (2002). Chemistry : the central science. Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-066997-0.
↑ Cotton, F. Albert (1990). Chemical applications of group theory (3rd ed.). New York: Wiley. pp. 102. ISBN 0471510947. OCLC 19975337. https://archive.org/details/isbn_9780471510949/page/102.
↑ Albright, T. A.; Burdett, J. K.; Whangbo, M.-H. (2013). Orbital Interactions in Chemistry. Hoboken, N.J.: Wiley. ISBN 9780471080398.
↑ Hund, F. (1926). "Zur Deutung einiger Erscheinungen in den Molekelspektren" (in de). Zeitschrift für Physik (Springer Science and Business Media LLC) 36 (9–10): 657–674. doi:10.1007/bf01400155. ISSN 1434-6001. Bibcode: 1926ZPhy...36..657H.
↑ F. Hund, "Zur Deutung der Molekelspektren", Zeitschrift für Physik, Part I, vol. 40, pages 742-764 (1927); Part II, vol. 42, pages 93–120 (1927); Part III, vol. 43, pages 805-826 (1927); Part IV, vol. 51, pages 759-795 (1928); Part V, vol. 63, pages 719-751 (1930).
↑ Mulliken, Robert S. (May 1, 1927). "Electronic States and Band Spectrum Structure in Diatomic Molecules. IV. Hund's Theory; Second Positive Nitrogen and Swan Bands; Alternating Intensities". Physical Review (American Physical Society (APS)) 29 (5): 637–649. doi:10.1103/physrev.29.637. ISSN 0031-899X. Bibcode: 1927PhRv...29..637M.
↑ Mulliken, Robert S. (1928). "The assignment of quantum numbers for electrons in molecules. Extracts from Phys. Rev. 32, 186-222 (1928), plus currently written annotations". International Journal of Quantum Chemistry (Wiley) 1 (1): 103–117. doi:10.1002/qua.560010106. ISSN 0020-7608.
↑ Friedrich Hund and Chemistry, Werner Kutzelnigg, on the occasion of Hund's 100th birthday, Angewandte Chemie International Edition, 35, 573–586, (1996)
↑ Robert S. Mulliken's Nobel Lecture, Science, 157, no. 3785, 13-24. Available on-line at: Nobelprize.org
↑ Lennard-Jones, John (Sir) (1929). "The electronic structure of some diatomic molecules". Transactions of the Faraday Society 25: 668–686. doi:10.1039/tf9292500668. Bibcode: 1929FaTr...25..668L. https://www.chemteam.info/Chem-History/Lennard-Jones-1929/Lennard-Jones-1929.html.
↑ ^13.0 ^13.1 Miessler, G.L.; Tarr, Donald A. (2008). Inorganic Chemistry. Pearson Education. ISBN 978-81-317-1885-8. https://books.google.com/books?id=rBfolO_rhf8C.
↑ ^14.0 ^14.1 ^14.2 ^14.3 ^14.4 Catherine E. Housecroft, Alan G. Sharpe, Inorganic Chemistry, Pearson Prentice Hall; 2nd Edition, 2005, p. 29-33.
↑ Peter Atkins; Julio De Paula. Atkins’ Physical Chemistry. Oxford University Press, 8th ed., 2006.
↑ Yves Jean; François Volatron. An Introduction to Molecular Orbitals. Oxford University Press, 1993.
↑ Michael Munowitz, Principles of Chemistry, Norton & Company, 2000, p. 229-233.
↑ Atkins, Peter (2006). Inorganic chemistry (4. ed.). New York: W.H. Freeman. p. 208. ISBN 978-0-7167-4878-6.
↑ "Ionic bond". IUPAC Compendium of Chemical Terminology. 2009. doi:10.1351/goldbook.IT07058. ISBN 978-0-9678550-9-7. https://doi.org/10.1351/goldbook.IT07058.
↑ "5.3.3: Ionic Compounds and Molecular Orbitals" (in en). August 6, 2020. https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Inorganic_Chemistry_(LibreTexts)/05%3A_Molecular_Orbitals/5.03%3A_Heteronuclear_Diatomic_Molecules/5.3.03%3A_Ionic_Compounds_and_Molecular_Orbitals.
↑ Bondybey, V.E. (1984). "Electronic structure and bonding of Be2". Chemical Physics Letters 109 (5): 436–441. doi:10.1016/0009-2614(84)80339-5. Bibcode: 1984CPL...109..436B.
↑ ^22.0 ^22.1 Bredas, Jean-Luc (2014). "Mind the gap!". Mater. Horiz. 1 (1): 17–19. doi:10.1039/C3MH00098B.
↑ König, Burkhard (February 21, 1995). "Chemical Bonding. VonM. J. Winter. 90 S., ISBN 0-19-855694-2. – Organometallics 1. Complexes with Transition Metal-Carbon σ-Bonds. VonM. Bochmann. 91 S., ISBN 0-19-855751-5. – Organometallics 2. Complexes with Transition Metal-Carbon π-Bonds. VonM. Bochmann. 89 S., ISBN 0-19-855813-9. – Bifunctional Compounds. VonR. S. Ward. 90 S., ISBN 0-19-855808-2. – Alle aus der Reihe: Oxford Chemistry Primers, Oxford University Press, Oxford, 1994, Broschur, je 4.99 £" (in de). Angewandte Chemie 107 (4): 540. doi:10.1002/ange.19951070434. Bibcode: 1995AngCh.107..540K. https://onlinelibrary.wiley.com/doi/10.1002/ange.19951070434.
↑ Atkins, Peter W.; Friedman, Ronald (2011). Molecular quantum mechanics (5. ed.). Oxford: Oxford Univ. Press. ISBN 978-0-19-954142-3.
↑ Catherine E. Housecroft, Alan G, Sharpe, Inorganic Chemistry, Pearson Prentice Hall; 2nd Edition, 2005, ISBN 0130-39913-2, p. 41-43.
↑ Szabo, Attila; Ostlund, Neil S. (2012). Modern quantum chemistry: introduction to advanced electronic structure theory. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-69186-2.
↑ Gold, Victor, ed (2019) (in en). The IUPAC Compendium of Chemical Terminology: The Gold Book (4 ed.). Research Triangle Park, NC: International Union of Pure and Applied Chemistry (IUPAC). doi:10.1351/goldbook.k03411. https://goldbook.iupac.org/.

