Conical intersections four-electron systems

The H4 system is the prototype for many four-elecbon reactions [34]. The basic tetrahedral sfructure of the conical intersection is preserved in all four-electron systems. It arises from the fact that the four electrons are contributed by four different atoms. Obviously, the tefrahedron is in general not a perfect one. This result was found computationally for many systems (see, e.g., [37]). Robb and co-workers [38] showed that the structure shown (a tetraradicaloid conical intersection) was found for many different photochemical transformations. Having the form of a tetrahedron, the conical intersection can exist in two enantiomeric structures. However, this feature is important only when chiral reactions are discussed. 

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 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). 

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. 

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. 

Computing a reliable hypersurface of the potential energy (PES) for the motion of nuclei, represents an extremely difficult task for today s computers, even for systems of four atoms. In principle routine calculations are currently performed for three-atomic (and, of course, two-atomic) systems. The technical possibilities are discussed by J. Hinze, A. Alijah and L. Wolniewicz, Pol. J. Chem. 72 (1998) 1293, in which the most accurate calculations are also reported (for the hJ system). Analysis of conical intersections is only occasionally carried out, because the problem pertains to mostly unexplored electronic excited states. 

Prom Table 1, it is seen that there is more than one type of conical intersection. For molecules with even (odd) / = 2 (77 = 3 or 5). The origin of the differences is, as explained by Mead, time reversal symmetry. For odd (even) and are (are not) linearly independent. Here T is the time reversal operator. See appendix A. Thus for odd (even) electron systems, the intersection of two potential energy surfaces requires the degeneracy of four (two) electronic states. As discussed below, this leads to the values of 77 reported in Table 1. 

The examples collected for this survey of femtosecond nonadiabatic dynamics at conical intersections illustrate the interesting interplay of coherent vibrational motion, vibrational energy relaxation and electronic transitions within a fully microscopic quantum mechanical description. It is remarkable that irreversible population and phase relaxation processes are so clearly developed in systems with just three or four nuclear degrees of freedom. 

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