NONSEPARABLE DYNAMICS

Here is the generalized Laplace operator, defined by equation (37), while R is the hyperradius (equation (5)). In a later chapter of this book. Professor Fano will discuss the application of the hyperspherical method to nonseparable dynamical problems. Here we shall only note that if mass-weighted coordinates axe used, the Schrodinger equation for any system interacting through Coulomb forces can be written in the form ... 

The most general problem should be that of a particle in a nonseparable potential, linearly coupled to an oscillator heat bath, when the dynamics of the particle in the classically accessible region is subject to friction forces due to the bath. However, this multidimensional quantum Kramers problem has not been explored as yet. 

In TD-DFT, the wave function is antisymmetrized and therefore, nonfactorizable or entangled. However, as said above, it is not entangled from a dynamical point of view because the quantum forces originated from a nonseparable quantum potential as in Equation 8.34 are not taken into account. 

We can understand better this asymptotics by using the Markov chain language. For nonseparated constants a particle in has nonzero probability to reach and nonzero probability to reach A, . The zero-one law in this simplest case means that the dynamics of the particle becomes deterministic with probability one it chooses to go to one of vertices A, A3 and to avoid another. Instead of branching, A2 A and A2 A3, we select only one way either A2 A] or A2 A3. Graphs without branching represent discrete dynamical systems. 

Such Hamiltonian mappings are generated by a Poincare surface of section transverse to the orbits of the flow. Thus, v(q) plays the role of a potential function for the motion perpendicular to the periodic orbit. Note that the mapping takes into account the nonseparability of the dynamics. 

The orientation of bonds is strongly affected by local molecular motions, and orientation CF reflect local dynamics in a very sensitive way. However, the interpretation of multimolecular orientation CF requires the knowledge of dynamic and static correlations between particles. Even in simple liquids this problem is not completely elucidated. In the case of polymers, the situation is even more difficult since particules i and j, which are monomers or parts of monomers may belong to the same chain or to different Chains. Thus, we believe that the molecular interpretation of monomolecular orientation experiments in polymer melts is easier, at least in the present early stage of study. Experimentally, the OACF never appears as the complicated nonseparated function of time and orientation given in expression (3), but only as correlation functions of spherical harmonics... 

We point out that similar analyses and results have been performed and obtained also by other authors [33, 35, 38 0]. The spectral lines at 86meV and 123 meV excitation energy in the theoretical spectrum correspond to excitation of the modes V6 and vi, respectively. The first spacing deviates from the harmonic frequency of mode V6 in Table 3 because of the JT effect, while the second coincides with that of mode vi because of the linear coupling scheme adopted. For higher excitation energies the lines represent an intricate mixture of the various modes because of the well-know nonseparability of modes in the multi-mode dynamical JT effect. Overall, the excitation of the various modes can be characterized as moderately weak. The total JT stabilization energy amounts to 930 cm and is dominated by the contribution of mode ve- The barrier to pseudorotation is of the order of 10 cm only, consistent with the fact that the theoretical spectrum of Fig. 3 is obtained within the LVC scheme (see Sect. 2.1 above). 

Nonisotopic methods have also been described. For example, a homogeneous (nonseparation) fluorescence polarization immunoassay for DHEA-S that uses a rabbit polyclonal antibody and a DHEA-fluorescein tracer is available. The measured polarization is inversely related to DHEA-S concentration. This fully automated system has a dynamic range of 1 to lOOOjJ-g/dL (0,03 to 27 Limoi/L), and interassay coefficients of variation are less than 10% over a broad concentration interval (25 to lOOOpg/dL 0.7 to 27pmol/L). Assay time is about 15 minutes for a single sample and 30 minutes for 20 samples. 

The adiabatic potential (11b.II) was first introduced by HIRSCHPELDER and WIGNER /6/ in the simplest case of a rectilinear (dynamically nonseparable) reaction coordinate and a quantized (high frequency) vibration normal to it. It is obviously also applicable, according to (11 a.II), to the case of a curvilinear reaction path and a classical (low frequency) y-vibration. The more general representation (9.II) of the adiabatic separation includes also the case of a quantized (enharmonic) vibration along the reaction coordinate in the reactant region of configuration space, in which E = E, pro-... 

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