Author: Harold Foppele

Source attribution: Physics:Quantum molecular orbital

[4] Prior to Mulliken, the word "orbital" was used only as an adjective, for example "orbital velocity" or "orbital wave function."^[1] Mulliken used orbital as a noun, when he suggested the terms "atomic orbitals" and "molecular orbitals" to describe the electronic structures of polyatomic molecules.^[2]^[3]

[1] "orbital". Dictionary by Merriam-Webster: America's most-trusted online dictionary. Merriam-Webster. https://www.merriam-webster.com/dictionary/orbital. Retrieved April 18, 2021.

[muller1932-2] 2.0 ^2.1 Mulliken, Robert S. (July 1932). "Electronic Structures of Polyatomic Molecules and Valence. II. General Considerations". Physical Review 41 (1): 49–71. doi:10.1103/PhysRev.41.49. Bibcode: 1932PhRv...41...49M.

[3] Brown, Theodore (2002). Chemistry : the central science. Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-066997-0.

[5] Cotton, F. Albert (1990). Chemical applications of group theory (3rd ed.). New York: Wiley. pp. 102. ISBN 0471510947. OCLC 19975337. https://archive.org/details/isbn_9780471510949/page/102.

[6] Albright, T. A.; Burdett, J. K.; Whangbo, M.-H. (2013). Orbital Interactions in Chemistry. Hoboken, N.J.: Wiley. ISBN 9780471080398.

[Hund1926-7] Hund, F. (1926). "Zur Deutung einiger Erscheinungen in den Molekelspektren" (in de). Zeitschrift für Physik (Springer Science and Business Media LLC) 36 (9–10): 657–674. doi:10.1007/bf01400155. ISSN 1434-6001. Bibcode: 1926ZPhy...36..657H.

[8] F. Hund, "Zur Deutung der Molekelspektren", Zeitschrift für Physik, Part I, vol. 40, pages 742-764 (1927); Part II, vol. 42, pages 93–120 (1927); Part III, vol. 43, pages 805-826 (1927); Part IV, vol. 51, pages 759-795 (1928); Part V, vol. 63, pages 719-751 (1930).

[Mulliken1927-9] Mulliken, Robert S. (May 1, 1927). "Electronic States and Band Spectrum Structure in Diatomic Molecules. IV. Hund's Theory; Second Positive Nitrogen and Swan Bands; Alternating Intensities". Physical Review (American Physical Society (APS)) 29 (5): 637–649. doi:10.1103/physrev.29.637. ISSN 0031-899X. Bibcode: 1927PhRv...29..637M.

[Mulliken1928-10] Mulliken, Robert S. (1928). "The assignment of quantum numbers for electrons in molecules. Extracts from Phys. Rev. 32, 186-222 (1928), plus currently written annotations". International Journal of Quantum Chemistry (Wiley) 1 (1): 103–117. doi:10.1002/qua.560010106. ISSN 0020-7608.

[11] Friedrich Hund and Chemistry, Werner Kutzelnigg, on the occasion of Hund's 100th birthday, Angewandte Chemie International Edition, 35, 573–586, (1996)

[12] Robert S. Mulliken's Nobel Lecture, Science, 157, no. 3785, 13-24. Available on-line at: Nobelprize.org

[13] Lennard-Jones, John (Sir) (1929). "The electronic structure of some diatomic molecules". Transactions of the Faraday Society 25: 668–686. doi:10.1039/tf9292500668. Bibcode: 1929FaTr...25..668L. https://www.chemteam.info/Chem-History/Lennard-Jones-1929/Lennard-Jones-1929.html.

[Gary_L._Miessler_2004-14] 13.0 ^13.1 Miessler, G.L.; Tarr, Donald A. (2008). Inorganic Chemistry. Pearson Education. ISBN 978-81-317-1885-8. https://books.google.com/books?id=rBfolO_rhf8C.

[H&C-15] 14.0 ^14.1 ^14.2 ^14.3 ^14.4 Catherine E. Housecroft, Alan G. Sharpe, Inorganic Chemistry, Pearson Prentice Hall; 2nd Edition, 2005, p. 29-33.

