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

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.

💡 Research Summary

In this paper the authors introduce a novel approach to probing the vibrational spectrum of the highly nonlinear, floppy tri‑atomic molecule potassium cyanide (K‑CN) by treating the Planck constant ℏ as a continuously tunable parameter. The central idea follows Weyl’s semiclassical argument: a quantum state occupies a phase‑space volume proportional to ℏⁿ (n being the number of degrees of freedom). By gradually reducing ℏ toward zero, the quantum states are forced into ever smaller phase‑space cells, allowing them to “fit” into the regular islands of the underlying classical phase space. This provides a microscope‑like view of the classical Kolmogorov‑Arnold‑Moser (KAM) tori that are otherwise hidden when ℏ is fixed at its physical value (ℏ = 1 a.u.).

The molecular model retains only two vibrational degrees of freedom: the distance R from the C‑N centre of mass to the potassium atom and the bending angle θ. The C‑N bond is frozen at its equilibrium length (r_eq = 2.22 a.u.). The potential energy surface V(R,θ) is an analytic fit to high‑level ab‑initio data and possesses a deep triangular minimum near θ ≈ π/2 and a shallow linear minimum at θ = 0; the opposite linear configuration (θ = π) is a saddle. Classical dynamics are explored via Poincaré surfaces of section at several energies. At very low energies (E < 65 cm⁻¹) the motion is regular, organized around a 1:1 resonance and a 1:2 asymmetric resonance. Already at E ≈ 290 cm⁻¹ the phase space becomes overwhelmingly chaotic, with only tiny islands persisting near the K‑N‑C saddle and around the linear configurations at higher energies (1200–5400 cm⁻¹).

Quantum eigenvalues and eigenfunctions are obtained with the Discrete Variable Representation–Distributed Gaussian Basis (DVR‑DGB) method. Approximately 1 000 basis functions are used to converge the lowest 300 vibrational levels for a set of ℏ values ranging from 0.10 to 3.00 a.u. (finer grids are required for ℏ < 0.5 a.u.). For each ℏ the authors construct a correlation diagram of vibrational energies versus ℏ. Two complementary representations are presented.

The “adiabatic” diagram (left panel of Fig. 1) displays a dense “spaghetti” of level curves that avoid crossing in accordance with the von Neumann–Wigner non‑crossing rule. The prevalence of avoided crossings (ACs) and the strong level repulsion are interpreted through the Bohigas‑Giannoni‑Schmit conjecture as signatures of underlying classical chaos. Notably, the regular region near ℏ → 0 is extremely narrow, indicating that even in the semiclassical limit the system remains largely chaotic.

The “diabatic” diagram (right panel) is obtained by analytically removing the ACs using a simple two‑mode model: a harmonic oscillator (HO) describing the stretching motion and a hindered rotor (HR) describing the bending/hinge motion, separately for the two linear configurations (θ = 0 and θ = π). In this representation, families of hyperbolic curves in the lower‑left corner correspond to states localized near the K‑CN linear geometry, while nearly straight lines in the upper‑right correspond to states near the K‑NC linear geometry. Each curve is labeled by a pair of quantum numbers (n₁, n₂) that count quanta in the HO and HR modes, respectively.

To quantify the coupling between states as ℏ varies, the off‑diagonal Hellmann‑Feynman matrix elements ⟨m|∂/∂ℏ|n⟩ are evaluated via the relation ⟨m|∂/∂ℏ|n⟩ = (⟨m|∂Ĥ/∂ℏ|n⟩)/(Eₙ − Eₘ). Large values of these elements are found precisely at the avoided crossings, confirming that the ACs are the loci of strong quantum mixing.

A particularly striking result is the identification of a “quantum order‑to‑chaos transition” that appears as a frontier of scarred wavefunctions. As ℏ is reduced, wavefunctions become increasingly localized on the least unstable periodic orbits (POs) of the classical chaotic sea, forming scarred states. The boundary where scarred states first emerge coincides with the region in the diabatic diagram where the regular linear‑geometry families terminate, providing a clear visual marker of the transition.

In summary, the paper demonstrates that varying ℏ offers a powerful diagnostic tool for uncovering the hidden correspondence between classical invariant structures (KAM tori, stable periodic orbits) and quantum vibrational levels in a strongly nonlinear molecule. The method reveals diabatic families that are invisible in conventional fixed‑ℏ calculations, clarifies the role of avoided crossings in mediating quantum chaos, and establishes scarred functions as markers of the quantum order‑to‑chaos crossover. The authors argue that this approach is broadly applicable to other molecular systems with mixed phase‑space dynamics and may even be interpreted as mimicking isotopic mass variations, thereby linking computational insights with experimental possibilities.