Editing Covalent bond (section)

== Quantum mechanical description ==
After the development of quantum mechanics, two basic theories were proposed to provide a quantum description of chemical bonding: [[valence bond theory|valence bond (VB) theory]] and [[molecular orbital theory|molecular orbital (MO) theory]]. A more recent quantum description<ref>{{cite journal|last1=Cammarata|first1=Antonio|last2=Rondinelli|first2=James M.|title=Covalent dependence of octahedral rotations in orthorhombic perovskite oxides|journal=Journal of Chemical Physics|date=21 September 2014|volume=141|issue=11|pages=114704|doi=10.1063/1.4895967|pmid=25240365|bibcode=2014JChPh.141k4704C}}</ref> is given in terms of atomic contributions to the electronic density of states.

=== Comparison of VB and MO theories ===
The two theories represent two ways to build up the [[electron configuration]] of the molecule.<ref name="Quanta">{{cite book | title=Quanta: A Handbook of Concepts| publisher=Oxford University Press | year=1974 |pages=147–148 | first=P. W.|last= Atkins | isbn=978-0-19-855493-6}}</ref> For valence bond theory, the atomic [[orbital hybridisation|hybrid orbitals]] are filled with electrons first to produce a fully bonded valence configuration, followed by performing a linear combination of contributing structures ([[resonance (chemistry)|resonance]]) if there are several of them. In contrast, for molecular orbital theory, a [[linear combination of atomic orbitals]] is performed first, followed by filling of the resulting [[molecular orbital]]s with electrons.<ref name=":0" />

The two approaches are regarded as complementary, and each provides its own insights into the problem of chemical bonding. As valence bond theory builds the molecular wavefunction out of localized bonds, it is more suited for the calculation of [[bond energy|bond energies]] and the understanding of [[reaction mechanism]]s. As molecular orbital theory builds the molecular wavefunction out of delocalized orbitals, it is more suited for the calculation of [[ionization energy|ionization energies]] and the understanding of [[Absorption spectroscopy|spectral absorption bands]].<ref>James D. Ingle Jr. and Stanley R. Crouch, ''Spectrochemical Analysis'', Prentice Hall, 1988, {{ISBN|0-13-826876-2}}</ref>

At the qualitative level, both theories contain incorrect predictions. Simple (Heitler–London) valence bond theory correctly predicts the dissociation of homonuclear diatomic molecules into separate atoms, while simple (Hartree–Fock) molecular orbital theory incorrectly predicts dissociation into a mixture of atoms and ions. On the other hand, simple molecular orbital theory correctly predicts [[Hückel's rule]] of aromaticity, while simple valence bond theory incorrectly predicts that cyclobutadiene has larger resonance energy than benzene.<ref name="Modern Physical Organic Chemistry">{{cite book|last=Anslyn|first=Eric V.|title=Modern Physical Organic Chemistry|year=2006|publisher=University Science Books|isbn=978-1-891389-31-3}}</ref>

Although the wavefunctions generated by both theories at the qualitative level do not agree and do not match the stabilization energy by experiment, they can be corrected by [[configuration interaction]].<ref name="Quanta"/> This is done by combining the valence bond covalent function with the functions describing all possible ionic structures or by combining the molecular orbital ground state function with the functions describing all possible excited states using unoccupied orbitals. It can then be seen that the simple molecular orbital approach overestimates the weight of the ionic structures while the simple valence bond approach neglects them. This can also be described as saying that the simple molecular orbital approach neglects [[electron correlation]] while the simple valence bond approach overestimates it.<ref name="Quanta"/>

Modern calculations in [[quantum chemistry]] usually start from (but ultimately go far beyond) a molecular orbital rather than a valence bond approach, not because of any intrinsic superiority in the former but rather because the MO approach is more readily adapted to numerical computations. Molecular orbitals are orthogonal, which significantly increases the feasibility and speed of computer calculations compared to nonorthogonal valence bond orbitals.

