Theory of Radiationless Transitions

The probability of radiationless transitions between different states is particularly great when the potential energy surfaces of the corresponding states touch or come at least very close to each other. 

In the framework of the Born-Oppenheimer approximation, radiationless transitions from one surface to another are impossible. (See, e.g., Michl and BonaCit -Koutecky, 1990.) It is therefore necessary to go beyond the Born-Oppenheimer approximation and to include the interaction between different electronic molecular states through the nuclear motion in order to be able to describe such transitions. Using the time-dependent perturbation theory for the rate constant of a transition between a pair of states one arrives at 

In the case of internal conversion between states of equal multiplicity A = is the kinetic energy operator of the nuclei. In the case of weak coupling the matrix element of the perturbation operator can be split into an electronic part and a contribution due to the vibrational terms which, with the help of further simplifying assumptions, can be written as the Franck-Condon overlap integral  

In the case of intersystem crossing transitions between states of different multiplicity, an additional spin-orbit coupling term Ago has to be considered. From the perturbational expansion it follows that the contribution 

In radiationless transitions from the triplet state to the ground state of aromatic hydrocarbons, the excess electronic energy goes predominantly into the CH stretching vibrations. Being of high frequency (i) = 3,000 cm ), they are widely spaced and much smaller vibrational quantum numbers are required than for other normal modes of vibration whose wave numbers are at most half 

From the agreement between theory and experiment, the sys-tematics of excited-state kinetics can be rationalized. Certainly, recent experiments in which the details of the photochemical dynamics have been studied through rotational-state selection provide a great deal of insight into the excited-state kinetics of small molecules , just as high-resolution molecular electronic spectroscopy provides detailed information about the structure, and the various forms of interactions, in the excited states. 

Emission of a photon from an electronically excited state is referred to as luminescence. Fluorescence and phosphorescence can be differentiated depending on whether the transition is between states of equal or different multiplicity and hence spin-allowed or spin-forbidden. (Cf. Section 5.1.1.) Thus, for molecules with singlet ground states fluorescence constitutes a pathway for deactivating excited singlet states whereas phosphorescence is observed in the deactivation of triplet states. 

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In the theory of radiationless transitions as covered in this paper (6,4g), the two non BO terms are treated as perturbations (not externally plied, but arising as imperfections within this model of molecular structure) that can induce transitions between unperturbed states each of which is taken to be a specific Bom-Oppenheimer product state ... 

In Chapters 4 and 5 we made use of the theory of radiationless transitions developed by Robinson and Frosch.(7) In this theory the transition is considered to be due to a time-dependent intramolecular perturbation on non-stationary Bom-Oppenheimer states. Henry and Kasha(8) and Jortner and co-workers(9-12) have pointed out that the Bom-Oppenheimer (BO) approximation is only valid if the energy difference between the BO states is large relative to the vibronic matrix element connecting these states. When there are near-degenerate or degenerate zeroth-order vibronic states belonging to different configurations the BO approximation fails. 

Instead of the quantity given by Eq. (15), the quantity given by Eq. (10) was treated as the activation energy of the process in the earlier papers on the quantum mechanical theory of electron transfer reactions. This difference between the results of the quantum mechanical theory of radiationless transitions and those obtained by the methods of nonequilibrium thermodynamics has also been noted in Ref. 9. The results of the quantum mechanical theory were obtained in the harmonic oscillator model, and Eqs. (9) and (10) are valid only if the vibrations of the oscillators are classical and their frequencies are unchanged in the course of the electron transition (i.e., (o k = w[). It might seem that, in this case, the energy of the transition and the free energy of the transition are equal to each other. However, we have to remember that for the solvent, the oscillators are the effective ones and the parameters of the system Hamiltonian related to the dielectric properties of the medium depend on the temperature. Therefore, the problem of the relationship between the results obtained by the two methods mentioned above deserves to be discussed. 

