^{1}and Martin Gruebele

^{2,a)}

### Abstract

A fourth-order resonance Hamiltonian is derived from the experimental normal-mode Hamiltonian of . The anharmonic vibrational state space constructed from the effective Hamiltonian provides a realistic model for vibrational energy flow from bright states accessible by pulsed laser excitation. We study the experimentally derived distribution of dilution factors as a function of energy. This distribution characterizes the dynamics in the long-time limit. State space models predict that should be bimodal, with some states undergoing facile intramolecular vibrational energy redistribution (small ), while others at the same total energy remain “protected” . The bimodal distribution is in qualitative agreement with analytical and numerical local density of states models. However, there are fewer states protected from energy flow, and the protected states begin to fragment at higher energy, shifting from to . We also examine how dilution factors are distributed in the vibrational state space of and how the power law specifying the survival probability of harmonic initial states correlates with the dilution factor distribution of anharmonic initial states.

This work was supported by a creativity extension to NSF Grant No. CHE 99-86670 of one of the authors (M.G.) and NSF Grant No. CHE 0315243 of the other author (E.S.).

I. INTRODUCTION

II. COMPUTATIONAL METHODS

A. Hamiltonian

B. Dynamics

III. RESULTS AND DISCUSSION

A. Effective Hamiltonian

B. Energy flow dynamics

C. Dilution factor distribution

IV. CONCLUSIONS

### Key Topics

- Oscillators
- 6.0
- Numerical modeling
- 5.0
- Diffusion
- 4.0
- Networks
- 3.0
- Probability theory
- 3.0

## Figures

Three-dimensional state space showing basis states (circles) on the energy shell. In a harmonic basis, the energy shell is a hyperpolyhedron. In an anharmonic basis, the energy shell is concave because edge states (whose energy is concentrated in a few modes) require more quanta to reach a given energy than interior states. Two pairs of states separated by and 4 in state space are shown connected by resonances (arrows). A vibrational wave packet initially localized at a bright state (open circles) spreads on a manifold of dimension (dotted curve).

Three-dimensional state space showing basis states (circles) on the energy shell. In a harmonic basis, the energy shell is a hyperpolyhedron. In an anharmonic basis, the energy shell is concave because edge states (whose energy is concentrated in a few modes) require more quanta to reach a given energy than interior states. Two pairs of states separated by and 4 in state space are shown connected by resonances (arrows). A vibrational wave packet initially localized at a bright state (open circles) spreads on a manifold of dimension (dotted curve).

Dilution factor distributions predicted by the analytical Leitner-Wolynes model, and by the local random matrix Bose statistics triangle rule model. is a coupling parameter dependent on the average coupling and local density of states from a bright state to states away in state space. is the local number of coupled states defined in the text. The IVR threshold is approached when and .

Dilution factor distributions predicted by the analytical Leitner-Wolynes model, and by the local random matrix Bose statistics triangle rule model. is a coupling parameter dependent on the average coupling and local density of states from a bright state to states away in state space. is the local number of coupled states defined in the text. The IVR threshold is approached when and .

Differences between the second-order and fourth-order energies as a function of the sixth-order energy.

Differences between the second-order and fourth-order energies as a function of the sixth-order energy.

Survival probabilities of the state for two matrix size truncations of the effective fourth-order Hamiltonian. are the integrals excluding the initial peak (scaled by for readability) yielding the dilution factor.

Survival probabilities of the state for two matrix size truncations of the effective fourth-order Hamiltonian. are the integrals excluding the initial peak (scaled by for readability) yielding the dilution factor.

[(A) and (B)] Survival probabilities of harmonic and fourth-order effective Hamiltonian initial states with the nominal quantum number assignments shown. [(C)–(E)] Three representative decays of harmonic initial states in the 7500–8900 energy range, with different values of [Eq. (1)] from the threshold to the maximum dimension of the IVR energy flow manifold.

[(A) and (B)] Survival probabilities of harmonic and fourth-order effective Hamiltonian initial states with the nominal quantum number assignments shown. [(C)–(E)] Three representative decays of harmonic initial states in the 7500–8900 energy range, with different values of [Eq. (1)] from the threshold to the maximum dimension of the IVR energy flow manifold.

Dilution factor distributions . (B) (middle diagonal) Probability distribution (binned into increments) as a function of the total vibrational energy (in increments). The second-order (solid bars) and fourth-order (dashed bars) results are both shown. At , the normalized asymptotic form from Eq. (2) is also shown. [(A) and (C)] Dilution factor distributions at 4000 and for all states and for “bright” states and .

Dilution factor distributions . (B) (middle diagonal) Probability distribution (binned into increments) as a function of the total vibrational energy (in increments). The second-order (solid bars) and fourth-order (dashed bars) results are both shown. At , the normalized asymptotic form from Eq. (2) is also shown. [(A) and (C)] Dilution factor distributions at 4000 and for all states and for “bright” states and .

Three projections of dilution factors in the six-dimensional state space onto three anharmonic normal-mode quantum numbers. The smallest dots correspond to , the largest to . The dashed boxes indicate quantum number combinations where protected states appear. Because these are projections of a six-dimensional space, states are distributed in three-dimensional space up to a two-dimensional surface, not just on the two-dimensional surface as in Fig. 1.

Three projections of dilution factors in the six-dimensional state space onto three anharmonic normal-mode quantum numbers. The smallest dots correspond to , the largest to . The dashed boxes indicate quantum number combinations where protected states appear. Because these are projections of a six-dimensional space, states are distributed in three-dimensional space up to a two-dimensional surface, not just on the two-dimensional surface as in Fig. 1.

## Tables

Fourth-order corrected harmonic frequencies, anharmonic constants, and largest explicit resonances for the state. (A full listing is available in the supplementary material.)

Fourth-order corrected harmonic frequencies, anharmonic constants, and largest explicit resonances for the state. (A full listing is available in the supplementary material.)

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