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 (KEO) for centrally-connected penta-atomic molecules such as methane and its isotopomers is derived for two sets of internal valence coordinates: the polyspherical coordinates and the bond-angle valence coordinates. The vibrational KEO including the pseudopotential term is discussed for two forms of the modified Jacobian. For the rotational and vibration-rotation coupling KEO, results for two schemes of embedding the bodyfixed coordinate system are presented: the bond embedding and the bisector embedding. Full expressions 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 vibrationrotation KEO expressions derived in this work, are discussed.