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Notes: It has been shown by Heller that a nonstationary wave packet resulting from a Franck–Condon transition evolves on the potential energy surface of the final electronic state and propagates through phase space at a rate which can be determined from the autocorrelation function ↓C(t)↓2=↓〈(0)||(t)〉↓2. Since C(t) can be obtained by Fourier transformation of an optical spectrum S(E), i.e., from an observable quantity, it is possible to derive from an experimental measurement information concerning the density operator of a so-called dynamical statistical ensemble (DSE). This density operator, denoted ρav, represents a statistical mixture of the eigenstates of the system with weights determined by the dynamics of the system. It becomes diagonal after a so-called break time TB. Its measure, according to a definition due to Stechel, can be interpreted as an effective number of states (denoted N) that significantly contribute to the dynamics. The break time TB represents the finite period of time allowed to expand in the phase space and after which no further progress can be made. Therefore, the number N∞ of phase space cells which are accessed after a very long interval of time (or in practice after the break time) remains limited.Information on the validity of statistical theories of unimolecular reactions is contained in the fraction F of the available phase space which is eventually explored. In order to assess the representativity of the sampling, it is necessary to account for the selection rule which requires all the states counted in N∞ to belong to the totally symmetric representation. It is also appropriate to estimate the role played by Fermi resonances and similar vibrational interactions which bring about energy flow into zero-order antisymmetric modes. A method to carry out the necessary partitionings is suggested. The functions NT and RT, and the quantities TB, N∞, N *, and F have been determined from experimental data in three cases. In each case, the rate RT=dNT/dT starts from an initial value of zero, increases up to a maximum which is reached after a time of the order of 10−14 s, and then exhibits an overall decrease upon which oscillations are superimposed. For state X˜ 2B1 of H2O+, TB(approximately-equal-to)2.4×10−14 s and F(approximately-equal-to)0.3. The wave packet never accesses that part of the phase space that corresponds to the excitation of antisymmetric vibrations. For state X˜ 2B3u of C2H+4, TB(approximately-equal-to)1.6×10−13 s and F(approximately-equal-to)5×10−4. This fraction raises to 6×10−3 if measured with respect to the effectively available phase space. When the spectrum consists of a discrete part followed by a dissociation continuum, the method can be extended to study the behavior of the bound part of the wave packet only. This has been applied to state B˜ 2∑+ of HCN+ which is characterized by a very irregular spectrum. This case offers an example of complete occupation of phase space after a break time which is of the order of 2×10−13 s.

Notes: At low energies, methylnitrite ions dissociate via two channels giving rise to CH3O+NO+ or to [CH3O+]+NO fragments. Peculiar characteristics have been detected in the dissociation of energy-selected parent ions, viz., remarkably low rate constants, one of which is found to remain insensitive to an increase of the internal energy, and large isotope effect. These peculiarities are accounted for by a statistical, nonadiabatic model. Ab initio calculations, confirmed by multipolar expansions reveal that the potential energy curves which correlate to these two dissociation asymptotes cross. The crossing takes place at a large value along the reaction coordinate R, indicating a long-range interaction. Production of the CH3O+NO+ fragments results from a simple bond cleavage taking place on a single diabatic surface. On the other hand, production of [CH3O+]+NO fragments is brought about by a transition from one diabatic surface to the other. It leads to deformed methoxy ions which immediately rearrange to the much more stable H2COH+ structure. The nonadiabatic rate constant has been calculated by a statistical method. The contribution of each channel is weighted by a transmission coefficient which is equal to the nonadiabatic transition probability. Implementation of this statistical treatment requires partitioningthe set of degrees of freedom as follows: {R, y, d, v}. It is necessary to withdraw the isomerization mode y from the statistical treatment, because the equilibrium positions along this coordinate are very different in each electronic state. Physically, this means that the reaction involves tunneling from one surface to the other along the displaced degree of freedom y. The large isotope effect has a double origin: part of it (a factor of ∼9) results from tunneling along y; the remainder (an additional factor of 3) comes from the usual RRKM-like effect on the densities of states. The degrees d constitute a set of four low-energy bending modes which form a sink for the internal energy. The nonadiabatic transition probability is determined by the off-diagonal matrix element V12(Rc). Effective potential energy curves have been calculated by extending the Quack and Troe method to nonadiabatic reactions. It turns out that the excitation of the set d leads to a decrease of V12(Rc) and hence to a decrease of the nonadiabatic transmission coefficient. This accounts for the weak dependence of the rate constants kCH3O+ and kCD3O+ vs the energy (at least above a certain energy threshold).

