Exact vibration-rotation kinetic energy operators in two sets of valence coordinates for centrally connected penta-atomic molecules

The exact vibration-rotation quantum mechanical kinetic energy operator (KE O) for centrally-connected penta-atomic molecules such as methane and its i sotopomers is derived for two sets of internal valence coordinates: the pol yspherical coordinates and the bond-angle valence coordinates. The vibratio nal KEO including the pseudo-potential term is discussed for two forms of t he modified Jacobian. For the rotational and vibration-rotation coupling KE G, results for two schemes of embedding the body-fixed coordinate system ar e presented: the bond embedding and the bisector embedding. Full expression s for the bisector embedding are too complicated to give in detail, but the working connection between the two embedding schemes is given. The future applications, including the perturbative and variational calculation of the vibrations and/or rotations for centrally connected penta-atomic molecules using the vibration-rotation KEO expressions derived in this work, are dis cussed.