Document Type
Undergraduate Research Award
Publication Date
4-2026
Abstract
High-resolution molecular spectroscopy requires an effective Hamiltonian whose operator content is both complete, containing every term allowed by molecular symmetry and nonredundant, free of any algebraically dependent operators that would cause ill-conditioned parameter fits. Traditional derivations based on Van Vleck contact transformations satisfy neither criterion automatically. This paper develops a rigorous, algorithmic pipeline that guarantees both properties simultaneously. Starting from the permutation– inversion (PI) group GPI of a molecule (Longuet-Higgins, 1963), we apply Molien’s theorem (Molien, 1897) to the symplectic normal-coordinate representation to obtain the vibrational generating function Φvib(t); integrate over the Haar measure of SO(3) (Weyl, 1946) to obtain the rotational generating function Φrot(u); and combine both via the Hadamard (coefficient-wise) product to enumerate every symmetry-allowed rovibrational operator at each bi-degree (n,m). Algebraic redundancies (syzygies) are bounded above by the Todd-class arithmetic genus (Hirzebruch, 1978) and removed exactly by Gr¨obner-basis reduction (Buchberger, 1965; Cox et al., 2015). The resulting minimal operator basis is assembled into a Watson-form effective Hamiltonian whose Jacobian is guaranteed non-singular (Watson, 1977). A complete worked example for sulfur dioxide (SO2, C2v) reproduces the published spectroscopic constants (Flaud et al., 1981).
Recommended Citation
Ajo, Leandro, "Non-Redundant Rovibrational Hamiltonians by Molien Generating Functions and Gröbner-Basis Reduction" (2026). University Libraries Undergraduate Research Award. 1.
https://ir.library.louisville.edu/ugresearchaward/1
Included in
Algebra Commons, Algebraic Geometry Commons, Physical Chemistry Commons, Quantum Physics Commons
Comments
I would like to express my sincere gratitude to the University Libraries for offering the Undergraduate Research Award and for their consistent support throughout the research process. Their resources were vital in establishing the foundation of this project. I am also deeply thankful to the International Symposium on Molecular Spectroscopy for providing me with the opportunity to present my work, which allowed me to engage with the broader scientific community and refine my findings. Finally, I owe a great deal of thanks to my advisor and mentor, Dr. Jinjun Liu. His guidance and expertise in the University of Louisville Laser Labs were indispensable to the success of this project and my development as a researcher.