^{1,a)}, Fabien Gatti

^{2,b)}, David Lauvergnat

^{3,c)}and Hans-Dieter Meyer

^{4,d)}

### Abstract

Quantum-dynamical full-dimensional (15D) calculations are reported for the protonated water dimer using the multiconfiguration time-dependent Hartree (MCTDH) method. The dynamics is described by curvilinear coordinates. The expression of the kinetic energy operator in this set of coordinates is given and its derivation, following the polyspherical method, is discussed. The potential-energy surface (PES) employed is that of Huang *et al.* [J. Chem. Phys.122, 044308 (2005)]. A scheme for the representation of the PES is discussed which is based on a high-dimensional model representation scheme, but modified to take advantage of the mode-combination representation of the vibrational wave function used in MCTDH. The convergence of the PES expansion used is quantified and evidence is provided that it correctly reproduces the reference PES at least for the range of energies of interest. The reported zero point energy of the system is converged with respect to the MCTDH expansion and in excellent agreement ( below) with the diffusion Monte Carlo result on the PES of Huang *et al.* The highly fluxional nature of the cation is accounted for through use of curvilinear coordinates. The system is found to interconvert between equivalent minima through wagging and internal rotation motions already when in the ground vibrational state, i.e., . It is shown that a converged quantum-dynamical description of such a flexible, multiminima system is possible.

The authors thank Professor J. Bowman for providing the potential-energy routine, M. Brill for the help with the parallelized code, and the Scientific Supercomputing Center Karlsruhe for generously providing computer time. O.V. is grateful to the Alexander von Humboldt Foundation for financial support. Travel support by the Deutsche Forschungsgemeinschaft (DFG) is also gratefully acknowledged.

I. INTRODUCTION

II. BRIEF DESCRIPTION OF THE MCTDH METHOD

III. SYSTEM HAMILTONIAN

A. Kinetic energy operator for

B. Hierarchical representation of the potential using mode combination

C. Potential-energy surface for the cation

IV. RESULTS AND DISCUSSION

A. Quantum-dynamical calculations

B. Properties of the ground state of the system

C. Quality of the PES expansion

V. SUMMARY AND CONCLUSION

### Key Topics

- Wave functions
- 16.0
- Polymers
- 11.0
- Protons
- 11.0
- Vibrational states
- 10.0
- Angular momentum
- 8.0

## Figures

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.

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.

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.

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.

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.

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.

## Tables

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.

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 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).

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 .

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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