[16] Peter Atkins; Julio De Paula. Atkins’ Physical Chemistry. Oxford University Press, 8th ed., 2006.

[17] Yves Jean; François Volatron. An Introduction to Molecular Orbitals. Oxford University Press, 1993.

[18] Michael Munowitz, Principles of Chemistry, Norton & Company, 2000, p. 229-233.

[19] Atkins, Peter (2006). Inorganic chemistry (4. ed.). New York: W.H. Freeman. p. 208. ISBN 978-0-7167-4878-6.

[20] "Ionic bond". IUPAC Compendium of Chemical Terminology. 2009. doi:10.1351/goldbook.IT07058. ISBN 978-0-9678550-9-7. https://doi.org/10.1351/goldbook.IT07058.

[21] "5.3.3: Ionic Compounds and Molecular Orbitals" (in en). August 6, 2020. https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Inorganic_Chemistry_(LibreTexts)/05%3A_Molecular_Orbitals/5.03%3A_Heteronuclear_Diatomic_Molecules/5.3.03%3A_Ionic_Compounds_and_Molecular_Orbitals.

[22] Bondybey, V.E. (1984). "Electronic structure and bonding of Be2". Chemical Physics Letters 109 (5): 436–441. doi:10.1016/0009-2614(84)80339-5. Bibcode: 1984CPL...109..436B.

[Bredas2014-23] 22.0 ^22.1 Bredas, Jean-Luc (2014). "Mind the gap!". Mater. Horiz. 1 (1): 17–19. doi:10.1039/C3MH00098B.

[24] König, Burkhard (February 21, 1995). "Chemical Bonding. VonM. J. Winter. 90 S., ISBN 0-19-855694-2. – Organometallics 1. Complexes with Transition Metal-Carbon σ-Bonds. VonM. Bochmann. 91 S., ISBN 0-19-855751-5. – Organometallics 2. Complexes with Transition Metal-Carbon π-Bonds. VonM. Bochmann. 89 S., ISBN 0-19-855813-9. – Bifunctional Compounds. VonR. S. Ward. 90 S., ISBN 0-19-855808-2. – Alle aus der Reihe: Oxford Chemistry Primers, Oxford University Press, Oxford, 1994, Broschur, je 4.99 £" (in de). Angewandte Chemie 107 (4): 540. doi:10.1002/ange.19951070434. Bibcode: 1995AngCh.107..540K. https://onlinelibrary.wiley.com/doi/10.1002/ange.19951070434.

[25] Atkins, Peter W.; Friedman, Ronald (2011). Molecular quantum mechanics (5. ed.). Oxford: Oxford Univ. Press. ISBN 978-0-19-954142-3.

[26] Catherine E. Housecroft, Alan G, Sharpe, Inorganic Chemistry, Pearson Prentice Hall; 2nd Edition, 2005, ISBN 0130-39913-2, p. 41-43.

[27] Szabo, Attila; Ostlund, Neil S. (2012). Modern quantum chemistry: introduction to advanced electronic structure theory. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-69186-2.

[28] Gold, Victor, ed (2019) (in en). The IUPAC Compendium of Chemical Terminology: The Gold Book (4 ed.). Research Triangle Park, NC: International Union of Pure and Applied Chemistry (IUPAC). doi:10.1351/goldbook.k03411. https://goldbook.iupac.org/.

