Inchemical bonds, anorbital overlap is the concentration oforbitals on adjacent atoms in the same regions of space. Orbital overlap can lead to bond formation. The general principle for orbital overlap is that, the greater the greater the over between orbitals, the greater is the bond strength.Linus Pauling explained the importance of orbital overlap in the molecularbond angles observed through experimentation; it is the basis fororbital hybridization. Ass orbitals are spherical (and have no directionality) andp orbitals are oriented 90° to each other, a theory was needed to explain why molecules such asmethane (CH₄) had observed bond angles of 109.5°.^[1] Pauling proposed that s and p orbitals on the carbon atom can combine to formhybrids (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.^[2]

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

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

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.

Theoverlap matrix is asquare matrix, used inquantum chemistry to describe the inter-relationship of a set ofbasis vectors of aquantum system, such as an atomic orbitalbasis set used in molecular electronic structure calculations. In particular, if the vectors areorthogonal to one another, the overlap matrix will be diagonal. In addition, if the basis vectors form anorthonormal set, the overlap matrix will be theidentity matrix. The overlap matrix is alwaysn×n, wheren is the number of basis functions used. It is a kind ofGramian matrix.

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

${\displaystyle {\mathbf{S}}_{jk}=⟨b_j|b_k⟩=\int {\mathrm{\Psi }}_j^{\ast }{\mathrm{\Psi }}_{k}d\tau }$

where

${\displaystyle |b_j⟩}$ is thej-th basisket (vector), and

${\displaystyle {\mathrm{\Psi }}_j}$ is thej-thwavefunction, defined as :${\displaystyle {\mathrm{\Psi }}_j(x)=⟨x|b_j⟩}$.

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 elements less 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.

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

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

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