Lifting of degeneracies

When the processes involve a zero-field resonance, one has to add the ingredient of lifting of degeneracy. This means that we have to consider the dynamics starting (or ending) near the conical intersection in a direction not parallel to the Cl = 0-plane. This can be seen in Fig. 6 where the surfaces of Fig. 2 have been redrawn for positive detunings (case of a one-photon resonance). When the dynamics starts this way, it is characterized by two adiabatic paths, one on each surface. They will lead in general to coherent superpositions of states. 

Figure 6. Surfaces of eigenener-gies (in units of A n ) as functions of fi/ Ajfl and A/ A for positive A. The coefficients 1/2 refer to the equal sharing of the lifting of degeneracy in the A = 0-plane.

All the other directions lead to a lifting of degeneracy with a sharing of population into the two surfaces. The one-photon process gives in the Q direction, or equivalently in the field-amplitude direction, the following ... 

These two cases are produced by two different sequences respectively (a) first the Stark pulse and next the pump pulse (referred to as Stark-pump sequence) and (b) first the pump pulse and next the Stark pulse (referred to as pump-Stark sequence). The phases associated to the superposition of states resulting from these two different sequences are not identical. For the Stark-pump sequence, we start (at time f,) with the lifting of degeneracy /(f,)) = v /+), which leads at the final time tf to (up to an irrelevant global phase)... 

For the pump-Stark sequence, we start with the lifting of degeneracy (r (r j) = cos( /2) /+) — sin((/2) v / ). Using the adiabatic transport for each branch, the state solution reads at the end... 

This can be analyzed with the help of Fig. 8, where we have taken AP = 0. The intuitive pump-Stokes sequence induces first a lifting of degeneracy with equal sharing between the dressed states /+) (the upper one, associated to the eigenenergy X+) and v / ) (the lower one, associated to the eigenenergy X ) initially connected to 1) and 2). If we assume that the peak pump field amplitude is beyond the conical intersection, then the branches / ) and v(/+), respectively, connect —12) and 3) at the end. When As < 0 (as in Fig. 8), this leads at the end of the process to the coherent superposition with a dynamical phase (up to an irrelevant global phase)... 

The various consequences of the spin-orbit coupling for the electronic structure are demonstrated in Figure 5.1 for the dispersion relation ,-( ) of Ni (Ebert era/. 1997b). Note that the spin-orbit coupling gives rise to a lifting of degeneracies (e.g. at A and... 

Bl.15.2). The lifting of degeneracy of the spin states at zero field is called zero-field splitting (ZFS) and it is common for systems with S>1. For triplet states of organic molecules (S= 1) the ZFS arises from the dipolar interaction of the two magnetic moments of the electron spins with each other. The interaction is described by an additional term... 

A complete or partial lifting of degeneracies between excited molecular states and the breaking of selection rules which hold for transitions in the free molecules, because now not just the symmetry of the molecules, but that of the whole crystal is the determining factor. For example, the 0,0 transition Si <- So at a wavenumber of 48 000 cm is forbidden in the benzene molecule for optical dipole radiation by reasons of symmetry in benzene crystals, however, it is allowed. 

Fig. 3.33 Lifting of degeneracy of porphyrin HOMOs on (M) metal chelation and (D) dication formation bi, b2, Cj, and C2 represent the MOs in Figure 3.30. On metal chelation (M), b, c, and C2 are raised in energy. On dication formation (D), h , Ci, and C2 are lowered in energy.

The original degeneracy is lifted by the presence of the metal surface and the resulting surface symmetry electron density of states is sketched in Fig. 33 c (the figure is the result of a Bethe lattice approximation calculation). It is essential to consider this lifting of degeneracy and the creation of asymmetric orbital combinations, since some authors, e.g., Banholzer et al. [100], have erroneously ignored this. As a result,... 

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