_{2}

^{1}and Michael E. Kellman

^{1,a)}

### Abstract

We present a two-dimensional model for isomerization in the hydroperoxyl radical (HO_{2}). We then show that spectroscopic fitting Hamiltonians are capable of reproducing large scale vibrational structure above isomerization barriers. Two resonances, the 2:1 and 3:1, are necessary to describe the pertinent physical features of the system and, hence, a polyad-breaking Hamiltonian is required. We further illustrate, through the use of approximate wave functions, that inclusion of additional coupling terms yields physically unrealistic results despite an improved agreement with the exact energy levels. Instead, the use of a single diagonal term, rather than “extra” couplings, yields good fits with realistic results. Insight into the dynamical nature of isomerization is also gained through classical trajectories. Contrary to physical intuition the bend mode is not the initial “reaction mode,” but rather isomerization requires excitation in both the stretch and bend modes. The dynamics reveals a Farey tree formed between the 2:1 and 3:1 resonances with the prominent 5:2 (2:1 + 3:1) feature effectively dividing the tree into portions. The 3:1 portion is associated with isomerization, while the 2:1 portion leads to “localization” and perhaps dissociation at higher energies than those considered in this work. Simple single resonance models analyzed on polyad phase spheres are able to account in a qualitative way for the spectral, periodic orbit, and wave function patterns that we observe.

The authors are very grateful to Professor Hua Guo both for the use of his potential and fruitful discussions about the HO_{2} system. This work was supported by the US Department of Energy Basic Energy Sciences program under Contract No. DE-FG02-05ER15634. Computer resources were provided by the University of Oregon high performance computer cluster.

I. INTRODUCTION

II. METHOD

A. Isomerizationmodel

B. Spectroscopic Hamiltonians

III. RESULTS

A. Energy levels and wave functions

B. Comparison of exact and fit dynamics

C. Qualitative models for spectral and wave function patterns

IV. SUMMARY

### Key Topics

- Wave functions
- 55.0
- Isomerization
- 33.0
- Speed of sound
- 14.0
- Phase space methods
- 12.0
- Bifurcations
- 10.0

## Figures

Plot of the model potential energy surface with contours representing 2000 cm^{−1} equipotential lines. The O–O bond length is fixed to the equilibrium value. The oxygen atoms are symmetrically placed on the x axis about zero.

Plot of the model potential energy surface with contours representing 2000 cm^{−1} equipotential lines. The O–O bond length is fixed to the equilibrium value. The oxygen atoms are symmetrically placed on the x axis about zero.

Comparison of various fits with the energy zero defined as the ZPE level. The isomerization barrier is located at 11 950 cm^{−1}. Several states are labeled based on a visual inspection of the *ab initio* wave functions.

Comparison of various fits with the energy zero defined as the ZPE level. The isomerization barrier is located at 11 950 cm^{−1}. Several states are labeled based on a visual inspection of the *ab initio* wave functions.

Comparison of squared wave functions for the |3 2〉 state from several resonant fits, all of which yield good results. The physical coordinates are those of Fig. 1; however, due to the symmetry of the potential, we present a view encompassing only the *X* _{ H } > 0 portion. The red “X” in the exact plot indicates the location of the transition state. The wave functions for the effective Hamiltonians were constructed using a harmonic basis with frequencies obtained from the zero order fit.

Comparison of squared wave functions for the |3 2〉 state from several resonant fits, all of which yield good results. The physical coordinates are those of Fig. 1; however, due to the symmetry of the potential, we present a view encompassing only the *X* _{ H } > 0 portion. The red “X” in the exact plot indicates the location of the transition state. The wave functions for the effective Hamiltonians were constructed using a harmonic basis with frequencies obtained from the zero order fit.

A comparison of squared wave functions for the |0 11〉 state. The result illustrates the negative effects on the fit by adding too many off-diagonal coupling terms. shows that addition of a single third order diagonal coupling term to a minimal set of off-diagonal resonance couplings yields physically realistic wave functions and good agreement with the energy level.

A comparison of squared wave functions for the |0 11〉 state. The result illustrates the negative effects on the fit by adding too many off-diagonal coupling terms. shows that addition of a single third order diagonal coupling term to a minimal set of off-diagonal resonance couplings yields physically realistic wave functions and good agreement with the energy level.

A comparison of squared wave functions for the |0 13_{I}〉 state where the “I”designates an isomerizing bend state. Although yields physically realistic results up to 13 500 cm^{−1}, above that energy poor results are seen. Good results are obtained by including additional couplings () or a selected third order diagonal term (). The former option produces poor results for some levels (see Fig. 4), while the latter yields good results for all states below 15 000 cm^{−1}.

