^{1}, H. Ushiyama

^{2}, K. Takatsuka

^{2}and O. Kühn

^{3,a)}

### Abstract

Based on the *Cartesian Reaction Surface* framework we construct a four-dimensional potential for the tropolone derivative 3,7-dichlorotropolone, a molecule with an intramolecular hydrogen bond. The reduced configuration space involves the in-plane hydrogen atom coordinates, a symmetric O–O vibrational mode, and an antisymmetric mode related to deformations of the seven-membered ring. The system is characterized in terms of quantum mechanical computations of the low-lying eigenstates as well as a classical and semiclassical analysis of spectra obtained via Fourier transforming autocorrelation functions. For the semiclassical analysis we utilize the amplitude-free correlation function method [K. Hotta and K. Takatsuka, J. Phys. A36, 4785 (2003)]. Our results demonstrate substantial anharmonic couplings leading to highly correlated wave functions even at moderate energies. Furthermore, the importance of dynamical tunneling in tropolone is suggested since many low-lying states—including the ground state—lie above the classical saddle point but nevertheless appear as split pairs.

This work was supported in parts by the Deutsche Forschungsgemeinschaft (Project No. Ma 515/19-1), the Fonds der Chemischen Industrie (O.K.), and by a Grant-in-Aid from the Ministry of Education, Science, and Culture of Japan (H.U. and K.T.). The authors are grateful to Professor J. Manz (Berlin) for discussions and support of their work, and they thank Dr. H.-D. Meyer (Heidelberg) for sharing with them his expertise in the MCTDH method. The authors are also grateful to the support from DFG and JSPS for the German-Japanese cooperative research project.

I. INTRODUCTION

II. FOUR-DIMENSIONAL REACTION SURFACE

III. CHARACTERIZATION OF THE EIGENSTATES

A. Dynamical tunneling for low lying states

B. Higher excited states

IV. CLASSICAL AND SEMICLASSICAL ANALYSIS

A. Classical power spectra

B. AFC-II

V. SUMMARY AND OUTLOOK

### Key Topics

- Tunneling
- 61.0
- Wave functions
- 18.0
- Phase space methods
- 11.0
- Semiclassical theories
- 9.0
- Zero point energy
- 8.0

## Figures

(a) Reference geometry. (b) Reorganization energy for individual skeleton modes relative to the total reorganization energy at the minimum. The displacement vectors of the two most important skeleton modes are indicated as well.

(a) Reference geometry. (b) Reorganization energy for individual skeleton modes relative to the total reorganization energy at the minimum. The displacement vectors of the two most important skeleton modes are indicated as well.

Contour plots of the 4D PES: (a) The case when all skeleton modes are relaxed ; (b) 2D cut through the 4D PES with , where *min* denotes the right-hand minimum configuration. The contour line spacing is and the highest contour line is at .

Contour plots of the 4D PES: (a) The case when all skeleton modes are relaxed ; (b) 2D cut through the 4D PES with , where *min* denotes the right-hand minimum configuration. The contour line spacing is and the highest contour line is at .

Reduced densities of low-lying state doublets with respect to coordinate axes (in ) of the 4D model. The remaining DOF have been integrated out. The assignment is (a) ground state, (b) fundamental of the vibration, (c,d) first overtone of the vibration, (e,f) fundamental of the OH-bend vibration, (g,h) second overtone of the vibration, and (i,j) fundamental of the antisymmetric vibration. The frequencies (in ) are given in each panel; frequencies in parenthesis belong to ungerade states that are not shown.

Reduced densities of low-lying state doublets with respect to coordinate axes (in ) of the 4D model. The remaining DOF have been integrated out. The assignment is (a) ground state, (b) fundamental of the vibration, (c,d) first overtone of the vibration, (e,f) fundamental of the OH-bend vibration, (g,h) second overtone of the vibration, and (i,j) fundamental of the antisymmetric vibration. The frequencies (in ) are given in each panel; frequencies in parenthesis belong to ungerade states that are not shown.

Logarithm of the tunneling splitting vs where as discussed in the text. Curves for the six lowest lying states are given . The solid lines are results for a fitting of the function to the data. The dimensionless values are (a) , (6.4, 9.2), and (5.3, 9.3); and (b) (6.4, 9.6), (3.7, 8.3), and (5.6, 6.8) (from top to bottom).

