Transition, radiative theory

The following conclusion of the theory (1 ) is extremely important. The radiative transition 2 > Sq in a sandwich dimer is forbidden. In case of a dimer of 04 symmetry, the transition 2 (4Eg) > Sg (A g) is forbidden because of parity. There is no principle difference in the splitting nature of 2 and states for sandwich type dimers with lesser than D4h symmetry and the 2 > Sq transition remains quasi forbidden. This makes it possible to explain low P2 values obtained in (1 ) by a decrease of the 2 > Sg transition radiative probability, i.e., by decreasing or 2 > Sq fluorescence quantum yield in dimeric TTA complexes. In the case of non-sandwich dimer structures with location of subunits in one plane, the So state also is split into two states (high 202y and low 2B3g). However, two radiative transitions S2(B2y)... 

Temperature dependence of the fluorescence quantum yields and fluorescence lifetimes of frans-4,4 -di-fert-butylstilbene in n-hexane and n-tetradecane allowed to define the index of refraction dependence of the radiative rate constants, kf= (3.9 — 1.8) X 10 s, and fluorescence lifetime [78]. This relationship was used to calculate torsional relaxation rate constants ktp> for traws-4,4 -dimethyl- and frans-4,4 -di-ferf-butylstilbene in the n-alkane solvent series. It was found that activation parameters for ktp, based on Eyring s transition state theory, adhered to the medium-enhanced thermodynamic barrier model relationship, AHtp = AHt + aEr, and to the isokinetic relationship. The isokinetic relationship between the activation parameters for the parent frans-stilbene led to an isokinetic temperature of P = 600K and brings it into agreement with the isokinetic temperature for activation parameters based on estimated microviscosities, qp, experienced by stilbene in its torsional motion. The authors concluded that only microviscosities raflier than shear viscosities, q, can be employed in the expression ktp = ktSq — b, when a = b. These data clearly indicated the important role of the media dynamics in the stilbene cis-trans photoisomerization. 

Section BT1.2 provides a brief summary of experimental methods and instmmentation, including definitions of some of the standard measured spectroscopic quantities. Section BT1.3 reviews some of the theory of spectroscopic transitions, especially the relationships between transition moments calculated from wavefiinctions and integrated absorption intensities or radiative rate constants. Because units can be so confusing, numerical factors with their units are included in some of the equations to make them easier to use. Vibrational effects, die Franck-Condon principle and selection mles are also discussed briefly. In the final section, BT1.4. a few applications are mentioned to particular aspects of electronic spectroscopy. 

As seen in the radiationless process, intercombinational radiative transitions can also be affected by spin-orbit interaction. As stated previously, spin-orbit coupling serves to mix singlet and triplet states. Although this mixing is of a highly complex nature, some insight can be gained by first-order perturbation theory. From first-order perturbation theory one can write a total wave function for the triplet state as... 

The master equation approach considers the state of a spur at a given time to be composed of N. particles of species i. While N is a random variable with given upper and lower limits, transitions between states are mediated by binary reaction rates, which may be obtained from bimolecular diffusion theory (Clifford et al, 1987a,b Green et al., 1989a,b, 1991 Pimblott et al., 1991). For a 1-radi-cal spur initially with Ng radicals, the probability PN that it will contain N radicals at time t satisfies the master equation (Clifford et al., 1982a)... 

The polydiacetylene crystals (1-4) most strikingly corroborate these conjectures. Along this line of thought is also shown that this electron-phonon interaction is intimately interwoven with the polymerisation process in these materials and plays a profound role there. We make the conjecture that this occurs through the motion of an unpaired electron in a non-bonding p-orbital dressed with a bending mode and guided by a classical intermolecular mode. Such a polaron type diffusion combined with the theory of non radiative transitions explains the essentials of the spectral characteristics of the materials as well as their polymerisation dynamics. ... 

Here we outline a dynamical description (42) of the polymerisation of the polydiacetylenes. The approach relies much on the one used (43,44) in the theory of non radiative transitions in crystals and the soliton description of the defects in the lD-or-ganic semiconductors. 

Group theory can also be applied to determine whether an optical transition is allowed in a particular optical center. As we showed in Section 5.3, the probability of a radiative transition between two given states, (initial) and (final), is proportional to... 

In the theory of radiative transition, the dipole moment operator p. couples the two electronic energy states and the Franck-Condon overlap... 

Specifically, the collision-induced absorption and emission coefficients for electric-dipole forbidden atomic transitions were calculated for weak radiation fields and photon energies Ha> near the atomic transition frequencies, utilizing the concepts and methods of the traditional theory of line shapes for dipole-allowed transitions. The example of the S-D transition induced by a spherically symmetric perturber (e.g., a rare gas atom) is treated in detail and compared with measurements. The case of the radiative collision, i.e., a collision in which both colliding atoms change their state, was also considered. 

