Group-theoretical formulation of an Eckart-frame kinetic energy operator in curvilinear coordinates for polyatomic molecules

A new procedure is presented for building a general kinetic energy operator expressed as a polynomial series expansion of symmetry-adapted curvilinear coordinates for semirigid polyatomic molecules. As a starting point, the normal-mode Watson kinetic energy part is considered and then transformed into its curvilinear counterpart. An Eckart molecular fixed-frame is thus implicitly used. To this end, we exploit symmetry at all stages of the calculation and show how group-theoretically based methods and Γ-covariant tensors help properly invert nonlinear polynomials for the coordinate changes. Such a linearization procedure could also be useful in different contexts. Unlike the usual normal mode approach, the potential part initially expressed in curvilinear coordinates is not transformed in this work, making convergence of the Hamiltonian expansion generally faster. For dimensionality reduction, the final curvilinear kinetic and potential parts are expanded in terms of irreducible tensor operators when doubly and triply degenerate vibrations are involved. The procedure proposed here is general and can be applied to arbitrary Abelian and non-Abelian point groups. Illustrative examples will be given for the H₂S (⁠$C2v$⁠), H₂CO (⁠$C2v$⁠), PH₃ (⁠$C3v$⁠), and SiH₄ (T_d) molecules.