Two-electron potential energy

For the system under consideration, with only two electronic potential-energy surfaces, the conservation of population implies that... 

D14.2 The Franck-Condon principle states that because electrons are so much lighter than nuclei, an electronic transition occurs so rapidly compared to vibrational motions that the internuclear distance is relatively unchanged as a resu It of the transition. This implies that the most probable transitions vf <— vj are vertical. This vertical line will, however, intersect any number of vibrational levels Vf in the upper electronic state. Hence transitions to many vibrational states of the excited state will occur with transition probabilities proportional to the Frank-Condon factors which are in turn proportional to the overlap integral of the wavefunctions of the initial and final vibrational states. A vibrational progression is observed, the shape of which is determined by the relative horizontal positions of the two electronic potential energy curves. The most probable transitions are those to excited vibrational states with wavefunctions having a large amplitude at the internuclear position Re. 

Finally, in brief, we demonstrate the influence of the upper adiabatic electronic state(s) on the ground state due to the presence of a Cl between two or more than two adiabatic potential energy surfaces. Considering the HLH phase, we present the extended BO equations for a quasi-JT model and for an A -1- B2 type reactive system, that is, the geometric phase (GP) effect has been inhoduced either by including a vector potential in the system Hamiltonian or... 

Now, we examine the effect of vibronic interactions on the two adiabatic potential energy surfaces of nonlinear molecules that belong to a degenerate electronic state, so-called static Jahn-Teller effect. 

Describing complex wave-packet motion on the two coupled potential energy surfaces, this quantity is also of interest since it can be monitored in femtosecond pump-probe experiments [163]. In fact, it has been shown in Ref. 126 employing again the quasi-classical approximation (104) that the time-and frequency-resolved stimulated emission spectrum is nicely reproduced by the PO calculation. Hence vibronic POs may provide a clear and physically appealing interpretation of femtosecond experiments reflecting coherent electron transfer. We note that POs have also been used in semiclassical trace formulas to calculate spectral response functions [3]. 

The two physical interpretations of LSD and GGA are about equally plausible in a normal system. Buf especially in abnormal systems, these approximations may be more faithful to the alternative theory, because of the close relationship between Px=i r,r) and the electron-electron potential energy of Eq. (5). Thus, accurate total energies are expected to accompany accurate... 

This reaction profile also illustrates one of the other important challenges in the study of transition metal systems, namely that the metal-containing active site often has several accessible spin states. Specifically in the case of Fe(IV)=0, the triplet, quintet, and septet spin states. Consequently, the reaction can, in principle, proceed on different electronic potential energy surfaces and it is necessary to test all possibilities when exploring a reaction surface. This has been labeled two-state reactivity and has been elaborated by Shaik, Schwarz, Schroder, and co-workers (36—40). In the case of TauD, the results show that the reaction is only feasible on the quintet surface, in agreement with earlier DFT studies (11,41 —45). 

In Ref. [4] we have studied an intense chirped pulse excitation of a molecule coupled with a dissipative environment taking into account electronic coherence effects. We considered a two state electronic system with relaxation treated as diffusion on electronic potential energy surfaces with respect to the generalized coordinate a. We solved numerically equations for the density matrix of a molecular system under the action of chirped pulses of carrier frequency a> with temporal variation of phase [Pg.131]

The theory stems from the writer s work on simple electron transfer reactions of conventional reactants (5). A simple electron transfer reaction is defined as one in which no bonds are broken or formed during the redox step such a reaction might be preceded or followed by bondbreaking or bond-forming steps in a several-step reaction mechanism. Other chemical reactions involve rupture or formation of one or several chemical bonds, and only a few coordinates suffice to establish their essential features. In simple electron transfers in solution, on the other hand, numerous coordinates play a role. One cannot then use the usual two-coordinate potential energy contour diagram (4) to visualize the... 

The photodissociation of methyl nitrite in the first absorption band, CH30N0(Si) — CH3O + NO(n, j), exemplifies indirect photodissociation (Hennig et al. 1987). Figure 1.11 shows the two-dimensional potential energy surface (PES) of the S electronic state as a function of the two O-N bonds. All other coordinates are frozen at the equilibrium values in the electronic ground state. Although these two modes suffice to illustrate the overall dissociation dynamics, a more realistic picture... 

Fig. 2. The model of an electron transfer reaction as the intersection of two parabolic potential energy wells. The abcissa x represents a reaction coordinate such a diagram is only one two-dimensional section through a multidimensional space...

In this sense, the control of electronic transitions of wavepackets using short quadratically chirped laser pulses of moderate intensity is a very promising method, for two reasons. First, only information about the local properties of the potential energy surface and the dipole moment is required to calculate the laser pulse parameters. Second, this method has been demonstrated to be quite stable against variations in pulse parameters and wavepacket broadening. However, controlling of some types of excitation processes, such as bond-selective photodissociation and chemical reaction, requires the control of wavepacket motion on adiabatic potential surfaces before and/or after the localized wavepacket is made to jump between the two adiabatic potential energy surfaces. 

It is of considerable importance to note that the density-potential relationship (3) of the TF theory follows from a variational principle for the total energy. To see this, we note first that the classical electrostatic potential energy U consists of the sum of two terms in an atomic ion, the electron-nuclear potential energy Ken and the electron-electron potential energy Kee. We can write... 

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