Displaced Potential Surfaces

In this section, we provide an alternative derivation of the nonradiative decay rate for a statistically large molecule under the simplifying assumption that the normal modes are parallel and their frequencies are the same in the two electronic states 

Taking into account that z l = e Equation 4.99 becomes 

Taking the explicit expression (3.75) for the single-promoting mode generating function, the expression for the nonradiative decay rate in the simple displaced potential surface model can be written as 

Under conditions that will be given later. Equation 4.104 is essentially independent of the vibrational relaxation width y and therefore valid when y = 0. Therefore, in writing Equation 4.104 we have made use of the integral representation of the 6-function (3.78) to insure conservation of energy. 

As follows, we will treat Equation 4.101 in the strong coupling limit 

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Eq. (4.129b) has the form of the expression (4.99) for the generating functions for electronic relaxation between two vertically displaced potential surfaces, except with replaced by —Wa, by —z and Aj by — and exhibits the same t dependence as the latter. We emphasize, however, that the effect of replacing the AJ by — A changes the temperature dependence of w r manifested through dramatically in comparison to the expected for electronic relaxation processes [130]. Now, expression (4.129a) is expanded in terms of the associated MID for different vibrational occupation numbers ni,W2, , Un> Si according to (4.79). This may be... 

The anharmonic modes for both the a symmetric and 67 asymmetric CH stretching vibrations have been explored. In order to perform a reasonable anharmonic treatment, we had to take into account the stretching of the bonds to larger elongations than for the harmonic description where displacements can be confined close to the equilibrium geometry. Consequently, correlation effects were included in the determination of the potential surface. The electronic calculations were carried out at the MP2 level, which insures a good description of the CH bond potential towards dissociation. A double zeta... 

Fig. 9a-c. Relative positions of LS and HS potential energy surfaces for complexes showing spin-state equilibria associated with different amounts of geometric reorganization a no intersection of potential surfaces b intersection accompanied by moderate displacement of the minima of potential surfaces c (avoided) interseetion accompanied by sizeable displacement of the minima of potential surfaces. AEq = AG° is the difference of zero-point energies of LS and HS states, E = AG h and jlj... 

The plasma potential is the maximum value with which ions can be accelerated from the edge of the sheath towards the substrate, located at the grounded electrode. This may cause ion bombardment, which may induce ion-surface interactions such as enhancement of adatom diffusion, displacement of surface atoms, trapping or sticking of incident ions, sputtering, and implantation see Section 1.6.2.1. 

Because we are concerned only with the analysis of the absorption spectra of P band and B band, we consider the excitonic interactions among P, BL, and BM shown in Fig. 8. Here (oti, ot2,0C3,014) represent the diagonal matrix elements, while (p, (314, P23, P34) represent the off-diagonal matrix elements [67]. As shown in Introduction, a main feature of the P band is that its absorption maximum shows a pronounced temperature shift [42,52], According to the displaced oscillator model, the absorption maximum is independent of T. Although the distortion effect of potential surfaces will introduce some temperature shift, the effect cannot be as large as that shown in Fig. 2. 

Now we are in a position to calculate the absorption coefficient with displaced harmonic potential surfaces using Eq. (2.38). Substituting the... 

Next we shall discuss how to theoretically construct the three-dimensional fs time-resolved spectra. For this purpose, recent fs time-resolved spectra reported by Scherer s group for Rb. sphaeroides R26 with A,cxcltatlon = 800 nm at room temperature are shown in Fig. 18 is considered [39]. They have used a laser pulse of 30 fs. To theoretically construct the fs time-resolved spectra, we need the potential surfaces for displaced surfaces we need the vibrational frequencies go, and their Huang-Rhys factors S,. For bacterial photosynthetic RCs, these physical constants are given in Table I. In addition to the potential surfaces, we need interactions between different electronic states which are shown in... 

In the framework of DECP, the first pump pulse establishes a new potential surface, on which the nuclei start to move toward the new equilibrium. The nuclei gain momentum and reach the classical turning points of their motion at t = nT and t = (n + l/2)T. The second pump pulse then shifts the equilibrium position, either away from (Fig. 3.10b) or to the current position of the nuclei (Fig. 3.10c). The latter leads to a halt of the nuclear motion. Because photo-excitation of additional electrons can only shift the equilibrium position further in the same direction, the vibrations can only be stopped at their maximum displacement [32]. 

Thus we conclude that the potential surfaces involve large displacements in specific normal coordinates of the molecule, not small displacements on many coordinates. 

The decrease in < 0 < (t) > depends on the slope of the potential surface at the point at which the wavepacket is initially placed. The slope of the potential in the Qy direction is steeper on the positive Qx side of the surface than on the negative Qx side. When the slope in the Qy dimension is large, the Qy part of the two-dimensional wavepacket will rapidly change its shape and < 010 (t) > will decrease rapidly. Therefore, for a positive Qx displacement in the coupled potential, the autocorrelation function will decrease more rapidly than it would for a negative displacement. 

The physical insight obtained from the time-domain point of view allows simple qualitative predictions about the widths of the progressions to be made. Two potential surfaces (such as the coupled and uncoupled surfaces in Fig. 2) can be compared in terms of the slope of steepest descent at the position of the initial wavepacket at t = 0. The steeper the slope, the faster the initial decrease of < (j) 10(t) > in the time domain and the broader the spectrum in the frequency domain. This insight is important in developing a strategy to fit experimental spectra. In the example above, a vibronic progression in an experimental spectrum that is broader than that expected from a harmonic potential surface (Fig. 3a) requires that the wavepacket be displaced in the positive Qx direction in the coupled potential surface. 

Fig. 4. Autocorrelation functions plotted versus time for the spectra described in Fig. 3. The autocorrelation function shown by the middle curve corresponds to the reference spectrum (Fig. 3a), k.j, = Ocm-1, the autocorrelation function shown by the lowest curve corresponds to a positive displacement in the quadratically coupled potential surface, and the autocorrelation function shown by the top curve corresponds to a negative displacement in the quadratically coupled potential surface...

A simple starting point for fitting an experimental spectrum is the use of uncoupled harmonic potential surfaces. The theoretical spectra are calculated by using Eqs. (2, 6, 7 and 9). The adjustable parameters in the fitting procedure are the displacements A along the normal coordinates and the force constants kx, ky> and kxy. The k s defining the potential surface are obtained from the emission spectrum and/or from vibrational spectra. The values of E00 and T that appear in Eq. (2) are obtained from the experimental spectrum. 

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