Classical Electron Dynamics

Consider the equation of motion for an electron at rest when an electric field is suddenly 

Assume a collision occurs at time t. The average drift velocity can be found from 

However, this model assumes that the electrons come to a complete stop after each collision. A better model can be obtained by writing Newton s law with a drag term. 

Putting this i d hito Equation 18.7, the mobility and conductivity can be written as 

Note that since mobility and conductivity, which are material properties, depend on the collision time, then this collision time must be independent of the applied field if Ohm s law is to be obeyed. Why should the collision time be a material property The mean free path for electrons to travel before collisions would seem fo be a much more reasonable material property since it is determined by the size and number density of atoms in the structure. But the mean free path A is the velocity times the collision time t and since i d is directly proportional to tE, the collision time would be inversely proportional to the square root of the field, in violation of Ohm s law. This paradox can be resolved (at least temporarily) by assuming that the thermal velocities of the electrons are much higher than i d so that A = r(i th d) I t th- Let us now check to see if this assumption is valid. 

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In the Car and Parrinello (1985) scheme, ion dynamics is combined with a fictitious classical electron dynamics, with nuclei assigned real masses and the electron wave functions arbitrary fictitious masses. One starts the molecular-dynamics simulation at high temperature and cools progressively to zero temperature to find the ground state of both electrons and ions simultaneously. Although this approach at first seems strange and unphysical, it has yielded excellent results for amorphous Si (Car and Parrinello, 1988) and recently for SiOj (Allan and Teter, 1987) and S clusters (Hohl et al., 1988) and will probably play an important role in the future development of the field. 

In this minimal END approximation, the electronic basis functions are centered on the average nuclear positions, which are dynamical variables. In the limit of classical nuclei, these are conventional basis functions used in moleculai electronic structure theoiy, and they follow the dynamically changing nuclear positions. As can be seen from the equations of motion discussed above the evolution of the nuclear positions and momenta is governed by Newton-like equations with Hellman-Feynman forces, while the electronic dynamical variables are complex molecular orbital coefficients that follow equations that look like those of the time-dependent Hartree-Fock (TDHF) approximation [24]. The coupling terms in the dynamical metric are the well-known nonadiabatic terms due to the fact that the basis moves with the dynamically changing nuclear positions. 

It should be noted that there is a limited number of works on classical relativistic dynamical chaos (Chernikov et.al., 1989 Drake and et.al., 1996 Matrasulov, 2001). However, the study of the relativistic systems is important both from fundamental as well as from practical viewpoints. Such systems as electrons accelerating in laser-plasma accelerators (Mora, 1993), heavy and superheavy atoms (Matrasulov, 2001) and many other systems in nuclear and particle physics are essentially relativistic systems which can exhibit chaotic dynamics and need to be treated by taking into account relativistic dynamics. Besides that interaction with magnetic field can also strengthen the role of the relativistic effects since the electron gains additional velocity in a magnetic field. 

The Car-Parrinello approach combines an electronic structure method with a classical molecular dynamics scheme and thus unifies two major fields of computational chemistry, which have hitherto been essentially orthogonal. Through this unification a... 

Tab. 1.1 Comparison of the properties of quantum chemical electronic structure calculations (QC methods), classical molecular dynamics (Classical MD) based on empirical force fields and first-principles molecular dynamics (ab initio MD) simulations.

When addressing problems in computational chemistry, the choice of computational scheme depends on the applicability of the method (i.e. the types of atoms and/or molecules, and the type of property, that can be treated satisfactorily) and the size of the system to be investigated. In biochemical applications the method of choice - if we are interested in the dynamics and effects of temperature on an entire protein with, say, 10,000 atoms - will be to run a classical molecular dynamics (MD) simulation. The key problem then becomes that of choosing a relevant force field in which the different atomic interactions are described. If, on the other hand, we are interested in electronic and/or spectroscopic properties or explicit bond breaking and bond formation in an enzymatic active site, we must resort to a quantum chemical methodology in which electrons are treated explicitly. These phenomena are usually highly localized, and thus only involve a small number of chemical groups compared with the complete macromolecule. 

Considering the semiclassical description of nonadiabatic dynamics, only the mapping approach [99, 100] and the equivalent formulation that is obtained by requantizing the classical electron analog model of Meyer and Miller [112] appear to be amenable to a numerical treatment via an initial-value representation [114, 116, 117, 121, 122]. Other semiclassical formulations such as Pechukas path-integral formulation [45] and the various connection... 

Equation (28) is still exact. To introduce the classical-path approximation, we assume that the nuclear dynamics of the system can be described by classical trajectories that is, the position operator x is approximated by its mean value, namely, the trajectory x t). As a consequence, the quantum-mechanical operators of the nuclear dynamics (e.g., Eh (x)) become classical functions that depend parametrically on x t). In the same way, the nuclear wave functions dk x,t) become complex-valued coefficients dk x t),t). As the electronic dynamics is evaluated along the classical path of the nuclei, the approximation thus accounts for the reaction of the quantum DoE to the dynamics of the classical DoF. 

Classical molecular dynamics (MD) implementing predetermined potentials, either empirical or derived from independent electronic structure calculations, has been used extensively to investigate condensed-matter systems. An important aspect in any MD simulation is how to describe or approximate the interatomic interactions. Usually, the potentials that describe these interactions are determined a priori and the full interaction is partitioned into two-, three-, and many-body contributions, long- and short-range terms, etc., for which suitable analytical functional forms are devised. Despite the many successes with classical MD, the requirement to devise fixed potentials results in several serious problems... 

We start our discussion of laser-controlled electron dynamics in an intuitive classical picture. Reminiscent of the Lorentz model [90, 91], which describes the electron dynamics with respect to the nuclei of a molecule as simple harmonic oscillations, we consider the electron system bound to the nuclei as a classical harmonic oscillator of resonance frequency co. Because the energies ha>r of electronic resonances in molecules are typically of the order 1-10 eV, the natural timescale of the electron dynamics is a few femtoseconds to several hundred attoseconds. The oscillator is driven by a linearly polarized shaped femtosecond... 

Next we model the laser-driven electron dynamics quantum mechanically to reveal analogies and differences to the simple classical model. In view of the SPODS mechanism, which is based on resonant interactions, we consider only two quantum states at hrst, the ground state and the resonantly excited state. Eor this purpose, we briehy recapitulate the relevant equations for a two-state system driven on... 

A quantum mechanical theory is in principle needed to describe molecular phenomena in both few-atom and many-atom systems. In some cases a single electronic state is involved, and it is possible to gain valuable insight using only classical molecular dynamics, which can be relatively easy to apply even for a system of many atoms. A quantum mechanical description of molecular phenomena is however clearly needed for electronic states, insofar these have pronounced wavemechanical properties. The need for a quantum description of nuclear motions in molecular dynamics is less apparent, but it is required in some important situations. If we consider a generic interaction between two species A(a) and B(j3) leading to formation of two others, C(7) and D(6), all of them in the specified quantum states, so that... 

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