^{1}and Richard M. Stratt

^{1}

### Abstract

Given the limited intermolecular spaces available in dense liquids, the large amplitudes of highly excited, low frequency vibrational modes pose an interesting dilemma for large molecules in solution. We carry out molecular dynamics calculations of the lowest frequency (“warping”) mode of perylene dissolved in liquid argon, and demonstrate that vibrational excitation of this mode should cause identifiable changes in local solvation shell structure. But while the same kinds of solvent structural rearrangements can cause the non-equilibrium relaxation dynamics of highly excited diatomic rotors in liquids to differ substantially from equilibrium dynamics, our simulations also indicate that the non-equilibrium vibrational energy relaxation of large-amplitude vibrational overtones in liquids should show no such deviations from linear response. This observation seems to be a generic feature of large-moment-arm vibrational degrees of freedom and is therefore probably not specific to our choice of model system: The lowest frequency (largest amplitude) cases probably dissipate energy too quickly and the higher frequency (more slowly relaxing) cases most likely have solvent displacements too small to generate significant nonlinearities in simple nonpolar solvents. Vibrational kinetic energy relaxation, in particular, seems to be especially and surprisingly linear.

This work was supported by the U.S. National Science Foundation (NSF) (Grant No. CHE-0809385). We are grateful to Mark Maroncelli for extensive discussions concerning the vibrational dynamics and spectroscopy of the low-frequency modes of perylene dissolved in liquids. We are also pleased to thank Guohua Tao and Crystal Nguyen for helpful discussions about the design and implementation of the calculations reported here.

I. INTRODUCTION

II. MODELS AND METHODS

A. A simple representation of a rotating and vibrating perylene molecule in solution

B. Molecular dynamics simulation

III. VIBRATIONAL ENERGY FLUCTUATIONS AT EQUILIBRIUM

IV. NON-EQUILIBRIUM ASPECTS OF VIBRATIONAL ENERGY RELAXATION

A. Non-equilibrium preparation and the initial transient response

B. Solvent structural dynamics

C. Vibrational relaxation

V. CONCLUDING REMARKS

### Key Topics

- Solvents
- 108.0
- Correlation functions
- 9.0
- Rotational dynamics
- 6.0
- Energy transfer
- 5.0
- Coherence
- 4.0

##### B01F1/00

## Figures

The perylene molecule and its 1a_{u} (“warping”) vibrational mode. (Upper picture) The equilibrium molecular geometry giving the numbering convention for the H atoms and showing how the mode rotates the two naphthalene halves about the molecular-frame X axis. (Lower picture) A schematic view of the normal mode (from a different angle) showing our molecular-frame coordinate axes and defining the vibrational angle *θ* between the two naphthalene planes (shown in red), the “lever arm” for the vibration *R*, and what we refer to as the amplitude of the vibration *A*.

The perylene molecule and its 1a_{u} (“warping”) vibrational mode. (Upper picture) The equilibrium molecular geometry giving the numbering convention for the H atoms and showing how the mode rotates the two naphthalene halves about the molecular-frame X axis. (Lower picture) A schematic view of the normal mode (from a different angle) showing our molecular-frame coordinate axes and defining the vibrational angle *θ* between the two naphthalene planes (shown in red), the “lever arm” for the vibration *R*, and what we refer to as the amplitude of the vibration *A*.

Schematic portrayal of the real parts of the instantaneous normal mode solvent bands of Ar-like liquids with solvent masses *m* _{ v } = 0.1, 1.0, 10 *m* _{Ar} (roughly to scale). The arrows show the locations of two of the solute frequencies Ω considered in this paper.

Schematic portrayal of the real parts of the instantaneous normal mode solvent bands of Ar-like liquids with solvent masses *m* _{ v } = 0.1, 1.0, 10 *m* _{Ar} (roughly to scale). The arrows show the locations of two of the solute frequencies Ω considered in this paper.

Equilibrium vibrational energy dynamics for the perylene warping mode in liquid argon when different frequencies (Ω/(2*π*c) = 12, 48, and 192 cm^{−1}) are assigned to the mode. Shown here are the normalized total-vibrational-energy-fluctuation correlation functions, *C*(*t*), and their kinetic- and potential energy components *C* _{KK}(*t*), *C* _{VV}(*t*), and *C* _{KV}(*t*), (denoted by E, K, V, and KV, respectively).

