Surface crossings avoided crossing

It is clearly seen that the adiabatic surfaces exhibit avoided crossing at the angles for which the hopping terms vanish, i.e. cp = t[/2 for the mono-dimensional case and and the four points (, = Irr / 2, = nH) for the bidimensional case. 

The experiment is illustrated in figure B2.5.9. The initial pump pulse generates a localized wavepacket in the first excited state of Nal, which evolves with time. The potential well in the state is the result of an avoided crossing with the ground state. Every time the wavepacket passes this region, part of it crosses to the lower surface before the remainder is reflected at the outer wall of the potential. The crossing leads to... 

Figure 3.1 Avoided crossing of potential energy surfaces for LiF...

In order to calculate q (Q) all possible quantum states are needed. It is usually assumed that the energy of a molecule can be approximated as a sum of terms involving translational, rotational, vibrational and electronical states. Except for a few cases this is a good approximation. For linear, floppy (soft bending potential), molecules the separation of the rotational and vibrational modes may be problematic. If two energy surfaces come close together (avoided crossing), the separability of the electronic and vibrational modes may be a poor approximation (breakdown of the Bom-Oppenheimer approximation. Section 3.1). 

Fig. 28. Schematic of potential energy surfaces of the vinoxy radical system. All energies are in eV, include zero-point energy, and are relative to CH2CHO (X2A//). Calculated energies are compared with experimentally-determined values in parentheses. Transition states 1—5 are labelled, along with the rate constant definitions from RRKM calculations. The solid potential curves to the left of vinoxy retain Cs symmetry. The avoided crossing (dotted lines) which forms TS5 arises when Cs symmetry is broken by out-of-plane motion. (From Osborn et al.67)...

Fig. 2. Avoided crossing of the So and Sj hypersurfaces. Vertical arrows indicate initial excitation additional arrows show how the surfaces govern nuclear motion afterwards...

It is now clear why the abrupt changes in the physical nature of a state, encountered in our convention whenever there is a weakly avoided crossing, are inconvenient they will induce jumps from one surface to... 

An (avoided) crossing of So and Si need not even always correspond to a minimum in the excited state surface 64a>. Fig. 1 shows that this will depend on the slopes of the surfaces. Even such crossings which do not actually lead to a minimum in the excited state surface will be called funnels since they can provide efficient return to So- Further, if S2 lies very close to Si, vibronic coupling may make it impossible to describe the nuclear motion as specifically due to only one or the other hypersurface and both will have to be considered. Then, whether Si or S2 undergoes an avoided crossing with So may be immaterial — a funnel is still present and return of the molecule to So probable. 

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