Sloped conical intersection

Fig. 1. Conical intersection surface topologies (top), and Renner-Teller surface topologies (bottom). Top left is a generic circular cone, such as is obtained from a Jahn-Teller problem involving only the linear vibronic coupling. Top right is a sloped conical intersection obtained in a general vibronic coupling problem where all three linear vibronic coupling constants are different. Bottom left to right show type-1, -II, -III Renner-Teller surfaces. These are obtained when only second-order vibronic coupling is included.

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). 

Fig. 8. Schematic representation of the potential surfaces leading to photoisomerisation of (BQA)PtMe2I from mer to fac isomer via a sloped conical intersection at / -like geometries. Shown to the right are the branching space vectors the gradient difference (gd=x1), and the derivative coupling (dc=x2). The primary orbitals involved in the electronic transition are shown to the left [Adapted from Ref. (110) with permission].

It is on the fac (product) side that we find a non-adiabatic radiationless decay channel back to the ground state via a sloped conical intersection (Fig. 8) that connects the excited and ground state fac species. To investigate this region we have used CASSCF. The shear system size here presents difficulties... 

Figure 4 Opening of a fast radiationless decay channel via conical intersection for (a) a barrier controlled reaction, (b) a barrierless path, and (c) an uphill path without transition state (sloped conical intersection). M" is an excited state intermediate and FC is a Franck-Condon point.

As reported in the introduction, we have indicated that in some situations there is no transition state connecting an excited state intermediate (M ) to the conical intersection point (sloped conical intersections, see Figures 4c and 8b). In such situations, mechanistic information associated with surface crossings must be obtained by locating the lowest lying intersection point along the n — 2 intersection space of the molecule. 

Figure 2.8 Potential energy surface along the two degeneracylifting coordinates Xj and Xg for a model sloped conical intersection.

The position and local topology of a conical intersection is crucial in determining the photochemical reactivity of a molecule. A peaked cmiical intersection occurs when both PES have orthogonal gradients near the intersection (such as that in Fig. 2). Such topology leads to very efficient transitions from the excited state to the ground state. An example of such a conical intersection will be presented below for several systems. A sloped conical intersection occurs when... 

Fig. 7 Schematic of mer-fac photoizomerization in (BQA)PtMe2l involving localized jot excitation on the BQA ligand, followed by relaxation to Sj /ac minimum, followed by radiationless deactivation through a sloped conical intersection connecting the So and Si states, (reproduced from reference [76])...

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.]...

For the mechanistic studies made, this protocol is able to give information about how dynamical properties affect the evolution of a photochemical reaction, but is not accurate enough for quantitative results. The information obtained relates to aspects of the surface such as the relative steepness of regions on the lower slopes of the conical intersection, and the relative width of alternative channels. 

Figure 7. Two-dimensional cut of the ground- and excited-state adiabatic potential energy surfaces of Li + H2 in the vicinity of the conical intersection. The Li-EL distance is fixed at 2.8 bohr, and the ground and excited states correspond to Li(2,v) + H2 and Lit2/j ) + H2, where the p orbital in the latter is aligned parallel to the H2 molecular axis, y is the angle between the H-H intemuclear distance, r, and the Li-to-H2 center-of-mass distance. Note the sloped nature of the intersection as a function of the H-H distance, r, which occurs because the intersection is located on the repulsive wall. (Figure adapted from Ref. 140.)...

Figure 8. Li + H2 Ground-state population as a function of time for a representative initial basis function (solid line) and the average over 25 (different) initial basis functions sampled (using a quasi-classical Monte Carlo procedure) from the Lit2/j) + H2(v — 0, j — 0) initial state at an impact parameter of 2 bohr. Individual nonadiabatic events for each basis function are completed in less than a femtosecond (solid line) and due to the sloped nature of the conical intersection (see Fig. 7), there is considerable up-funneling (i.e., back-transfer) of population from the ground to the excited electronic state. (Figure adapted from Ref. 140.)...

The import of diabatic electronic states for dynamical treatments of conical intersecting BO potential energy surfaces is well acknowledged. This intersection is characterized by the non-existence of symmetry element determining its location in nuclear space [25]. This problem is absent in the GED approach. Because the symmetries of the cis and trans conformer are irreducible to each other, a regularization method without a correct reaction coordinate does not make sense. The slope at the (conic) intersection is well defined in the GED scheme. Observe, however, that for closed shell structures, the direct coupling of both states is zero. A configuration interaction is necessary to obtain an appropriate description in other words, correlation states such as diradical ones and the full excited BB state in the AA local minimum cannot be left out the scheme. 

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... 

For instance Cr(CO)6+ is formed only during LI. The time-dependent behavior of the ion yields of Cr(CO)6+ is presented in Fig. 13. Deconvolution of the time-dependent ion yield with the instrument function derived from the Xe+ signal provides a measure of the time constant (ij) of 12.5 0.05 fs for the LI level (Table 2). This represents the time it takes for the excited Cr(CO)6 to cross to the repulsive surface through the conical intersection close to the Franck-Condon state. At the Franck-Condon point with Oh symmetry, the only coordinates with nonzero slope are the totally symmetric alg M-C stretch or the Jahn-Teller-active vibrations which have eg or t2g symmetry [32], The time taken for a wavepacket to travel from any... 

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