Multi-mode dynamics

As demonstrated by the power spectra in Figs. 12.2a and 12.3b, regulation of the blood flow to the individual nephron involves several oscillatory modes. The two dominating time scales are associated with the period Tsiow 30—40 s of the slow TGF-mediated oscillations and the somewhat shorter time scale Tjast 5—10 s defined by the myogenic oscillations of the afferent arteriolar diameter. The two modes interact because they both involve activation of smooth muscle cells in the arteriolar wall. Our model describes these mechanisms and the coupling between the two modes, and it also reproduces the observed multi-mode dynamics. We can, therefore, use the model to examine some of the phenomena that can be expected to arise from the interaction between the two modes. 

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). 

Because of limited space we confine ourselves here to the presentation of this single prototypical multi-mode dynamical JT system. However, investigations along... 

The vibronic coupling Hamiltonian provides a realistic model for accurately describing the short-time multi-mode dynamics in nonadiabatic systems. 

Eiding J, Schneider R, Domcke W, Kdppel H, von Niessen W (1991) Ab initio investigation of the multi-mode dynamical Jahn-TeUer effect in the ground state of the benzene cation. Chem Phys Lett 177 345... 

The fabrication and characterization of a fiber optic pH sensor based on evanescent wave absorption was presented by Lee63. The unclad portion of a multi-mode optical fibre was coated with the sol-gel doped with pH sensitive dye. The sensitivity of the device increased when the multiple sol-gel coatings were used in the sensing region. The dynamic range and the temporal response of the sensor were investigated for two different dyes -bromocresol purple and bromocresol green. 

Time-resolved fluorescence of coumarin C522 was determined in water and in host-guest complex with p-cyclodextrin, representing free aqueous and cavity restricted environments, respectively. Experimental fluorescence clearly showed faster dynamics in a case of water. The time parameters of monoexponential fit for water and p-cyclodextrin at 500 nm and 520 nm were determined to be 1.37 ps and 2.02 ps, and 2.97 ps and 7.14 ps, respectively. Multi-mode Brownian oscillator model, as an attempt to simulate the solvation dynamics, supported these fluorescence dynamics results. 

Apart from the heat bath mode, the harmonic potential surface model has been used for the molecular vibrations. It is possible to include the generalized harmonic potential surfaces, i.e., displaced-distorted-rotated surfaces. In this case, the mode coupling can be treated within this model. Beyond the generalized harmonic potential surface model, there is no systematic approach in constructing the generalized (multi-mode coupled) master equation that can be numerically solved. The first step to attack this problem would start with anharmonicity corrections to the harmonic potential surface model. Since anharmonicity has been recognized as an important mechanism in the vibrational dynamics in the electronically excited states, urgent realization of this work is needed. 

H. Koppel, W. Domcke and L. S. Cederbaum, Multi-mode molecular dynamics beyond the Born-Oppenheimer approximation, Adv. Chem. Phys., 57 (1984) 59-246. 

It is of interest that two additional papers on the subject of Reactor Dynamics in this volume include problems involving expansions in space dependent modes, first, the paper on Tem perature coefficients and stability by Harvey Brooks, and, second, the paper on System kinetics by T. A. Welton. Brooks is also interested in representations of the neutron density with the aid of a multi-mode analysis, but his problem is more complicated than that of this section because of feedback considerations. However, he confines his detailed analysis to a case in which the fundamental mode is dominant and where the effect of higher modes can be treated by a perturbation method. Welton s multi-mode analysis is peculiar to the aqueous homogeneous reactor and bears little resemblance to the corresponding problems treated by Brooks and this writer. Neither Brooks nor Welton appear to be interested in graphical representations of their results. 

The computations have shown that a multi-mode treatment is essential for correctly describing the dynamics in the presence of a conical intersection. 

The advent of MCTDH has made it possible to solve the (second-order) vibronic coupling Hamiltonian of small to medium sized molecules (5-12 atoms, say), including all internal degrees of freedom. In fact, it is the combination of the vibronic coupling model with MCTDH which is numerically so successful. The vibronic coupling model provides a realistic multi-mode Hamiltonian, and this Hamiltonian is, from its ansatz, in the product form advantageous for MCTDH. MCTDH then solves the dynamics problem accurately and efficiently. 

Koppel H, Domcke W, Cederbaum LS (1984) Multi-mode molecular dynamics beyond the born-oppenheimer approximation. Adv Chem Phys 57 59... 

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