Anisotropic harmonic oscillator

The discussion has been focussed on the hydrogen atom, but some of the results are applicable also for the confinement by a dihedral angle of any system with a central potential, or simply with rotational symmetry around the z-axis, including isotropic and anisotropic harmonic oscillators. 

As discussed above (see Section 4.2.2), in the SE-SCM method we need to introduce appropriate empirical potentials. As will become apparent from our results, one-center potentials (like the one-center modified, anisotropic harmonic oscillator) are not adequate... 

In their initial paper, Condon, Altar, and Eyring chose to examine the effect of a perturbing potential of the form A xyz, where is a constant, on a three-dimensional anisotropic harmonic oscillator. Their purpose was to show that an electron moving in a properly dissymmetric potential field could give rise to optical activity. The harmonic oscillator model was not essential to their main thesis, but was chosen in order to have basis functions with which to carry through the perturbation calculations in detail. Nevertheless, they did also show how the harmonic potential field could be adapted to describe chromophoric electrons in actual molecules. 

The eigenvalue problem for the simple cos y potential of Eq. (4) can be solved easily by matrix diagonalization using a basis of free-rotor wave functions. For practical purposes, however, it is also useful to have approximate analytical expressions for the channel potentials V,(r). The latter can be constructed by suitable interpolation between perturbed free-rotor and perturbed harmonic oscillator eigenvalues in the anisotropic potential for large and small distances r, respectively. Analogous to the weak-field limit of the Stark effect, for linear closed-shell dipoles at large r, one has [7]... 

As a second example we analyze the anisotropic charge-permanent dipole potential where SACM and PST differ from each other. Here, for demonstration, we only consider the low-energy perturbation and the high-energy harmonic oscillator limits. In the former limit, the adiabatic channel potential curve for the lowest channel j = m = 0 has the form... 

In anisotropic materials, the electronic bonds may have different polarizabilities for different directions (you may think of different, orientation-dependent spring constants for the electronic harmonic oscillator). Remembering that only the E-vector of the light interacts with the electrons, we may use polarized light to test the polarizability of the material in different directions, lno is one of the most important electro-optic materials and we use it as an example. The common notations are shown in Figure 4.7. If the E-vector is in plane with the surface of the crystal, the wave is called a te wave. In this example, the te wave would experience the ordinary index na of LiNbOs (nG 2.20). If we rotate the polarization by 90°, the E-ve ctor will be vertical to the surface and the wave is called tm. In lno, it will experience the extraordinary index ne 2.29. Therefore these two differently polarized waves will propagate with different phase velocities v c/n. In the example of Figure 4.7, the te mode is faster than the tm mode. 

If the mean-square amplitude of libration has been determined from a TLS analysis of the anisotropic displacement parameters, the force constant (for a harmonic oscillator) is... 

Figure 14.12a demonstrates the Rietveld refinement pattern for time-of-flight (TOF) neutron diffraction data measured at room temperature for LiFeP04. Fitting was satisfactory (/ p = 2.66%, Rf - 0.46%, 5=1.34) with accurately refined atomic positions as well as anisotropic atomic displacement parameters for all atoms under the classical harmonic oscillation model. 

In the same way, the contribution from the first-order quadrupolar interaction can be successfully removed by applying fast MAS. However, the resulting spectrum measured even at a very high-spinning speed is only partially narrowed and still exhibits some remaining anisotropic broadening the residual is narrowed by about V3 of the original width (Fig. 10). Similarly, Eq. (11) describes harmonic oscillation P4 which indicates that, to get rid of... 

The ab initio potentials used in solid nitrogen are from Refs. [31] and [32]. They have been respresented by a spherical expansion, Eq. (3), with coefficients up to = 6 and Lg = 6 inclusive, which describe the anisotropic short-range repulsion, the multipole-multipole interactions and the anisotropic dispersion interactions. They have also been fitted by a site-site model potential, Eq. (5), with force centers shifted away from the atoms, optimized for each interaction contribution. In the most advanced lattice dynamics model used, the TDH or RPA model, the libra-tions are expanded in spherical harmonics up to / = 12 and the translational vibrations in harmonic oscillator functions up to = 4, inclusive. 

The model described above can be generalised to treat anisotropic diffusive motions which might be relevant in particular systems. For an anisotropic overdamped harmonic oscillator, with independent motions... 

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... 

Fig. 3. Nanostructured composite of p-nitroaniline (pNA, top left) and AIPO4-5. Top right Anisotropic absorption of the composite The crystals appear yellow only when the electric field vector E of the linearly polarized light oscillates in the direction of the transition dipole moment of the blue absorption of the pNA molecules. When E and these dipole moments are perpendicular to each other, no absorption occurs in the visible region, so that the pNA/AlPO -S crystals appear colourless. Center Experiment to measure the second harmonic generation (SHG) of the pNA/AIP04-5 composite. Bottom right The measured SHG effect is a function of cos q>, where q> is the rotation angle of the polarization rotator. This result is in agreement with the expectation if it is assumed that the pNA molecules are aligned in the channels and their dipoles all possess the same orientation. After the work of Caro, Marlow and co-workers [30-32],...

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