Bound-state nuclear dynamics

In this way r is singled out as the fast electronic coordinate. In the absence of prior knowledge we can, equally well, designate R as the fast coordinate. This gives rise to an alternative adiabatic breakdown of the Schrodinger equation, and exemplifies the nonuniqueness associated with the adiabatic separation of bound-state nuclear dynamics. 

A further important property of a MQC description is the ability to correctly describe the time evolution of the electronic coefficients. A proper description of the electronic phase coherence is expected to be particularly important in the case of multiple curve-crossings that are frequently encountered in bound-state relaxation dynamics [163]. Within the limits of the classical-path approximation, the MPT method naturally accounts for the coherent time evolution of the electronic coefficients (see Fig. 5). This conclusion is also supported by the numerical results for the transient oscillations of the electronic population, which were reproduced quite well by the MFT method. Similarly, it has been shown that the MFT method in general does a good job in reproducing coherent nuclear motion on coupled potential-energy surfaces. 

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. 

From a theoretical perspective, the object that is initially created in the excited state is a coherent superposition of all the wavefunctions encompassed by the broad frequency spread of the laser. Because the laser pulse is so short in comparison with the characteristic nuclear dynamical time scales of the motion, each excited wavefunction is prepared with a definite phase relation with respect to all the others in the superposition. It is this initial coherence and its rate of dissipation which determine all spectroscopic and collisional properties of the molecule as it evolves over a femtosecond time scale. For IBr, the nascent superposition state, or wavepacket, spreads and executes either periodic vibrational motion as it oscillates between the inner and outer turning points of the bound potential, or dissociates to form separated atoms, as indicated by the trajectories shown in Figure 1.3. 

Bound-state photoabsorption, direct molecular dynamics, nuclear motion Schrodinger equation, 365-373... 

Bound and Quasi-Bound States, it was shown in the last section how the problem of the evaluation of photofragment energy distributions can be reduced to the analysis of nuclear dynamics. Indeed, it is necessary to evaluate integrals containing the nuclear wavefunctions (see eqs. 28, 34, and 57). 

The approach described for polyatomic photodissociation as a quantum transition can be generalized to describe the dynamics of chemical reactions. Polyatomic photodissociation is a transition from a quasi-bound or bound state to a bound-continuous dissociative state. By extension then, a chemical reaction is a transition from one bound-continuous state (reactants) to another (products). The state of reactants (products) is analogous to the dissociative state and, hence, the results described in Section III can be used to define the nuclear wavefunctions of reactants and of products. Following this analogy, a chemical reaction can be treated as a quantum transition reactants - products, enabling the evaluation of product energy distributions (63,33). 

In order to simplify the evaluation of overlap integrals between bound and continuum wavefunctions, it is advisable (although not necessary) to describe both wavefunctions by the same set of coordinates. Usually, the calculation of continuum, i.e., scattering, states causes far more problems than the calculation of bound states and therefore it is beneficial to use Jacobi coordinates for both nuclear wavefunctions. If bound and continuum wavefunctions are described by different coordinate sets, the evaluation of multi-dimensional overlap integrals requires complicated coordinate transformations (Freed and Band 1977) which unnecessarily obscure the underlying dynamics. 

The calculations in part (b) may be of two types the determination of the nuclear energy levels for bound states of the system, i.e. the quantized vibrational and rotational levels of the system, or the study of the dynamics of the chemical changes described by the surface in terms of quantum reactive scattering or classical trajectory calculations. 

Gallina et al. [20] introduced the hyperspherical symmetrical parametrization in a particle-physics context, as did Zickendraht later [21, 22], At the same time, F.T. Smith [23] gave the definitions of internal coordinates following Fock s work already mentioned [16], Clapp [24, 25] and others and established, for the symmetrical and asymmetrical parametrization, the basic properties and the notation we follow. Since then, applications have been extensive, especially for bound states. For example, the symmetrical coordinates have often been used in atomic [26], nuclear [27] and molecular [28-31] physics. This paper accounts for modem applications, with particular reference to the field of reaction dynamics, in view of the prominent role played by these coordinates for dealing with rearrangement problems. 

The adiabatic approximation, which is to be applied to bound-state dynamics, involves the separation of fast and slow motions. However, unlike the original application of the Bom-Oppenheimer approximation,77 in which fast electronic motion is separated from the slower nuclear one, there is no clear separation of time scales for nuclear motions. Nevertheless, there is now ample evidence to show that there are always states, to be termed adiabatic, which are accurately described by an adiabatic approximation.66,69-71 78... 

This is an eigenvalue equation for the nuclei degrees of freedom submitted to the model potential generated by the exact electronic wave function Yj(p aoi). Depending upon the particular mass vector, the electronic potential may sustain bound nuclear dynamical states of diverse frequencies. The nuclear stationary states Xik(R) in the inertial frame are obtained as functions relative to the electronic attractor potential Ej(R). 

The computation described above is completely classical the nuclear motion is assumed to be well described by Newton s equations. The extent to which classical mechanics provides a useful description of intramolecular energy flow is another focus of current research in this area. As one example of the validity of classical mechanics, consider the bound-state dynamics of a three-atom system confined to a line, that is. A—B—C. Computations on the case where the A—B and B—C bonds are anharmonic have been performed using both classical and quantum... 

Schaefer et al. (19) studied the interphase microstructure of ternary polymer composites consisting of polypropylene, ethylene-propylene-diene-terpolymer (EPDM), and different types of inorganic fillers (e.g., kaolin clay and barium sulfate). They used extraction and dynamic mechanical methods to relate the thickness of absorbed polymer coatings on filler particles to mechanical properties. The extraction of composite samples with xylene solvent for prolonged periods of time indicated that the bound polymer around filler particles increased from 3 to 12 nm thick between kaolin to barium sulfate filler types. Solid-state Nuclear Magnetic Resonance (NMR) analyses of the bound polymer layers indicated that EPDM was the main constituent adsorbed to the filler particles. Without doubt, the existence of an interphase microstructure was shown to exist and have a rather sizable thickness. They proceeded to use this interphase model to fit a modified van der Poel equation to compute the storage modulus G (T) and loss modulus G"(T) properties. 

The nuclear dynamics in an isolated resonant state can be described in analogy to that in a bound electronic state by the Born-Oppenheimer approximation using the above mentioned complex potential surface. This approximation has, however, a number of drawbacks not encountered in the case of bound states. First, the resonant state is embedded in the continuum, and this continuum of electronic states may also contribute to the nuclear dynamics and to the process to be described. In scattering processes, the contribution of the continuum is denoted as background scattering. 

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