Correspondence between classical and quantum resonances

TL;DR

The paper addresses how classical resonances in the CN-Li↔Li-CN isomerization system manifest quantum-mechanically through a correlation diagram of eigenenergies versus $\hbar$. Classical resonances appear as island chains in Poincaré sections, while quantum resonances show up as series of avoided crossings whose energies extrapolate to the classical bifurcation energies in the $\hbar\to0$ limit. A DVR-DGB quantum calculation across a range of $\hbar$ values enables identification of six quantum-resonance series and their connection to corresponding classical resonances, with a semiclassical theory developed via quadratic and cubic expansions (both coupled and decoupled) to provide analytic expressions for the resonance energies in the semiclassical limit. The results yield a quantitative bridge between classical bifurcations and quantum resonances, offering insight into the order-to-chaos transition and the role of scars, while highlighting the frontier of scars as a challenging case for further semiclassical refinement.

Abstract

Bifurcations take place in molecular Hamiltonian nonlinear systems as the excitation energy increases, this leading to the appearance of different classical resonances. In this paper, we study the quantum manifestations of these classical resonances in the isomerizing system CN-Li$\leftrightarrows$Li-CN. By using a correlation diagram of eigenenergies versus Planck constant, we show the existence of different series of avoided crossings, leading to the corresponding series of quantum resonances, which represent the quantum manifestations of the classical resonances. Moreover, the extrapolation of these series to $\hbar=0$ unveils the correspondence between the bifurcation energy of classical resonances and the energy of the series of quantum resonances in the semiclassical limit $\hbar\to0$. Additionally, in order to obtain analytical expressions for our results, a semiclassical theory is developed.

Figures (5)

• Figure 1: (a) Energy profile along the minimum energy path connecting minima and saddles of the potential energy surface. (b) Potential energy surface represented as contour plots spaced 1000 cm$^{-1}$. The minimum energy path has also been plotted superimposed in thick line.
• Figure 2: Composite Poincaré surfaces of section, defined along the minimum energy path, for energies of 1500 cm$^{-1}$ (a) and 2300 cm$^{-1}$ (b), below and above, respectively, the order-chaos threshold energy. The chains of islands corresponding to the main resonances have been specifically represented, namely, 1:6, 1:7, 1:8, 1:9, 1:10, again 1:10, and 1:8 (in the chaotic sea) classical resonances. Gray region represents the energetically forbidden region.
• Figure 3: Correlation diagram of eigenenergies versus Planck constant. On grounds of graphical clarity, energy is divided by Planck constant. Different series of avoided crossings, corresponding to 1:6 ($\hbox{\FiveStarOpen}$), 2:14 ($\bigtriangleup$), 1:8 ($\lozenge$), 2:18 ($\bigtriangledown$), 1:10 ($\square$), and 1:8 ($\bigcirc$) quantum resonances, have been marked.
• Figure 4: (a) Magnification of the region of the correlation diagram depicted in Fig. \ref{['fig:correlation_diagram']} where take place the avoided crossings corresponding to the pair of 1:10 resonances with $k=4$ (see Table \ref{['tab:resonances']}). The involved states have been represented in thick line. The corresponding wavefunctions at $\hbar=0.9$ a.u. (value between both avoided crossings) are depicted in panel (b) for the upper state, with quantum numbers $(n_1,n_2)=(1,8)$, and in panel (c) for the lower state, with quantum numbers $(n_1,n_2)=(0,18)$. In both cases, the minimum energy path and the corresponding eigenenergy contour have been plotted superimposed in thick line and thin line, respectively.
• Figure 5: Series of avoided crossing points, corresponding to 1:6 ($\hbox{\FiveStarOpen}$), 2:14 ($\bigtriangleup$), 1:8 ($\lozenge$), 2:18 ($\bigtriangledown$), 1:10 ($\square$), and 1:8 ($\bigcirc$) quantum resonances. The points belonging to the same series have been connected with straight lines. The open symbols plotted at $\hbar = 0$ are the result of a linear extrapolation. The filled symbols plotted at $\hbar = 0$ represent the bifurcation energies of the corresponding 1:6 ($\bigstar$), 1:7 ($\blacktriangle$), 1:8 ($\blacklozenge$), 1:9 ($\blacktriangledown$), 1:10 ($\blacksquare$), again 1:10 ($\blacksquare$), 1:8 ($\hbox{\huge$\bullet$}$), and again 1:8 ($\hbox{\huge$\bullet$}$) classical resonances.