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@@ Line 1: / Line 1: @@
 {{Short description|Wave-like behavior of an electron in a molecule}}
+{{Quantum matter backlink|Molecules}}
+{{Quantum article nav|previous=Physics:Quantum chemical bond|previous label=Chemical bond|next=Physics:Quantum degenerate orbitals|next label=Degenerate orbitals}}
+<!-- PageTitle: Physics:Quantum molecular orbital -->
-{{Quantum book backlink|Molecules}}
-<!-- PageTitle: Physics:Quantum molecular orbital -->
-{{Quantum matter backlink|Molecules}}
 {{See also|Physics:Quantum Molecular orbital theory|Physics:Quantum Molecular orbital diagram}}
 {{Use mdy dates|date=October 2025}}
 <div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">
@@ Line 16: / Line 15: @@
 '''Molecular orbitals''' describe how [[Physics:Quantum atoms/electron|electrons]] can be spread across an entire [[Physics:Quantum molecular structure|molecule]] rather than belonging to a single atom. In quantum terms, they are wave-like solutions used to model bonding, antibonding, electron density, and molecular energy levels.
-In [[Physics:Quantum chemistry|chemistry]], a '''molecular orbital''' is a [[Function (mathematics)|mathematical function]] describing the location and [[Physics:Quantum Matter wave|wave-like]] behavior of an [[Physics:Quantum atoms/electron|electron]] in a [[Physics:Quantum molecular structure|molecule]]. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region.
+In [[Physics:Quantum chemistry|chemistry]], a '''molecular orbital''' is a mathematical function describing the location and [[Physics:Quantum Matter wave|wave-like]] behavior of an [[Physics:Quantum atoms/electron|electron]] in a [[Physics:Quantum molecular structure|molecule]]. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region.
-The terms ''atomic orbital'' and ''molecular orbital''{{efn|Prior to Mulliken, the word "orbital" was used only as an [[adjective]], for example "orbital velocity" or "orbital wave function."<ref>{{cite web|title=orbital |publisher=[[Merriam-Webster]] |work=Dictionary by Merriam-Webster: America's most-trusted online dictionary |url=https://www.merriam-webster.com/dictionary/orbital |accessdate=April 18, 2021}}</ref> Mulliken used orbital as a [[noun]], when he suggested the terms "atomic orbitals" and "molecular orbitals" to describe the electronic structures of polyatomic molecules.<ref name="muller1932"/><ref>{{cite book | last = Brown | first = Theodore | title = Chemistry : the central science | publisher = Prentice Hall | location = Upper Saddle River, NJ | year = 2002 | isbn = 0-13-066997-0 }}</ref>}} were introduced by [[Biography:Robert S. Mulliken|Robert S. Mulliken]] in 1932 to mean ''one-electron orbital wave functions''.<ref name="muller1932">{{cite journal| last=Mulliken | first=Robert S. | title=Electronic Structures of Polyatomic Molecules and Valence. II. General Considerations |date=July 1932| journal=[[Physical Review]] | volume=41 | issue=1 | pages=49–71 | bibcode = 1932PhRv...41...49M | doi = 10.1103/PhysRev.41.49}}</ref> At an elementary level, they are used to describe the ''region'' of space in which a function has significant amplitude.
+The terms ''atomic orbital'' and ''molecular orbital''{{efn|Prior to Mulliken, the word "orbital" was used only as an adjective, for example "orbital velocity" or "orbital wave function."<ref>{{cite web|title=orbital |publisher=Merriam-Webster |work=Dictionary by Merriam-Webster: America's most-trusted online dictionary |url=https://www.merriam-webster.com/dictionary/orbital |accessdate=April 18, 2021}}</ref> Mulliken used orbital as a noun, when he suggested the terms "atomic orbitals" and "molecular orbitals" to describe the electronic structures of polyatomic molecules.<ref name="muller1932"/><ref>{{cite book | last = Brown | first = Theodore | title = Chemistry : the central science | publisher = Prentice Hall | location = Upper Saddle River, NJ | year = 2002 | isbn = 0-13-066997-0 }}</ref>}} were introduced by Robert S. Mulliken in 1932 to mean ''one-electron orbital wave functions''.<ref name="muller1932">{{cite journal| last=Mulliken | first=Robert S. | title=Electronic Structures of Polyatomic Molecules and Valence. II. General Considerations |date=July 1932| journal=Physical Review | volume=41 | issue=1 | pages=49–71 | bibcode = 1932PhRv...41...49M | doi = 10.1103/PhysRev.41.49}}</ref> At an elementary level, they are used to describe the ''region'' of space in which a function has significant amplitude.
 In an isolated atom, electron locations are described by [[Physics:Quantum atoms/orbital|atomic orbitals]]. When multiple atoms combine chemically into a molecule by forming a chemical bond, the electrons' locations are determined by the molecule as a whole, so the atomic orbitals combine to form molecular orbitals. The electrons from the constituent atoms occupy these molecular orbitals.
@@ Line 26: / Line 25: @@
 Molecular orbitals are commonly divided into three types: ''[[Physics:Quantum bonding molecular orbital|bonding orbitals]]'', which have an energy lower than the atomic orbitals that formed them and promote chemical bonding; ''[[Physics:Quantum antibonding molecular orbital|antibonding orbitals]]'', which have higher energy and oppose bonding; and ''[[Physics:Quantum non-bonding orbital|non-bonding orbitals]]'', which have approximately the same energy as their constituent atomic orbitals and have little effect on bonding.
-<div style="float:right; border:1px solid #e0d890; background:#fff8cc; padding:6px; margin:0 0 1em 1em; width:420px;">
-<div style="font-size:90%;">
-Simplified molecular orbital diagram showing bonding and antibonding orbitals formed from atomic orbitals.
-</div>
-</div>
 </div>
 <div style="width:300px;">
-[[File:Molecular_orbital_schema.jpg|thumb|280px|Quantum molecular orbital.]]
+[[File:Molecular_orbital_schema.jpg|thumb|280px|Simplified molecular orbital diagram showing bonding and antibonding orbitals formed from atomic orbitals.]]
 </div>
@@ Line 46: / Line 40: @@
