Oscillator anisotropic

There exist a variety of extensions of the basic shell model. One variation for molecular systems uses an anisotropic oscillator to couple the core and shell charges,thus allowing for anisotropic polarizability in nonspherical systems. Other modifications of the basic shell model that account for explicit environment dependence include a deformable or breathing shelF ° and shell models allowing for charge transfer between neighboring sites. 

For a comparison of the classical trajectories and the above-mentioned quantum mechanical wave functions of the anisotropic oscillator in a magnetic field we refer the reader to the literature [7]. 

Example 1.9 (Anisotropic Oscillator) A perturbation of the central force problem illustrates the consequences of a breakdown of integrability. Consider the system with energy... 

The Anisotropic Oscillator is included as an example in the MD.M package, see http // MolecularDynatnics.info... 

Fig. 2.5 The error in energy in a numerically computed trajectory (Verlet method) with stepsize h = 0.00001, for the anisotropic oscillator, showing that it remains bounded by 1.5 x 10 " in very long runs...

Example 5.6 (Anisotropic Oscillator) The anisotropic oscillator was used in Chap. 1 to illustrate the concept of a chaotic dynamical system. Fix, arbitrarily, a small set of initial conditions for the anisotropic pendulum at energy Eq, with positions in a small interval on the positive x axis, 1.05 < x < 1.15 and y = 0. For each point (x, 0) taken from this set select the initial velocities as x = 0, y = [2(Eo — U(x, 0))] /. The set y = 0, y = 0 is invariant under the differential equations of the perturbed central force problem, since the force acts in the radial direction our choice of initial conditions ensures that trajectories explore the larger H = Eq energy surface. 

Meath W J and Kumar A 1990 Reliable isotropic and anisotropic dipole dispersion energies, evaluated using constrained dipole oscillator strength techniques, with application to interactions involving H2, N2 and the rare gases Int. J. Quantum Chem. Symp. 24 501... 

Pack R T 1978 Anisotropic potentials and the damping of rainbow and diffraction oscillations in differential cross-sections Chem. Phys. Lett. 55 197... 

The Drude oscillators are typically treated as isotropic on the atomic level. However, it is possible to extend the model to include atom-based anisotropic polarizability. When anisotropy is included, the harmonic self-energy of the Drude oscillators becomes... 

Silicon is a model for the fundamental electronic and mechanical properties of Group IV crystals and the basic material for electronic device technology. Coherent optical phonons in Si revealed the ultrafast formation of renormalized quasiparticles in time-frequency space [47]. The anisotropic transient reflectivity of n-doped Si(001) featured the coherent optical phonon oscillation with a frequency of 15.3 THz, when the [110] crystalline axis was parallel to the pump polarization (Fig. 2.11). Rotation of the sample by 45° led to disappearance of the coherent oscillation, which confirmed the ISRS generation,... 

Fig. 2.12. Left transient anisotropic reflectivity change of the (001) surface of single crystal type Ha diamond. Inset shows the FT spectrum of the oscillation, demonstrating a narrow peak of the optical phonon at 40THz. Right pump and probe polarizations to detect the optical phonon. Adapted from [50]...

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