Bixon-Jortner model

In the Sharf and Fischer treatment the real manifold is subdivided into n idealized submanifolds where each one conforms with the Bixon-Jortner model. By taking suitable linear combinations of the n idealized submanifolds the problem may be reduced to that of a discrete state and two manifolds, both of which carry intensity but only one of which is associated with a non-zero interaction. The Bixon-Jortner procedure applied to this situation leads to a Fano lineshape superposed on a constant background absorption. 

There are but few models of irreversibility which can be solved exactly. Such a model was originally introduced by Ref. [Friedrichs 1948] but it also goes under the name the Fano model, [Fano 1961 Cohen-Tannoudji 1992], It offers the theory of a single quantum state imbedded in a continuum or a quasi-continuum. It has been used in molecular physics as the Bixon-Jortner model [Bixon 1968] and in solid state physics as an Anderson model [Anderson 1961]. 

In intermediate or small systems, their population dynamic behaviors often exhibit nonexponential decay or even oscillatory decay like the vibrational relaxation of C6H5NH2 in Sect. 5.2. To show how the density matrix method can be applied to study these systems, the Bixon-Jortner model is considered in this section. For this purpose, we consider the following model (see Fig. 4.2). 0) and /)(i = 1, ) are the eigenstates of the Hamiltonian Ho. For simplicity, we assume that only the perturbation matrix elements between 0) and /) states are nonzero. That is. 

Fig. 4.2 Bixon-Jortner model for decay from 0) state...

The main reason for choosing the treatment of vibrationai reiaxation of (H20)2 and C6H5NH2 is to show that the quantum chemistry programs can now provide the anharmonic vibrationai potentiais so that the first-principie caicuiation of vibrational relaxation has become possible. Their dynamical behaviors may be described by the density matrix method through the Bixon-Jortner model (see Sect. 4.3). 

In a recent work, Atkins and Stannard (1977) have proposed two mechanisms to describe the magnetic quenching effect by an enhancement of radiationless decay. They describe the radiationless decay by the Bixon-Jortner model (Bixon and Jornter, 1968) ... 

Next, two models designed to deal with nuclear tunneling will be discussed the PKS model and the Bixon-Jortner model. The Bixon-Jortner model (1968) provided the conceptual basis for the understanding of radiationless transitions in excited electronic and vibrational states. 

At lower temperatures, the log k versus 1/T curve no longer decreases linearly but bends with a decreasing slope (Figure 10.19). This is due to nuclear tunneling, as described in the Bixon-Jortner model. What is stated here holds for all dimensions, of course, not just in the one-dimensional systems. 

Figure 5. Results of simulation using Sumi-Marcus Jortner-Bixon hybrid model, (a) AGO = -0.50 eV. (b) AGq = -0.33 eV, T = 273. 293. 313, 333, 353, and 373 K.

In the Bixon-Jortner (BJ) model [14a[, a steady-state form of the type given in Eq. 3 is employed for Ad, where A = a/h)c is the single-mode quantal non-adiabatic TST expression given by Eqs. 31 and 51, with /(,] = /, (the solvent reorganization energy) and given by Eq. 49, and where a is given by the... 

Several theoretical works investigating population transfer via a continuum have been reported. In the earliest model [238], the continuum was treated as a Bixon-Jortner-type [235] unbounded set of equally spaced quasibound states. In such a model complete population transfer was shown to be possible [238]. Soon thereafter it was however pointed out [239] that complete population transfer via a true continuum is not generally achievable. Several works have tried to maximize the transfer efficiency, by employing structured continua auto-ionizing states [240] or in the case of LICS by controlling the Stark shifts, the area of laser pulses, and the effect of... 

Figure 6.1 The Bixon and Jortner model [11] to describe the intrastate coupling and intramolecular vibrational redistribution.

In this book we shall write the Hamiltonian as an (algebraic) operator using the appropriate Lie algebra. We intend to illustrate by many applications what we mean by this cryptic statement. It is important to emphasize that one way to represent such a Hamiltonian is as a matrix. In this connection we draw attention to one important area of spectroscopy, that of electronically excited states of larger molecules,4 which is traditionally discussed in terms of matrix Hamiltonians, the simplest of which is the so-called picket fence model (Bixon and Jortner, 1968). A central issue in this area of spectroscopy is the time evolution of an initially prepared nonstationary state. We defer a detailed discussion of such topics to a subsequent volume, which deals with the algebraic approach to dynamics. 

U. Even In a recent series of papers [M. Bixon and J. Jortner], using a model Hamiltonian quantum treatment, it is shown that all multipole contributions to l mixing are negligible when compared with / mixing by low external fields. Thus the long lifetimes associated with ZEKE states are attributed (in atoms and in molecules) to the external fields alone. 

Farver, O. and Pecht, I. (1999) Copper protein as model system for investigating intramolecular electron transfer processes, in Jortner, J. and Bixon. (eds.), Advances in Chemical Physics 107, Part 1, John Wiley Sons. NY., pp. 555-590. 

Most polaron models consider only electron transfer steps parallel and antiparallel to the applied field. Van der Auweraer et al. (1994) derived an expression for the mobility that takes into account isotropic hopping in three dimensions. The treatment is based on the Marcus theoiy (Marcus, 1964, 1968, 1984 Kester et al., 1974 Jortner, 1976 Sumi and Marcus, 1986 Jortner and Bixon, 1988) and assumes that energetic and positional disorder can be neglected. 

The Sumi-Marcus model treats both the solvent and intramolecular mode classically. However, the actual system should have not only the classical modes but also quantum mechanical high-frequency modes. Jortner and Bixon have developed an ET model which introduces the effect of the quantum mechanical high-frequency modes [4]. In this model, the reactant surface crosses not only with the vibrational ground state of the product but also with vibrationally excited states of the product. The reaction of oxazines in DMA is activationless, which means that the reaction is not in the inverted region but very close to it. We use a hybrid model of Sumi-Marcus and Jortner-Bixon developed by Walker et al. [15]. In this model, the intramolecular vibration is separated into the quantum mechanical high-frequency mode and classical low-frequency mode ... 

Before discussing the completely general cases, it is instructive to consider the simple model of Bixon and Jortner io,-i3-is) jn this model, the states 0,i are taken to be equally spaced in energy, (spacing is e) and the vsl = v are all the... 

An important achievement of the early theories was the derivation of the exact quantum mechanical expression for the ET rate in the Fermi Golden Rule limit in the linear response regime by Kubo and Toyozawa [4b], Levich and co-workers [20a] and by Ovchinnikov and Ovchinnikova [21], in terms of the dielectric spectral density of the solvent and intramolecular vibrational modes of donor and acceptor complexes. The solvent model was improved to take into account time and space correlation of the polarization fluctuations [20,21]. The importance of high-frequency intramolecular vibrations was fully recognized by Dogonadze and Kuznetsov [22], Efrima and Bixon [23], and by Jortner and co-workers [24,25] and Ulstrup [26]. It was shown that the main role of quantum modes is to effectively reduce the activation energy and thus to increase the reaction rate in the inverted... 

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