Internal conversion refers to radiationless transition between states of the same multiplicity, whereas intersystem crossing refers to such transitions between states of different multiplicities. The difference between the electronic energies is vested as the vibrational energy of the lower state. In the liquid phase, the vibrational energy may be quickly degraded into heat by collision, and in any phase, the differential energy is shared in a polyatomic molecule among various modes of vibration. The theory of radiationless transitions developed by Robinson and Frosch (1963) stresses the Franck-Condon factor. Jortner et al. (1969) have extensively reviewed the situation from the photochemical viewpoint. 

Hamiltonians expressed in matrix forms have been extensively employed in the theory of radiationless transitions of electronically excited states of larger molecules (Bixon and Jortner, 1968 Schlag et al., 1971 Freed, 1972 Nitzan et al., 1972 Avouris et al., 1977 Jortner and Levine, 1981 Felker and Zewail, 1988 Seel and Domcke, 1991). 

The theory of radiationless transition considers the transition to occur in two steps (i) horizontal transition from one energy state to the other at the isocnergetic point, for the two combining states and (ii) vibrational relaxation of the lower energy state. The step (i) is the rate determining step. The rate constant is given by the theory of Robinson and Frosch as,... 

In order to apply the general considerations we have outlined to the theory of radiationless transitions it is necessary to consider ... 

Freed and Jortner226 have reworked the formal theory of radiationless transitions described in this paper. They carefully account for the difference between distinguishable and indistinguishable levels, and allow for variable coupling of the sparse system to the dense system of states. Of course, only certain vibrational modes in the dense manifold have the appropriate symmetries to couple to the sparse manifold and thereby contribute to the radiationless transition. Freed and Jortner take this into account in the fashion in which the zero-order manifolds of the molecule are classified. 

Since the writing of this review, Englman and Jortner have presented a new formulation of the theory of radiationless transitions.227 Their treatment rests on the assumptions that the molecular vibrations are harmonic and that the normal modes and their frequencies are the same in the initial and final states, except for displacements in the origins. They consider the nonradiative transition rate in the conventional form... 

Let us assume the availability of a useful body of quantitative data for rates of decay of excited states to give new species. How do we generalize this information in terms of chemical structure so as to gain some predictive insight For reasons explained earlier, I prefer to look to the theory of radiationless transitions, rather than to the theory of thermal rate processes, for inspiration. Radiationless decay has been discussed recently by a number of authors.16-22 In this volume, Jortner, Rice, and Hochstrasser 23 have presented a detailed theoretical analysis of the problem, with special attention to the consequences of the failure of the Born-Oppenheimer approximation. They arrive at a number of conclusions with which I concur. Perhaps the most important is, "... the theory of photochemical processes outlined is at a preliminary stage of development. Extension of that theory should be of both conceptual and practical value. The term electronic relaxation has been applied to the process of radiationless decay. 

Electron transfer from the excited states of Fe(II) to the H30 f cation in aqueous solutions of H2S04 which results in the formation of Fe(III) and of H atoms has been studied by Korolev and Bazhin [36, 37]. The quantum yield of the formation of Fe(III) in 5.5 M H2S04 at 77 K has been found to be only two times smaller than at room temperature. Photo-oxidation of Fe(II) is also observed at 4.2 K. The actual very weak dependence of the efficiency of Fe(II) photo-oxidation on temperature points to the tunneling mechanism of this process [36, 37]. Bazhin and Korolev [38], have made a detailed theoretical analysis in terms of the theory of radiationless transitions of the mechanism of electron transfer from the excited ions Fe(II) to H30 1 in solutions. In this work a simple way is suggested for an a priori estimation of the maximum possible distance, RmSiX, of tunneling between a donor and an acceptor in solid matrices. This method is based on taking into account the dependence... 

Comparison of the Marcus Model with the Theory of Radiationless Transitions Is the Marcus Inverted Region Related to the Energy Gap Law .126... 

