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{{Short description|Wave-like behavior of an electron in a molecule}}
{{See also|Physics:Quantum Molecular orbital theory|Physics:Quantum Molecular orbital diagram}}
{{Use mdy dates|date=October 2025}}

{{Quantum matter backlink|Molecules}}

'''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.

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.

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.

Mathematically, molecular orbitals are approximate solutions to the [[Physics:Quantum Schrödinger equation|Schrödinger equation]] for electrons in the field of the molecule's [[Physics:Quantum atoms/nucleus|atomic nuclei]]. They are usually constructed by [[Physics:Quantum molecular orbital#Linear combinations of atomic orbitals (LCAO)|combining]] [[Physics:Quantum atoms/orbital|atomic orbitals]] or [[Physics:Quantum Orbital hybridisation|hybrid orbital]]s from each atom of the molecule, or other molecular orbitals from groups of atoms. They can be quantitatively calculated using the [[Physics:Quantum Multi-electron atoms#Hartree and Hartree–Fock methods|Hartree–Fock]] or self-consistent field (SCF) methods.

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.

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[[File:Molecular_orbital_schema.jpg|400px]]
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Simplified molecular orbital diagram showing bonding and antibonding orbitals formed from atomic orbitals.
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=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 [[Physics:Quantum atoms/electron configuration|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 [[Physics:Quantum molecular orbital#Linear combinations of atomic orbitals (LCAO)|linear combination of atomic orbitals]] using the LCAO-MO method. This gives a simple model of bonding in molecules and is central to [[Physics:Quantum molecular orbital theory|molecular orbital theory]].

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>

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.

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 [[Physics:Quantum molecular orbital#Linear combinations of atomic orbitals (LCAO)|Linear combination of atomic orbitals molecular orbital method]] ansatz. In this approach, molecular orbitals are expressed as [[Physics:Quantum molecular orbital#Linear combinations of atomic orbitals (LCAO)|linear combination]]s of atomic orbitals.<ref>{{Cite book|title=Orbital Interactions in Chemistry|last1=Albright|first1=T. A.|last2=Burdett|first2=J. K.|last3=Whangbo|first3=M.-H.|publisher=Wiley|year=2013|isbn=9780471080398|location=Hoboken, N.J.}}</ref>

==Linear combinations of atomic orbitals (LCAO)==
{{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>

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:

:<math>\Psi = c_a \psi_a + c_b \psi_b</math>
:<math>\Psi^* = c_a \psi_a - c_b \psi_b</math>

where <math>\Psi</math> and <math>\Psi^*</math> are the wavefunctions for bonding and antibonding molecular orbitals, <math>\psi_a</math> and <math>\psi_b</math> are atomic wavefunctions from atoms a and b, and <math>c_a</math> and <math>c_b</math> 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.<ref name="Gary L. Miessler 2004">{{cite book | last1=Miessler | first1=G.L. |last2=Tarr |first2=Donald A. | title=Inorganic Chemistry | publisher=Pearson Education | year=2008 | isbn=978-81-317-1885-8 | url=https://books.google.com/books?id=rBfolO_rhf8C}}</ref>

==Bonding, antibonding, and nonbonding MOs==

When [[Physics:Quantum atoms/orbital|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 [[Physics:Quantum Node|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===
{{Further|Physics:Quantum sigma bond}}

A molecular orbital with σ symmetry can result from interaction between two s orbitals or between two pz 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 px or py 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.

==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 σg, while an antibonding σ orbital may be σu. For π orbitals, the bonding and antibonding gerade/ungerade assignments depend on the phase relationship of the interacting p orbitals.<ref name = H&C />

==MO diagrams==
{{main|Physics:Quantum molecular orbital diagram}}

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 [[Physics:Quantum Postulates#Pauli exclusion principle|Pauli exclusion principle]] and [[Physics:Quantum number#Aufbau principle and Hund's rules|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.<ref>{{cite book|last1=Atkins |first1=Peter |display-authors=etal |title=Inorganic chemistry|date=2006|publisher=W.H. Freeman|location=New York|isbn=978-0-7167-4878-6|page=208|edition= 4.}}</ref>

=Bonding in molecular orbitals=

==Orbital degeneracy==
{{main|Physics:Quantum degenerate orbitals}}

Molecular orbitals are degenerate when they have the same energy. For example, in homonuclear diatomic molecules, molecular orbitals derived from px and py atomic orbitals may form two degenerate bonding orbitals and two degenerate antibonding orbitals.<ref name="Gary L. Miessler 2004"/>

==Ionic bonds==
{{main|Physics:Quantum ionic bonding}}

In an ionic bond, oppositely charged [[Physics:Quantum atoms/ion|ions]] are held together by electrostatic attraction.<ref>{{Cite book|chapter-url=https://doi.org/10.1351/goldbook.IT07058|doi = 10.1351/goldbook.IT07058|chapter = Ionic bond|title = IUPAC Compendium of Chemical Terminology|year = 2009|isbn = 978-0-9678550-9-7}}</ref> 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.<ref>{{Cite web |date=August 6, 2020 |title=5.3.3: Ionic Compounds and Molecular Orbitals |url=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 |access-date=June 6, 2024 |website=Chemistry LibreTexts |language=en}}</ref>

==Bond order==
{{main|Physics:Quantum 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.<ref name = H&C /><ref>{{cite journal | last1 = Bondybey | first1 = V.E. | year = 1984 | title = Electronic structure and bonding of Be2 | journal = Chemical Physics Letters | volume = 109 | issue = 5| pages = 436–441 | doi = 10.1016/0009-2614(84)80339-5 | bibcode = 1984CPL...109..436B }}</ref>

==HOMO and LUMO==
{{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.

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}}

=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.<ref name = H&C />

===H2===

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[[File:H2OrbitalsAnimation.gif|300px]]
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Bonding and antibonding molecular orbitals of H2 formed from hydrogen 1s atomic orbitals.
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In the hydrogen molecule H2, two hydrogen 1s [[Physics:Quantum atoms/orbital|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===

[[Chemistry:Dilithium|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.<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==

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.<ref>{{Cite book |last1=Atkins |first1=Peter W. |title=Molecular quantum mechanics |last2=Friedman |first2=Ronald |date=2011 |publisher=Oxford Univ. Press |isbn=978-0-19-954142-3 |edition=5. |location=Oxford}}</ref>

===HF===

In hydrogen fluoride, HF, overlap between the hydrogen 1s orbital and the fluorine 2pz 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.<ref>Catherine E. Housecroft, Alan G, Sharpe, Inorganic Chemistry, Pearson Prentice Hall; 2nd Edition, 2005, {{ISBN|0130-39913-2}}, p. 41-43.</ref>

=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 [[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>

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>

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also/Matter}}

=Notes=
{{notes}}

=References=
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Molecular orbital|1}}

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