Vibrational dynamics formal expression

A novel data analysis procedure is described, based on a variational solution of the Schrddinger equation, that can be used to analyze gas electron diffraction (GED) data obtained from molecular ensembles in nonequilibrium (non-Boltzmann) vibrational distributions. The method replaces the conventional expression used in GED studies, which is restricted to molecules with small-amplitude vibrations in equilibrium distributions, and is important in time-resolved (stroboscopic) GED, a new tool developed to study the nuclear dynamics of laser-excited molecules. As an example, the new formalism has been used to investigate the structural and vibrational kinetics of C=S, using stroboscopic GED data recorded during the first 120 ns following the 193 nm photodissociation of CS2. Temporal changes of vibrational population are observed, which can... 

In the following sections we show how the quantum-classical Liouville equation and quantum-classical expressions for reaction rates can be deduced from the full quantum expressions. The formalism is then applied to the investigation of nonadiabatic proton transfer reactions in condensed phase polar solvents. A quantum-classical Liouville-based method for calculating linear and nonlinear vibrational spectra is then described, which involves nonequilibrium dynamics on multiple adiabatic potential energy surfaces. This method is then used to investigate the linear and third-order vibrational spectroscopy of a proton stretching mode in a solvated hydrogen-bonded complex. 

An alternative route is based on time-dependent approaches, where the standard statistical mechanics formalism relies on Fourier transform of the time correlation of vibrational operators [54—57]. These approaches can provide a complete description of the experimental spectrum, that is, the characterization of the real molecular motion consisting of many degrees of freedom activated at finite temperature, often strongly coupled and anharmonic in namre. However, computation of the exact quantum dynamics evolution of the nuclei on the ab initio potential surface is as prohibitive as the quantum/stationary-state approaches. In fact, even a semiclassical description of the time evolution of quanmm systems is usually computationally expensive. Therefore, time correlation methods for realistic systems are usually carried out by sampling of the nuclear motion in the classical phase space. In this context, summation over i in Eq. 11.1 is a classical ensemble average furthermore, the field unit vector e can be averaged over all directions of an isotropic fluid, leading to the well-known expression... 

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