Translation-rotation invariance for N-particle systems: Internal coordinates and search for stationary points in reduced spaces

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Abstract
Using the translational and rotational invariance of the energy, we show that for an N-body system there exists a subset of 3N−6 Cartesian coordinates such that the derivatives of the energy of a given order with respect to the members of this subset form an independent and complete set. That is, all other derivatives can be calculated knowing these independent derivatives. We further show that this subset of coordinates can be chosen to be a bonafide set of internal coordinates. Since these coordinates are a subset of the 3N Cartesian coordinates, they are orthonormal and uniform. Applications have been made of such internal coordinates in algorithms which search for stationary points on potential energy surfaces. It is shown that the surface walking algorithms are exactly separable for these coordinates. Thus the problem can be reexpressed in terms of (3N−6) Cartesian variables without the annoying zero eigenvalues of the Hessian (of the energy) matrix corresponding to the translational and rotational eigenvalues.