Conical intersections three-electron systems

Conical intersections between electronically adiabatic potential energy surfaces are not only possible but actually quite frequent, if not prevalent, in polyatomic systems. Some examples are triatomic systems whose isolated atoms have 2S ground states [1,2] such as H3 and its isotopomers (DH2, HD2, HDT, etc.), LiH2 and its isotopomers, and tri-alkali systems such as Na3 and LiNaK. Many other kinds of polyatomic molecules also display such intersections. The reason is that they have three or more internal nuclear motion degrees of freedom, and only two independent relations between electronic Ham-... 

It is dear that an equilateral triangle of three 1 s electrons (where the conditions ofEqn (3.8) hold) must lie on a conical intersection. As we have indicated previously, there are many conical intersections in polyene systems that fit this triangular arrangement of electrons including (to cite a few examples), ergosterol, sigmatropic rearrangement in but-l-ene, butadiene... 

Although this reaction appears to involve only two electrons, it was shown by Mulder [57] that in fact two jc and two ct elections are required to account for this system. The three possible spin pairings become clear when it is realized that a pair of carbene radicals are formally involved. Figure 14. In practice, the conical intersection defined by the loop in Figme 14 is high-lying, so that often other conical intersections are more important in ethylene photochemistry. Flydrogen-atom shift products are observed [58]. This topic is further detailed in Section VI. 

We illustrate the method for the relatively complex photochemistry of 1,4-cyclohexadiene (CHDN), a molecule that has been extensively studied [60-64]. There are four it electrons in this system. They may be paired in three different ways, leading to the anchors shown in Figure 17. The loop is phase inverting (type i ), as every reaction is phase inverting), and therefore contains a conical intersection Since the products are highly strained, the energy of this conical intersection is expected to be high. Indeed, neither of the two expected products was observed experimentally so far. 

THE cvcLOBUTADENE-TETRAHEDRANE SYSTEM. A related reaction is the photoisomerization of cyclobutadiene (CBD). It was found that unsubstituted CBD does not react in an argon matrix upon irradiation, while the tri-butyl substituted derivative forms the corresponding tetrahedrane [86,87]. These results may be understood on the basis of a conical intersection enclosed by the loop shown in Figure 37. The analogy with the butadiene loop (Fig. 13) is obvious. The two CBDs and the biradical shown in the figure are the three anchors in this system. With small substituents, the two lobes containing the lone electrons can be far... 

The spin in quantum mechanics was introduced because experiments indicated that individual particles are not completely identified in terms of their three spatial coordinates [87]. Here we encounter, to some extent, a similar situation A system of items (i.e., distributions of electrons) in a given point in configuration space is usually described in terms of its set of eigenfunctions. This description is incomplete because the existence of conical intersections causes the electronic manifold to be multivalued. For example, in case of two (isolated) conical intersections we may encounter at a given point m configuration space four different sets of eigenfunctions (see Section Vni). 

While the simulations of the pyrazine system discussed in the previous sections of this chapter have employed a three-mode model (Model I), the semiclassical simulations we will present here are based on two different models a four-mode model and a model including all 24 normal modes of the pyrazine molecule. Let us first consider the four-mode model of the S1-S2 conical intersection in pyrazine which was developed by Domcke and coworkers [269]. In addition to the three modes considered in Model I, it takes into account another Condon-active mode (V9a). Figure 37 shows the modulus of the autocorrelation function [cf. Eq. (24)] of this model after photoexcitation to the S2 electronic state. The exact quantum results (full line) are compared to the... 

Abstract In this article we present a survey of the various conical intersections which govern potential transitions between the three lower electronic states for the title molecular system. It was revealed that these three states, for a given fixed HH distanee, Rhhj usually form four conical intersections two between the two lower states and two between the two upper states. One of the four is the well-known equilateral Dsh ci and the others are, essentially, C2V cis One of them is located on the symmetry line perpendicular to the HH axis (like the Dsh ci) and the other two are located on both sides of this symmetry line and in this way form the twin C2V cis. The study was carried out for two Rnn-values, namely, Rhh O.74 and 0.4777 A. 

To understand the structure of conical intersections in more detail, it is useful to consider the electronic states (of the same multiplicity) involved in the crossing. It is possible to study this problem mathematically, using concepts from group theory that have been developed for the study of Jahn-Teller systems. However, our aim is to make the concept useful to understanding photochemical mechanisms by presenting it in a more intuitive manner. We describe the crossing of two states that involve three electrons and three centers in a simplified version of the classic three-electron problem (Hg). ... 

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