Equilibrium vibrational energy dynamics for the perylene warping mode in liquid argon when different frequencies (Ω/(2*π*c) = 12, 48, and 192 cm^{−1}) are assigned to the mode. Shown here are the normalized total-vibrational-energy-fluctuation correlation functions, *C*(*t*), and their kinetic- and potential energy components *C* _{KK}(*t*), *C* _{VV}(*t*), and *C* _{KV}(*t*), (denoted by E, K, V, and KV, respectively).

The solution-phase equilibrium vibrational kinetic energy dynamics referred to in Fig. 3, viewed in the frequency domain. The results plotted are for three choices of solvent mass (*m* _{ v } = 0.1, 1.0, and 10.0 *m* _{Ar}), and for both Ω/(2*π*c) = 48 and 192 cm^{−1}. Arrows indicate where peaks would be located in the absence of solvent.

The solution-phase equilibrium vibrational kinetic energy dynamics referred to in Fig. 3, viewed in the frequency domain. The results plotted are for three choices of solvent mass (*m* _{ v } = 0.1, 1.0, and 10.0 *m* _{Ar}), and for both Ω/(2*π*c) = 48 and 192 cm^{−1}. Arrows indicate where peaks would be located in the absence of solvent.

Non-equilibrium relaxation of the average vibrational potential energy of perylene in liquid argon following a 1998 cm^{−1} = 24 k_{B}T excitation of the warping mode. Results are plotted for three values of the warping frequency, Ω. Note the difference in scales between the upper and lower panels.

Non-equilibrium relaxation of the average vibrational potential energy of perylene in liquid argon following a 1998 cm^{−1} = 24 k_{B}T excitation of the warping mode. Results are plotted for three values of the warping frequency, Ω. Note the difference in scales between the upper and lower panels.

Non-equilibrium solvent structural relaxation following excitation of the perylene warping mode in liquid argon. Here, we choose the warping-mode frequency to be Ω/(2*π*c) = 12 cm^{−1}. This figure shows the evolution with time of the average number of solvent atoms in the innermost (r_{H–Ar} < *σ* _{Ar–H}) and outermost (*σ* _{Ar–H} < r_{H–Ar} < 1.5 *σ* _{Ar–H}) halves of the first solvation shells of the perylene “corner” H atoms.^{89} (Upper panel) Results for different initial vibrational energies *E* _{ vib }(0): 125, 1998, and 3996 cm^{−1} (1.5, 24, and 48 k_{B}T) with a normal Ar-mass solvent. (Lower panel) Results for different solvent masses (*m* _{ v } = 0.1, 1.0, and 10.0 *m* _{Ar}) with a common initial vibrational energy *E* _{ vib }(0) = 1998 cm^{−1}. The two large-dashed horizontal lines in each panel indicate the equilibrium number of solvent atoms in each region.

Non-equilibrium solvent structural relaxation following excitation of the perylene warping mode in liquid argon. Here, we choose the warping-mode frequency to be Ω/(2*π*c) = 12 cm^{−1}. This figure shows the evolution with time of the average number of solvent atoms in the innermost (r_{H–Ar} < *σ* _{Ar–H}) and outermost (*σ* _{Ar–H} < r_{H–Ar} < 1.5 *σ* _{Ar–H}) halves of the first solvation shells of the perylene “corner” H atoms.^{89} (Upper panel) Results for different initial vibrational energies *E* _{ vib }(0): 125, 1998, and 3996 cm^{−1} (1.5, 24, and 48 k_{B}T) with a normal Ar-mass solvent. (Lower panel) Results for different solvent masses (*m* _{ v } = 0.1, 1.0, and 10.0 *m* _{Ar}) with a common initial vibrational energy *E* _{ vib }(0) = 1998 cm^{−1}. The two large-dashed horizontal lines in each panel indicate the equilibrium number of solvent atoms in each region.

Non-equilibrium solvent structural relaxation following excitation of the perylene warping mode in liquid argon. The details of the graph are the same as those shown in Fig. 6, except that here the warping mode frequency is chosen to be Ω/(2*π*c) = 48 cm^{−1}.

Non-equilibrium solvent structural relaxation following excitation of the perylene warping mode in liquid argon. The details of the graph are the same as those shown in Fig. 6, except that here the warping mode frequency is chosen to be Ω/(2*π*c) = 48 cm^{−1}.

