Editing Molecular orbital (section)

==Qualitative discussion==
For an imprecise, but qualitatively useful, discussion of the molecular structure, the molecular orbitals can be obtained from the "[[Linear combination of atomic orbitals molecular orbital method]]" [[ansatz]]. Here, the molecular orbitals are expressed as [[linear combination]]s of [[atomic orbital]]s.<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|Linear combination of atomic orbitals}}
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=1 May 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 [[linear combination of atomic orbitals]] or "LCAO" approximation for molecular orbitals was introduced in 1929 by [[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 }}</ref> His ground-breaking paper showed how to derive the electronic structure of the [[fluorine]] and [[oxygen]] molecules from quantum principles. This qualitative approach to molecular orbital theory is part of the start of modern [[quantum chemistry]].
Linear combinations of atomic orbitals (LCAO) can be used to estimate the molecular orbitals that are formed upon bonding between the molecule's constituent atoms. Similar to an atomic orbital, a Schrödinger equation, which describes the behavior of an electron, can be constructed for a molecular orbital as well. Linear combinations of atomic orbitals, or the sums and differences of the atomic wavefunctions, provide approximate solutions to the [[Hartree–Fock method|Hartree–Fock equations]] which correspond to the independent-particle approximation of the molecular [[Schrödinger equation]]. For simple diatomic molecules, the wavefunctions obtained are represented mathematically by the equations

:<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 molecular wavefunctions for the bonding and antibonding molecular orbitals, respectively, <math>\psi_a</math> and <math>\psi_b</math> are the atomic wavefunctions from atoms a and b, respectively, and <math>c_a</math> and <math>c_b</math> are adjustable coefficients. These coefficients can be positive or negative, depending on the energies and symmetries of the individual atomic orbitals. As the two atoms become closer together, their atomic orbitals overlap to produce areas of high electron density, and, as a consequence, molecular orbitals are formed between the two atoms. The atoms are held together by the electrostatic attraction between the positively charged nuclei and the 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 atomic orbitals interact, the resulting molecular orbital can be of three types: bonding, antibonding, or nonbonding.

[[Bonding molecular orbital|Bonding MO]]s:
* Bonding interactions between atomic orbitals are constructive (in-phase) interactions.
* Bonding MOs are lower in energy than the atomic orbitals that combine to produce them.
[[Antibonding molecular orbital|Antibonding MO]]s:
* Antibonding interactions between atomic orbitals are destructive (out-of-phase) interactions, with a [[Node (physics)|nodal plane]] where the wavefunction of the antibonding orbital is zero between the two interacting atoms
* Antibonding MOs are higher in energy than the atomic orbitals that combine to produce them.
[[Non-bonding orbital|Nonbonding MO]]s:
* Nonbonding MOs are the result of no interaction between atomic orbitals because of lack of compatible symmetries.
* Nonbonding MOs will have the same energy as the atomic orbitals of one of the atoms in the molecule.

===Sigma and pi labels for MOs===<!-- this section-title is linked from some redirect pages...do not change it here without also updating them -->
The type of interaction between atomic orbitals can be further categorized by the molecular-orbital symmetry labels σ (sigma), π (pi), δ (delta), φ (phi), γ (gamma) etc. These are the Greek letters corresponding to the atomic orbitals s, p, d, f and g respectively. The number of nodal planes containing the internuclear axis between the atoms concerned is zero for σ MOs, one for π, two for δ, three for φ and four for γ.

====σ symmetry====
{{Further|Sigma bond}}
A MO with σ symmetry results from the interaction of either two atomic s-orbitals or two atomic p<sub>z</sub>-orbitals. An MO will have σ-symmetry if the orbital is symmetric with respect to the axis joining the two nuclear centers, the internuclear axis. This means that rotation of the MO about the internuclear axis does not result in a phase change. A σ* orbital, sigma antibonding orbital, also maintains the same phase when rotated about the internuclear axis. The σ* orbital has a nodal plane that is 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|Pi bond}}
A MO with π symmetry results from the interaction of either two atomic p<sub>x</sub> orbitals or p<sub>y</sub> orbitals. An MO will have π symmetry if the orbital is asymmetric with respect to rotation about the internuclear axis. This means that rotation of the MO about the internuclear axis will result in a phase change. There is one nodal plane containing the internuclear axis, if [[Atomic orbital#Real orbitals|real orbitals]] are considered.

