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# 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

Ψj

is the j-th wavefunction, defined as

$\Psi_j(x)=\left \langle x | b_j \right \rangle$.