 Most present-day methods in [[Physics:Quantum computational chemistry|computational chemistry]] begin by calculating the molecular orbitals of the system. A molecular orbital describes the behavior of one electron in the electric field generated by the nuclei and by an average distribution of the other electrons. If two electrons occupy the same orbital, the [[Physics:Quantum Multi-electron atoms#Pauli principle and exchange symmetry|Pauli principle]] requires that they have opposite spin. This is an approximation: highly accurate descriptions of molecular electronic wave functions do not rely only on orbitals, as in [[Physics:Quantum Multi-electron atoms#Configuration interaction|configuration interaction]].
-Molecular orbitals are usually delocalized over the whole molecule. If a molecule has symmetry elements, its nondegenerate molecular orbitals are either symmetric or antisymmetric with respect to those symmetries. For example, applying a symmetry operation '''S''' to a molecular orbital ψ can leave it unchanged or reverse its sign: '''S'''ψ = ±ψ. In planar molecules, orbitals may be symmetric ([[Physics:Quantum sigma bond|sigma]]) or antisymmetric ([[pi bond|pi]]) with respect to reflection in the molecular plane. More generally, molecular orbitals form bases for the [[Physics:Quantum methods/quantum gate#Representation|irreducible representation]]s of the molecule's [[Physics:Quantum chromodynamics#Symmetry groups|symmetry group]].<ref>{{Cite book|url=https://archive.org/details/isbn_9780471510949/page/102|title=Chemical applications of group theory|last=Cotton|first=F. Albert|date=1990|publisher=Wiley|isbn=0471510947|edition=3rd|location=New York|pages=[https://archive.org/details/isbn_9780471510949/page/102 102]|oclc=19975337|url-access=registration}}</ref>
+Molecular orbitals are usually delocalized over the whole molecule. If a molecule has symmetry elements, its nondegenerate molecular orbitals are either symmetric or antisymmetric with respect to those symmetries. For example, applying a symmetry operation '''S''' to a molecular orbital ψ can leave it unchanged or reverse its sign: '''S'''ψ = ±ψ. In planar molecules, orbitals may be symmetric ([[Physics:Quantum sigma bond|sigma]]) or antisymmetric (pi) with respect to reflection in the molecular plane. More generally, molecular orbitals form bases for the [[Physics:Quantum methods/quantum gate#Representation|irreducible representation]]s of the molecule's [[Physics:Quantum chromodynamics#Symmetry groups|symmetry group]].<ref>{{Cite book|url=https://archive.org/details/isbn_9780471510949/page/102|title=Chemical applications of group theory|last=Cotton|first=F. Albert|date=1990|publisher=Wiley|isbn=0471510947|edition=3rd|location=New York|pages=[https://archive.org/details/isbn_9780471510949/page/102 102]|oclc=19975337|url-access=registration}}</ref>
 The delocalized nature of molecular orbitals distinguishes molecular orbital theory from [[Physics:Quantum valence bond theory|valence bond theory]], where bonds are often treated as localized electron pairs. The two descriptions are complementary. Localized molecular orbitals can also be formed by mathematical transformations of canonical orbitals; these localized orbitals resemble bonds in a Lewis structure, but their individual energy levels no longer have the same direct physical meaning.
 == Formation of molecular orbitals ==
-Molecular orbitals arise from allowed interactions between [[Physics:Quantum atoms/orbital|atomic orbitals]]. Such interactions are allowed when the symmetries of the atomic orbitals are compatible. The strength of the interaction depends on [[Orbital overlap|orbital overlap]] and on how close the atomic orbitals are in energy.
+Molecular orbitals arise from allowed interactions between [[Physics:Quantum atoms/orbital|atomic orbitals]]. Such interactions are allowed when the symmetries of the atomic orbitals are compatible. The strength of the interaction depends on orbital overlap and on how close the atomic orbitals are in energy.
 The number of molecular orbitals formed must equal the number of atomic orbitals combined to form the molecule.
@@ Line 61: / Line 55: @@
 {{main|Physics:Quantum linear combination of atomic orbitals}}
-Molecular orbitals were first introduced by [[Biography:Friedrich Hund|Friedrich Hund]]<ref name="Hund1926">{{cite journal | last=Hund | first=F. | title=Zur Deutung einiger Erscheinungen in den Molekelspektren |trans-title=On the interpretation of some phenomena in molecular spectra | journal=Zeitschrift für Physik | publisher=Springer Science and Business Media LLC | volume=36 | issue=9–10 | year=1926 | issn=1434-6001 | doi=10.1007/bf01400155 | pages=657–674 | bibcode=1926ZPhy...36..657H | s2cid=123208730 | language=de}}</ref><ref>F. Hund, "Zur Deutung der Molekelspektren", ''Zeitschrift für Physik'', Part I, vol. 40, pages 742-764 (1927); Part II, vol. 42, pages 93–120 (1927); Part III, vol. 43, pages 805-826 (1927); Part IV, vol. 51, pages 759-795 (1928); Part V, vol. 63, pages 719-751 (1930).</ref> and [[Biography:Robert S. Mulliken|Robert S. Mulliken]]<ref name="Mulliken1927">{{cite journal | last=Mulliken | first=Robert S. | title=Electronic States and Band Spectrum Structure in Diatomic Molecules. IV. Hund's Theory; Second Positive Nitrogen and Swan Bands; Alternating Intensities | journal=[[Physical Review]] | publisher=American Physical Society (APS) | volume=29 | issue=5 | date=May 1, 1927 | issn=0031-899X | doi=10.1103/physrev.29.637 | pages=637–649| bibcode=1927PhRv...29..637M }}</ref><ref name="Mulliken1928">{{cite journal | last=Mulliken | first=Robert S. | title=The assignment of quantum numbers for electrons in molecules. Extracts from Phys. Rev. 32, 186-222 (1928), plus currently written annotations | journal=International Journal of Quantum Chemistry | publisher=Wiley | volume=1 | issue=1 | year=1928 | issn=0020-7608 | doi=10.1002/qua.560010106 | pages=103–117}}</ref> in 1927 and 1928.<ref>[[Friedrich Hund]] and Chemistry, [[Werner Kutzelnigg]], on the occasion of Hund's 100th birthday, ''[[Angewandte Chemie International Edition]]'', 35, 573–586, (1996)</ref><ref>[[Robert S. Mulliken]]'s Nobel Lecture, ''[[Science (journal)|Science]]'', 157, no. 3785, 13-24. Available on-line at: [http://nobelprize.org/nobel_prizes/chemistry/laureates/1966/mulliken-lecture.pdf Nobelprize.org]</ref> The [[Physics:Quantum molecular orbital#Linear combinations of atomic orbitals (LCAO)|linear combination of atomic orbitals]] or LCAO approximation for molecular orbitals was introduced in 1929 by [[Biography:John Lennard-Jones|Sir John Lennard-Jones]].<ref>{{cite journal| url=https://www.chemteam.info/Chem-History/Lennard-Jones-1929/Lennard-Jones-1929.html |last1=Lennard-Jones |first1=John (Sir) |author1-link=John Lennard-Jones |title=The electronic structure of some diatomic molecules |journal=Transactions of the Faraday Society |volume=25 |pages=668–686 |date=1929|doi=10.1039/tf9292500668 |bibcode=1929FaTr...25..668L |url-access=subscription }}</ref>
