Aspects of the Eckart frame ro-vibrational kinetic energy operator

A most decisive part of developing the Eckart frame ro-vibrational kinetic energy operator (KEO) in terms of curvilinear internal coordinates is the introduction of vibrational displacements obeying the Eckart conditions. It is examined here, how such displacements may be constructed by orthogonal and nonorthogonal transformations, and how they can be employed in developing the KEO. In particular, it is shown analytically, by term by term comparison, that the Eckart KEOs obtained by employing Pickett's method and the gateway Hamiltonian method are equivalent. That is, although it may appear in the form of very different expressions, the general, exact ro-vibrational Eckart frame KEO is unique. As side result, it has been shown that the simple formula of the pseudo-potential derived by Watson for the exact KEO given in normal coordinates applies to any Eckart frame KEO, irrespective of the choice of the vibrational coordinates.