Exact Bethe Ansatz for the vibron model: a critical reassessment of q-deformation and parameterization

The vibron model classifies energy spectra through U <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"> <mml:mrow> <mml:mo>→</mml:mo> </mml:mrow> </mml:math> SO <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m3"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> irreducible representation (irrep) branching, enabling exploration of quantum phase transitions between vibrational and rotational regimes. We advance a solvable Bethe Ansatz framework, deriving exact solutions for the vibron Hamiltonian and addressing discrepancies in prior studies. By applying algebraic techniques, we demonstrate that precise control parameters—rather than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m4"> <mml:mrow> <mml:mi>q</mml:mi> </mml:mrow> </mml:math> -deformation—are essential for accurate spectral predictions. This study critically reassesses existing computational methods, emphasizing parameter optimization to improve alignment with experimental data. Our refined approach accurately reproduces the procedure for obtaining reproducible molecular energy spectra, elucidating the impact of control parameters on quantum phase transitions.