Intrinsic coordinate system

As a first example the states of any pure rotor series should all have maximum probability for r, = r2 and 0l2 = n, and the probability densities for members of a given rotor series should have very similar spatial distributions in their internal coordinate system (called the intrinsic coordinate system in the context of nuclear physics). The total wavefunctions of different rotor states in any series should differ primarily only in the parts that describe the rotation of the figure axis in space these parts do not affect the distributions in their internal coordinate systems. At a higher level of approximation, the distributions for the states of a given series may be expected to differ a little because of centrifugal distortion such differences, of course, are apparent in the internal coordinate system. 

The intrinsic coordinate system is associated with the path of a particle. Considering a point M in the flrrid that rrroves in time, its path follows the crrrve OM (t). The movement of the particle along its path can be characterized by the... 

Figure 2.1. Definition of the intrinsic coordinate system associated with the path of a fluid particle for a steady plane flow...

At every point M, the intrinsic coordinate system comprises the unit vectors t and . It is important because velocity is tangential to the path ... 

In the intrinsic coordinate system, the Euler equations for a steady flow are written along the tangential and normal directions, respectively, as ... 

Figure 2.4. Representation of the intrinsic coordinate system in the vicinity of a point around which the flow is parallel...

This chapter provides analytical solutions to mass transfer problems in situations commonly encountered in the pharmaceutical sciences. It deals with diffusion, convection, and generalized mass balance equations that are presented in typical coordinate systems to permit a wide range of problems to be formulated and solved. Typical pharmaceutical problems such as membrane diffusion, drug particle dissolution, and intrinsic dissolution evaluation by rotating disks are used as examples to illustrate the uses of mass transfer equations. 

Sidebar 10.3 outlines the useful analogy to normal-mode analysis of molecular vibrations, where the null modes correspond to overall translations or rotations of the coordinate system that lead to spurious alterations of coordinate values, but no real internal changes of interatomic distances. For this reason, the internal metric M( of (10.29) is the starting point for analyzing intrinsic state-related (as opposed to size-related) aspects of a given physical system of interest. 

A series of relationships have been derived between the stationary coordinate system (the scientist in his or her laboratory) and a moving (intrinsic, invariant) coordinate system that can be compared to classical calculations of dynamic variables (Table 1.3). 

The variables q and q represent generalized coordinates of particle positions and velocities. The advantage of generalized dynamic variables is their independence of coordinate system. This is of special importance when dealing with phenomena in which the motion of material particles is not observed directly, for instance in the study of electricity. Parameters, other than particle positions (e.g. currents) are observed here, although the behaviour of the system is really controlled by the motion of electrons which remains intrinsically concealed. The values assumed by the descriptive parameters, called generalized coordinates, are those connected with the position coordinates of the electrons. Another important application is in statistical mechanics. 

Fig. 3.7 Variance during rotation of the coordinate system for data sets with different intrinsic dimensionality (polar coordinates, values calculated in steps of 1 degree). (A corr. coeif. 0.955, variances xi 6.39, X2 6.86, PCI 12.95, PC2 0.30) and B corr. coeff. 0.136, variances xi 6.39, x 5.24, PCI 6.79, PC2 4.85).

When the true intrinsic rank of a data matrix (the number of factors) is properly determined, the corresponding eigenvectors form an orthonormal set of basis vectors that span the space of the original data set. The coordinates of a vector a in an m-dimensional space (for example, a 1 x m mixture spectrum measured at m wavelengths) can be expressed in a new coordinate system defined by a set of orthonormal basis vectors (eigenvectors) in the lower-dimensional space. Figure 4.14 illustrates this concept. The projection of a onto the plane defined by the basis vectors x and y is given by a. To find the coordinates of any vector on a normalized basis vector, we simply form the inner product. The new vector a, therefore, has the coordinates a, = aTx and a2 = aTy in the two-dimensional plane defined by x and y. 

With the intrinsic local coordinate system as in Fig. 11, the value of... 

Following a center-of-mass fixed coordinate system tied to an air mass, we use intrinsic coordinates to avoid artificial diffusion in the horizontal direction. Physical diffusion, therefore, is distinct and identifiable because the moving control volume can be allowed to undergo mass exchange with a neighboring air mass in a prescribed fashion. The question of horizontal spatial resolution is answered by a selection of source grid size, and the vertical resolution is set by the choice of the interval size in the z-direction. 

The ZA(f) are chosen so that they tend to 1 in the vicinity of atom A but drop to zero in the direction of all other nuclei. Thus, even for integrals involving atomic basis functions of two different atoms the integrand of each contribution la has no more than one singular point. The integration can be further simplified by suitable transformations to intrinsic coordinates, e.g. elliptic-hyperbolic coordinates for diatomic molecules or spherical coordinates for polyatomic systems. 

The actual path mapped out by the MEP on the PES is dependent on coordinate system. However, changes in coordinate system do not alter the nature of the stationary points on the PES (i.e. minima, TSs, etc.). One coordinate system, mass-weighted Cartesian coordinates (see Section 10.2.3), is especially significant for reaction dynamics, and the MEP in this coordinate system is known as the intrinsic reaction coordinate (IRC) [162]. In this section, we use the terms MEP, IRC, steepest descent path, and reaction path synonymously. 

In the case of H2O, we obtain nine frequencies, six of which are near zero. These six frequencies must be eliminated from the results because they correspond to the translational and rotational motions of the molecule as a whole (3V-6 rule). Thus, the remaining three frequencies correspond to the normal modes of the H2O molecule. It should be noted that vibrational frequencies are intrinsic of individual molecules and do not depend on the coordinate system chosen (R or Xm)-... 

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