Molecular orbital-conical intersection

At this stage, we wish to emphasize that a point (molecular geometry) on a conical intersection hyperline has a well-defined electronic structure (illustrated in Figure 9.6 or Eq. 9.2 with T = 0) and a well-defined geometry. Of course, the four electrons in four Is orbitals shown in Figure 9.6 is a very simple example, but we believe it is useful in order to be able to appreciate the generality of the conical intersection construct. In more complex systems, the conical intersection hyperline concept persists, but the rationalization may be less obvious. 

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.)...

Localized molecular orbital/generalized valence bond (LMO/GVB) method, direct molecular dynamics, ab initio multiple spawning (AIMS), 413-414 Longuet-Higgins phase-change rule conical intersections ... 

Figure 2c shows the electronic structure of graphene described by a simple tight-binding Hamiltonian the electronic wavefunctions from different atoms overlap. However, such an overlap between the Pz(it) orbital and the Px and Py orbitals is zero by symmetry. Thus, the Pz electrons form the 71 band, and they can be treated independently from the other valence electrons. The two sub-lattices lead to the formation of two bands, n and Jt, which intersect at the corners of the Brillouin zone. This yields the conical energy spectrum (Dirac cone, inset in Fig. 2c) near the points K and K, which are called Dirac points. The bottom cone (equivalent to the HOMO molecular orbital) is fully occupied, while the top cone (equivalent to the LUMO molecular orbital) is empty. The Fermi level Ep is chosen as the zero-energy reference and lies at the Dirac point. Consequently, graphene is a special semimetal or zero-band-gap semicondutor, whose intrinsic Fermi surface is reduced to the six points at the corners of the two-dimensional Brillouin zone. 

The central mechanistic feature in most photochemical mechanisms is the conical intersection. Thus we hope to present some thoughts about how to predict and rationalize the molecular and electronic structure of such mechanistic features using VB ideas. It turns out that one can derive analytical results for n orbitals with n electrons so we shall develop the main ideas with reference to the photochemistry of some simple model systems such as the cycloaddition of two ethylene molecules and the radiationless decay of benzene. Once one allows zwitterionic systems, lone pairs and heteroatoms, the same principles apply but analytical results are not available so easily and one must be content with a qualitative analysis at the moment. 

Figure 3.4 Conical intersection occurring in the ring opening of diarylethenes. The arrows correspond to the branching space vectors Xi X2 treated as if they were molecular vibrations. The red arrows correspond to the three carbon atoms associated with a conical intersection involving 3 orbitals with 3 electrons. (For interpretation of the references to color in this figure legend, the reader is referred to the online version of this book.)...

In this review, we have shown how some very simple ideas that originate in simple VB theory can be used to predict and rationalize the types of molecular structures encountered at conical intersections (and the branching space) that occur in organic photoreactivity. The simple coupling of 3 or 4 orbitals with 3 or 4 electrons can be used within an analytical formulation. 

---