+Molecular orbitals were first introduced by Friedrich Hund<ref name="Hund1926">{{cite journal | last=Hund | first=F. | title=Zur Deutung einiger Erscheinungen in den Molekelspektren |trans-title=On the interpretation of some phenomena in molecular spectra | journal=Zeitschrift für Physik | publisher=Springer Science and Business Media LLC | volume=36 | issue=9–10 | year=1926 | issn=1434-6001 | doi=10.1007/bf01400155 | pages=657–674 | bibcode=1926ZPhy...36..657H | s2cid=123208730 | language=de}}</ref><ref>F. Hund, "Zur Deutung der Molekelspektren", ''Zeitschrift für Physik'', Part I, vol. 40, pages 742-764 (1927); Part II, vol. 42, pages 93–120 (1927); Part III, vol. 43, pages 805-826 (1927); Part IV, vol. 51, pages 759-795 (1928); Part V, vol. 63, pages 719-751 (1930).</ref> and Robert S. Mulliken<ref name="Mulliken1927">{{cite journal | last=Mulliken | first=Robert S. | title=Electronic States and Band Spectrum Structure in Diatomic Molecules. IV. Hund's Theory; Second Positive Nitrogen and Swan Bands; Alternating Intensities | journal=Physical Review | publisher=American Physical Society (APS) | volume=29 | issue=5 | date=May 1, 1927 | issn=0031-899X | doi=10.1103/physrev.29.637 | pages=637–649| bibcode=1927PhRv...29..637M }}</ref><ref name="Mulliken1928">{{cite journal | last=Mulliken | first=Robert S. | title=The assignment of quantum numbers for electrons in molecules. Extracts from Phys. Rev. 32, 186-222 (1928), plus currently written annotations | journal=International Journal of Quantum Chemistry | publisher=Wiley | volume=1 | issue=1 | year=1928 | issn=0020-7608 | doi=10.1002/qua.560010106 | pages=103–117}}</ref> in 1927 and 1928.<ref>Friedrich Hund and Chemistry, Werner Kutzelnigg, on the occasion of Hund's 100th birthday, ''Angewandte Chemie International Edition'', 35, 573–586, (1996)</ref><ref>Robert S. Mulliken's Nobel Lecture, ''Science'', 157, no. 3785, 13-24. Available on-line at: [http://nobelprize.org/nobel_prizes/chemistry/laureates/1966/mulliken-lecture.pdf Nobelprize.org]</ref> The [[Physics:Quantum molecular orbital#Linear combinations of atomic orbitals (LCAO)|linear combination of atomic orbitals]] or LCAO approximation for molecular orbitals was introduced in 1929 by Sir John Lennard-Jones.<ref>{{cite journal| url=https://www.chemteam.info/Chem-History/Lennard-Jones-1929/Lennard-Jones-1929.html |last1=Lennard-Jones |first1=John (Sir) |author1-link=John Lennard-Jones |title=The electronic structure of some diatomic molecules |journal=Transactions of the Faraday Society |volume=25 |pages=668–686 |date=1929|doi=10.1039/tf9292500668 |bibcode=1929FaTr...25..668L |url-access=subscription }}</ref>
 Linear combinations of [[Physics:Quantum atoms/orbital|atomic orbitals]] can be used to estimate the molecular orbitals formed when atoms bond. For simple diatomic molecules, the bonding and antibonding wavefunctions may be represented as:
@@ Line 92: / Line 86: @@
 ===σ symmetry===
-{{Further|Physics:Quantum sigma bond}}
 A molecular orbital with σ symmetry can result from interaction between two s orbitals or between two p<sub>z</sub> orbitals. It is symmetric with respect to rotation around the internuclear axis. A σ* antibonding orbital also has σ symmetry, but contains a nodal plane between the nuclei and perpendicular to the internuclear axis.<ref name = H&C>Catherine E. Housecroft, Alan G. Sharpe, ''Inorganic Chemistry'', Pearson Prentice Hall; 2nd Edition, 2005, p. 29-33.</ref>
 ===π symmetry===
-{{Further|Physics:Quantum pi bond}}
 A molecular orbital with π symmetry results from side-by-side interaction of p orbitals, such as p<sub>x</sub> or p<sub>y</sub> orbitals. A π orbital changes phase under rotation about the internuclear axis and has one nodal plane containing that axis. A π* antibonding orbital has an additional nodal plane between the nuclei.<ref name = H&C /><ref>Peter Atkins; Julio De Paula. ''Atkins’ Physical Chemistry''. Oxford University Press, 8th ed., 2006.</ref><ref>Yves Jean; François Volatron. ''An Introduction to Molecular Orbitals''. Oxford University Press, 1993.</ref><ref>Michael Munowitz, ''Principles of Chemistry'', Norton & Company, 2000, p. 229-233.</ref>
 ===δ symmetry===
-{{Further|Physics:Quantum delta bond}}
 A molecular orbital with δ symmetry results from interaction between suitable d orbitals. Such orbitals are important in some transition-metal complexes. A δ bonding orbital has two nodal planes containing the internuclear axis, while a δ* antibonding orbital also has a nodal plane between the nuclei.
 ===φ symmetry===
-{{Further|Physics:Quantum phi bond}}
 Theoretical chemists have proposed higher-order bonds such as φ bonds, which would correspond to overlap of f atomic orbitals. No established molecule is known to contain a φ bond.
@@ Line 163: / Line 153: @@
 {{main|Chemistry:HOMO and LUMO}}
-The highest occupied molecular orbital is called the HOMO, and the lowest unoccupied molecular orbital is called the LUMO. The energy difference between them is the [[Chemistry:HOMO and LUMO#gap|HOMO–LUMO gap]]. It is important in chemical reactivity, optical absorption, and electronic transport.
+The highest occupied molecular orbital is called the HOMO, and the lowest unoccupied molecular orbital is called the LUMO. The energy difference between them is the HOMO–LUMO gap. It is important in chemical reactivity, optical absorption, and electronic transport.
