Editing Molecular orbital (section)

===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>