Title: Eckart ro-vibrational Hamiltonians via the gateway Hamilton operator: theory and practice,
Abstract: Recently, a general expression for Eckart-frame Hamilton operators has been
obtained by the gateway Hamiltonian method ({\it J. Chem. Phys.} {\bf 142},
174107 (2015); {\it ibid.} {\bf 143}, 064104 (2015)). The kinetic energy
operator in this general Hamiltonian is nearly identical with that of the
Eckart-Watson operator even when curvilinear vibrational coordinates are
employed. Its different realizations correspond to different methods of
calculating Eckart displacements. There are at least two different methods for
calculating such displacements: rotation and projection. In this communication
the application of Eckart Hamiltonian operators constructed by rotation and
projection, respectively, is numerically demonstrated in calculating
vibrational energy levels. The numerical examples confirm that there is no need
for rotation to construct an Eckart ro-vibrational Hamiltonian. The application
of the gateway method is advantageous even when rotation is used, since it
obviates the need for differentiation of the matrix rotating into the Eckart
frame. Simple geometrical arguments explain that there are infinitely many
different methods for calculating Eckart displacements. The geometrical picture
also suggests that a unique Eckart displacement vector may be defined as the
shortest (mass-weighted) Eckart displacement vector among Eckart displacement
vectors corresponding to configurations related by rotation. Its length, as
shown analytically and demonstrated by way of numerical examples, is equal to
or less than that of the Eckart displacement vector one can obtain by rotation
to the Eckart frame.