=== Covalency from atomic contribution to the electronic density of states ===
Evaluation of bond covalency is dependent on the [[basis set (chemistry)|basis set]] for approximate quantum-chemical methods such as COOP (crystal orbital overlap population),<ref>{{Cite journal|last1=Hughbanks|first1=Timothy|last2=Hoffmann|first2=Roald|date=2002-05-01|title=Chains of trans-edge-sharing molybdenum octahedra: metal-metal bonding in extended systems|journal=Journal of the American Chemical Society|volume=105|issue=11|pages=3528–3537|doi=10.1021/ja00349a027}}</ref> COHP (Crystal orbital Hamilton population),<ref>{{Cite journal|last1=Dronskowski|first1=Richard|last2=Bloechl|first2=Peter E.|date=2002-05-01|title=Crystal orbital Hamilton populations (COHP): energy-resolved visualization of chemical bonding in solids based on density-functional calculations|journal=The Journal of Physical Chemistry|volume=97|issue=33|pages=8617–8624|doi=10.1021/j100135a014}}</ref> and BCOOP (Balanced crystal orbital overlap population).<ref>{{Cite journal|last1=Grechnev|first1=Alexei|last2=Ahuja|first2=Rajeev|last3=Eriksson|first3=Olle|date=2003-01-01|title=Balanced crystal orbital overlap population—a tool for analysing chemical bonds in solids|journal=Journal of Physics: Condensed Matter|volume=15|issue=45|pages=7751|doi=10.1088/0953-8984/15/45/014|issn=0953-8984|bibcode=2003JPCM...15.7751G|s2cid=250757642 }}</ref> To overcome this issue, an alternative formulation of the bond covalency can be provided in this way.

The [[Center of mass|mass center]] {{tmath|cm(n,l,m_l,m_s)}} of an atomic orbital <math>| n,l,m_l,m_s \rangle ,</math> with [[quantum number]]s {{tmath|n,}} {{tmath|l,}} {{tmath|m_l,}} {{tmath|m_s,}} for atom A is defined as

:<math>cm^\mathrm{A}(n,l,m_l,m_s)=\frac{\int\limits_{E_0}\limits^{E_1} E g_{|n,l,m_l,m_s\rangle}^\mathrm{A}(E) dE}{\int\limits_{E_0}\limits^{E_1} g_{|n,l,m_l,m_s\rangle}^\mathrm{A} (E)dE}</math>

where <math>g_{|n,l,m_l,m_s\rangle}^\mathrm{A}(E)</math> is the contribution of the atomic orbital <math>|n,l,m_l,m_s \rangle</math> of the atom A to the total electronic density of states {{tmath|g(E)}} of the solid

:<math>g(E)=\sum_\mathrm{A}\sum_{n, l}\sum_{m_l, m_s}{g_{|n,l,m_l,m_s\rangle}^\mathrm{A}(E)}</math>

where the outer sum runs over all atoms A of the unit cell. The energy window {{tmath|[E_0, E_1]}} is chosen in such a way that it encompasses all of the relevant bands participating in the bond. If the range to select is unclear, it can be identified in practice by examining the molecular orbitals that describe the electron density along with the considered bond.

The relative position {{tmath|C_{n_\mathrm{A}l_\mathrm{A},n_\mathrm{B}l_\mathrm{B} } }} of the mass center of <math>| n_\mathrm{A},l_\mathrm{A}\rangle</math> levels of atom A with respect to the mass center of <math>| n_\mathrm{B},l_\mathrm{B}\rangle</math> levels of atom B is given as

:<math>C_{n_\mathrm{A}l_\mathrm{A},n_\mathrm{B}l_\mathrm{B}}=-\left|cm^\mathrm{A}(n_\mathrm{A},l_\mathrm{A})-cm^\mathrm{B}(n_\mathrm{B},l_\mathrm{B})\right|</math>

where the contributions of the magnetic and spin quantum numbers are summed. According to this definition, the relative position of the A levels with respect to the B levels is

:<math>C_\mathrm{A,B}=-\left|cm^\mathrm{A}-cm^\mathrm{B}\right|</math>

where, for simplicity, we may omit the dependence from the principal quantum number {{tmath|n}} in the notation referring to {{tmath|C_{n_\mathrm{A}l_\mathrm{A},n_\mathrm{B}l_\mathrm{B} }.}}

In this formalism, the greater the value of {{tmath|C_\mathrm{A,B},}} the higher the overlap of the selected atomic bands, and thus the electron density described by those orbitals gives a more covalent {{chem2|A\sB}} bond. The quantity {{tmath|C_\mathrm{A,B} }} is denoted as the ''covalency'' of the {{chem2|A\sB}} bond, which is specified in the same units of the energy {{tmath|E}}.