(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 .
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).
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).
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) .
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).
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 .
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