Eigenvalues separated wells

Thus a wavepacket initiated in well A passes to well B by a curve crossing. Prof. Fleming showed an interesting case of persistent coherence in such a situation, despite the erratic pattern of the eigenvalue separations. An alternative, possibly more revealing approach, is to employ Stuckelberg-Landau-Zener theory, which relates the interference (i.e., coherence) in the two different wells via the area shown in Fig. 2. A variety of applications to time-independent problems may be found in the literature [1]. 

As a rule, in diemial unimolecular reaction systems at modest temperatures, is well separated from the other eigenvalues, and thus the time scales for incubation and relaxation are well separated from the steady-... 

More generally, the relaxation follows generalized first-order kinetics with several relaxation times i., as depicted schematically in figure B2.5.2 for the case of tliree well-separated time scales. The various relaxation times detemime the tiimmg points of the product concentration on a logaritlnnic time scale. These relaxation times are obtained from the eigenvalues of the appropriate rate coefficient matrix (chapter A3.41. The time resolution of J-jump relaxation teclmiques is often limited by the rate at which the system can be heated. With typical J-jumps of several Kelvin, the time resolution lies in the microsecond range. 

The search for eigenfunctions and eigenvalues in the example of the simplest difference problem. The method of separation of variables being involved in the apparatus of mathematical physics applies equelly well to difference problems. Employing this method enables one to split up an original problem with several independent variables into a series of more simpler problems with a smaller number of variables. As a rule, in this situation eigenvalue problems with respect to separate coordinates do arise. Difference problems can be solved in a quite similar manner. 

The quantum version of the Hamiltonian Eq. (7) has been studied for decades in both Physics and Chemistry in the 2-level limit. If the potential energy surface (PES) is represented as a quartic double well, then the energy eigenvalues are doublets separated by, roughly, the well frequency. When the mass of the transferred particle is small (e g. electron), or the barrier is very high, or the temperature is low, then only the lowest doublet is occupied this is the 2-level limit of the Zwanzig Hamiltonian. 

Multiscale ensembles of reaction networks with well-separated constants are introduced and typical properties of such systems are studied. For any given ordering of reaction rate constants the explicit approximation of steady state, relaxation spectrum and related eigenvectors ( modes ) is presented. In particular, we prove that for systems with well-separated constants eigenvalues are real (damped oscillations are improbable). For systems with modular structure, we propose the selection of such modules that it is possible to solve the kinetic equation for every module in the explicit form. All such solvable networks are described. The obtained multiscale approximations, that we call dominant systems are... 

The fast stage of relaxation of a complex reaction network could be described as mass transfer from nodes to correspondent attractors of auxiliary dynamical system and mass distribution in the attractors. After that, a slower process of mass redistribution between attractors should play a more important role. To study the next stage of relaxation, we should glue cycles of the first auxiliary system (each cycle transforms into a point), define constants of the first derivative network on this new set of nodes, construct for this new network an (first) auxiliary discrete dynamical system, etc. The process terminates when we get a discrete dynamical system with one attractor. Then the inverse process of cycle restoration and cutting starts. As a result, we create an explicit description of the relaxation process in the reaction network, find estimates of eigenvalues and eigenvectors for the kinetic equation, and provide full analysis of steady states for systems with well-separated constants. 

While Eq. (9.49) has a well-defined potential energy function, it is quite difficult to solve in the indicated coordinates. However, by a clever transfonnation into a unique set of mass-dependent spatial coordinates q, it is possible to separate the 3 Ai-dirncnsional Eq. (9.49) into 3N one-dimensional Schrodinger equations. These equations are identical to Eq. (9.46) in form, but have force constants and reduced masses that are defined by the action of the transformation process on the original coordinates. Each component of q corresponding to a molecular vibration is referred to as a normal mode for the system, and with each component there is an associated set of harmonic oscillator wave functions and eigenvalues that can be written entirely in terms of square roots of the force constants found in the Hessian matrix and the atomic masses. 

The product J Eg)coQ is equal to Eg for a harmonic oscillator potential truncated at = Eg, and to 2Eg for a Morse potential with dissociation energy equal to Eg. Equation (2.41) is the low-friction limit result of Kramers. There are other methods to derive the results obtained in the previous section. One is to look for the eigenvalue with smallest positive real part of the ojjerator L defined so that dP/dt = — LP is the relevant Fokker-Planck or Smoluchowski equation. Under the usual condition of time scale separation this smallest real part is the escapie rate for a single well potential. Another way uses the concept of mean passage time. For the one-dimensional Fokker-Planck equation of the form... 

Well-separated instantaneous eigenvalues. In this case, if the initial condition is an instantaneous eigenstate, the evolution in the adiabatic limit follows the corresponding branch. 

The above equation, although provides an exact relation could be useful in practice only if the pseudopotential can be reasonably well approximated without knowing the all-electron orbitals and eigenvalues e,. Such approximations are available to separate core and valence electrons. The Phillips-Kleinman formal route can also be used to separate electrons in different molecules64,65. This route is, however, not orbital-free. The environment needs to be described at the orbital-level. Therefore, this group of methods will not be discussed further here. 

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