No abstract

7539c is not very small, the decay of the IP can be reproduced approximately by a single exponential. In this case, the rate of C R deactivation of CIP and its solvation to form LIP are rather slow and may be close to the rate of CR deactivation and dissociation of LIP, which seems to make difficult the detection of the double-exponential decay of IP.(d) The C R rate constant of CIP (k,) is larger than that of LIP (k,) in the case of TCNB-toluene and -xylene complexes in acetonitrile solution where both kinds of CR rate constants have been obtained, in agreement with the general tendency observed thus far in the case of various donor-acceptor systems.I*I5 The energy gap dependence of the CR rate constant of CIP formed by excitation of TCNB complexes in acetonitrile solution including a wide range of aromatic hydrocarbon donors of various strength has been confirmed to agree qualitatively with the previous result that the C R rate increases exponentially with decrease of the energyIn this way, we have examined a wide range of aromatic hydrocarbon donors by changing their oxidation potentials, molecular structures including their size, etc., in the case of TCNB complexes in relation to their photoinduced CS and ionic dissociation mechanisms, and have established the existence of several different cases concerning the behaviors of IP by referring also to our previous results.615In the above discussions, we have assumed the existence of SSIP in addition to CIP in the course of ionic dissociation. The results in case a seem to indicate strongly the existence of such an intermediate state of ionic dissociation. In case c, however, the existence of such an intermediate state in the course of ionic dissociation is not so evident as in case a. Moreover, some model calculations21*22 on ion pair-solvent dipole systems indicate the existence of LIPS with more intervening solvents between ions in the pair than SSIP, that is, the existence of the multiple kinds of IPS in the course of the dissociation. Actually, we have obtained some results that indicated the existence of multiple kinds of IPS in the course of CR and d i s s o c i a t i~n ,~~ which will be discussed elsewhere. N.M. acknowledges the support by a Grant-in-Aid (No. 62065006) from the Japanese Ministry of Education, Science and Culture. Acknowledgment. ~ (21) Levesque, D.; Weis, J. J.; Patey, G. N. J. Chem. Phys. 1980, 72, 1887. (22) Salem, L. Electrons in Chemical Reactions: First Principles; Wi-(23) Mataga, N.; Kanda, Y.; Asahi, T.; Hirata, Y.; Okada, T., manuscript (24) Gould, I. R.; Moser, J. E.; Ege, D.; Farid, S. J. Am. Chem. Soc. 1988.Photophysical properties of cis-stilbene and several homologues are investigated in supersonic expansions as free molecules and in large Ar clusters. Assignment of sharp vibrational structure in the spectra of 1,2-diphenylcyclobutene to long progressions in phenyl twisting and bending vibrations shows that in the SI state cis-stilbene phenyl twist angles are more planar and the phenyl-ethylene bend angles are more bent than i...

We present a time evolution equation that provides a novel basis for the treatment of quantum systems in phase space and for the investigation of the quantum-classical correspondence. Through the use of a generalized Husimi transform, we obtain a phase space representation of the time-dependent Schrodinger equation directly from the coordinate representation. Such an equation governs the time evolution of densities such as the Husimi density entirely in phase space, without recourse to a coordinate or momentum representation. As an application of the phase-space Schrodinger equation, we compute the eigenfunctions of the harmonic oscillator in phase space, relate these to the Husimi transform of coordinate representation eigenstates, and investigate the coherent state, its time evolution, and classical limit (Ii-a) for the probability density generated by this state. Finally, we discuss our results as they relate to the quantumclassical correspondence, and quasiclassical trajectory simulations of quantum dynamics.

