Determination of the Eckart molecule-fixed frame by use of the apparatus of quaternion algebra

The problem of determining the Eckart molecule-fixed frame for an arbitrary molecule with nuclei displaced from the equilibrium positions is considered. The solution of the problem is formulated by minimizing the sum of mass-weighted squared deviations (MWSD) of the nuclei of a displaced configuration from the nuclei of the equilibrium configuration. A mathematical proof of the equivalence of Eckart conditions and the minimum of MWSD is given. It is shown that the extrema of the sum of MWSD coincide with eigenvalues of a special 4 x 4 symmetric matrix. Its minimal eigenvalue corresponds to the desired solution, and the respective eigenvector can be treated as the quaternion containing the necessary information for rotating the original coordinate system and aligning its axes with the molecule-fixed coordinate system. A detailed scheme for an efficient numerical implementation of the method is provided, and a numerical example is given. (C) 2014 AIP Publishing LLC.