Angular variable

Figure Cl.5.15. Molecular orientational trajectories of five single molecules. Each step in tire trajectory is separated by 300 ms and is obtained from tire fit to tire dipole emission pattern such as is shown in figure Cl.5.14. The radial component is displayed as sin 0 and tire angular variable as (ji. The lighter dots around tire average orientation represent 1 standard deviation. Reprinted witli pennission from Bartko and Dickson 11481. Copyright 1999 American Chemical Society.

Here g and go are a set of angular variables, which define a molecular orientation at instants of time 0 and t, respectively and ft is the orientation at instant t which was g0 at t = 0. By difference of arguments we mean the difference of turns. In the molecular frame (MS), where the axes are oriented along the main axes of the inertia tensor, [Pg.86]

As a simple model which takes into account valence and deformation vibrations of a molecule imbedded in the condensed phase, we consider a diatomic molecule with two degrees of freedom corresponding to valence (in the radial variable r) and torsional (in the angular variable [Pg.94]

From the momentum density one can find the radial momentum density by integration over the angular variables Of in momentum space ... 

The proof depends on Proposition A.3 of Appendix A, which ensures that all solutions of the Schrodinger equation can be approximated by linear combinations of solutions where the radial and angular variables have been... 

If molecules are involved, isotropic potential functions are in general not adequate and angular dependences reflecting the molecular symmetries may have to be accounted for. In general, up to five angular variables may be needed, but in many cases the anisotropies may be described rigorously by fewer angles. We must refer the reader to the literature for specific answers (Maitland et al., 1981) and mention here merely that much of what will interest us below can be modeled in the framework of the isotropic interaction approximation. 

The detailed geometry of the dodecahedron is described by two angular variables, A and b, the angles the M—A and M—B bonds make with the fourfold inversion axis, and the bond length ratio MA/MB (Figure 76). 

G. Gaigalas, Integration over spin-angular variables in atomic physics, Lithuanian J. Phys., 39, 79-105 (1999). 

Equation 3.48 is of course the same equation as we have solved before, e.g. for the particle in a box. Its solutions are simple sine and cosine functions of boundary conditions for the wavefunction are therefore different from those for the particle in a box. There is no requirement that iff must be zero anywhere instead, it must be single valued, which means for any 0,... 

Angular Variables and the Ligand Field Potential Matrix.116... 

The key question is what significance can be attached to parameters, specifically AOM parameters and angular variables, derived from spectroscopic data. As these examples were intended to illustrate, it is useful to divide this question into two parts, whether or not a calculation is actually performed this way (1) can the spectroscopically independent elements of the ligand field potential matrix be uniquely determined from the data (2) can the AOM parameters and angular variables be uniquely determined from the spectroscopically independent elements of the potential matrix ... 

The obvious answer to the second part is that the number of spectroscopically independent elements in the ligand field potential matrix is the upper limit to the number of AOM parameters and angular variables that may be determined from them. No matter how many experimental peak positions have been measured, it is the form of the ligand field potential matrix that determines how much information can be extracted. If there are more parameters than spectroscopically independent matrix elements, it may still be possible to... 

When the number of AOM parameters plus angular variables is greater than the number of spectroscopically independent elements in the potential matrix it may still be possible to determine all the parameters by placing restrictions on their values, provided the restrictions can be justified. An examination of the propagated uncertainties will reveal whether this device is successful or not. Our experience has been that the uncertainties in the angular values are unacceptably high when there are too few spectroscopically independent elements in the potential matrix. 

The transverse Kronecker symbols 5A in (17) and (21) guarantee the normalization of A. This implies that only two of the (17) are independent. For the following calculations it turned out to be useful to guarantee the normalization of the director by introducing two angular variables 9 and 0 to describe the director ... 

Consequently, (17) has to be replaced using angular variables. Denoting the right hand side of (17) with 7 , this can be done the following way ... 

We write the solution as the vector X = (6,(j),u,vx,vy,i ,P,) consisting of the angular variables of the director, the layer displacement, the velocity field, the pressure, and the modulus of the (nematic or smectic) order parameter. For a spatially homogeneous situation the equations simplify significantly and the desired solution Xo can directly be found (see Sect. 3.1). To determine the region of stability of Xq we perform a linear stability analysis, i.e., we add a small perturbation Xi to... 

Here we focus on the longitudinal situation and assume that the imposed fields are collinearly directed along the anisotropy axis n. Then the set of the angular variables reduces to the polar angle fi of e with respect to it. Setting cos ) (e n) = x, at Hp = const for the equilibrium distribution function of the particle magnetic moment, one gets... 

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