Jacobi description of the system. The vector connects the two centers of mass of the water monomers. The vector connects the center of mass of the water dimer with the central proton.
Definition of the angles for the system. The angles and describe the rotation of the water monomers around the vector , or equivalently around the axis of the or frame. In the dynamical calculations only the torsion angle appears and the polar coordinates of the proton are replaced by Cartesian ones. See Appendix for details.
Number of grid points needed for the representation of the clusters of the PES expansion. is the number of coordinates making a mode. 10 grid points per coordinates and 15 coordinates are assumed. A horizontal line is drawn at , which tentatively signals the maximum practical number of points both regarding their calculation and the use of the grids in the dynamical calculations.
Geometries of the ten reference points used in the PES expansion. The view is along the O–H–O axis. Hence only the closest of the two oxygen and the four hydrogens can be seen. The difference between the geometries in the left column and each geometry at the right column is a rotation of along . Equivalently, the pairs of structures (a,b), (c,j), (e,h), (g,f), and (i,d) are related to a permutation of hydrogen atoms of one of the monomers. The following coordinates are identical for all reference points: , , , and , . Only coordinates , , and differ at the ten reference points.
For the ground vibrational-state probability density along selected coordinates and integration over the rest: probability density along the proton-transfer coordinate (a), along the internal rotation coordinate, (b) and on the 2D space spanned by the wagging and coordinates (c). The dotted line in (b) corresponds to a ten times enlarged scale. It indicates that the probability density at is not vanishing.
Definition of the one-dimensional grids. denotes the number of grid points and , the location of the first and the last point. The DVRs are defined in Appendix B of Ref. 24.
Comparison of the zero point energy (ZPE) of the cation calculated by various approaches on the PES by Huang et al. (Ref. 7): diffusion Monte Carlo (DMC), normal-mode analysis (harmonic), vibrational CI single reference (VCI-SR), and reaction path (VCI-RP) as published in Ref. 12 and MCTDH results. denotes the difference to the DMC result. The converged MCTDH result is obtained with 10 500 000 configurations. Compare with Table III.
Comparison of the zero point energy (ZPE) of the cation between different MCTDH calculation with ascending number of configurations. The values are given with respect to the diffusion Monte Carlo result, (Ref. 12).
Expectation value of the different terms of the potential expansion (central column) and square root of the expectation value of the potential squared (right column). All energies in . The combined modes read , , , , and .
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