The set of center motions

Theorem 7.2. (BirkhofF) The Poisson-stable trajectories are dense everywhere in the set of center motions. 

In the preceding sections, we have discussed the set of center motions. In essence, we have found that it is the closure of the set of Poisson-stable trajectories. It does not exclude the case where the latter ones may simply be periodic orbits. But if there is a single Poisson-stable unclosed trajectory, then by virtue of Birkhoff s theorem in Sec. 1.2, there is a continuum of Poisson-stable trajectories. As for the rest of the trajectories in the center, it is known that the set of points which are not Poisson-stable is the union of not more... 

Since the actual motion of the Mossbauer drive, as for any frequency transmission system, can show phase shifts relative to the reference signal, the ideal folding point (FP) of the raw data in terms of channel numbers may be displaced from the center at channel number (N — l)/2 (= 255.5 in the example seen earlier). The folding routine must take this into account. Phase shift and FP depend on the settings of the feedback loop in the drive control unit. Therefore, any change of the spectrometer velocity tuning requires the recording of a new calibration spectrum. 

The translational motion corresponds to a displacement of the molecule as a whole in an arbitrary direction it can be depicted by a single vector showing the displacement of the center of mass. Let this vector have components x,y,z. We showed in Section 9.3 that under any symmetry operation, each of the functions x,y,z is transformed into a linear combination of x,y, and z. Hence (Section 9.6) the set of functions x,y,z forms a basis for some three-dimensional representation of the molecular point group we shall call this representation rtran8. [The representation (9.25) is... 

Equivalent considerations for nonstatic, sheared systems demonstrate the kinematical possibility of such shearing motions. This requires, inter alia, that the distance between any two sphere centers remains larger than 2a. The static viewpoint can be generalized to such circumstances as follows Rather than considering the lattice deformation, it suffices to examine the deformed collision sphere. The latter body 3 is defined as the set of points... 

In addition to the electronically adiabatic representation described by (4) and (5) or, equivalently (57) and (58), other representations can be defined in which the adiabatic electronic wave function basis set used in expansions (4) or (58) is replaced by some other set of functions of the electronic coordinates rel or r. Let us in what follows assume that we have separated the motion of the center of mass G of the system and adopted the Jacobi mass-scaled vectors R and r defined after (52), and in terms of which the adiabatic electronic wave functions are i] l,ad(r q) and the corresponding nuclear wave function coefficients are Xnd (R). The symbol q(R) refers to the set of scalar nuclear position coordinates defined after (56). Let iKil d(r q) label that alternate electronic basis set, which is allowed to be parametrically dependent on q, and for which we will use the designation diabatic. We now proceed to define such a set. LetXn(R) be the nuclear wave function coefficients associated with those diabatic electronic wave functions. As a result, we may rewrite (58) as... 

The foregoing discussion of impulse and momentum applies only when no change in rotational motion is involved. There is an analogous set of equations for angular impulse and impulse momentum. The angular momentum about an axis through the center of mass is defined as... 

The assignment of (hr) - 5) vibrational modes for a linear molecule and (hr) - 6) vibrational modes for a nonlinear molecule comes from a consideration of the number of degrees of freedom in the molecule. It requires hr) coordinates to completely specify the position of all t) atoms in the molecule, and each coordinate results in a degree of freedom. Three coordinates (x, y, and z) specify the movement of the center of mass of the molecule in space. They set the translational degrees of freedom, since translational motion is associated with movement of the molecule as a whole. Two internal coordinates (angles) are required to specify the orientation of the axis of a linear molecule during rotation, while three angles are required for a nonlinear... 

The total number of spatial coordinates for a molecule with Q nuclei and N electrons is 3(Q + N), because each particle requires three cartesian coordinates to specify its location. However, if the motion of each particle is referred to the center of mass of the molecule rather than to the external spaced-fixed coordinate axes, then the three translational coordinates that specify the location of the center of mass relative to the external axes may be separated out and eliminated from consideration. For a diatomic molecule (Q = 2) we are left with only three relative nuclear coordinates and with 3N relative electronic coordinates. For mathematical convenience, we select the center of mass of the nuclei as the reference point rather than the center of mass of the nuclei and electrons together. The difference is negligibly small. We designate the two nuclei as A and B, and introduce a new set of nuclear coordinates defined by... 

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

In fact, the diffusion constant in solutions has the form of an Einstein diffusion of hard spheres with radius Re. For a diffusing chain the solvent within the coil is apparently also set in motion and does not contribute to the friction. Thus, the long-range hydrodynamic interactions lead, in comparison to the Rouse model, to qualitatively different results for both the center-of-mass diffusion—which is not proportional to the number of monomers exerting friction - as well as for the segment diffusion - which is considerably accelerated and follows a modified time law t2/3 instead of t1/2. 

---