Nonredundant internal coordinates

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).]... 

In practice, the following procedure is carried out to calculate the frequencies and generalized normal mode eigenvectors in redvmdant internal coordinates, where nonredundant internal coordinates are simply a special case and may be used in the same manner. 

In the GVFF model used in NMA of molecules, the problem of redundancy is solved by the choice of independent linear combinations of the internal coordinates. This choice permits one to work with well-defined (nonredundant) sets of the coordinates and corresponding force constants, even though they may not reflect our view of the physics of the interatomic interactions in the system. If we wish to use the knowledge about the chemical bonding in the... 

Figure 3. Frequency uncertainty test for benzocyclobutadiene according to FiF/6-31G(d,p) calculations. The correlation diagrams correspond to correlations between normalized amplitudes and frequency differences = cOp - with cOp being a normal mode frequency of a molecular fragment (t)p and 0) being a normal mode frequency. Amplitudes AvAF, AvPF, CvAF, CvPF are employed in connection with adiabatic internal frequencies CO3 and c-vector frequencies using a nonredundant set of internal coordinates (a - d) or a strongly redundant set (e - h). In all cases, points that have A/ = 0 for all tests within a given row of diagrams are removed.

---