Rotational invariance, orbital connections

One may also wish to impose an additional requirement on the connection, namely that it is translationally and rotationally invariant. This may seem to be a trivial requirement. However, a connection is conveniently defined in terms of atomic Cartesian displacements rather than in terms of a set of nonredundant internal coordinates. This implies that each molecular geometry may be described in an infinite number of translationally and rotationally equivalent ways. The corresponding connections may be different and therefore not translationally and rotationally invariant. In other words, the orbital basis is not necessarily uniquely determined by the internal coordinates when the connections are defined in terms of Cartesian coordinates. Conversely, a rotationally invariant connection picks up the same basis set regardless of how the rotation is carried out and so the basis is uniquely defined by the internal coordinates. [For a discussion of translationally and rotationally invariant connections, see Carlacci and Mclver (1986).]... 

Sources of atomic data for determining the Hu are discussed at length below. There are two contributions to the Hij(i j). The neighbor atom potential introduces terms connecting members of a set of p, d, or f orbitals on the same atom if the neighbor atom is not in the direction of a coordinate axis from the reference atom. These Hij terms are necessary to preserve rotational invariance. For atomic orbitals on different atoms,... 

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