The Marcus Inverted Region (MIR) is that part of the function of rate constant versus free energy where a chemical reaction becomes slower as it becomes more exothermic. It has been observed in many thermal electron transfer processes such as neutralization of ion pairs, but not for photoinduced charge separation between neutral molecules. The reasons for this discrepancy have been the object of much controversy in recent years, and the present article gives a critical summary of the theoretical basis of the MIR as well as of the explanations proposed for its absence in photoinduced electron transfer. The role of the solvent receives special attention, notably in view of the possible effects of dielectric saturation in the field of ions. The relationship between the MIR and the theories of radiationless transitions is a topic of current development, although in the Marcus-Hush Model electron transfer is treated as a thermally activated process. 

There are however essential differences between the Marcus model and the theory of radiationless transitions. In the former, the decrease of the rate constant in the inverted region results from an activation barrier which must be overcome by thermal energy, whereas the rates of radiationless transitions are in principle temperature independent. As implied in [14], there is no normal region in the case of nonradiative transitions, a no bell-shaped curve is expected from the plot of the rate constant against the energy gap. 

The temperature dependence of the rates of e.t. processes is therefore the obvious criterion for a choice between the Marcus model and a model of e.t. as a type of radiationless transition. In this respect, it must, however, be noted that an apparent temperature effect may appear in the theories of radiationless transitions, if the... 

The above issues are addressed in the book because of numerous newly published results since the publication of the well-known surveys on the theory of radiationless transitions. Therefore, along with a description of the traditional approaches to the theory of radiationless transitions (Chapter 2), these effects are discussed starting from the pioneering to contemporary studies (Chapter 3). 

The theory of radiationless transitions (or electronic relaxation) based on the BOA approximation as a basis set was originally proposed by Huang and Rhys [29] and applied to color centers and later modified and extended by Lin and Bersohn [30] to molecular systems in photochemistry and photophysics. Notice that for the IC a b, the IC rate constant is given by... 

The details of the mechanism of decay of states in alkanes retain their interest. The effect of deuterium on fluorescence lifetimes has been discussed in terms of the theory of radiationless transitions. Analysis of fluorescence line shapes and Raman excitation profiles of tetradesmethyl-p-carotene in isopentane has been carried out at 190 and 230K . Solvation occurs over a time scale of about 100 fs whilst vibrational relaxation has a time scale of about 250 fs. The kinetics of the interaction of alcohols with the excited state of triethylamine shows involvement of a charge transfer exciplex . Ionizing radiation is a means of exciting saturated hydrocarbons and the complexity of three component systems containing saturated hydrocarbons, aromatic solvent, and fluorescent solute has been examined. ... 

Rice and co-workershave proposed a different approach to vibrational relaxation in liquids based on analogies with the theory of radiationless transitions. Kushick and Rice considered the case in which an initially excited vibrational level 1) is coupled to a continuum 2, c) where 2> is a second vibrational state and )> is a continuous variable representing solvent states whose translational energy has been increased by . The coupling is chosen, for mathematical convenience, to be the square root of a Lorentzian function of e ... 

Thus where for the overall reaction is known (it being zero for the symmetrical complex just mentioned), the reorganization energy A can be calculated and thence the frequency factor A. In the theory of radiationless transitions, A is identified with the frequency of the vibrational mode which causes the electron transfer thus A x 10 s . In the transition state theory, equation (40) is replaced by... 

When the fluorescence quantum yield of a highly excited valence state is small, as is usual, the hfetime of the excited state is determined primarily by the rates of its radiationless transitions. The theory of radiationless transitions in the excited electronic states of polyatomic molecules was developed several decades ago and is not reviewed in detail. Readers are referred to several excellent comprehensive reviews for further information [9,10]. Here we provide only those materials needed to understand the photophysics of the highly excited electronic states of those polyatomic molecules that exhibit unusually long lifetimes. 

Theory of radiationless transitions in an isolated molecule. /. Chem. Phys., 49, 610. 

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