Orientational probability density

Figure 2 shows calculated [55] orientational probability density functions P(cos0), for several of the readily obtained iJKM> states. (Here 0 is the angle between the dipole moment i and the electric field e). 

FIGURE 2. Calculated orientational probability density functions for the selected JKM states of a symmetric top molecule [adapted from Ref. 55]. Average values, , are as Mows 222>, 0.667 111>, 0.500 212>, 0.333 i313>, 0.250. 

The orientation of the coordinate frame R fixed in a particular rod with respect to the laboratory frame L is specified by the Euler rotation, = (afiy), as before. Under the preceding assumptions, a diffusion equation for the probability density [/(if, /)] is derived,<29) namely,... 

In essence, it is the probability density of the two nuclei to have relative separation Since the orientation of the molecule is not fixed (nuclei are not fixed any more if we deal with an non-BO approach), gi( ) is a spherically symmetric function. The plots of gi ( ) are presented in Figs. 1-4-. It should be noted that all the correlation functions shown are normalized in such a way that... 

Quadratic discriminant analysis (QDA) is a probabilistic parametric classification technique which represents an evolution of EDA for nonlinear class separations. Also QDA, like EDA, is based on the hypothesis that the probability density distributions are multivariate normal but, in this case, the dispersion is not the same for all of the categories. It follows that the categories differ for the position of their centroid and also for the variance-covariance matrix (different location and dispersion), as it is represented in Fig. 2.16A. Consequently, the ellipses of different categories differ not only for their position in the plane but also for eccentricity and axis orientation (Geisser, 1964). By coimecting the intersection points of each couple of corresponding ellipses (at the same Mahalanobis distance from the respective centroids), a parabolic delimiter is identified (see Fig. 2.16B). The name quadratic discriminant analysis is derived from this feature. 

Fig. 6.77. Calculations done using the statistical mechanical theory of electrolyte solutions. Probability density p(6,r) for molecular orientations of water molecules (tetrahedral symmetry) as a function of distance rfrom a neutral surface (distances are given in units of solvent diameter d = 0.28 nm) (a) 60H OH bond orientation and (b) dipolar orientation, (c) Ice-like arrangement found to dominate the liquid structure of water models at uncharged surfaces. The arrows point from oxygen to hydrogen of the same molecule. The peaks at 180 and 70° in p(0OH,r) for the contact layer correspond to the one hydrogen bond directed into the surface and the three directed to the adjacent solvent layer, respectively, in (c). (Reprinted from G. M. Tome and G. N. Patey, ElectrocNm. Acta 36 1677, copyright 1991, Figs. 1 and 2, with permission from Elsevier Science.

Let us proceed to the derivation of the pertinent Fokker-Planck equation. The probability density W(e,n,t) of various orientations of the magnetic moment and the easy magnetization axis of a ferroparticle must satisfy the conservation law... 

In the case of righthanded circular polarized i -type excitation, when orientation of angular momenta takes place and the probability density of the angular momenta distribution is proportional to (1 — cos )2/2 (see (2.13)), only alignment of internuclear axes occurs, described by the probability density, which is proportional to (1/2)[1 + (sin20r)/2j. 

The connection between the covariant cyclic and cartesian coordinates of the vector J yields Eq. (A.6), whilst (A.5) makes it possible to form the vector itself out of the components (J)q. As follows from (2.18), the components of the multipole moment pq characterize the preferred orientation of the angular momentum J in the molecular ensemble. Fig. 2.3(a, b) shows the probability density p(0, [Pg.30]

The distributions of both segment and bond densities allow us to calculate any structural property of the mono-layer. The bond orientational probabilities (forward, lateral, and backward) as well as the bond order parameter of a chain as a function of bond number along the chain are given below. 

V. BOND DENSITY PROFILE AND BOND ORIENTATION PROBABILITY... 

Nematic phases are characterised by a uniaxial symmetry of the molecular orientation distribution function f(6), describing the probability density of finding a rod with its orientation between 6 and 6 + d0 around a preferred direction, called the director n (see Fig. 15.49). An important characteristic of the nematic phase is the order parameter (P2), also called the Hermans orientation function (see also the discussion of oriented fibres in Sect. 13.6) ... 

What does a H 2 for a 2p orbital look like The probability density plot is no longer spherically symmetrical. This time the shape is completely different—the orbital now has an orientation in space and it has two lobes. Notice also that there is a region where there is no electron density between the two lobes—another nodal surface. This time the node is a plane in between the two lobes and so it is known as a nodal plane. One representation of the 2p orbitals is a three-dimensional plot, which gives a clear idea of the true shape of the orbital. 

This particular solution of the rotational diffusion equation can be interpreted as the transition probability that is, the probability density for a rod to have orientation u at time t, given that it had an orientation Uo initially. 

We suppose that a small probing held Fj, having been applied to the assembly of dipoles in the distant past (f = —oo) so that equilibrium conditions have been attained at time t = 0, is switched off at t = 0. Our starting point is the fractional Smoluchowski equation (172) for the evolution of the probability density function W(i), cp, t) for normal diffusion of dipole moment orientations on the unit sphere in configuration space (d and (p are the polar and azimuthal angles of the dipole, respectively), where the Fokker-Planck operator LFP for normal rotational diffusion in Eq. (8) is given by l j p — l j /> T L where... 

---