Surfaces orthogonal basis

The most serious shortcoming of the foregoing approach is within the procedure for transforming the basis. It is valid only for a symmetric reaction and has been carried out setting v equal to its value at the reaction path, in order to define a matching surface. Furthermore the computed transformation matrix was only approximately orthogonal. 

The valence-bond approach plays a very important role in the qualitative discussion of chemical bonding. It provides the basis for the two most important semi-empirical methods of calculating potential energy surfaces (LEPS and DIM methods, see below), and is also the starting point for the semi-theoretical atoms-in-molecules method. This latter method attempts to use experimental atomic energies to correct for the known atomic errors in a molecular calculation. Despite its success as a qualitative theory the valence-bond method has been used only rarely in quantitative applications. The reason for this lies in the so-called non-orthogonality problem, which refers to the difficulty of calculating the Hamiltonian matrix elements between valence-bond structures. 

For a basis set of (re) atomic orbitals which are singly noded in the plane perpendicular to the radial vector (see Fig. 16b), the required Harmonics are the Vector Surface Harmonics146). The two p11 (or d") atomic orbitals at each cluster vertex (i) behave as a pair of orthogonal unit vectors which are tangential to the surface of the sphere, jrf and jif are defined as Jt-symmetry orbitals on the ith cluster atom, pointing in the direction of increasing 0 and cj> respectively, as shown in Fig. 18. At the poles of the cluster sphere these vectors may be related to Cartesian vectors as follows ... 

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