The Eckart-frame hamiltonian of a triatomic molecule in terms of the bond lengths and the intrabond angle

Eckart-Sayvetz conditions revisited

It is shown that vibrational displacements satisfying the Eckart-Sayvetz conditions can be constructed by projection of unconstrained displacements. This result has a number of interesting direct and indirect ramifications: (i) The normal coordinates corresponding to an electronic state or an isotopologue of a molecule are transformed to those of another state or isotopologue by a linear and, in general, non-orthogonal transformation. (ii) Novel interpretation of axis switching. (iii) One may enhance the separation of rotational-large-amplitude internal motions and the vibrational motions beyond that offered by the standard use of the Eckart-Sayvetz conditions. (iv) The rotational-vibrational Hamiltonian given in terms of curvilinear internal coordinates may be derived with elementary mathematical tools while taking into account the Eckart conditions with or without enhancement.

Body frames in the separation of collective angles in quantum N-body problems

The application of the concept of body-fixed reference frames, proposed by C. Eckart [Phys. Rev. 47, 552 (1935)], to the problem of the separation of three collective angles in quantum N-body problems is analyzed based on the technique recently developed by Meremianin and Briggs [Phys. Rep. 384, 121 (2003)]. Special attention is paid to the body frame defined by the "second Eckart condition" which minimizes vibro-rotational couplings near the equilibrium position. The important case of the Eckart frame for three-body systems is considered in detail. The connection of the basis vectors of the Eckart frame with Jacobi vectors is derived. All results of this work are valid for an arbitrary choice of internal (body-frame) coordinates.

The molecular Hamiltonian

The molecular Hamiltonian represents one of the most basic concepts in spectroscopy and molecular reaction dynamics. Its derivation is notoriously difficult owing to the use of a rotating reference frame which, in turn, is necessary to define the concept of vibration and rotation. In this article, we review the construction of the molecular Hamiltonian in normal mode and in internal coordinates. For normal mode coordinates, the Watson Hamiltonian including its modification for linear molecules is derived using an approach based on classical mechanics and the Podolsky transformation. The method is subsequently used to derive the molecular Hamiltonian in terms of Jacobi and valence coordinates. Results are presented for the triatomic system and for the extension toward N-atom systems with N > or = 3.