Orbital overlap

Orbital overlap was an idea first introduced by Linus Pauling to explain the molecular bond angles observed through experimentation and is the basis for the concept of orbital hybridisation. s orbitals are spherical and have no directionality while p orbitals are oriented 90° to one another. A theory was needed therefore to explain why molecules such as methane (CH₄) had observed bond angles of 109.5°. Pauling's theory was that bonds are created by the overlap of adjacent atomic orbitals of two atoms. The greater the overlap of the two orbitals, the stronger the bond. The resulting bond is known as an sp hybrid and it exhibits strength greater than that of an s or p orbital.

A quantitative measure of the overlap of two atomic orbitals on atoms A and B is their overlap integral, defined as $\mathbf {S} _{AB}=\int \Psi _{A}^{*}\Psi _{B}dV$ , where the integration extends over all space. The star on the first orbital wavefunction indicates the complex conjugate of the function, which in general may be complex-valued.

Overlap matrix

The overlap matrix is a square matrix, used in quantum chemistry to describe the inter-relationship of a set of basis vectors of a quantum system, such as an atomic orbital basis set used in molecular electronic structure calculations. In particular, if the vectors are orthogonal to one another, the overlap matrix will be diagonal. In addition, if the basis vectors form an orthonormal set, the overlap matrix will be the identity matrix. The overlap matrix is always n×n, where n is the number of basis functions used. It is a kind of Gramian 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 the j-th basis ket (vector), and

$\Psi _{j}$ is the j-th wavefunction, 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 the off-diagonal elements 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; that is to say, the eigenvalues are all strictly positive.

See also

• Roothaan equations
• Hartree–Fock method

References

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