Orbital overlap: Difference between revisions

In [[chemical bond]]s, an '''orbital overlap''' is the concentration of [[atomic orbital|orbitals]] on adjacent atoms in the same regions of space. Orbital overlap can lead to bond formation. The[[Linus Pauling]] explained the importance of orbital overlap was emphasized by [[Linus Pauling]] to explainin the molecular [[bond angle]]s observed through experimentation; andit is the basis for the concept of [[orbital hybridisationhybridization]]. SinceAs ''s''&nbsp;orbitals are spherical (and have no directionality) whileand ''p'' orbitals are oriented 90° to oneeach anotherother, a theory was needed to explain why molecules such as [[methane]] (CH₄) had observed bond angles of 109.5°.<ref>Anslyn, Eric V./Dougherty, Dennis A. (2006). ''Modern Physical Organic Chemistry''. University Science Books.</ref> Pauling proposed that s and p orbitals on the carbon atom can combine to form hybrids (sp³ in the case of methane) which are directed toward the hydrogen atoms. The carbon hybrid orbitals have greater overlap with the hydrogen orbitals, and can therefore form stronger C–H bonds.<ref>Pauling, Linus. (1960). ''The Nature Of The Chemical Bond''. Cornell University Press.</ref>

A quantitative measure of the overlap of two atomic orbitals Ψ_A and Ψ_B on atoms A and B is their '''overlap integral''', defined as

where the integration extends over all space. The star on the first orbital wavefunction indicates the function's [[complex conjugate]] of the function, which in general may be [[complex-valued]].

In particular, if the set is normalized (though not necessarily orthogonal) then the diagonal elements will be identically 1 and the magnitude of the [[off-diagonal elementselement]]s less than or equal to one with equality if and only if there is linear dependence in the basis set as per the [[Cauchy–Schwarz inequality]]. Moreover, the matrix is always [[positive-definite matrix|positive definite]]; that is to say, the eigenvalues are all strictly positive.