Molecular Chaos Mapped with New Diagrams Reveals Hidden Order in Potassium Cyanide

Scientists have long sought to understand the complex vibrational behaviour of highly nonlinear molecules, and a new study utilising variable parameter correlation diagrams offers significant insight into these systems. H. Párraga, F. J. Arranz, and R. M. Benito, alongside F. Borondo et al., demonstrate the utility of this approach by examining the vibrational spectrum of potassium cyanide (K-CN). Their research reveals how classical structures, specifically Kolmogorov-Arnold-Moser tori, manifest as emerging diabatic states within the correlation diagrams, a phenomenon obscured by conventional constant-Planck analyses. This methodology successfully unveils a transition from order to chaos, presenting it as a frontier of scarred functions and providing a novel means of characterising molecular dynamics.

This technique reveals hidden classical structures, specifically Kolmogorov-Arnold-Moser tori, as emerging diabatic states in the quantum levels correlation diagram, structures that would otherwise remain obscured when using a fixed value for Planck’s constant.

The research focuses on the K-CN molecule, a system known for its complex and chaotic dynamics, and demonstrates a pathway to understanding the transition from order to chaos through the identification of a frontier of scarred functions. The work builds upon established correlation diagrams, traditionally used to rationalize molecular rovibrational states based on real-valued parameters like geometrical distances or angles.
Instead, researchers artificially varied the Planck constant, ħ, to effectively implement a microscopic lens focusing on classical regular structures embedded within chaotic regions of the molecular phase space. By reducing ħ, quantum states are confined to smaller phase space volumes, allowing for detailed examination of their dynamical characteristics within the regular classical region.

This approach provides a unique perspective on the interplay between quantum and classical behaviour in molecular systems. Classical molecular vibrations are understood through nonlinear dynamics, where molecules are viewed as Hamiltonian systems of coupled oscillators governed by the Kolmogorov-Arnold-Moser and Poincaré-Birkhoff theorems.

At low energies, motion occurs around stable configurations with regular dynamics confined to invariant tori. As excitation increases, these tori can be destroyed, leading to stochasticity and a transition to local modes. The study identifies how these classical structures manifest in the quantum realm, specifically through the emergence of diabatic states within the correlation diagrams.

Furthermore, the research unveils a quantum transition from order to chaos, visualized as a frontier of scarred functions. Scarring refers to the localization of quantum probability density along the less unstable periodic orbits of a classically chaotic system, providing insights into the organization of chaotic dynamics. This method, applied to the K-CN molecule, establishes a framework for understanding the dynamics of floppy triatomic molecules and offers a means to assess the validity of this approach for other molecular systems exhibiting mixed classical dynamics.

Revealing Kolmogorov-Arnold-Moser tori through Planck constant variation in K-CN vibrational spectra

Correlation diagrams of vibrational energy levels, utilising the Planck constant as a variable, served as the primary tool to investigate vibrational molecular states within the K-CN molecule. This work focused on revealing how regular classical structures, specifically Kolmogorov-Arnold-Moser tori, manifest as emerging diabatic states in these correlation diagrams, a phenomenon obscured when using a fixed Planck constant.

By varying the Planck constant, the research implemented a microscopic approach to focus on classical regular structures embedded within chaotic regions of the molecular phase space. Classically, the molecular vibrations were understood through nonlinear dynamics, treating the molecule as a Hamiltonian system of coupled nonlinear oscillators.

The study detailed how, at low energies, motion occurs around stable equilibrium configurations, validating the harmonic approximation and resulting in regular dynamics confined to invariant tori. As excitation increases, the Kolmogorov-Arnold-Moser theorem predicts the destruction of certain tori, creating bands of stochasticity bounded by surviving structures.

These bands are explored by trajectories, with resonant tori being particularly susceptible to destruction as dictated by the Poincaré-Birkhoff theorem. Quantum mechanically, the research examined the topology and nodal patterns of wavefunctions to ascertain dynamical characteristics, a method effective for near-integrable systems but limited in cases of strong mixing.

The study leveraged the phenomenon of scarring, observing the localization of quantum probability density along periodic orbits within classically chaotic systems, extending this concept to non-stationary wavefunctions. Phase space pictures were constructed using quasiprobability density functions, like Wigner and Husimi functions, to reveal information about the dynamics of the states through the positioning of maxima and zeros. The K-CN molecule, previously identified as exhibiting early onset chaos, was chosen for this theoretical investigation, utilising energy level correlation diagrams with a varying Planck constant to unveil the dynamical characteristics of its vibrational eigenstates.

KCN Vibrational Chaos and Quantum State Calculations via Ray Eigenvector Expansion

The vibrational dynamics of the KCN molecule were investigated using a two-degree-freedom model, revealing a transition to widespread chaos at an energy of 145cm−1, corresponding to the quantum ground state. Poincaré surfaces of section demonstrate this early transition, detailing the evolution of the KCN phase space with increasing energy.

Classical trajectories were calculated by numerically integrating Hamilton’s equations of motion, providing insight into the system’s dynamical behaviour. Quantum vibrational states were obtained using the Discrete Variable Representation, Distributed Gaussian Basis method, employing a basis set of approximately 1000 ray eigenvectors.

Calculations converged to within 1cm−1 for approximately 300 low-lying eigenfunctions, computed for Planck constants ranging from 0.10 to 3.00 atomic units. For lower values of ħ, specifically between 0.05 and 0.10 a.u., the number of rays was increased to 120 to maintain computational accuracy. A correlation diagram of vibrational energy levels was constructed, utilizing the Planck constant as a variable parameter.

This approach mirrors the effect of varying isotopic masses within the molecular Hamiltonian. Analysis of this diagram reveals emerging diabatic states corresponding to regular classical structures, specifically Kolmogorov-Arnold-Moser tori, which are otherwise hidden when a fixed value of the Planck constant is used.

Coupling matrix elements, ⟨m|∂/∂ħ|n⟩, were evaluated using the off-diagonal Hellmann-Feynman theorem, quantifying the interaction between eigenstates as the parameter ħ is varied. The research identifies a quantum transition from order to chaos, manifested as a frontier of scarred functions within the correlation diagrams. This work provides a detailed understanding of the highly nonlinear dynamics of the KCN molecule and the relationship between classical and quantum vibrational states.

Planck constant variation elucidates K-CN vibrational structure and quantum chaos transitions

Scientists have demonstrated how variations in the Planck constant can reveal hidden structures within the vibrational energy levels of the K-CN molecule. By treating the Planck constant as a variable parameter in correlation diagrams, regular classical structures known as Kolmogorov-Arnold-Moser tori emerge as diabatic states.

These structures would otherwise remain obscured when using a fixed value for the Planck constant, providing a new method for analysing chaotic dynamical systems. Furthermore, the research unveils a transition from order to chaos, appearing as a frontier of scarred functions within the correlation diagrams.

Analysis of eigenstates reveals that those initially localized on classical resonances at higher Planck constant values evolve towards more regular states as the Planck constant decreases. This transition is particularly clear for certain eigenstates, suggesting a connection to underlying 1:2 quantum resonances and the boundaries between ordered and chaotic regions in molecular systems.

The authors acknowledge that some results are not entirely conclusive, particularly regarding the precise identification of resonant states. However, the presented evidence supports the interpretation of observed coupling curves as originating from the lowest lying 1:2 quantum resonance associated with scarred functions.

Future research could focus on extending these correlation diagram techniques to other highly nonlinear molecular systems, potentially refining the understanding of quantum chaos and its manifestation in molecular dynamics. These findings offer a valuable tool for investigating the interplay between classical and quantum behaviour in complex molecular systems.