Global bending quantum number and the absence of monodromy in the $\text{HCN}\leftrightarrow \text{CNH}$ molecule

We introduce and analyze a model system based on a deformation of a spherical pendulum that can be used to reproduce large amplitude bending vibrations of flexible triatomic molecules with two stable linear equilibria. On the basis of our model and the recent vibrational potential [J. Chem. Phys. 115, 3706 (2001)], we analyze the HCN∕CNH isomerizing molecule. We find that HCN∕CNH has no monodromy and introduce the second global bending quantum number for this system at all energies where the potential is expected to work. We also show that LiNC∕LiCN is a qualitatively different system with monodromy.

Figure 1

Image and fibers of the energy-momentum map $\mathcal{E}\mathcal{M}$ of the spherical pendulum; see Chap. IV.3 of Ref. 4.

Figure 2

Energy-momentum diagram for the spherical pendulum. Dots show quantum levels, bold solid lines represent relative equilibria, opaque circle marks the position $(l,h)=(0,1)$ of the unstable (“upper”) equilibrium. Rectangles show successive consistent definitions of local quantum numbers along the contour which goes around $(0,1)$. Both axes are in atomic units but the quantization step is $\hslash ∕8$.

Figure 3

Spherical pendulum (left) and flexible triatomic molecule HCN (right) with fixed bond lengths H-CN $\left(R\right)$, C-N $\left(r\right)$ and bending angle $\gamma$.

Figure 4

Possible choices of quantum numbers for the spherical pendulum (left to right): “vibrational” $n$, “rotational” $j$, and “mixed” $n+\ell$; cf. Fig. 2. Fine solid lines join states with the same quantum number which is given at the end of the sequence.

Figure 5

Image of the classical energy-momentum map (light and dark shaded area) for the quadratic spherical pendulum defined in Sec. 2C.

Figure 6

Inverse images of the points in the upper boundary $[{\alpha }_{-},{\alpha }_{+}]$ of leaf $B$ of the $\mathcal{E}\mathcal{M}$ map image of the quadratic spherical pendulum in Fig. 5.

Figure 7

Quantum and classical energy-momentum diagram for the quadratic spherical pendulum defined in Sec. 2C. Rectangles show successive consistent definitions of local quantum numbers along the contour which goes around the line of singular values of the main leaf $A$, cf. Fig. 5. Both axes are in atomic units but the quantization step is $\hslash ∕8$.

Figure 8

Image of the classical energy-momentum map (shaded areas) of the quadratic aspherical pendulum with asphericity $ϵ=\frac{1}{2}$ and quadratic potential constants $c_0$, $c_1$, $c_2$ used in Sec. 2C; cf. Figs. 5, 10.

Figure 9

Geometry and reaction path minimum potential energy of the HCN∕CNH system computed for the ab initio potential surface in Refs. 21, 22. Atoms C, N, and H are sized according to their covalent radii.

Figure 10

Image of the energy-momentum map (shaded areas) of the isomerizing HCN∕CNH system computed for pure bending states with ${\nu }_1={\nu }_3=0$ using the classical analog of the Hamiltonian (3).

Figure 11

Energy-momentum diagram for the ${\nu }_1={\nu }_3=0$ states of the HCN∕CNH system computed using the Hamiltonian (3). Shades of gray distinguish HCN, CNH, and delocalized level regions respectively, cf. Fig. 10. The regions are bordered by the energies of classical relative equilibria shown by bold solid lines. Filled circles show levels attributed to the HCN minimums while hollow circles represent either CNH or delocalized states depending on the region. A few larger black circles mark levels assigned in Ref. 22. Fine solid lines join states with the same quantum numbers $n_{\text{HCN}}$, $n_{\text{CNH}}$, or $j$, whose values are given at the end of the respective sequences. The diagram is symmetric with respect to $\ell \leftrightarrow -\ell$, and only half of it is shown.

Figure 12

Minimum stretch distance $R$ in the LiNC-NCLi system computed for the ab initio potential in Ref. 24.

Figure 13

Energy-momentum diagram for the pure bending states of LiNC∕NCLi computed using the ab initio potential in Ref. 24. Hollow circles show levels attributed to the NCLi minimum; filled circles represent LiNC levels which become delocalized states at higher energies; bold solid lines show energies of classical relative equilibria. The diagram is symmetric with respect to $\ell \leftrightarrow -\ell$ and only half of it is shown.

Figure 14

The flow of the two vector fields $X_H$ and $X_L$ on a regular 2-torus ${\mathbb{T}}_{(l,h)}^2$; the flow $X_L$ is periodic while $X_H$ is not.

Figure 15

Example of the local lattice of quantum states in a domain of the regular values of the energy-momentum map $\mathcal{E}\mathcal{M}$ of an $\text{SO}\left(2\right)$ symmetric system with two degree of freedoms in Appendix . Left panels show the initial “germ” cell; the right panel illustrates propagating this cell in order to define quantum numbers over a larger domain.

Figure 16

Two possible plots of a pinched torus; cf. Chap. IV.3, Fig. 3.5 on p. 163 of Ref. 4. Both representations are equivalent in the four-dimensional phase space.

Figure 17

The $\{p_z=0\}$ plane projections of the reduced spaces (shaded area) $P_{\ell =1}$ (left) and $P_{\ell =0}$ (right), and of the $h$-level sets for the spherical pendulum (solid lines $a$, $b$, $c$, $d$, $e$) and its quadratic deformation (dashed lines, label $f$).