Bixon-Jortner coupling

The present formula Eq. (126) is tested in comparison with the Bixon-Jortner perturbation theory in the weak electronic coupling regime [109]. The Arrhenius plot is shown in Fig. 23, where the electronic coupling Had is taken... 

Figure 23. Arrhenius plot of the electron transfer rate. The electronic coupling strength is TIad = 0.0001 a.u. Solid line-Bixon-Jortner perturbation theory Ref. [109]. FuU-circle present results of Eq. (26 ). Dashed line-results of Marcus s high temperature theory [Eq.(129)]. Taken from Ref. [28].

Figure 25. Electron-transfer rate the electronic coupling strength at T = 500 K for the asymmetric reaction (AG = —3ffl2, oh = 749 cm ). Solid line-present full dimensional results with use of the ZN formulas. Dotted line-full dimensional results obtained from the Bixon-Jortner formula. Filled dotts-effective ID results of the quantum mechanical flux-flux correlation function. Dashed line-effective ID results with use of the ZN formulas. Taken from Ref. [28].

The exponential decay of IVR on the ground-state potential surface is a crude but useful description of that process in many molecules. It may be justified on theoretical grounds by considering the Bixon Jortner level coupling scheme. In (3.11), ojs is the central frequency for the vibrational feature s. 

M. Bixon and J. Jortner. Coupling of protein modes to electron transfer in bacterial photosynthesis. J. Phys. Chem., 90 3795-3800, 1986. 

The dynamics of inter- vs intrastrand hole transport has also been the subject of several theoretical investigations. Bixon and Jortner [38] initially estimated a penalty factor of ca. 1/30 for interstrand vs intrastrand G to G hole transport via a single intervening A T base pair, based on the matrix elements computed by Voityuk et al. [56]. A more recent analysis by Jortner et al. [50] of strand cleavage results reported by Barton et al. [45] led to the proposal that the penalty factor depends on strand polarity, with a factor of 1/3 found for a 5 -GAC(G) sequence and 1/40 for a 3 -GAC(G) sequence (interstrand hole acceptor in parentheses). The origin of this penalty is the reduced electronic coupling between bases in complementary strands. 

Finally we shall derive the equation used by Bixon and Jortner. Suppose that an intramolecular vibrational mode, say Qi, plays a very important role in electron transfer. To this mode, we can apply the strong-coupling approximation (or the short-time approximation). From Eq. (3.40), we have... 

Bixon and Jortner, Rosch and Voityuk, Olofsson and Larsson, Berlin, Burin, Siebbeles, and Ratner, Orlandi and their coworkers [4, 8, 19-31] have explored the energetics and base-base interactions associated with hole and electron transfer in DNA base-pair stacks. They find nearest-neighbor coupling inter-... 

Various theories have been proposed for horizontal transfer at the isoenergetic point. Gouterman considered a condensed system and tried to explain it in the same way as the radiative mechanism. In the radiative transfer, the two energy states are coupled by the photon or the radiation field. In the nonradiative transfer, the coupling is brought about by the phonon field of the crystalline matrix. But this theory is inconsistent with the observation that internal conversion occurs also in individual polyatomic molecules such as benzene. In such cases the medium does not actively participate except as a heat sink. This was taken into consideration in theories proposed by Robinson and Frosch, and Siebrand and has been further improved by Bixon and Jortner for isolated molecules, but the subject is still imperfectly understood. 

The sum in the denominator represents the total coupling width of all the low-/ states to the continuum (or continua). The predictions of Bixon and Jortner are in semiquantitative agreement with the MQDT results presented here for example they would predict a mean lifetime of approximately 18 ns for the long-lived states in the region of n = 90 in the high-field limit. 

In what follows we study the coupling matrix elements between vibronic states following the treatment of Lin,88 and of Bixon and Jortner.8 To begin, we display the coupling matrix elements in terms of intramolecular normal coordinates (see eq. (4-10)) ... 

One of the interesting consequences of eqs. (11-25) and (11-26) is the dependence of the probability of the molecule being in a given nonstationary state on the time correlations in the coupled radiation field. In most experimental studies the radiation field employed consists of a superposition of many frequencies with random phases. It is convenient to represent that form of field in terms of a correlation function d>(t, t"), which is defined in eq. (6-16). Introducing, because of the polychromaticity of the radiation field, the averages of eqs. (11-25) and (11-26), choosing the same representation for the field correlation function as did Bixon and Jortner, and using the conservation of probability, we find for the probability of dissociation of the molecule the relation ... 

Figure 2.4. The energy-gap dependence of the nuclear Franck-Condon factor, which incorporates the role of the high-frequency intramolecular modes. Sc = A/2 is the dimensionless electron-vibration coupling, given in terms which reduce replacement (A) between the minimum of the nuclear potential surfaces of the initial and final electronic states. (Bixon and Jortner, 1999) Reproduced with permission.

In the golden rule approach, developed by Jortner, Bixon and others (Kestner et al., 1974 Ulstrap and Jortner, 1975 Jortner, 1976 Siders and Marcus, 1981a and 1981b Bixon and Jortner, 1982), D and A are treated as weakly coupled but distinct entities and ET as a nonadiabatic radiationless transition between them governed by Fermi s golden rule, which may be written in the form... 

The derivation of this formula, outlined in Appendix 16A, is limited by the assumption that the electronic coupling can be described in a two-state framework. For a discussion of this point and extension to more general situations see the paper by Bixon and Jortner cited at the end of this chapter. 

It is interesting to note the formal similarity, but the conceptual difference, of results (176) and (177) and those derived by Bixon and Jortner (1968) in their pioneering study of radiationless molecular transitions in the statistical limit. They used the previously mentioned uniform model (see Section II,E,l,b), with the assumptions of equal level spacings (5e, = D) and equal interstate couplings (v, = v). If y/D 1, expression (173b) becomes... 

Figure 7.16 Simplified level scheme studied by Bixon and Jortner. The levels , in electronic state B are equally separated with spacing e the level is offset from the photon-excited level Eg by an arbitrary energy a. The coupling matrix element = k is as-...

The criterion that vp I now implies that a critical number of vibrations is required to make an irreversible radiationless relaxation process possible. We may consider a hypothetical case in which the energy gap between the vibrationless electronic states is leV (8066 cm ) that is, state s> decays into a set of final states, -> which have 1 eV of excess vibrational energy. For IC and ISC, typical values of the coupling v may be taken to be 10 and 10 cm S respectively [11]. A table of products vp calculated by Bixon and Jortner for nonlinear molecules with N atoms in which all (3N — 6) vibrational modes oscillate with frequency 1000 cm is shown below the densities of states p were evaluated according to the method of Haarhoff [13] ... 

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