Vibronic manifolds

We consider a general dissipative environment, using a three-manifold model, consisting of an initial ( ), a resonant ( r ), and a final ( / ) manifold to describe the system. One specific example of interest is an interface system, where the initial states are the occupied states of a metal or a semiconductor, the intermediate (resonance) states are unoccupied surface states, and the final (product) states are free electron states above the photoemission threshold. Another example is gas cell atomic or molecular problems, where the initial, resonant, and final manifolds represent vibronic manifolds of the ground, an excited, and an ionic electronic state, respectively. 

These two processes can be treated on equal footing. A schematic representation of vibronic levels of an electron transfer system is shown in Fig. 10. In other words, we are dealing with two vibronic manifolds, iv for D A and / for D+A or DA. For convenience, hereafter we shall use the following notation ... 

To simplify the expression of the thermal average rate W, f given by Eq. (3.32), we shall assume that both potential energy surfaces of the two excited vibronic manifolds of the DA system consist of a collection of harmonic oscillators, that is,... 

We consider a model for the pump-probe stimulated emission measurement in which a pumping laser pulse excites molecules in a ground vibronic manifold g to an excited vibronic manifold 11 and a probing pulse applied to the system after the excitation. The probing laser induces stimulated emission in which transitions from the manifold 11 to the ground-state manifold m take place. We assume that there is no overlap between the two optical processes and that they are separated by a time interval x. On the basis of the perturbative density operator method, we can derive an expression for the time-resolved profiles, which are associated with the imaginary part of the transient linear susceptibility, that is,... 

A vibronic coupling model for mixed-valence systems has been developed over the last few years (1-5). The model, which is exactly soluble, has been used to calculate intervalence band contours (1, 3, 4, 5), electron transfer rates (4, 5, 6) and Raman spectra (5, 7, 8), and the relation of the model to earlier theoretical work has been discussed in detail (3-5). As formulated to date, the model is "one dimensional (or one-mode). That is, effectively only a single vibrational coordinate is used in discussing the complete ground vibronic manifold of the system. This is a severe limitation which, among other things, prevents an explicit treatment of solvent effects which are... 

A complete set of vibronic energies (E ) and eigenfunctions ( r) > the ground vibronic manifold, are associated with the potential surfaces of eq 12. These are obtained as solutions... 

If the system under consideration possesses non-adiabatic electronic couplings within the excited-state vibronic manifold, the latter approach no longer is applicable. Recently, we have developed a simple model which allows for the explicit calculation of RF s for electronically nonadiabatic systems coupled to a heat bath [2]. The model is based on a phenomenological dissipation ansatz which describes the major bath-induced relaxation processes excited-state population decay, optical dephasing, and vibrational relaxation. The model has been applied for the calculation of the time and frequency gated spontaneous emission spectra for model nonadiabatic electron-transfer systems. The predictions of the model have been tested against more accurate calculations performed within the Redfield formalism [2]. It is natural, therefore, to extend this... 

In the treatment of radiationless transitions presented above, we have mainly considered the case of a closed channel decaying into a single open channel, which latter consists of the dense vibronic manifold of some one electronic state (statistical limit). That description is obviously incomplete, since both radiative and nonradiative decay processes occur simultaneously. Clearly, a complete theoretical description of the radiationless transition... 

Next, the calculation of molecular absorption spectra in dense media is presented. For the transition between the two vibronic manifolds ar -> bu, Eq. (4.34) becomes [41,42]... 

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... 

Here a model for the pump-probe time-resolved measurement of a system with two vibronic manifolds and m embedded in a heat bath is considered. The two vibronic manifolds are coupled by the interaction H. ... 

A pumping laser excites the system from the ground vibronic manifold g to the excited vibronic manifold n. After excitation, a probing laser is applied to induce transitions from the manifold to the manifold g via stimulated emission and/or to higher excited manifolds via induced absorption. This work shall focus on the pump-probe time-resolved stimulated emission experiment. In this case, an expression for the time-resolved profiles is derived in terms of the imaginary part of the transient susceptibility X (copu,copr, x). In the adiabatic approximation and the Condon approximation, it has been shown that [18,21]... 

