Radiationless transitions Born-Oppenheimer

Understand that intermolecular radiationless transitions of excited states are caused by a breakdown of the Born-Oppenheimer approximation. 

The interest aroused by the field of radiationless transitions in recent years has been enormous, and several reviews have been published 72-74) Basically, the ideas of Robinson and Frosch 75) who used the concepts on non-stationary molecular states and time-dependent perturbation theory to calculate the rate of transitions between Born-Oppenheimer states, are still valid, although they have been extended and refined. The nuclear kinetic energy leads to an interaction between different Born-Oppenheimer states and the rate of radiationless transitions is given by... 

Prepared State. Here the Hamiltonian H is the time-independent molecular Hamiltonian. Both H0 and T are time independent. The initial prepared state is an eigenket to H0 and thus is nonstationary with respect to H = H0 + T. One example is provided by considering H0 as the spin-free Hamiltonian 77sp and the perturbation T as a spin interaction. A second example is provided by considering H0 as the spin-free Born-Oppenheimer Hamiltonian and T as a spin-free nonadiabatic perturbation. In the first example spin-free symmetry is not conserved but double-point group symmetry may be. In the second example point-group symmetry is not conserved, but spin-free symmetry is. The initial prepared state arises from some other time-dependent process as, for example, radiative absorption which occurs at a rate very much faster than the rate at which our prepared state evolves. Mechanisms for radiationless transitions in excited benzene may involve such prepared states, as is discussed in Section XI. 

Finally, Freed and Jortner discuss, in general terms, the influence of external perturbations on radiationless processes. They show under what conditions the external perturbation has either no effect, a small, or a large effect on the radiationless transitions in the statistical, intermediate, and resonance coupling limits, respectively. An interesting aspect of their analysis is the demonstration that the widely used Born-Oppenheimer and molecular eigenstate basis sets provide complimentary pictures, and hence are completely equivalent. 

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. 

The Born-Oppenheimer (BO) description is not exact. The deviation from the BO approximation can be treated as an additional nonadiabatic interaction. This interaction does not depend on time and can be the origin of radiationless transitions. Moreover, the nonadiabatic interaction is a main mechanism for one kind of indirect photodissociation, namely, photopredissociation of Type I (electronic predissociation). 

Chapter 3 describes radiationless transitions in the tunneling electron transfers in multi-electron systems. The following are examined within the framework of electron Green s function approach the dependence on distance, the influence of crystalline media, and the effect of intermediate particles on the tunneling transfer. It is demonstrated that the Born-Oppenheimer approximation for the wave function is invalid for longdistance tunneling. 

This section briefly introduces the generalized coupled master equation within the Born-Oppenheimer adiabatic (BOA) approximation. In this case, the non-adiabatic processes are treated as the vibronic transitions between the vibronic manifolds. Three types of the rate constant are then introduced to specify the nature of the transitions depending on whether the electronically excited molecular system achieves its vibrational thermal equilibrium or not. The radiationless transitions can occur between two... 

Diabatic photoreaction Within the Born-Oppenheimer approximation, a reaction beginning on one excited state potential-energy surface and ending, as a result of radiationless transition, on another surface, usually that of the ground state. Also called non-adiabatic. 

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... 

Fig. 1. The molecular energy level model used to discuss radiationless transitions in polyatomic molecules. 0O, s, and S0,S are vibronic components of the ground, an excited, and a third electronic state, respectively, in the Born-Oppenheimer approximation. 0S and 0 and 0j are assumed to be allowed, while transitions between j0,j and the thermally accessible 00 are assumed to be forbidden. The f 0n are the molecular eigenstates...

Radiationless transitions (IC and ISC) represent a conversion of electronic energy of an initial, excited state to vibrational energy in a lower-energy electronic state (cf. Figure 2.1). Within the Born Oppenheimer approximation (Section 1.3), radiationless... 

In Chapters 4 and 5 we made use of the theory of radiationless transitions developed by Robinson and Frosch." In this theory the transition is considered to be due to a time-dependent intramolecular perturbation on non-stationary Born-Oppenheimer states. Henry and Kasha > and Jortner and co-workers< > have pointed out that the Born-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. 

There are molecules which, in the Born-Oppenheimer approximation, can be bound only in excited states, such as the "excimer" molecules (HeH), (NeH), (ArH), and (HeF). These excited bound states decay to the dissociative continuum of the ground state either via radiative transitions or via radiationless transitions, each having its characteristic probability. Suppose we look at the collision of the (ground repulsive state) free atoms in the... 

The population probabilities Pn t) defined in Eqs. (8)-(13) should not be confused with the population probabilities which have been considered in the extensive earlier literature on radiationless transitions in polyatomic molecules, see Refs. 28 and 29 for reviews. There the population of a single bright (i.e. optically accessible from the electronic ground state) zero-order Born-Oppenheimer (BO) level is considered. Here, in contrast, we define the electronic population as the sum of all vibrational level populations within a given (diabatic or adiabatic) electronic state. These different definitions are adapted to different regimes of time scales of the system dynamics. If nonadiabatic interactions are relatively weak, and radiationless transitions relatively slow, the concept of zero-order BO levels is useful the populations of these levels can be prepared and probed using suitable laser pulses (typically of nanosecond duration). If the nonadiabatic transitions occur on femtosecond time scales, the preparation of individual zero-order BO levels is no longer possible. The total population of an electronic state then becomes the appropriate concept for the interpretation of time-resolved experiments. ° ... 

The Born-Oppenheimer approximation is generally a good one and is used in most molecular problems. However, it breaks down in many cases such as in avoided crossing regions, where transitions between potential energy surfaces can occur. The neglected interactions in this approximation are responsible for many physical processes of interest e.g. predissociation, coilisions or radiationless transitions. It is still possible to use this approximation if we take into account the neglected terms. The first theoretical calculation of the adiabatic correction was performed for the hJ molecule [45] in 1941. For the heteronuclear molecules, the first theoretical calculation of the adiabatic correction for the HeH+ molecuie [30,31] with two eiectrons, then for more compiex systems such as LiH, KH, RbH and CsH... 

Chapter 6 deals with the time evolution of radiative decaying states of polyatomic molecules with special emphasis on radiative decay in internal conversion. The decay of a manifold of closely spaced coupled states is handled by the Green s function formalism, where the matrix elements are displayed in an energy representation that involves either the Born-Oppenheimer or the molecular eigenstate basis set. The features of radiationless transitions in large, medium-sized, and small molecules are elucidated, deriving general expressions for the radiative decay times and for the fluorescence quantum yields. 

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