Molecular eigenstates decay

The initial state of the excited system has been represented as a superposition of the (time-independent) molecular eigenstates, each of which is a superposition of BO basis functions. The decay process is then described in terms of the time evolution of the amplitudes of the molecular eigenstates. The general theory of quantum mechanics implies that the decay of the state (10-4) will exhibit interference effects. 

From theoretical discussions involving the molecular eigenstates picture questions have arisen as to whether particular quantum mechanical interference effects can be observed by the use of suitably monochromatic radiation for excitation of the molecules 13>. (See Sect. 7.) Of course, it is also necessary to settle the controversies as to whether the BO or molecular eigenstates are correct, and if the former is indeed correct, which particular version of the BO approximation is to be employed for the calculation of nonradiative decay rates. 

Fig. 4. (a) A schematic representation of the small molecule limit. The states are the same as those represented in Fig. 1. The molecular eigenstates approximately diagonalize the effective molecular Hamiltonian (20), and each carries only a portion of the original oscillator strength to s can therefore decay radiatively to

Under these conditions the singlet amplitude is distributed according to a Lorentzian distribution over the molecular eigenstates. Exciting with a broad (white) laser (or at least with a laser that completely spans the interaction width), one then sees in the fluorescence first the Fourier transform of the Lorentzian distribution, that is, an exponential decay. The density of /c> was, however, not taken to be so high as to dilute the singlet amplitude effectively to zero. It was taken to be intermediate, which meant that each ME still had enough radiative probability so as to radiate independently,... 

Chopping their CW intracavity doubled ring laser with an electrooptical modulator van Herpen et al.9 obtained 40 nsec pulses with a spectral width of about 25 MHz. This allowed them to study the decay of a single molecular eigenstate. Figure 6 shows a typical result of such an experiment. As might... 

In the absence of accidental degeneracies among the molecular eigenstates, // , (2.5) or (2.6) are the appropriate solutions of the Schrodinger equation for the isolated molecular system in a real world where spontaneous emission processes are admissible. Assuming that the and < >, do not radiate to any set of common levels, it is possible to evaluate the radiative decay rates of the molecular eigenstates, as... 

This short time apparent decay is a particularly interesting intramolecular dephasing process, which merely accounts for the fact that the individual molecular eigenstates in the summation in (2.21) all have slightly different energies, . When the molecule is initially prepared in the nonstationary state or with some other set of a ,, there is a coherent superposition of all of these molecular eigenstates with relative phases which are fixed. However, because of the differences between the energies, , these phases become different at a subsequent time, the exp[ —... 

It should be stressed that the wave-packet picture of photophysical relaxation and photochemical reaction dynamics described in this chapter is substantially different from the traditional concepts in this area. In contrast to the established picture of radiationless transitions in terms of interacting tiers of zero-order molecular eigenstates, the dynamics is rationalized in terms of local properties of PE surfaces such as slopes, barriers and surface intersections, a view which now becomes widely accepted in photochemistry. This picture is firmly based on ah initio electronic-structure theory, and the molecular relaxation d3mamics is described on the basis of quantum mechanics, replacing previously prevaUing kinetic models of electronic decay processes. Such a more detailed and rigorous description of elementary photochemical processes appears timely in view of the rich and specific information on ultrafast chemical processes which is provided by modern time-resolved spectroscopy. " ... 

Figure 6.2 Schematic picture of the effective manifold. Diagonalization of the effective Hamiltonian. The zero-order states s) and /) Hamiltonian results in a set of independently are characterized by the energies Ej and E/, decaying generalized molecular eigenstates respectively, and bythe decay widths Yj and y . j), characterized by energies and decay...

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. 

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. 

The experimentally achievable localized excitations are typically described by one of the zero-order basis states (see Section 3.2), which are eigenstates of a part of the total molecular Hamiltonian. Localization can be in a part of the molecule or, more abstractly, in state space . The localized excitations are often described by extremely bad quantum numbers. The evolution of initially localized excitations is often more complex and fascinating than an exponential decay into a nondescript bath or continuum in which all memory of the nature of the initial excitation is monotonically lost. The terms in the effective Hamiltonian that give birth to esoteric details of a spectrum, such as fine structure, lambda doubling, quantum interference effects (both lineshapes and transition intensity patterns), and spectroscopic perturbations, are the factors that control the evolution of an initially localized excitation. These factors convey causality and mechanism rather than mere spectral complexity. 

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