Figure 6.1 shows simulation of the population dynamics of the two vibronic manifolds. The populations pbv>b and pcw>cw(T) after excitation are calculated for (a) the weak coupling case and (b) the strong coupling case. Figure 6.1 clearly shows the population transfer between the two electronic states due to the creation of the vibronic coherence. 

Figure 6. Schematic molecular level structure for electron transfer processes in an isolated molecule. Excitation So(D-A) — S2[(D-A) ] selects the vibronic level(s), which undergo(es) intramolecular charge separation (denoted by horizontal arrow) to the Si(D+-A ) vibronic manifold quasidegenerate with it. Excitation So — Si selects the vibronic levels of the charge-transfer singlet state, which undergo intramolecular charge recombination (denoted by a horizontal arrow) to the ground-state vibronic manifold. Radiative electron transfer exemphfied by the CT fluorescence is labeled with a broken arrow. Adapted from Refs. [103a-d].

Fig. 18.1 A dressed-state model that is used in the text to describe absorption, emission, and elastic (Rayleigh) and inelastic (Raman) light scattering. g) and. v> represent particular vibronic levels associated with the lower (1) and upper (2) electronic states, respectively. These are levels associated with the nuclear potential surfaces of electronic states 1 and 2 (schematically represented hy the parabolas). Rj are radiative continua— 1 -photon-dressed vibronic levels of the lower electronic states. The quasi-continuum L represents a nonradiative channel—the high-energy regime of the vibronic manifold of electronic state 1. Note that the molecular dipole operator /t couples ground (g) and excited (s) molecular states, but the ensuing process occurs between quasi-degenerate dressed states g,k and 5,0).

Problem 18.2. A well-known result from the theory of optical absorption lineshapes is that the integrated lineshape associated with the transition between two quantum levels is equal, up to known numerical factors, to the squared radiative coupling element between these levels. For example, using Eq. (18.9) or (18.10) yields / dcoLlai ) o< /zi,2l. Show that, under the Condon approximation, the integrated absorption lineshape of an overall transition between two vibronic manifolds of two electronic states 1 and 2 is also proportional to the squared radiative electronic coupling l/xp2p. 

We need one more extension of Eq. (6). It may well be that both the states s> and k) are in their turn coupled to different states of a manifold /> (see Fig. 1). For instance, if s> is an excited singlet state S, fc> could denote the vibronic manifold of the electronic triplet state T, and /> would then be the vibronic manifold of the ground state S0. 

The picosecond kinetics of tetracene dianions have been studied using a new extension of picosecond spectroscopy methods.100 The rise times of the Stokes-shifted fluorescence from rhodamine B, rhodamine 6G, and erythrosine dissolved in water have been investigated using picosecond techniques. Figure 4 schematically indicates the situation following excitation. The best fit to the data corresponds to a relaxation time within the vibronic manifold of S of <1 ps.101 Although these fast spectroscopic techniques provide direct means of examining the behaviour of short-lived species, indirect methods are more convenient and often quite successful. Such is the case for the determination of rf from calculated radiative rate constants and measured Of values for a series of cyanine dyes.102... 

The excited state Hamiltonian for the generalized vibronic manifold of N electronic excited states and M vibrational modes can be written as... 

The reaction center of Rps. viridis is a chromophore-protein complex. Four bacteriochlorophylls and two bacteriopheophytins are anchored in the protein matrix of three pol)q)eptides. The Qy vibronic manifolds of the six chromophores are of major interest in our investigation on spectroscopy and electron transfer dynamics of the photo reaction center. 

The laser functions according to a four-level scheme similar in nature to that of a (molecular) dye laser. Excitation is from the lowest ro-vibronic level in the electronic ground state 2 into the E excited-state manifold. Fast relaxation to the lowest ro-vibronic level in this state occurs, where population inversion accumulates due to the long lifetime of that level of 3.5 ps. After radiative transitions into the high part of the ro-vibronic manifold of 2,... 

To further investigate the structure of the vibronic manifold of the molecule dressed by the control field at energies close to the onset of the S2 state, we have calculated the absorption spectrum of the molecule in this energy range as a function of the intensity of the control field. Specifically, the time-dependent Schrbdinger equation (TDSE)... 

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