^{1,a)}, Xinchuan Huang

^{2}, Stuart Carter

^{2,b)}and Joel M. Bowman

^{2,c)}

### Abstract

The vibrations of and are investigated using diffusion Monte Carlo (DMC) and vibrational configuration-interaction approaches, as implemented in the program MULTIMODE. These studies use the potential surface recently developed by Huang *et al.* [ J. Am. Chem. Soc.126, 5042 (2004)]. The focus of this work is on the vibrational ground state and fundamentals which occur between 100 and . In most cases, excellent agreement is obtained between the fundamental frequencies calculated by the two approaches. This serves to demonstrate the power of both methods for treating this very anharmonic system. Based on the results of the MULTIMODE and DMC treatments, the extent and nature of the couplings in and are investigated.

We thank Professor Mark A. Johnson for stimulating discussions on these systems. S.C. would like to thank Nicholas C. Handy for drawing his attention to the role of one-dimensional integrals in the evaluation of matrix elements involving intrinsic four-mode potentials which can be accurately reproduced by polynomials in the normal modes. The National Science Foundation is thanked for the generous support of this work through Grant No. CHE-0200968/CHE-0515627 by one of the authors (A.B.M.) and Grant No. CHE-0131482/CHE-0446527 by another author (J.M.B.). One of the authors (J.M.B.) also thanks the ONR/DURIP for funding computational resources. Another author (S.C.) thanks the U.S. Office of Naval Research for continued support of this work under Grant No. N00014-01-1-0235.

I. INTRODUCTION

II. THE SYSTEM

A. Global potential-energysurface

B. Coordinates

III. DIFFUSION MONTE CARLO (DMC)

A. Excited states

B. Determining properties from DMC

IV. REACTION PATH MULTIMODE (MM-RP)

A. Method

B. Numerical issues

V. RESULTS

A. Ground state

B. Fundamentals in

1. Torsion, OO stretch, wag, and rock

2. OH stretches

3. Bridging hydrogen modes

C. Effects of deuteration

D. Comments on the new global potential

VI. SUMMARY AND CONCLUSIONS

### Key Topics

- Wave functions
- 29.0
- Excited states
- 14.0
- Normal modes
- 11.0
- Zero point energy
- 11.0
- Ground states
- 10.0

## Figures

The structures of the four lowest-energy stationary points on the HBB potential (see Ref. 14) (a) the potential minimum, (b) the saddle point, (c) the *trans* saddle point, and (d) the *cis* saddle point. The coordinates that will be used to describe the excited states are indicated in panel (a).

The structures of the four lowest-energy stationary points on the HBB potential (see Ref. 14) (a) the potential minimum, (b) the saddle point, (c) the *trans* saddle point, and (d) the *cis* saddle point. The coordinates that will be used to describe the excited states are indicated in panel (a).

The normal-mode vectors at the saddle point in . The numbers in each panel indicate the corresponding harmonic frequency. For clarity totally symmetric modes are given in the left column, and modes that have symmetry are in the right column.

The normal-mode vectors at the saddle point in . The numbers in each panel indicate the corresponding harmonic frequency. For clarity totally symmetric modes are given in the left column, and modes that have symmetry are in the right column.

Plot of the dependence of five harmonic frequencies along the torsion path used to define the Hamiltonian used for the VCI calculations for (a) and (b) . Using the notation in Fig. 2, the solid line shows the OO stretch frequency, the dotted- and short-dashed lines give the wag and rock frequencies, while the dot-dot-dashed and long-dashed lines give the frequencies of and . In the frequencies of the OO stretch and wag flip their energy ordering over this range of angles and we plotted both frequencies with thick solid lines. To illustrate the underlying, uncoupled (diabatic) frequencies, we have sketched them with thin lines.

Plot of the dependence of five harmonic frequencies along the torsion path used to define the Hamiltonian used for the VCI calculations for (a) and (b) . Using the notation in Fig. 2, the solid line shows the OO stretch frequency, the dotted- and short-dashed lines give the wag and rock frequencies, while the dot-dot-dashed and long-dashed lines give the frequencies of and . In the frequencies of the OO stretch and wag flip their energy ordering over this range of angles and we plotted both frequencies with thick solid lines. To illustrate the underlying, uncoupled (diabatic) frequencies, we have sketched them with thin lines.

Plot of the energies of the OO stretch fundamental as a function of the position of the node, obtained from a DMC simulation (gray curves), as well as fifth-order fits to the raw data (black curves). The point where the two black curves cross provides the position of the node and the energy of the state, relative to the global minimum of the potential.

Plot of the energies of the OO stretch fundamental as a function of the position of the node, obtained from a DMC simulation (gray curves), as well as fifth-order fits to the raw data (black curves). The point where the two black curves cross provides the position of the node and the energy of the state, relative to the global minimum of the potential.

Plot of the projection of the wave functions for the ground state onto (a) the hydrogen transfer mode and (b) the torsion coordinate . In both plots, the results for are shown with a solid line; the dashed line provides the results for . The arrows show the values of these coordinates at the minimum-energy configuration of .

Plot of the projection of the wave functions for the ground state onto (a) the hydrogen transfer mode and (b) the torsion coordinate . In both plots, the results for are shown with a solid line; the dashed line provides the results for . The arrows show the values of these coordinates at the minimum-energy configuration of .

Slice through the VCI states at (a) 532 and (b) plotted as functions of the two normal modes that correlate to the wag and OO stretch in Fig. 2. For these plots all other normal modes are set equal to zero and .

Slice through the VCI states at (a) 532 and (b) plotted as functions of the two normal modes that correlate to the wag and OO stretch in Fig. 2. For these plots all other normal modes are set equal to zero and .

Projections of excited states for the rock [(a)–(c)], [(d)–(f)], and [(g)–(i)] fundamentals. In the left column, the probability amplitudes are projected onto the (solid lines), (dashed lines), and displacements (dotted lines) of the bridging hydrogen atom. In cases where the distributions obtained from the parts of the wave function with positive and negative amplitudes differ, the separate distributions are plotted with thin lines. The plots in the central column show projections of the probability amplitudes onto one of the OOH bends with a thick solid black line. The contributions to this distribution from the parts of the wave function with positive and negative amplitudes are plotted with thin lines. In the right column, projections of the probability amplitudes onto are plotted.

Projections of excited states for the rock [(a)–(c)], [(d)–(f)], and [(g)–(i)] fundamentals. In the left column, the probability amplitudes are projected onto the (solid lines), (dashed lines), and displacements (dotted lines) of the bridging hydrogen atom. In cases where the distributions obtained from the parts of the wave function with positive and negative amplitudes differ, the separate distributions are plotted with thin lines. The plots in the central column show projections of the probability amplitudes onto one of the OOH bends with a thick solid black line. The contributions to this distribution from the parts of the wave function with positive and negative amplitudes are plotted with thin lines. In the right column, projections of the probability amplitudes onto are plotted.

## Tables

Average values and widths of projections of the ground-state wave functions for and onto the internal coordinates defined in Fig. 1.

Average values and widths of projections of the ground-state wave functions for and onto the internal coordinates defined in Fig. 1.

Energies of the fundamentals of and obtained from DMC and VCI calculations. (The details of the VCI calculations are given in the text.)

Energies of the fundamentals of and obtained from DMC and VCI calculations. (The details of the VCI calculations are given in the text.)

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