A π* orbital, pi antibonding orbital, will also produce a phase change when rotated about the internuclear axis. The π* orbital also has a second 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|Delta bond}}
A MO with δ symmetry results from the interaction of two atomic d<sub>xy</sub> or d<sub>x<sup>2</sup>-y<sup>2</sup></sub> orbitals. Because these molecular orbitals involve low-energy d atomic orbitals, they are seen in [[transition metal|transition-metal]] complexes. A δ bonding orbital has two nodal planes containing the internuclear axis, and a δ* antibonding orbital also has a third nodal plane between the nuclei.

====φ symmetry====<!-- this section-title is linked from some redirect pages...do not change it here without also updating them -->
{{Further|Phi bond}}
{{multiple image
| footer    = Suitably aligned f atomic orbitals overlap to form phi molecular orbital (a phi bond)
| width     = 120
| image1    = Phi-bond-f-orbitals-2D.png
| image2    = Phi-bond-boundary-surface-diagram-2D.png
| caption   = Suitably aligned f atomic orbitals can overlap to form a phi molecular orbital (a phi bond)
}}
Theoretical chemists have conjectured that higher-order bonds, such as phi bonds corresponding to overlap of f atomic orbitals, are possible. There is no known example of a molecule purported to contain a phi bond.

===Gerade and ungerade symmetry===
For molecules that possess a center of inversion ([[centrosymmetry|centrosymmetric molecules]]) there are additional labels of symmetry that can be applied to molecular orbitals.
Centrosymmetric molecules include:
* [[Homonuclear molecule|Homonuclear]] diatomics, X<sub>2</sub>
* [[Octahedral molecular geometry|Octahedral]], EX<sub>6</sub>
* [[square planar molecular geometry|Square planar]], EX<sub>4</sub>.
Non-centrosymmetric molecules include:
* [[Heteronuclear molecule|Heteronuclear]] diatomics, XY
* [[Tetrahedral molecular geometry|Tetrahedral]], EX<sub>4</sub>.
If inversion through the center of symmetry in a molecule results in the same phases for the molecular orbital, then the MO is said to have gerade (g) symmetry, from the German word for even.
If inversion through the center of symmetry in a molecule results in a phase change for the molecular orbital, then the MO is said to have ungerade (u) symmetry, from the German word for odd.
For a bonding MO with σ-symmetry, the orbital is σ<sub>g</sub> (s'&nbsp;+&nbsp;s<nowiki>''</nowiki> is symmetric), while an antibonding MO with σ-symmetry the orbital is σ<sub>u</sub>, because inversion of s'&nbsp;–&nbsp;s<nowiki>''</nowiki> is antisymmetric.
For a bonding MO with π-symmetry the orbital is π<sub>u</sub> because inversion through the center of symmetry for would produce a sign change (the two p atomic orbitals are in phase with each other but the two lobes have opposite signs), while an antibonding MO with π-symmetry is π<sub>g</sub> because inversion through the center of symmetry for would not produce a sign change (the two p orbitals are antisymmetric by phase).<ref name = H&C />

