Using correlation diagrams to study the vibrational spectrum of highly nonlinear floppy molecules: The K-CN case

TL;DR

This work uses $\hbar$-dependent correlation diagrams to study the vibrational spectrum of the highly nonlinear KCN molecule, linking quantum level structure to classical mixed dynamics. By combining a two-degree-of-freedom vibrational model with DVR–DGB quantum calculations across $\hbar$ and constructing adiabatic and diabatic level schemes, the authors reveal regular KAM-like tori as diabatic states embedded in a chaotic sea and identify a quantum frontier of scarred states arising from a $1:2$ resonance. The analysis shows widespread level repulsion in the adiabatic diagram and a clear diabatic structure that maps onto stable/unstable classical orbits, including hinge states around linear configurations. The results demonstrate a practical framework for diagnosing order–chaos transitions in floppy molecules and provide insight into how quantum signatures of classical phase-space structures emerge in vibrational spectra.

Abstract

The correlation diagrams of vibrational energy levels considering the Planck constant as a variable parameter have proven as a very useful tool to study vibrational molecular states, and more specifically in relation to the quantum manifestations of chaos in such dynamical systems. In this paper, we consider the highly nonlinear K-CN molecule, showing how the regular classical structures, i.e., Kolmogorov-Arnold-Moser tori, existing in the mixed classical phase space appear in the quantum levels correlation diagram as emerging diabatic states, something that remains hidden when only the actual value of the Planck constant is considered. Additionally, a quantum transition from order to chaos is unveiled with the aid of these correlation diagrams, where it appears as a frontier of scarred functions.

Figures (7)

• Figure 1: (Left) Correlation diagram for the KCN adiabatic vibrational energy levels vs.$\hbar$. The emergence of diabatic states formed by interaction between eigenstates at the avoided crossings is observed. Consider, for example, the hyperbolic "curves" specially visible in the lower left part of the figure, and also the (approximately) straight lines at the top right of it (see text for details). (Right) Correlation diagram for the KCN diabatic vibrational energy levels, obtained from Eq. (\ref{['eq:HO.model']}) (solid lines), and Eq. (\ref{['eq:HR.model']}) (dashed lines), vs.$\hbar$. Magenta/lighter and blue/darker curves correspond to states around the K-CN ($\theta=0$) and K-NC ($\theta=\pi$ rad.) linear configurations, respectively. All curves are labeled by the corresponding diabatic quantum numbers $(n_1,n_2)$. In both graphs energy in the vertical axis has been scaled with $\hbar$ to obtain clearer plots which would appear otherwise too crowded near the the origin(see text for details).
• Figure 2: Some adiabatic eigenstates of KCN that can be assigned to harmonic oscillator states on the K-CN linear configuration at $\theta=0$. The probability density is represented in a color/gray scale. The minimum energy path and the corresponding eigenenergy contour have also been represented as blue thick line and black thin line, respectively. Appropriate diabatic quantum numbers $(n_1,n_2)$ [as in Eq. (\ref{['eq:HO.model']})] are given in each plot. The values of $\hbar$ at which each of these states have been calculated is reported in Table \ref{['tab:eigenstates.parameters']}. The horizontal and vertical axis span the ranges $[0,\pi]$ rad and $[4,8]$ a.u., respectively.
• Figure 3: Same as Fig. \ref{['fig:wf.harmonic.minimum']} for harmonic oscillator states on the K-NC colinear configuration at $\theta=\pi$ rad. The values of $\hbar$ at which each of these states have been calculated is reported in Table \ref{['tab:eigenstates.parameters']}.
• Figure 4: Same as Fig. \ref{['fig:wf.harmonic.minimum']} for hinge eigenstates around K-NC (top) and K-CN (bottom) linear configurations. The values of $\hbar$ at which each of these states have been calculated is reported in Table \ref{['tab:eigenstates.parameters']}. The involved stable periodic orbits have been represented as red/lighter thick lines.
• Figure 5: (Bottom) Coupling between eigenstates $|3\rangle$ and $|4\rangle$ in the range $\hbar\in[0.1,0.5]$ a.u. The value of the mixing angle $\xi$, corresponding to the area under the curve, is also indicated. In the upper panels, the wavefunctions corresponding to the eigenstates $|m\rangle_\hbar$ at $\hbar=0.1$ a.u. (left column), $\hbar=0.5$ a.u. (right column), and the unmixed states obtained from Eqs. (\ref{['eq:orthogonal.transformation']}) and (\ref{['eq:mixing.angle']}) (middle column) are represented in color/gray scale. The minimum energy path and the potential contour corresponding to the eigenenergies (expectation energy $\langle\widehat{H}\rangle$, in the unmixed cases) have also been represented as blue thick line and black thin line, respectively. The involved 1:2 periodic orbits have been represented as red/lighter thick lines. Appropriate quantum numbers $(n_1,n_2)$ are also indicated. Axes are the same as in Fig. \ref{['fig:wf.regular.frontier']}.