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 .
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.
[(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 .
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.
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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