It is very important, in the theory of quantum relaxation processes, to understand how an atomic or molecular excited state is prepared, and to know under what circumstances it is meaningful to consider the time development of such a compound state. It is obvious, but nevertheless important to say, that an atomic or molecular system in a stationary state cannot be induced to make transitions to other states by small terms in the molecular Hamiltonian. A stationary state will undergo transition to other stationary states only by coupling with the radiation field, so that all time-dependent transitions between stationary states are radiative in nature. However, if the system is prepared in a nonstationary state of the total Hamiltonian, nonradiative transitions will occur. Thus, for example, in the theory of molecular predissociation4 it is not justified to prepare the physical system in a pure Born-Oppenheimer bound state and to force transitions to the manifold of continuum dissociative states. If, on the other hand, the excitation process produces the system in a mixed state consisting of a superposition of eigenstates of the total Hamiltonian, a relaxation process will take place. Provided that the absorption line shape is Lorentzian, the relaxation process will follow an exponential decay. 

Arnold et al.24 have calculated radiative lifetimes for the various collision complexes of singlet molecular oxygen on the basis of a collision time of 10"13 sec. The data for wavelengths and transition probabilities are presented in Table III. A recent paper25 describes the theory of double electronic transitions, and gives calculated oscillator strengths for the oxygen systems. 

Values of the radiative rate constant fcr can be estimated from the transition probability. A suggested relationship14 57 is given in equation (25), where nt is the index of refraction of the medium, emission frequency, and gi/ga is the ratio of the degeneracies in the lower and upper states. It is assumed that the absorption and emission spectra are mirror-image-like and that excited state distortion is small. The basic theory is based on a field wave mechanical model whereby emission is stimulated by the dipole field of the molecule itself. Theory, however, has not so far been of much predictive or diagnostic value. 

The results of experimental research have also stimulated the appearance of theoretical papers devoted to the analysis of an elementary act of electron tunneling reactions in terms of the theory of non-radiative electron transitions in condensed media. It has been shown that this theory allows one to explain virtually all the known experimental data on electron tunneling reactions. 

Fig. 15.10 Cross sections as a function of rf field strength for the first four orders of sideband resonances of the K 29s + K 27d radiative collisions in a 4 MHz rf field, (a) The zero-photon resonant collision cross section, (b) the +1 sideband resonance, (c) the —2 sideband resonance, and (d) the +3 sideband resonance. The solid line shows the experimental data, the bold line indicates the prediction the Floquet theory, and the dashed fine is the result of numerical integration of the transition probability (from ref. 17).

First approximation theory leads to certain wave mechanical selection rules on the basis of which a radiative electronic transition may be classified as allowed (high probability) or forbidden (vanishingly low probability). Some forbidden transitions are indeed too weak to observe easily but in actual practice with polyatomic molecules the selection rules often break down sufficiently to permit reasonably strong absorption processes to occur. The following kinds of transition are forbidden... 

Modified thermal (Bates 1983) or phase space (Herbst 1985c) calculations of radiative association rates indicate, as expected, an inverse temperature dependence and a direct dependence on the complexity of the reaction partners. Thus, if theory is to be believed, the importance of radiative association is enhanced by complex molecules reacting in cold clouds. Let us consider two important examples in the synthesis of interstellar methane (Huntress and Mitchell 1979). Although methane can only be observed with difficulty via radioastronomical methods (by centrifugal distortion induced rotational transitions) because it does not possess a permanent dipole moment, its synthesis is an important one because methane is a precursor to more complex hydrocarbons which can be and have been detected. This synthesis can proceed via the following series of normal and radiative association reactions, most of which have been studied in the laboratory ... 

The temperature dependence of non-radiative transitions, caused by linear diagonal and quadratic non-diagonal vibronic interactions, is also investigated on the basis of the non-perturbative quantum theory. It was found that the usual increase of the transition rate with temperature does not hold near some critical values of the non-diagonal interaction and temperature. At these critical values the rate is high (comparable to the mean phonon frequency) and its temperature dependence has a maximum. The results may be important for understanding the mechanisms of catalysis in chemical reactions. 

In this paper, the multiphonon relaxation of a local vibrational mode and the non-radiative electronic transitions in molecular systems and in solids are considered using this non-perturbative theory. Results of model calculations are presented. According to the obtained results, the rate of these processes exhibits a critical behavior it sharply increases near specific (critical) value(s) of the interaction. Also the usual increase of the non-radiative transition rate with temperature is reversed at certain value of the non-diagonal interaction and temperature. For a weak interaction, the results coincide with those of the perturbation theory. 

To sum up, we have developed a general non-perturbative method that allows one to calculate the rate of relaxation processes in conditions when perturbation theory is not applicable. Theories describing non-radiative electronic transitions and multiphonon relaxation of a local mode, caused by a high-order anharmonic interaction have been developed on the basis of this method. In the weak coupling limit the obtained results agree with the predictions of the standard perturbation theory. 

---