Orbital overlap

Inchemical bonds,anorbital overlapis the concentration oforbitalson adjacent atoms in the same regions of space. Orbital overlap can lead to bond formation.Linus Paulingexplained the importance of orbital overlap in the molecularbond anglesobserved through experimentation; it is the basis fororbital hybridization.Assorbitals are spherical (and have no directionality) andporbitals are oriented 90° to each other, a theory was needed to explain why molecules such asmethane(CH₄) had observed bond angles of 109.5°.Pauling proposed that s and p orbitals on the carbon atom can combine to form hybrids (sp³in 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.

A quantitative measure of the overlap of two atomic orbitals Ψ_Aand Ψ_Bon atoms A and B is theiroverlap integral,defined as

$\mathbf {S} _{\mathrm {AB} }=\int \Psi _{\mathrm {A} }^{*}\Psi _{\mathrm {B} }\,dV,$

where the integration extends over all space. The star on the first orbital wavefunction indicates the function'scomplex conjugate,which in general may becomplex-valued.

Overlap matrix

Theoverlap matrixis asquare matrix,used inquantum chemistryto describe the inter-relationship of a set ofbasis vectorsof aquantumsystem, such as an atomic orbitalbasis setused in molecular electronic structure calculations. In particular, if the vectors areorthogonalto one another, the overlap matrix will be diagonal. In addition, if the basis vectors form anorthonormalset, the overlap matrix will be theidentity matrix.The overlap matrix is alwaysn×n,wherenis the number of basis functions used. It is a kind ofGramian matrix.

In general, each overlap matrix element is defined as an overlap integral:

$\mathbf {S} _{jk}=\left\langle b_{j}|b_{k}\right\rangle =\int \Psi _{j}^{*}\Psi _{k}\,d\tau$

where

$\left|b_{j}\right\rangle$is thej-th basisket(vector), and

$\Psi _{j}$is thej-thwavefunction,defined as:$\Psi _{j}(x)=\left\langle x|b_{j}\right\rangle$.

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

See also

• Roothaan equations
• Hartree–Fock method
• Pi bond
• Sigma bond

References

Quantum Chemistry: Fifth Edition,Ira N. Levine, 2000