-For Hartree–Fock calculations where [[Chemistry:Koopmans' theorem|Koopmans' theorem]] applies, the HOMO–LUMO gap can be interpreted as an approximation to the difference between vertical ionization potential and vertical electron affinity. This interpretation must be used carefully, because the HOMO–LUMO gap is not always the same as the optical gap, fundamental gap, or bulk-material band gap.<ref name=Bredas2014>{{cite journal |last1=Bredas |first1=Jean-Luc |title=Mind the gap! |journal=Mater. Horiz. |date=2014 |volume=1 |issue=1 |pages=17–19 |doi=10.1039/C3MH00098B}}</ref>{{rp|pp=17–18}}<ref name=Bredas2014/>{{rp|p=18}}
+For Hartree–Fock calculations where Koopmans' theorem applies, the HOMO–LUMO gap can be interpreted as an approximation to the difference between vertical ionization potential and vertical electron affinity. This interpretation must be used carefully, because the HOMO–LUMO gap is not always the same as the optical gap, fundamental gap, or bulk-material band gap.<ref name=Bredas2014>{{cite journal |last1=Bredas |first1=Jean-Luc |title=Mind the gap! |journal=Mater. Horiz. |date=2014 |volume=1 |issue=1 |pages=17–19 |doi=10.1039/C3MH00098B}}</ref>{{rp|pp=17–18}}<ref name=Bredas2014/>{{rp|p=18}}
 == Examples ==
@@ Line 189: / Line 179: @@
 ===Li<sub>2</sub>===
-[[Chemistry:Dilithium|Dilithium]] Li<sub>2</sub> forms from overlap of atomic orbitals on two lithium atoms. Each lithium atom contributes three electrons, and the molecular orbital filling gives a bond order of one.<ref>{{Cite journal |last=König |first=Burkhard |date=February 21, 1995 |title=Chemical Bonding. VonM. J. Winter. 90 S., ISBN 0-19-855694-2. – Organometallics 1. Complexes with Transition Metal-Carbon σ-Bonds. VonM. Bochmann. 91 S., ISBN 0-19-855751-5. – Organometallics 2. Complexes with Transition Metal-Carbon π-Bonds. VonM. Bochmann. 89 S., ISBN 0-19-855813-9. – Bifunctional Compounds. VonR. S. Ward. 90 S., ISBN 0-19-855808-2. – Alle aus der Reihe: Oxford Chemistry Primers, Oxford University Press, Oxford, 1994, Broschur, je 4.99 £ |url=https://onlinelibrary.wiley.com/doi/10.1002/ange.19951070434 |journal=Angewandte Chemie |language=de |volume=107 |issue=4 |pages=540 |doi=10.1002/ange.19951070434|bibcode=1995AngCh.107..540K |url-access=subscription }}</ref>
+Dilithium Li<sub>2</sub> forms from overlap of atomic orbitals on two lithium atoms. Each lithium atom contributes three electrons, and the molecular orbital filling gives a bond order of one.<ref>{{Cite journal |last=König |first=Burkhard |date=February 21, 1995 |title=Chemical Bonding. VonM. J. Winter. 90 S., ISBN 0-19-855694-2. – Organometallics 1. Complexes with Transition Metal-Carbon σ-Bonds. VonM. Bochmann. 91 S., ISBN 0-19-855751-5. – Organometallics 2. Complexes with Transition Metal-Carbon π-Bonds. VonM. Bochmann. 89 S., ISBN 0-19-855813-9. – Bifunctional Compounds. VonR. S. Ward. 90 S., ISBN 0-19-855808-2. – Alle aus der Reihe: Oxford Chemistry Primers, Oxford University Press, Oxford, 1994, Broschur, je 4.99 £ |url=https://onlinelibrary.wiley.com/doi/10.1002/ange.19951070434 |journal=Angewandte Chemie |language=de |volume=107 |issue=4 |pages=540 |doi=10.1002/ange.19951070434|bibcode=1995AngCh.107..540K |url-access=subscription }}</ref>
 ==Heteronuclear diatomics==
@@ Line 202: / Line 192: @@
 Quantitative molecular orbital calculations require orbitals that allow the electronic wavefunction to converge toward the full configuration-interaction limit. The most common starting point is the [[Physics:Quantum Multi-electron atoms#Hartree and Hartree–Fock methods|Hartree–Fock method]], which expresses molecular orbitals as eigenfunctions of the [[Physics:Quantum fock state|Fock operator]].
-In practical calculations, molecular orbitals are expanded as linear combinations of basis functions, often [[Gaussian function]]s centered on the atomic nuclei. The resulting equations for the coefficients form a generalized eigenvalue equation known as the [[Chemistry:Roothaan equations|Roothaan equations]]. Quantum chemical programs such as [[Software:Spartan (chemistry software)|Spartan]] can perform such molecular orbital calculations.<ref>{{Cite book |last1=Szabo |first1=Attila |title=Modern quantum chemistry: introduction to advanced electronic structure theory |last2=Ostlund |first2=Neil S. |date=2012 |publisher=Dover Publications, Inc |isbn=978-0-486-69186-2 |location=Mineola, New York}}</ref>
+In practical calculations, molecular orbitals are expanded as linear combinations of basis functions, often Gaussian functions centered on the atomic nuclei. The resulting equations for the coefficients form a generalized eigenvalue equation known as the Roothaan equations. Quantum chemical programs such as Spartan can perform such molecular orbital calculations.<ref>{{Cite book |last1=Szabo |first1=Attila |title=Modern quantum chemistry: introduction to advanced electronic structure theory |last2=Ostlund |first2=Neil S. |date=2012 |publisher=Dover Publications, Inc |isbn=978-0-486-69186-2 |location=Mineola, New York}}</ref>
 Experimental techniques such as ultraviolet photoelectron spectroscopy and X-ray photoelectron spectroscopy measure ionization energies rather than orbital energies directly. Ionization energies can sometimes be related approximately to orbital energies through Koopmans' theorem, but the agreement can be poor in some molecules.<ref>{{Cite book |url=https://goldbook.iupac.org/ |title=The IUPAC Compendium of Chemical Terminology: The Gold Book |date=2019 |publisher=International Union of Pure and Applied Chemistry (IUPAC) |editor-last=Gold |editor-first=Victor |edition=4 |location=Research Triangle Park, NC |language=en |doi=10.1351/goldbook.k03411}}</ref>
@@ Line 209: / Line 199: @@
 {{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also/Matter}}
-== Notes ==
+= Notes =
-{{notes}}
+{{reflist|group=lower-alpha}}
+= References =
-=References=
 {{reflist|3}}
 {{Author|Harold Foppele}}
-{{Sourceattribution|Molecular orbital|1}}
+{{Sourceattribution|Physics:Quantum molecular orbital|1}}