Non-equilibrium solvent structural relaxation following excitation of the perylene warping mode in liquid argon. The details of the graph are the same as those shown in Fig. 6, except that here the warping mode frequency is chosen to be Ω/(2*π*c) = 192 cm^{−1}.

Non-equilibrium solvent structural relaxation following excitation of the perylene warping mode in liquid argon. The details of the graph are the same as those shown in Fig. 6, except that here the warping mode frequency is chosen to be Ω/(2*π*c) = 192 cm^{−1}.

Non-equilibrium vibrational energy relaxation following excitation of the perylene warping mode in liquid argon. The warping mode frequency is chosen to be Ω/(2*π*c) = 12 cm^{−1}. (Top panel) Comparison of the non-equilibrium total vibrational energy response functions *S*(*t*) for different initial vibrational energies *E* _{ vib }(0): 125, 1998, and 3996 cm^{−1} (1.5, 24, and 48 k_{B}T) (dashed lines), with the thermal, equilibrium, total vibrational energy fluctuation correlation function *C*(*t*) (solid line). (Bottom three panels) Comparison of the non-equilibrium total and kinetic energy vibrational energy response functions *S*(*t*) and *KS*(*t*) (for initial vibrational energy *E* _{ vib }(0) = 1998 cm^{−1}) (dashed lines) with thermal equilibrium vibrational energy fluctuation correlation functions for different solvent masses *m* _{ v } (solid lines). The equilibrium results plotted are the total (*C*(*t*)), kinetic energy (*K*(*t*)≡*C* _{KK}(*t*)), and potential energy (*V*(*t*)≡*C* _{VV}(*t*)) correlation functions. Note that each of these lower three panels show five different curves; some of these curves are similar enough to one another that they are difficult to distinguish on this scale.

Non-equilibrium vibrational energy relaxation following excitation of the perylene warping mode in liquid argon. The warping mode frequency is chosen to be Ω/(2*π*c) = 12 cm^{−1}. (Top panel) Comparison of the non-equilibrium total vibrational energy response functions *S*(*t*) for different initial vibrational energies *E* _{ vib }(0): 125, 1998, and 3996 cm^{−1} (1.5, 24, and 48 k_{B}T) (dashed lines), with the thermal, equilibrium, total vibrational energy fluctuation correlation function *C*(*t*) (solid line). (Bottom three panels) Comparison of the non-equilibrium total and kinetic energy vibrational energy response functions *S*(*t*) and *KS*(*t*) (for initial vibrational energy *E* _{ vib }(0) = 1998 cm^{−1}) (dashed lines) with thermal equilibrium vibrational energy fluctuation correlation functions for different solvent masses *m* _{ v } (solid lines). The equilibrium results plotted are the total (*C*(*t*)), kinetic energy (*K*(*t*)≡*C* _{KK}(*t*)), and potential energy (*V*(*t*)≡*C* _{VV}(*t*)) correlation functions. Note that each of these lower three panels show five different curves; some of these curves are similar enough to one another that they are difficult to distinguish on this scale.

Non-equilibrium vibrational energy relaxation following excitation of the perylene warping mode in liquid argon. The details of the graph are the same as those shown in Fig. 9, except that the warping mode frequency here is chosen to be Ω/(2*π*c) = 48 cm^{−1}.

Non-equilibrium vibrational energy relaxation following excitation of the perylene warping mode in liquid argon. The details of the graph are the same as those shown in Fig. 9, except that the warping mode frequency here is chosen to be Ω/(2*π*c) = 48 cm^{−1}.

Non-equilibrium vibrational energy relaxation following excitation of the perylene warping mode in liquid argon. The details of the graph are the same as those shown in Fig. 9, except that the warping mode frequency here is chosen to be Ω/(2*π*c) = 192 cm^{−1}.

Non-equilibrium vibrational energy relaxation following excitation of the perylene warping mode in liquid argon. The details of the graph are the same as those shown in Fig. 9, except that the warping mode frequency here is chosen to be Ω/(2*π*c) = 192 cm^{−1}.

## Tables

Hamiltonian parameters.

Hamiltonian parameters.

Average amplitudes of vibrational motion.^{a,b}

Average amplitudes of vibrational motion.^{a,b}

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