Eckart axis conditions, Gauss' principle of least constraint, and the optimal superposition of molecular structures

The relation of the Eckart axis conditions for polyatomic vibrating molecules to the problem of optimal superposition of molecular structures has been pointed out recently [J. Chem. Phys. 122, 224105 (2005)]. Here, it is shown that both problems are intimately related to Gauss' principle of least constraint, for which a concise derivation is presented. In the context of this article, Gauss' principle leads to a rotational superposition problem of the unconstrained atomic displacements and the corresponding displacements due to a molecular rigid-body motion. The Eckart axis conditions appear here as necessary conditions for a minimum of the constraint function. The importance of Eckart's problem for extracting the internal motions of macromolecules from simulated molecular dynamics trajectories is pointed out, and it is shown how the case of coarse-grained sampled trajectories can be treated.