Page 178 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

CORE-VALENCE SEPARATION IN THE STUDY OF ATOMIC CLUSTERS 161

To build up in the cluster function (1) we use the functions
all of which satisfy the strong orthogonality condition in the sense of to (2), but do not
satisfy the strong orthogonality needed for (1) We therefore consider the linear
combination

and require that this function be a linear combination of the functions in each sphere
SP. This condition can be only approximatively satisfied and it is useful to have a
measure of the goodness of the approximation. To obtain such a criterion we consider the
quantity

This quantity measures the error in the orthogonality of f i to all the group functions.
Since the urij are arbitrary coefficients, we can put and so obtain

As noted in previous papers [7–11], by considering the matrix

we can find the minimum of (15) by diagonalizing the matrix. The eigenvectors,
ordered according to the corresponding increasing eigenvalues, give functions less and
less orthogonal to the "core" orbitals. The associated eigenvalues give us a measure of the
goodness of the functions obtained. One must keep only functions corresponding to
eigenvalues smaller than some chosen threshold.
In this way we obtain n functions

with the property