^{1}, Tomoyuki Hayashi

^{2}, Wei Zhuang

^{2}and Shaul Mukamel

^{2,a)}

### Abstract

The effects of hydrogen-bond forming and breaking kinetics on the linear and coherent third-order infrared spectra of the OH stretch of HOD in are described by Markovian, not necessarily Gaussian, fluctuations and simulated using the stochastic Liouville equations. Slow fluctuations are represented by a collective electrostatic coordinate, whereas fast frequency fluctuations are described using either a second collective electrostatic coordinate or a four-state jump (FSJ) model for hydrogen-bonding configurations. Parameters for both models were obtained using a molecular-dynamics trajectory calculated using the TIP4P force field combined with an electrostatic*ab initio* map. The asymmetry of the photon-echo spectra (larger linewidth on the blue side than on the red side) predicted by the FSJ is in better agreement with recent experiments.

The support of the National Institutes of Health Grant No. (GM59230-005) and the National Science Foundation Grant No. (CHE-0446555) and the Air Force Office of Scientific Research (FA9550-04-1-0332) is gratefully acknowledged. We are grateful to Professor James L. Skinner for providing his simulation data and to Professor Andrei Tokmakoff for sharing his experimental photon-echo peak shift data.

I. INTRODUCTION

II. FLUCTUATING HAMILTONIAN FOR THE OH STRETCH

III. THE COLLECTIVE ELECTROSTATIC COORDINATE (CEC) MODEL

IV. THE FOUR-STATE JUMP-MODEL FOR FAST HYDROGEN-BOND FLUCTUATIONS

V. THE PHOTON-ECHO SPECTRA

VI. DISCUSSION

### Key Topics

- Hydrogen bonding
- 37.0
- Electrostatics
- 18.0
- Correlation functions
- 13.0
- Ab initio calculations
- 11.0
- Linewidths
- 9.0

## Figures

Energy-level diagram for the OH stretch in HOD. The model has five vibrational parameters (two transition frequencies and three transition dipole moments) relevant for the infrared spectra.

Energy-level diagram for the OH stretch in HOD. The model has five vibrational parameters (two transition frequencies and three transition dipole moments) relevant for the infrared spectra.

Static distribution of the OH stretch transition frequencies (, , and ) and anharmonicity ().

Static distribution of the OH stretch transition frequencies (, , and ) and anharmonicity ().

Autocorrelation functions of the transition frequencies (, , and ) and the anharmonicity .

Autocorrelation functions of the transition frequencies (, , and ) and the anharmonicity .

Scatter plot showing the correlation between the anharmonicity and the fundamental frequency .

Scatter plot showing the correlation between the anharmonicity and the fundamental frequency .

Upper panel: The vertical lines marked with stars show the center position of the four states of the FSJ model. The height of each line marks the probability of the state. The Gaussian lines on top illustrate the slowly changing frequency distribution added on top of each state (solid: I, dotted: II, dashed: III, and dash-dotted: IV). Middle panel: The solid line is the distribution of the collective electrostatic coordinate. The dashed line on top is the Gaussian distribution corresponding to the combination of the Gaussian distributions of the coordinates and . The distributions of these coordinates are the dotted and dash-dotted lines, respectively. Lower panel: solid line: The correlation function of the collective electrostatic coordinate; dashed line: the biexponential fit. The dotted line is the FSJ frequency correlation function.

Upper panel: The vertical lines marked with stars show the center position of the four states of the FSJ model. The height of each line marks the probability of the state. The Gaussian lines on top illustrate the slowly changing frequency distribution added on top of each state (solid: I, dotted: II, dashed: III, and dash-dotted: IV). Middle panel: The solid line is the distribution of the collective electrostatic coordinate. The dashed line on top is the Gaussian distribution corresponding to the combination of the Gaussian distributions of the coordinates and . The distributions of these coordinates are the dotted and dash-dotted lines, respectively. Lower panel: solid line: The correlation function of the collective electrostatic coordinate; dashed line: the biexponential fit. The dotted line is the FSJ frequency correlation function.

Distribution of the fundamental and overtone frequencies for the four hydrogen-bond species. I: solid, II: dotted, III: dash-dotted, IV: dash-dot-dotted, and total: dashed line. The integrated intensity of each curve gives the abundance of each species.

Distribution of the fundamental and overtone frequencies for the four hydrogen-bond species. I: solid, II: dotted, III: dash-dotted, IV: dash-dot-dotted, and total: dashed line. The integrated intensity of each curve gives the abundance of each species.

Kinetic scheme for the four hydrogen-bonding configurations. Reactions A and C hydrogen bonds on oxygen, while B and D hydrogen bonds on hydrogen.

