Franck-Condon classical

This model permits one to immediately relate the bath frequency spectrum to the rate-constant temperature dependence. For the classical bath (PhoOc < 1) the Franck-Condon factor is proportional to exp( —with the reorganization energy equal to... 

The height of the potential barrier separating the initial and final states of the nuclear subsystem decreases and, hence, the Franck-Condon factor increases (Fig. 6). In the classical limit, this results in a decrease of the activation free energy. 

Figure 9. Schematic reprsentation of a classical trajectory moving on the Si and So energy surfaces of the H2—(CH) -NHt trans cis photoisomerization, starting near the planar Franck-Condon geometry. The geometric coordinates are (a) torsion of the C2—C3 and C3 C4 bonds and (b) asymmetric stretching coupled with pyramidalization. Both Si and So intersect at a conical intersection (Si/S0 Cl) located near the minimum of the Si surface (Min-C ) where the C2C3C4N5 torsion angle is 104°. [Reproduced with permission from [87], Copyright 2000 Amercian Chemical Society],...

Box 2.3 Classical and quantum mechanical description of the Franck-Condon principle31... 

We next consider the expression for k in the classical formalism. According to the Franck-Condon principle, internuclear distances and nuclear velocities do not change during the actual electron transfer. This requirement is incorporated into the classical electron-transfer theories by postulating that the electron transfer occurs at the intersection of two potential energy surfaces, one for the reactants... 

An additional concern arises in regard to any differences which may exist between the classical theory and the quantum-mechanical approach in the calculation of the Franck-Condon factors for symmetrical exchange reactions. In fact, the difference is not very large. For a frequency of 400 cm for metal-ligand totally symmetric vibrational modes, one can expect... 

Fig. 12.2 Left The ground (X, solid line), excited (6, dashed line) and dissociative [a1g(3II), dotted line] electronic state potentials of the iodine molecule. The arrow indicates the electronic excitation. The initial excited wave packet is located in the Franck-Condon region near to the inner classical turning point of the B state. The transition from the B to the a state is forbidden by symmetry in the isolated molecule but becomes allowed when the molecule is placed in a solvent.

A semiclassical description is well established when both the Hamilton operator of the system and the quantity to be calculated have a well-defined classical analog. For example, there exist several semiclassical methods for calculating the vibrational autocorrelation function on a single excited electronic surface, the Fourier transform of which yields the Franck-Condon spectmm [108, 109, 150, 244]. In particular, semiclassical methods based on the initial-value representation of the semiclassical propagator [104-111, 245-248], which circumvent the cumbersome root-search problem in boundary-value-based semiclassical methods, have been successfully applied to a variety of systems (see, for example, Refs. 110, 111, 161, and 249 and references therein). The mapping procedure introduced in Section VI results in a quantum-mechanical Hamiltonian with a well-defined classical limit, and therefore it... 

First of all, consider the case when all normal vibrations are classical. This takes place if the condition a)k -4 T works well for all frequencies. In the classical case the probability of tunneling can be calculated with the help of the general formula (18) using the Franck-Condon approximation and the well-known [10] properties of quasi-classical wave functions. We will not dwell upon the details of transition from the quantum description to the... 

When the quasi-diatomic Franck-Condon model was compared with the experimental results it was found that it could predict the observed vibrational distribution as well as the observation that the translational energy is much greater than the rotational energy. The theory could not, however, predict the observed proportionality between the average rotational energy and the available energy. A simple classical description of the impulsive dissociation of a rotating molecule does predict this observed linear proportionality. 

In the analysis of the data the authors applied a model that consisted of a Franck-Condon transition to three different surfaces. Once the molecule was on one of these surfaces it was allowed to undergo a classical trajectory to a final product. 

Nevertheless, this simple propagation method provides an intriguing picture of the evolution of the quantum mechanical wavepacket, at least for short times. It readily demonstrates that for short times the center of the wavepacket follows essentially a classical trajectory ( Ehrenfest s theorem, Cohen-Tannoudji, Diu, and Laloe 1977 ch.III). Figure 4.2 depicts an example the evolution of the two-dimensional wavepacket follows very closely the classical trajectory that starts initially with zero momenta at the Franck-Condon point. 

Fig. 6.9. Left-hand side Vibrational excitation function N(ro) and weighting function W(ro) versus the initial oscillator coordinate ro for three values of the coupling parameter e. The equilibrium separation of the free BC molecule is f = 0.403 A and the equilibrium value within the parent molecule is re = 0.481 A. Right-hand side Final vibrational state distributions P(n) for fixed energy E the quantum mechanical and the classical distributions are normalized to the same height at the maxima. The classical distributions are obtained with the help of (6.32). The lowest part of the figure contains also the pure Franck-Condon (FC) distribution (

Figure 8.2 depicts a typical potential energy surface (PES) for a symmetric molecule ABA with intramolecular bond distances R and R2] the ABA bond angle is assumed to be 180° (collinear configuration). The PES is symmetric with respect to the line defined by Ri = R2 it has a saddle point at short distances and decreases monotonically from the saddle point out into the two identical product channels A + BA and AB + A (see also Figure 7.18). The shaded area indicates the Franck-Condon (FC) region accessed via photon absorption and the two arrows illustrate the main dissociation paths for the quantum mechanical wavepacket or, equivalently, a swarm of classical trajectories. Because no barrier obstructs dissociation, the majority of trajectories immediately evanesce in either one of the two product channels without ever returning to the vicinity of the FC point.

Figure 2.2. Scheme of nuclear potentials in the ground electronic state E°(Q) and the excited electronic state E (Q). In the excited state, the frequency changes (i20->S2r) and the equilibrium point is shifted. The classical relaxation energy to the new nuclear configuration in the excited state is the Franck-Condon energy Efc and characterizes the linear exciton-vibration coupling. 

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