Planar rotator model

So far we have considered the planar rotator model. However, the above equations can readily be generalized to rotation in space. Here, the space coordinate is the polar angle i) and the Fokker-Planck operator for normal rotational diffusion assumes the form [8] [cf. Eq. (80)]... 

We now recall that the classical planar rotator model may be used as a model of superfluid He4, 0 being the phase of the condensate wave function, S being related to the superfluid density ps as S = ps(hjm)2, m being the mass of a He4 atom. Thus one can have superfluid-normal fluid transition in d = 2 dimensions, despite the lack of conventional long range order This conclusion seems to be corroborated by experiments on He4 films (Bishop and Reppy, 1978). 

There exists many generalizations and variants of the isotropic planar rotator model, eq. (159). Here we only mention the anisotropic planar rotor mode (Mouritsen and Berlinsky, 1982 Harris et al., 1984)... 

In the first period, which ended with a review [18], the complex susceptibility x (0)) was expressed through the law of motion of the particles perturbed by a.c. external field E(t). The results of these calculations rigorously coincide with those obtained, for example, in Refs. 22 and 23, respectively, for the planar and spatial extended diffusion model (compare with our Ref. 18, pp. 65 and 68). The most important results of this period are (i) the planar confined rotator model [ 17, p. 70 20], which has found a number of applications in our and other [24—31] works (ii) the composite so-called confined rotator-extended diffusion model. However, this approach had no perspectives because of troublesome calculations of the susceptibility x ( )-... 

This idea has led long ago [20, 21, 45, 46] to a simple model termed the confined rotator model, where the dielectric response was found due to free planar librations performed without friction during the lifetime of a molecule in a site of near order. These librations occur between two reflective walls, the... 

It is concluded [217] that an interpretation of the ideal herringbone transition within the anisotropic-planar-rotor model (2.5) as a weak first-order transition seems most probable, especially since previous assignments [56, 244] can be rationalized. This phase transition is fluctuation-driven in the sense of the Landau theory because the mean-field theory [141] yields a second-order transition. Assuming that defects of the -v/3 lattice and additional fluctuations due to full rotations and translations in three dimensions are not relevant and only renormalize the nonuniversal quantities, these assignments should be correct for other reasonable models and also for experiment [217]. 

A diatomic molecule constrained to rotate on a flat surface can be modeled as a planar... 

When the IRP is traced, successive points are obtained following the energy gradient. Because there is no external force or torque, the path is irrotational and leaves the center of mass fixed. Sets of points coming from separate geometry optimizations (as in the case of the DC model) introduce the additional problem of their relative orientation. In fact, the distance in MW coordinates between adjacent points is altered by the rotation or translation of their respeetive referenee axes. The problem of translation has the trivial solution of centering the referenee axes at the eenter of mass of the system. On the other hand for non planar systems, the problem of rotations does not have an analytical solution and must be solved by numeiieal minimization of the distanee between sueeessive points as a funetion of the Euler angles of the system [16,24]. 

Sulfur dioxide is an example of a simple Lewis base that carries two sets of inequivalent n-pairs, one set on each O atom. The n-pair model (in which the tt bonding pairs are not drawn and are ignored here) is shown in Fig. 10. The geometries of S02 HF [126,127], S02 HC1 [28,126] and S02- C1F [70] have all been determined from investigations of their rotational spectra. Each molecule is planar and belongs to the Cs point group. Scale drawings for S02 HC1 and S02- C1F are displayed in Fig. 10. 

Wiberg split the stabilization of the energy barrier into two parts (a) electrostatic energy in the planar form and (b) delocalization. Electrostatic stabilization lowers the energy of the planar form because the charge is spread over three atoms rather than being localized on one carbon in the rotated form. An estimation of the electrostatic stabilization was made by calculating a model, methane, for the localized anion and yielded a 23 kealmol-1... 

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