Lifetimes and wave functions of ozone metastable vibrational states near the dissociation limit in a full-symmetry approach

Energies and lifetimes (widths) of vibrational states above the lowest dissociation limit of ${}^{16}\mathrm{O}_3$ were determined using a previously developed efficient approach, which combines hyperspherical coordinates and a complex absorbing potential. The calculations are based on a recently computed potential energy surface of ozone determined with a spectroscopic accuracy [Tyuterev et al., J. Chem. Phys. 139, 134307 (2013)]. The effect of permutational symmetry on rovibrational dynamics and the density of resonance states in ${\mathrm{O}}_3$ is discussed in detail. Correspondence between quantum numbers appropriate for short- and long-range parts of wave functions of the rovibrational continuum is established. It is shown, by symmetry arguments, that the allowed purely vibrational ($J=0$) levels of ${}^{16}\mathrm{O}_3$ and ${}^{18}\mathrm{O}_3$, both made of bosons with zero nuclear spin, cannot dissociate on the ground-state potential energy surface. Energies and wave functions of bound states of the ozone isotopologue ${}^{16}\mathrm{O}_3$ with rotational angular momentum $J=0$ and 1 up to the dissociation threshold were also computed. For bound levels, good agreement with experimental energies is found: The rms deviation between observed and calculated vibrational energies is 1 ${\mathrm{cm}}^{-1}$. Rotational constants were determined and used for a simple identification of vibrational modes of calculated levels.

Figure 1

Ozone potential energy surface (NR_PES of Ref. [64]) as a function of the two hyperangles for several values of the hyperradius. In the plots, the hyperangles are represented in a polar coordinate system (see Fig. 6 of Ref. [80]): $\theta$ increases from the center of each plot to its edge; $\varphi$ is a cyclic variable (polar angle) changing from 0 to $2\pi$. The minimum of PES, situated near $\rho =4.2a_0$, is chosen as the origin. The electronic energy of dissociation to the atom and the diatomic molecule at equilibrium is at $9164{\mathrm{cm}}^{-1}$.

Figure 2

Hyperspherical adiabatic curves $U_a\left(\rho \right)$ of the $A_1$ irreducible representation and $J=0$ as a function of hyperradius obtained for ${}^{16}\mathrm{O}_3$. In this figure, the energy origin is chosen at the ground vibrational level of ${}^{16}\mathrm{O}_3$, situated $1443.524{\mathrm{cm}}^{-1}$ above the PES minimum. The $v_d=0,1,2$ labels indicate energies of the $\mathrm{O}+{\mathrm{O}}_2\left(v_d\right)$ asymptotic vibrational channels. Multiple HSA curves between the vibrational channels correspond to various rotational channels $j$ of the dissociating oxygen molecule. For the $A_1$ vibrational states, only even $j$ are allowed. Because $J=0$, the partial wave in each asymptotic channel $(v_d,j,l)$ is determined simply as $l=j$.

Figure 3

Comparison of the energies of band centers obtained in this study with the previous calculation [64] and experimental data [8, 15, 16, 17, 18, 67, 68, 69, 70, 71, 72] for two vibrational symmetries ($A_1$ and $A_2$ in the $D_{3h}$ group employed here, $A_1$ and $B_1$ in the $C_{2v}$ group employed in Ref. [64]). The difference between the present results and the previous calculation and experimental data is labeled as $D_{3h}-C_{2v}$ and $\exp -D_{3h}$, respectively.

Figure 4

The largest of the three rotational constants for $A_1$ vibrational states in the $D_{3h}$ group. Almost linear dependence of the rotational constant $A_v$ on the energy of the vibrational states permits an assignment of normal modes for low energy levels. Normal mode quantum numbers are specified for a few levels. Note that at high energy the normal mode assignment becomes “nominative” and is to be taken with caution because of strong anharmonic basis state mixing.

Figure 5

Wave functions of the $(7,0,0)$ (top panels) and the $(0,11,0)$ (bottom panels) levels as functions of the Jacobi coordinates $R,r,$ and $\gamma$. Only the vibrational (without the rotational) part of the wave functions is given. In the left panels, the dependence on $R$ and $\gamma$ is shown for a fixed value of $r=2.28a_0$, which is the equilibrium nuclear distance in the ${\mathrm{O}}_2$ molecule. The right panels show the $R,r$ dependence for fixed $\gamma ={40}^{\circ }$.

Figure 6

As Fig. 5, but for the pure antisymmetric stretching modes $(0,0,4)$ (top panels) and $(0,0,6)$ (bottom panels).

Figure 7

As Fig. 5, but for the $(0,12,0)$ bound (top panels) and $(0,13,0)$ resonance (bottom panels) vibrational states. The interval or variation of $R$ in the left-panel plots is larger than in Figs. 5 and 6 in order to demonstrate the long-range tail of the wave functions. Note that the state shown in the lower figure exists only if rotation is excited. Thus, the wave functions are just the vibrational parts of the total rovibrational functions.

Figure 8

The $A_v$ rotational constants for $A_2\left(D_{3h}\right)$ vibrational states. Normal mode quantum numbers are specified for a few levels. Above the dissociation threshold (vertical dotted red line), the vibrational levels are predissociated.

Figure 9

Widths, $\mathrm{\Gamma }$, of resonances of the $A_2\left(D_{3h}\right)$ vibrational levels. Vertical dotted lines indicate threshold energies for dissociation channels with different excitation of the oxygen molecule $v_d=0,1,$ and 2. Numerically, lifetimes $\tau$ in ps are related to the widths in ${\mathrm{cm}}^{-1}$ as $\tau \left[\mathrm{ps}\right]={\left(2\pi c\mathrm{\Gamma }\left[{\mathrm{cm}}^{-1}\right]\right)}^{-1}$, where $c$ is the speed of light in units of cm/ps, $c=0.0299792458$ cm/ps. Wave functions of the encircled levels are shown in Figs. 11 and 12.

Figure 10

Same as Fig. 5, but for the $(7,0,1)$ and $(0,10,1)$ wave functions, of $A_2\left(D_{3h}\right)$ overall vibrational symmetry.

Figure 11

Vibrational part of the wave functions of the $(8,0,1)$ (top panels) and $(9,0,1)$ (bottom panels) levels of $A_2\left(D_{3h}\right)$ vibrational symmetry. The calculated widths are $\mathrm{\Gamma }=0.016 {\mathrm{cm}}^{-1}$ for the $(8,0,1)$ level and and 1.7 ${\mathrm{cm}}^{-1}$ for $(9,0,1)$.

Figure 12

Vibrational part of the wave functions of two highly excited levels of the $A_2\left(D_{3h}\right)$ symmetries. They are vibrationally assigned as $(8,1,1)$ and $(8,2,1)$ via the normal mode decomposition using the contact transformation method of Ref. [96]. The calculated widths are $\mathrm{\Gamma }=0.36 {\mathrm{cm}}^{-1}$ for the function shown in the top panels and 0.8 ${\mathrm{cm}}^{-1}$ for the function shown in the bottom panels.

Figure 13

Energies of vibrational levels for $J=0$ and $J=1$. In addition to the symbol $\left\{J_{K_a{K}_c}\right\}$ of the rotational states of an asymmetric top molecule, irreducible representations of the vibrational part (${\mathrm{\Gamma }}^v$, below the graph) and rotation part (${\mathrm{\Gamma }}^r$, above the graph) of the wave function are specified. The corresponding resonances are shown in blue.