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. 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, the overlap matrix is defined as:

$\mathbf{S}_{jk}=\left \langle b_j|b_k \right \rangle=\int \Psi_j^* \Psi_k d\tau$

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

References

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