Kinetic scheme for the four hydrogen-bonding configurations. Reactions A and C hydrogen bonds on oxygen, while B and D hydrogen bonds on hydrogen.

Lifetime histograms for the four hydrogen-bond configurations. is the probability that the the configuration survives a specified time. Solid line: configuration I, small dashed line: II, large dashed line: III, and dash-dotted: IV. The dashed lines are fitted exponential decays.

Lifetime histograms for the four hydrogen-bond configurations. is the probability that the the configuration survives a specified time. Solid line: configuration I, small dashed line: II, large dashed line: III, and dash-dotted: IV. The dashed lines are fitted exponential decays.

The three Liouville space pathways contributing to the photon-echo signal. represents stimulated emission, represents ground-state bleach, and shows excited-state absorption.

The three Liouville space pathways contributing to the photon-echo signal. represents stimulated emission, represents ground-state bleach, and shows excited-state absorption.

Linear absorption calculated using the SLE. Dashed: CEC(i), dotted: FSJ, and solid: experiment (Ref. 19). The experimental spectrum is displaced to the blue for a better comparison of the line shapes.

Linear absorption calculated using the SLE. Dashed: CEC(i), dotted: FSJ, and solid: experiment (Ref. 19). The experimental spectrum is displaced to the blue for a better comparison of the line shapes.

(Color) Comparison of the photon-echo spectra calculated using the two SLE models. The full black line illustrates the diagonal, the dashed line is displaced above the diagonal. The red and blue lines show where the antidiagonal slices for Fig. 12 are taken on the red and blue sides, respectively.

(Color) Comparison of the photon-echo spectra calculated using the two SLE models. The full black line illustrates the diagonal, the dashed line is displaced above the diagonal. The red and blue lines show where the antidiagonal slices for Fig. 12 are taken on the red and blue sides, respectively.

Antidiagonal slices of the photon-echo spectra of Fig. 11 intersecting the diagonal. The slices are made at 3350 and .

Antidiagonal slices of the photon-echo spectra of Fig. 11 intersecting the diagonal. The slices are made at 3350 and .

(Color) Effect of the anharmonicity fluctuations. (a) CEC(ii) spectrum with fluctuating anharmonicity. (b) CEC(ii) spectrum with fixed anharmonicity. The vertical arrows indicate the anharmonic shift at . (c) The anharmonic shift vs . Black: fluctuating anharmonicity; red: fixed anharmonicity.

(Color) Effect of the anharmonicity fluctuations. (a) CEC(ii) spectrum with fluctuating anharmonicity. (b) CEC(ii) spectrum with fixed anharmonicity. The vertical arrows indicate the anharmonic shift at . (c) The anharmonic shift vs . Black: fluctuating anharmonicity; red: fixed anharmonicity.

(Color) Photon-echo spectra for different delay times calculated using CEC(ii) and CEC(iii).

(Color) Photon-echo spectra for different delay times calculated using CEC(ii) and CEC(iii).

Three-pulse photon-echo peak shift spectrum. Solid line: experiment (Ref. 19); dotted: CEC(ii); dot-dashed: CEC(iii); and dashed: simulated data reported in Ref. 28.

Three-pulse photon-echo peak shift spectrum. Solid line: experiment (Ref. 19); dotted: CEC(ii); dot-dashed: CEC(iii); and dashed: simulated data reported in Ref. 28.

## Tables

Statistical data for the fluctuating frequencies for the considered transitions.

Statistical data for the fluctuating frequencies for the considered transitions.

Number of hydrogen bonds to each atom (H, D, and O), average frequency , frequency spread , and abundance for the hydrogen-bonding configurations.

Number of hydrogen bonds to each atom (H, D, and O), average frequency , frequency spread , and abundance for the hydrogen-bonding configurations.

Frequencies ( and ), frequency distributions ( and ), anharmonicity dependence on the hydrogen-bond configuration, lifetime , and abundance .

Frequencies ( and ), frequency distributions ( and ), anharmonicity dependence on the hydrogen-bond configuration, lifetime , and abundance .

Reaction dynamics for hydrogen-bond breaking and forming.

Reaction dynamics for hydrogen-bond breaking and forming.

Antidiagonal linewidths for cuts through the diagonal peak in photon-echo spectrum on the red and blue sides (red and blue lines in Fig. 11). The asymmetry parameter is defined in Eq. (12).

Antidiagonal linewidths for cuts through the diagonal peak in photon-echo spectrum on the red and blue sides (red and blue lines in Fig. 11). The asymmetry parameter is defined in Eq. (12).

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