A comparison of squared wave functions for the |0 13_{I}〉 state where the “I”designates an isomerizing bend state. Although yields physically realistic results up to 13 500 cm^{−1}, above that energy poor results are seen. Good results are obtained by including additional couplings () or a selected third order diagonal term (). The former option produces poor results for some levels (see Fig. 4), while the latter yields good results for all states below 15 000 cm^{−1}.

Comparison of squared wave functions and classical POs from the exact potential surface (a–d) and the effective Hamiltonian (e). Strong correlations are seen in (a) and (b) for bend and combination states. Panel (c) shows a strong relation to the 2:1 PO, while panel (d) illustrates the clear difference between a local PO and an isomerizing bend wave function. Although a classical PO on the exact surface cannot be found to match the isomerizing bend, the model Hamiltonians (e) shows that a 3:1 PO matches with the fit wave functions for this state.

Comparison of squared wave functions and classical POs from the exact potential surface (a–d) and the effective Hamiltonian (e). Strong correlations are seen in (a) and (b) for bend and combination states. Panel (c) shows a strong relation to the 2:1 PO, while panel (d) illustrates the clear difference between a local PO and an isomerizing bend wave function. Although a classical PO on the exact surface cannot be found to match the isomerizing bend, the model Hamiltonians (e) shows that a 3:1 PO matches with the fit wave functions for this state.

Comparison of the SOS for the exact, , and Hamiltonians, respective columns. The first column is in the physical coordinates of Fig. 1 and the second and third columns are in the action-angle variables of the semiclassical fitting Hamiltonians, hence the different appearance of the SOS. However, colors are consistent between plots and identify similar features. At low energy the phase space is simple, while at intermediate energies several resonance islands have appeared. At high energy, chaos has destroyed several of the island chains. Note that due to the sectioning plane some of the islands are missing; hence, the identity of the resonances was determined from inspection of coordinate space trajectories.

Comparison of the SOS for the exact, , and Hamiltonians, respective columns. The first column is in the physical coordinates of Fig. 1 and the second and third columns are in the action-angle variables of the semiclassical fitting Hamiltonians, hence the different appearance of the SOS. However, colors are consistent between plots and identify similar features. At low energy the phase space is simple, while at intermediate energies several resonance islands have appeared. At high energy, chaos has destroyed several of the island chains. Note that due to the sectioning plane some of the islands are missing; hence, the identity of the resonances was determined from inspection of coordinate space trajectories.

A portion of the Farey sequence between 1/3 and 1/2, related to the resonances present in the system. Solid lines denote Farey neighbors (e.g., 1/3 and 1/2) and their branchings (e.g., 2/5), while vertical dotted lines divide the tree and show boundaries for which neighbors cannot exist (5/13 and 5/12). Blue members denote the 3:1 portion, while red members denote the 2:1 portion.

A portion of the Farey sequence between 1/3 and 1/2, related to the resonances present in the system. Solid lines denote Farey neighbors (e.g., 1/3 and 1/2) and their branchings (e.g., 2/5), while vertical dotted lines divide the tree and show boundaries for which neighbors cannot exist (5/13 and 5/12). Blue members denote the 3:1 portion, while red members denote the 2:1 portion.

The frequencies for the bend states (|0 *n*〉) and combination states (|1 *m*〉). The bend states show a typical anharmonic progression with no evidence of the approach to the barrier, while the combination states show a dip just above the barrier at 11 950 cm^{−1}.

The frequencies for the bend states (|0 *n*〉) and combination states (|1 *m*〉). The bend states show a typical anharmonic progression with no evidence of the approach to the barrier, while the combination states show a dip just above the barrier at 11 950 cm^{−1}.

Polyad phase spheres resulting from the 3:1 (top) and 2:1 (bottom) single resonance Hamiltonians. Stable fixed points associated with the stretch mode and the anharmonically perturbed bend mode are labeled as S1 and S2, respectively. The 3:1 coupling shows an additional stable fixed point, S3, associated with the isomerizing bend states. The 2:1 bifurcation results in a hyperbolic fixed point (H1) at the north pole of the 2:1 polyad phase sphere. There is also an unstable fixed point on the 3:1 sphere between S3 and the north pole.

Polyad phase spheres resulting from the 3:1 (top) and 2:1 (bottom) single resonance Hamiltonians. Stable fixed points associated with the stretch mode and the anharmonically perturbed bend mode are labeled as S1 and S2, respectively. The 3:1 coupling shows an additional stable fixed point, S3, associated with the isomerizing bend states. The 2:1 bifurcation results in a hyperbolic fixed point (H1) at the north pole of the 2:1 polyad phase sphere. There is also an unstable fixed point on the 3:1 sphere between S3 and the north pole.

## Tables

Optimized parameters of the spectroscopic Hamiltonians.

Optimized parameters of the spectroscopic Hamiltonians.

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