Nuclear dynamics levels

The formulation outlined above allows for a simple stochastic implementation of the deterministic differential equation (35). Starting with an ensemble of trajectories on a given adiabatic PES W, at each time step At we (i) compute the transition probability pk k, (h) compare it to a random number ( e [0,1], and (iii) perform a hop if pt t > C- In Ih se of a pure A -level system (i.e., in the absence of nuclear dynamics), the assumption (37) holds in general, and the stochastic modeling of Eq. (35) is exact. Considering a vibronic problem with coordinate-dependent however, it can be shown that the electronic... 

This level of theory outhned above is implemented in the ENDyne code [18]. The explicit time dependence of the electronic and nuclear dynamics permits illustrative animated representations of trajectories and of the evolution of molecular properties. These animations reveal reaction mechanisms and details of dynamics otherwise difficult to discern, making the approach particularly suitable for the study of the subtleties of contributions to the stopping cross section. 

Although the theory of photodissociation has not yet reached the level of sophistication of experiment, major advances have been made in recent years by many research groups. This concerns the calculation of accurate multi-dimensional potential energy surfaces for excited electronic states and the dynamical treatment of the nuclear motion on these surfaces. The exact quantum mechanical modelling of the dissociation of a triatomic molecule is nowadays practicable without severe technical problems. Moreover, simple but nevertheless realistic models have been developed and compared against exact calculations which are very useful for understanding the interrelation between the potential and the nuclear dynamics on one hand and the experimental observables on the other hand. 

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. 

The CT/ET free energy surface is the central concept in the theory of CT/ ET reactions. The surface s main purpose is to reduce the many-body problem of a localized electron in a condensed-phase environment to a few collective reaction coordinates affecting the electronic energy levels. This idea is based on the Born-Oppenheimer (BO) separation " of the electronic and nuclear time scales, which in turn makes the nuclear dynamics responsible for fluctuations of electronic energy levels (Eigure 1). The choice of a particular collective mode is dictated by the problem considered. One reaction coordinate stands out above all others, however, and is the energy gap between the two CT states as probed by optical spectroscopy (i.e., an experimental observable). 

The term Lamb shift of a single atomic level usually refers to the difference between the Dirac energy for point-like nuclei and its observable value shifted by nuclear and QED effects. Nuclear effects include energy shifts due to static nuclear properties such as the size and shape of the nuclear charge density distribution and due to nuclear dynamics, i.e. recoil correction and nuclear polarization. To a zeroth approximation, the energy levels of a hydrogen-like atom are determined by the Dirac equation. For point-like nuclei the eigenvalues of the Dirac equation can be found analytically. In the case of extended nuclei, this equation can be solved either numerically or by means of successive analytical approximation (see Rose 1961 Shabaev 1993). 

Effects of nuclear dynamics on electron tunneling in redox proteins have been an important question for the biological electron transfer community. While it has been understood how nuclear dynamics controls the Pranck-Condon factor, little was known until now about how the dynamics affects the tunneling matrix element. Our results show that, when tunneling is dominated by a single pathway tube, dynamical effects are small and Pathways level calculations provide reasonable results. The situation changes when several pathway tub are important and destructive interference exists among them. In this case dynamic amplification becomes important,... 

The integration of this set of coupled first-order differential equation can be done in a number of ways. Care must be taken since there are basically rather two different time scales involved, i.e. that of the nuclear dynamics and that of the normally considerably faster electron dynamics. It should be observed that this END takes place in a Cartesian laboratory reference frame, which means that the overall translation as well as overall rotation of the molecular system is included. This offers no complications since the equations of motion satisfy basic conservation laws and, thus, total momentum and angular momentum are conserved. At any time in the evolution of the molecular system can the overall translation be isolated and eliminated if so should be deemed necessary. This level of theory [16,19] is implemented in the program system ENDyne [20], and has been applied to atomic and molecular reactive collisions. Calculations of cross sections, differential as well as integral, yield results in excellent agreement with the best experiments. 

Inclusion of quantum effects on the nuclear dynamics can be accomplished by using Fermi s Golden Rule (134), which is really a manifestation of first-order time-dependent perturbation theory and conservation of energy during a transition. In this level of refinement, and formally allowing for the inclusion of solvent effects, the rate is given by... 

The translational motions and spin dynamics of conduction electrons in metals produce fluctuating local magnetic hyperfine fields. These couple to the nuclear magnetic moments, inducing transitions between nuclear spin levels and causing nuclear spin relaxation. The translational motions of electrons occur on a very rapid time scale in metals (<10 s), so the frequency spectrum of hyperfine field fluctuations is spread over a wide range of w-values. Only a small fraction of the spectral intensity falls at the relatively low nuclear resonance frequency (ojq 10 s ). Nevertheless, the interaction is so strong that this process is usually the dominant mode of relaxation for nuclei in metallic systems, either solid or liquid. 

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