Logarithm of the tunneling splitting vs where as discussed in the text. Curves for the six lowest lying states are given . The solid lines are results for a fitting of the function to the data. The dimensionless values are (a) , (6.4, 9.2), and (5.3, 9.3); and (b) (6.4, 9.6), (3.7, 8.3), and (5.6, 6.8) (from top to bottom).

Reduced densities of states no. 12–14 with respect to the coordinate axes (in ) of the 4D model. The frequencies (in ) are given in each panel. (a) State no. 12, not characterized in terms of harmonic oscillator states. (b) State no. 13, third overtone of the vibration. (c) State no. 14, combination excitation of single excited OH bend and single excited vibration.

Reduced densities of states no. 12–14 with respect to the coordinate axes (in ) of the 4D model. The frequencies (in ) are given in each panel. (a) State no. 12, not characterized in terms of harmonic oscillator states. (b) State no. 13, third overtone of the vibration. (c) State no. 14, combination excitation of single excited OH bend and single excited vibration.

Classical trajectories, initial conditions by normal mode sampling with rescaling: (a) Mean energy of the doublet with normal mode quantum numbers (0,0,2,0); (b) mean energy of the OH-bend doublet with normal mode quantum numbers (0,1,0,0).

Classical trajectories, initial conditions by normal mode sampling with rescaling: (a) Mean energy of the doublet with normal mode quantum numbers (0,0,2,0); (b) mean energy of the OH-bend doublet with normal mode quantum numbers (0,1,0,0).

Power spectra for and , respectively. For the Fourier transform we used a Gaussian window function with and the trajectories were propagated up to 15 ps. The initial conditions have been obtained by normal mode sampling and rescaling to the eigenenergy, and the fundamental transition frequencies resulting from the *quantum* calculation are indicated by arrows and labeled , and for , OH stretch, and OH bend, respectively. (a) Ensemble of 100 trajectories with normal mode quantum numbers: , (b) with (0,0,3,0), (c) with (0,0,0,1), and (d) *single* trajectory with quantum numbers (0,1,0,0).

Power spectra for and , respectively. For the Fourier transform we used a Gaussian window function with and the trajectories were propagated up to 15 ps. The initial conditions have been obtained by normal mode sampling and rescaling to the eigenenergy, and the fundamental transition frequencies resulting from the *quantum* calculation are indicated by arrows and labeled , and for , OH stretch, and OH bend, respectively. (a) Ensemble of 100 trajectories with normal mode quantum numbers: , (b) with (0,0,3,0), (c) with (0,0,0,1), and (d) *single* trajectory with quantum numbers (0,1,0,0).

Semiclassical spectra of the 4D DCTRN model for energies below as counted from the ground state level. (Note that the latter is above the saddle point which differs slightly from the quantum mechanical value of .) The different panels corresponds to different approximations as discussed in the text: (a) AFC II , (b) , and (c) .

Semiclassical spectra of the 4D DCTRN model for energies below as counted from the ground state level. (Note that the latter is above the saddle point which differs slightly from the quantum mechanical value of .) The different panels corresponds to different approximations as discussed in the text: (a) AFC II , (b) , and (c) .

The most contributing trajectories to the spectrum in Fig. 8(c) corresponding to (a) the ground state and (b) the first excited state. Two different types of the trajectories which contribute most to the fourth peak of Fig. 8 are shown in panels (c) and (d).

The most contributing trajectories to the spectrum in Fig. 8(c) corresponding to (a) the ground state and (b) the first excited state. Two different types of the trajectories which contribute most to the fourth peak of Fig. 8 are shown in panels (c) and (d).

## Tables

Parameters used for the MCTDH propagation. For all DOF we employed the sine-DVR representation. The grid extends from to with grid points. The boundaries are not included in the grid, since there the wave function vanishes by definition of the sine-DVR. The hydrogen mass and one atomic mass unit has been associated to, respectively, the and the degrees of freedom.

Parameters used for the MCTDH propagation. For all DOF we employed the sine-DVR representation. The grid extends from to with grid points. The boundaries are not included in the grid, since there the wave function vanishes by definition of the sine-DVR. The hydrogen mass and one atomic mass unit has been associated to, respectively, the and the degrees of freedom.

Quantum mechanical results for the mode selectivity of the tunneling splittings (values in ) for several fundamentals and overtones . The ground state tunneling splitting is .

Quantum mechanical results for the mode selectivity of the tunneling splittings (values in ) for several fundamentals and overtones . The ground state tunneling splitting is .

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