Physics:Quantum molecular orbital: Difference between revisions

Latest revision as of 00:42, 24 May 2026

Contents

Overview

Formation of molecular orbitals

Qualitative discussion

Linear combinations of atomic orbitals (LCAO)

Bonding, antibonding, and nonbonding MOs

Sigma and pi labels for MOs

σ symmetry

π symmetry

δ symmetry

φ symmetry

Gerade and ungerade symmetry

MO diagrams

Bonding in molecular orbitals

Orbital degeneracy

Ionic bonds

Bond order

HOMO and LUMO

Examples

Homonuclear diatomics

H₂

He₂

Li₂

Heteronuclear diatomics

HF

Quantitative approach

See also

Table of contents (84 articles)

Index

Full contents

Notes

References

Navigation menu

Physics:Quantum molecular orbital: Difference between revisions

Latest revision as of 00:42, 24 May 2026

Overview

Formation of molecular orbitals

Qualitative discussion

Linear combinations of atomic orbitals (LCAO)

Bonding, antibonding, and nonbonding MOs

Sigma and pi labels for MOs

σ symmetry

π symmetry

δ symmetry

φ symmetry

Gerade and ungerade symmetry

MO diagrams

Bonding in molecular orbitals

Orbital degeneracy

Ionic bonds

Bond order

HOMO and LUMO

Examples

Homonuclear diatomics

H2

He2

Li2

Heteronuclear diatomics

HF

Quantitative approach

See also

Table of contents (84 articles)

Index

Full contents

Notes

References

Navigation menu

Search

H₂

He₂

Li₂