Orthogonalization of the Basis

Consider the Hartree-Fock-Roothan (HFR) equations.

F C = S C \epsilon

If the basis functions where orthonormal the overlap matrix would be the unit matrix and the HFR equations would have the form of the usual matrix eigenvalue problem.

F C = C \epsilon

The basis sets used in quantum chemical calculations are generally not orthonormal. To put the HFR into the form of the standard matrix eigenvalue problem, the basis functions must be orthogonalized, which is a very time consuming step. The HFR equations could then be solved by simply diagonalizing the Fock matrix.

A more efficient way employed by almost all quantum chemical codes is to transform the equations by unitary transformation.

F^\prime C^\prime = C^\prime \epsilon

The primed matrices are are just the Fock matrix and expansion coefficients in the orthonormal basis.

The transformation matrix is derived from the overlap matrix.

In molecular calculations the eigenvalues of the overlap matrix are all positive. If negative eigenvalues appear, the overlap matrix is thought to be unphysical.

In periodic calculations employing the CCM a small overlap of an atomic orbital with another on a distant atom in the real unit cell is replaced by a bigger overlap with one on an equivalent atom in a repeated unit cell. If a small unit cell is chosen this may cause the overlap matrix to be non-positive definite and it’s eigenvalues are negative.

Symmetric Orthogonalization

Symmetric orthogonalization uses the inverse square root of the overlap matrix as transformation matrix.

X = S^{ -\frac{1}{2} }

If the eigenvalues of S are all positive, there is no difficulty of taking square roots. However if there is (near) linear dependency in the basis set, then some of the eigenvalues will approach zero and calculating the transformation matrix will involve dividing by nearly zero.

Therefore this orthogonalization procedure is not recommended and will not work in the CCM is employed.

Canonical Orthogonalization

X = U s^{ -\frac{1}{2} }

Choleksy Orthogonalization

LL^T=S X = transpose(inv(L))

The transformation matrix is calculated via the Cholesky decomposition. The overlap matrix has to be positive definite.

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