A quantum mechanical representation suitable for studying the time evolution of quantum densities in phase space is proposed and examined in detail. This representation on 2'2 (2) phase space is based on definitions of the operators P and Q in phase space that satisfy various correspondences for the Liouville equation in classical and quantum phase space, as well as quantum position and momentum 2'2 (1) spaces. The definitions presented here, P=p/2-ifti)/aq and Q=q/2+ifza/ap, are related to definitions that have been recently proposed [J. Chern. Phys. 93, 8862 (1990)]. The resulting quantum phase space representation shares many of the mathematical properties of usual representations in coordinate and momentum spaces. Within this representation, time evolution equations for complex-valued functions (wave functions) and their square magnitudes (distribution functions) are derived, and it is shown that the coordinate and momentum space time evolution equations can be recovered by a simple Fourier projection. The phase space quantum probability conservation equation obtained is a good illustration of the quantization rule that requires one to replace the classical Poisson bracket between the Hamiltonian and the probability density with the quantum commutator between the corresponding operators. The possible classical analogs to quantum probabilities densities are also considered and some of the present results are illustrated for the dynamics of the coherent state. Tr(pp') =0 and then considering the integrand in Eq.

No abstract

A general formulation of the vibrational kinetic energy operator expressed in internal bond-angle coordinates is presented. This formulation is based on Podolsky’s expression for the covariant form of the Laplace–Beltrami operator. When a valid set of internal bond-angle coordinates is employed, it is possible to adapt a systematic approach to solve for the Jacobian determinant governing the coordinate transformation from Cartesian coordinates. In the general case of an arbitrary N-atom system, this Jacobian always factorizes to a simple form. This allows one to evaluate all the terms that contribute to V̂′, the effective potential that arises from transforming the kinetic energy operator to internal coordinates. We discuss restrictions on the choice of internal vibrational coordinates that may be included in a valid set. We then provide tabular information from which the vibrational kinetic energy operator for any molecular system can be constructed directly with no matrix inversion or chain rule manipulation required.

The potential-energy surfaces of the 1 1Ag, 2 1Ag, and 1 1Bu states of trans-1,3,5-hexatriene (THT) are explored in the vicinity of the ground state equilibrium structure. The S0 geometry optimization and force field calculation have been carried out with the restricted Hartree–Fock plus Mo/ller–Plesset second-order perturbation theory method. Vibronic coupling constants for the normal coordinates of ag and bu symmetry were computed with the complete-active-space self-consistent-field (CASSCF) and single state multiconfigurational second-order perturbation theory (CASPT2) electronic structure models. The CASSCF/CASPT2 method unequivocally places the vertical excitation energy of the dark 2 1Ag “phantom state” below the 1 1Bu level and predicts an energy difference of ca. 0.5 eV. The results are consistent with time-resolved photoionization yield and photoelectron spectroscopy experiments that indicate the existence of a low lying S1–S2 conical intersection which induces rapid 1 1Bu→2 1Ag internal conversion on a time scale of 40 fs to 50 fs [Cyr and Hayden, J. Chem. Phys. 104, 771 (1996)]. Based on the vibronic coupling constants five totally symmetric vibrations with high Franck–Condon and/or tuning activity have been identified. The S1 and S2 states interact primarily via the two bu normal modes ν24 and ν26. Other ag and bu normal vibrations do not appear to couple significantly to the lowest lying π→π* transition. The modeling of the ultrafast relaxation processes following optical excitation of the 1 1Bu state of THT and the calculation of absorption and resonance Raman spectra are discussed in the following paper.

The effect of the nature of the interaction potential on cluster reaction rates The application of statistical theories to the decomposition kinetics of metal clusters requires the estimation of the vibrational frequency distributions. We adapt elastic theories developed for bulk metals and fine particles to generate a physically reasonable frequency distribution model for small metal clusters. Results obtained from this elastic cluster model compare favorably with previously reported experimental heat capacity data for fine particles. In addition, predictions of the present model are shown to correlate very well with experimentally determined trends in metal cluster cohesive energies. The elastic cluster model is then applied to the statistical unimolecular decay kinetics of metal clusters and compared with results found using earlier theoretical models. The present model predicts slower rates of decomposition in comparison with the other models. These results suggest that the binding energies extracted from experimental photodissociation and collision-induced dissociation measurements may have been slightly overestimated in the past.