Peaked conical intersections

Figure 2.6 Energy profiles at a peaked conical intersection along (a) gradient difference and (b) interstate coupling coordinates.

Figure 2.7 Potential energy surface along the two degeneracylifting coordinates Xj and X2 for a model peaked conical intersection.

Figure 2.10 Potential energy surface for the covalent-to-ionic state switch in tryptophan along a peaked conical intersection.

We apply our second-order Ehrenfest method to a model system benzene radical cation. Ionization of the neutral from the degenerate HOMO/HOMO-1 leads to the Jahn-Teller [15] effect in the cation. There is a peaked conical intersection between the two lowest-energy eigenstates Dq and Dy at geometries with D h symmetry. Figure 1 represents the surrounding moat of the conical intersection seen from above. It contains several valence bond (VB) resonance structures three equivalent quinoid structures... 

In Figure 1, we see that there are relative shifts of the peak of the rotational distribution toward the left from f = 12 to / = 8 in the presence of the geometiic phase. Thus, for the D + Ha (v = 1, DH (v, f) - - H reaction with the same total energy 1.8 eV, we find qualitatively the same effect as found quantum mechanically. Kuppermann and Wu [46] showed that the peak of the rotational state distribution moves toward the left in the presence of a geometric phase for the process D + H2 (v = 1, J = 1) DH (v = 1,/)- -H. It is important to note the effect of the position of the conical intersection (0o) on the rotational distribution for the D + H2 reaction. Although the absolute position of the peak (from / = 10 to / = 8) obtained from the quantum mechanical calculation is different from our results, it is worthwhile to see that the peak... 

The relative shift of the peak position of the rotational distiibution in the presence of a vector potential thus confirms the effect of the geometric phase for the D + H2 system displaying conical intersections. The most important aspect of our calculation is that we can also see this effect by using classical mechanics and, with respect to the quantum mechanical calculation, the computer time is almost negligible in our calculation. This observation is important for heavier systems, where the quantum calculations ai e even more troublesome and where the use of classical mechanics is also more justified. 

Conical intersections can be broadly classified in two topological types peaked and sloped [189]. These are sketched in Figure 6. The peaked case is the classical theoretical model from Jahn-Teller and other systems where the minima in the lower surface are either side of the intersection point. As indicated, the dynamics of a system through such an intersection would be expected to move fast from the upper to lower adiabatic surfaces, and not return. In contrast, the sloped form occurs when both states have minima that lie on the same side of the intersection. Here, after crossing from the upper to lower surfaces, recrossing is very likely before relaxation to the ground-state minimum can occur. 

Figure 6. Two-dimensional (top) and 3D (bottom) representations of a peaked (a) and sloped (b) conical intersection topology. There are two directions that lift the degeneracy the GD and the DC. The top figures have energy plotted against the DC while the bottom figures represent the energy plotted in the space of hoth the GD and DC vectors. At a peaked intersection, as shown at the bottom of (a), the probability of recrossing the conical intersection should be small whereas in the case of a sloped intersection [bottom of ( )l, this possibility should be high. [Reproduced from [84] courtesy of Elsevier Publishers.]...

Commonly, it is asserted that upward transitions from the lower adiabat to the upper one should be less likely than downward transitions because of the funneling property of the intersection [144,145]. This is clearly seen in the usual model conical intersection—as seen, for example, in Fig. 1 of Ref. 146, where there is (a) a well, or funnel, in the upper adiabat which guides the wavepacket to the intersection and (b) a peak on the lower adiabat which tends to guide the wavepacket away from the intersection. The potential energy surfaces shown in Fig. 7 differ from this canonical picture, and in particular it is not at all clear that the wavepacket on the lower adiabatic state will be funneled away from the intersection. For the conditions chosen in our calculations, we... 

In Fig. 1 (top right) we show a sloped conical intersection in the terminology of Ruedenberg et al (29). Here the cone is tilted due to the fact that the force (gradient) vectors on both the upper and lower surfaces point in the same direction. The first-order topology (sloped vs. peaked) controls the nature of the photochemical reaction dynamics, and whether reactants are regenerated or photoproducts are formed (23,24). 

Figure 8 Topological possibilities for conical intersections (characterized according to Ref. 46) (a) peaked, (b) sloped, and (c) intermediate conical intersections.

Ruedenberg s terminology peaked, sloped, and intermediate, as shown in Figure 8. Often the chemically relevant conical intersection point is located along a valley on the excited state potential energy surface (i.e., a peaked intersection). Figure 9 illustrates a two-dimensional model example that occurs in the photochemical trans —> cis isomerization of octatetraene.28 Here two potential energy surfaces are connected via a conical intersection. This intersection... 

In this configuration, two conical intersections involve the intermediate surface, one with the lower surface and the other one with the upper surface. This topology gives here more possibilities for transfer The combined choice of the pulse sequence and the ratio of the peak amplitudes allows the selective transfer into the two other states. 

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)... 

If we assume that the peak pump field amplitude is below the conical intersection, then the branches v / ), and /+), respectively, connect — 12) and 11 at the end. This leads to coherent superpositions between the states 11) and 2). 

Path (c) in Fig. 14 involves a crossing of one conical intersection of the two described above. The first resonance is crossed by the rising pulse 1 (with fl2 = 0). The second pulse is chosen with a smaller peak amplitude in order to avoid the passage through the resonance that would lead the system to the third level surface. This leads to an atomic population transfer, accompanied by absorption of one C02 photon, since the path ends at 2 0, — 1). This can be generalized for upper and lower paths The connectivity leads to 2 — 1 + k, —k), with k positive (pulse 1 before smaller pulse 2 amplitude) or negative or zero integer (pulse 2 before smaller pulse 1 amplitude). 

One important point about photochemical reactions is that state switches between excited states occur in the region of topographies such as the peaked intersection region sketched above (Figure 2.4b). To understand the role of conical intersections in state switches, it is useful to compare the two-dimensional picture of Figure 2.6 with the simple one-dimensional model that is normally used to describe this phenomenon, with the help of an avoided crossing. (One dimensional means here that only one nuclear coordinate, the reaction coordinate, is considered.)... 

In our calculated potential energy surfaces, we have shown that the avoided crossing, which is the transition state for ET in Marcus theory, is a region centered on a conical intersection. The topology is similar to the one shown for the model peaked intersection, and in general cases the reaction will take place on the lower surface of the double cone (Figure... 

The general description of the conical intersections for ET goes along the lines used to describe the model peaked intersection in Figure 2.7 and the seam of intersection in... 

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