Notes: The formalism of the resonance states is used to derive approximate expressions of the unimolecular law of decay resulting from a specific excitation. These expressions contain no cross terms and wash out the quantum interferences. We propose a method to relate them to an experimentally observable quantity, viz., the autocorrelation function C(t) obtained as the Fourier transform of a spectral profile, which is available even when the spectrum is poorly resolved. For a specific excitation, the exact initial rate of decay (valid up to the dephasing time T1) is equal to the initial slope of ||C(t)||2. The subsequent time evolution can be obtained by averaging ||C(t)||2 over its oscillations. This generates a function ||C(t)||2av whose area (from time T1 onwards) is directly related to an average decay lifetime. At times t〉T1, a good approximation to the average decay curve Pav(t) can be derived by multiplying ||C(t)||2av by an appropriate constant. The method is exemplified on various diatomic and triatomic models. As an application to a real system, we study the B˜ 2B2 state of H2O+ which is coupled to the A˜ 2A1 state via a conical intersection. State B˜ is found to undergo an ultrafast intramolecular relaxation with a lifetime of (1.6±0.2) 10−14 s.

Notes: The aim of the paper is to estimate the volume of phase space that is, in principle, available to a nonstationary wave packet during its intramolecular vibrational relaxation. For that purpose, use is made of the maximum entropy method, together with the concept of constrained ergodicity to construct two so-called reference ergodic systems. The first one concerns thermal excitation processes. In that case, the only two constraints that are imposed on the intramolecular dynamics arise from the normalization of the wave function and from the conservation of energy. These constraints affect the zeroth and first moments of the spectrum. The second reference system concerns a situation where, as an additional constraint, use is made of the information that the system has been prepared spectroscopically, i.e., by a specific excitation process, consisting in the coherent excitation of an initial pure state. Then, the second moment of the spectrum, denoted σ, is shown to provide the appropriate additional constraint. Translated into the time domain, the prior knowledge of the dynamics used as a constraint is limited to an infinitesimally brief period of time [0,dt] with the remaining evolution determined by the maximum entropy method. The spectroscopic reference system constructed in that way can be understood as the one that samples the maximal volume of phase space available to a wave packet having a specified average energy and being put in motion by a specified initial force.Closed-form expressions are obtained for the phase space volumes occupied by these two reference systems for various simple parametrizations of the function D(E) that expresses the density of states as a function of the internal energy (power laws or exponential increase). Thermal reference systems are found to sample a larger volume of phase space than their spectroscopic counterparts. The difference between these two cases depends critically on the value of σ, and also on the symmetry characteristics of the excitation process. In general, the volumes occupied by the reference systems, thermal as well as spectroscopic, can be expressed as ηEavD(Eav), where Eav is the (conserved) average energy of the wave packet and η is a correcting factor that depends on the functional form of D(E) and on the nature of the imposed constraints. In all cases studied, the value of η was found not to greatly differ from 1. The method has been applied to the analysis of three experimental photoelectron spectra presenting different spectral characteristics (X˜ 2A1 state of NH+3, X˜ 2B3 state of C2H+4, and the X˜ 2A″ state of C2H3F+). The fractional occupancy index F defined by Heller as the fraction of the available phase space eventually explored up to the break time TB could be determined. After a time of the order of 100 fs, F was found to be of the order of a few percent for thermal excitation. When the molecule presents some symmetry, the expansion of the wave packet is restricted to that part of phase space spanned by the totally symmetric wave functions. The use of this additional a priori knowledge increases the fractional index F. © 1996 American Institute of Physics.