===MO diagrams===
{{main|Molecular orbital diagram}}

The qualitative approach of MO analysis uses a molecular orbital diagram to visualize bonding interactions in a molecule. In this type of diagram, the molecular orbitals are represented by horizontal lines; the higher a line the higher the energy of the orbital, and degenerate orbitals are placed on the same level with a space between them. Then, the electrons to be placed in the molecular orbitals are slotted in one by one, keeping in mind the Pauli exclusion principle and Hund's rule of maximum multiplicity (only 2 electrons, having opposite spins, per orbital; place as many unpaired electrons on one energy level as possible before starting to pair them). For more complicated molecules, the wave mechanics approach loses utility in a qualitative understanding of bonding (although is still necessary for a quantitative approach).
Some properties:
* A basis set of orbitals includes those atomic orbitals that are available for molecular orbital interactions, which may be bonding or antibonding
* The number of molecular orbitals is equal to the number of atomic orbitals included in the linear expansion or the basis set
* If the molecule has some symmetry, the degenerate atomic orbitals (with the same atomic energy) are grouped in linear combinations (called '''symmetry-adapted atomic orbitals (SO)'''), which belong to the [[Representation theory of finite groups|representation]] of the [[symmetry group]], so the [[wave function]]s that describe the group are known as '''symmetry-adapted linear combinations''' ('''SALC''').
* The number of molecular orbitals belonging to one group representation is equal to the number of symmetry-adapted atomic orbitals belonging to this representation
* Within a particular [[Representation theory of finite groups|representation]], the symmetry-adapted atomic orbitals mix more if their atomic [[energy level]]s are closer.

The general procedure for constructing a molecular orbital diagram for a reasonably simple molecule can be summarized as follows:

1. Assign a point group to the molecule.

2. Look up the shapes of the SALCs.

3. Arrange the SALCs of each molecular fragment in order of energy, noting first whether they stem from ''s'', ''p'', or ''d'' orbitals
(and put them in the order ''s'' < ''p'' < ''d''), and then their number of internuclear nodes.

4. Combine SALCs of the same symmetry type from the two fragments, and from N SALCs form N molecular orbitals.

5. Estimate the relative energies of the molecular orbitals from considerations of overlap and relative energies of the parent orbitals, and draw the levels on a molecular orbital energy level diagram (showing the origin of the orbitals).

6. Confirm, correct, and revise this qualitative order by carrying out a molecular orbital calculation by using commercial software.<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|Degenerate orbital}}
Molecular orbitals are said to be degenerate if they have the same energy. For example, in the homonuclear diatomic molecules of the first ten elements, the molecular orbitals derived from the p<sub>x</sub> and the p<sub>y</sub> atomic orbitals result in two degenerate bonding orbitals (of low energy) and two degenerate antibonding orbitals (of high energy).<ref name="Gary L. Miessler 2004"/>

====Ionic bonds====
{{main|Ionic bond}}
In an ionic bond, oppositely charged [[ion]]s are bonded 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> It is possible to describe ionic bonds with molecular orbital theory by treating them as extremely [[polar bond]]s. Their bonding orbitals are very close in energy to the atomic orbitals of the [[anion]]. They are also very similar in character to the anion's atomic orbitals, which means the electrons are completely shifted to the anion. In computer diagrams, the orbitals are centered on the anion's core.<ref>{{Cite web |date=2020-08-06 |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=2024-06-06 |website=Chemistry LibreTexts |language=en}}</ref>

====Bond order====
{{main|Bond order}}
The bond order, or number of bonds, of a molecule can be determined by combining the number of electrons in bonding and antibonding molecular orbitals. A pair of electrons in a bonding orbital creates a bond, whereas a pair of electrons in an antibonding orbital negates a bond. For example, N<sub>2</sub>, with eight electrons in bonding orbitals and two electrons in antibonding orbitals, has a bond order of three, which constitutes a triple bond.

[[Bond strength]] is proportional to bond order—a greater amount of bonding produces a more stable bond—and [[bond length]] is inversely proportional to it—a stronger bond is shorter.

There are rare exceptions to the requirement of molecule having a positive bond order. Although Be<sub>2</sub> has a bond order of 0 according to MO analysis, there is experimental evidence of a highly unstable Be<sub>2</sub> molecule having a bond length of 245&nbsp;pm and bond energy of 10&nbsp;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|HOMO and LUMO}}
The highest occupied molecular orbital and lowest unoccupied molecular orbital are often referred to as the HOMO and LUMO, respectively. The difference of the energies of the HOMO and LUMO is called the HOMO-LUMO gap. This notion is often the matter of confusion in literature and should be considered with caution. Its value is usually located between the fundamental gap (difference between ionization potential and electron affinity) and the optical gap. In addition, HOMO-LUMO gap can be related to a bulk material [[band gap]] or transport gap, which is usually much smaller than fundamental gap.{{cn|date=June 2022}}