Hyperline

The same classification into winding numbers can be used in a system with N nuclear degrees of freedom, in which the Cl seam is an N — 2)-dimensional hyperline as in Fig. 1. For example, if we take N = 3, then the seam is a line the... 

Fig. 6.2 Theoretical Fe Mossbauer relaxation spectra for longitudinal relaxation with the indicated relaxation times and with a hyperline field that can assume the values 55 T. The symmetry direction of the axially symmetric EFG is assumed parallel to the magnetic hyperfine field. (Reprinted with permission from [9] copyright 1966 by the American Physical Society)...

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. 

Let us summarize briefly at this stage. We have seen that the point of degeneracy forms an extended hyperline which we have illnstrated in detail for a four electrons in four Is orbitals model. The geometries that lie on the hyperline are predictable for the 4 orbital 4 electron case using the VB bond energy (Eq. 9.1) and the London formula (Eq. 9.2). This concept can be nsed to provide nseful qualitative information in other problems. Thns we were able to rationalize the conical intersection geometry for a [2+2] photochemical cycloaddition and the di-Jt-methane rearrangement. 

Figure 9.9. A cartoon showing the conical intersection hyperline traced out by a degeneracypreserving coordinate X3. The system remains degenerate as one traverses the coordinate X3, but the energy and the shape of the double-cone must change in Xi X2. See color insert.

It is useful to consider the information contained in Figure 9.9 in a different way. In Figure 9.10, we show the conical intersection hyperline traced out by a coordinate... 

Figure 9.10. The conical intersection hyperline traced out by a coordinate X3 plotted in a space containing the coordinate X3 and one coordinate from the degeneracy-lifting space X1X2. See color insert.

The important points on a conical intersection hyperline are those where the reaction path meets with the seam (see Fig. 9.10)... 

In Figure 9.10, this second principle appears to be violated since the reaction path appears to pass through the hyperline adiabatically. However, we emphasize—as indicated by the double-cone insert—that as one passes through the hyperline, decay takes place in the coordinates X X2 and in general their VB structure does not change. This idea is obviously easier to appreciate in Figure 9.9. We shall use both Figures 9.9 and 9.10 as models in subsequent discussions but the reader needs to remember the conceptual limitations. 

We now proceed to look at three examples from recent work in some depth. In the first example, we wish to illustrate that a knowledge of the VB structure or of the states involved in photophysics and photochemistry rationalize the potential surface topology in an intuitively appealing way. We then proceed to look at an example where the extended hyperline concept has interesting mechanistic implications. Finally, we shall look at an example of how conical intersections can control electron transfer problems. 

Figure 9.22. The geometries in Figure 9.21 located in the cone which changes shape along the conical intersection hyperline (adapted from reference 14). See color insert.

We hope that the preceding discussions have developed the concept of a conical intersection as being as real as many other reactive intermediates. The major difference compared with other types of reactive intermediate is that a conical intersection is really a family of structures, rather than an individual structure. However, the molecular structures corresponding to conical intersections are completely amenable to computation, even if their existence can only be inferred from experimental information. They have a well-defined geometry. Like the transition state, the crucial directions governing dynamics can be determined andX2) even if there are now two such directions rather than one. As for a transition structure, the nature of optimized geometries on the conical intersection hyperline can be determined from second derivative analysis. 

In general, the g- and nuclear hyperline coupling matrices, g and A can be written in diagonal form with three principal values, i.e., gx, gy, g. and A,x, Aiy, Aiz. In textbooks on ESR6a 30,33 35 it is usually assumed that the same set of principal axes diagonalizes all the relevant matrices. While this is sometimes true, there are many instances where the principal axes are non-coincident.36... 

The spinodal represents a hypersurface within the space of external parameters where the homogeneous state of an equilibrium system becomes thermodynamically absolutely unstable. The loss of this stability can occur with respect to the density fluctuations with wave vector either equal to zero or distinct from it. These two possibilities correspond, respectively, to trivial and nontrivial branches of a spinodal. The Lifshitz points are located on the hyperline common for both branches. 

An electron-transfer protein containing a type-1 copper site that manifests itself by a strong visible absorption spectum. Another characteristic attributable to copper coordination by a cysteinyl sulfur is the EPR signal that displays an unusually small hyperline coupling to the copper nucleus. 

The type-1 blue copper proteins act as electron carriers azurin, plastocyanin, stellacyanin, umecyanin e.g. They are characterized by a rather strong LMCT (ligand to metal charge transfer) band near 600 nm and by small hyperline coupling constants A in EPR. Copper is bound to two imidazole groups of histidine and to two... 

Separation of pseudocontact and contact (including any ligand centered pseudocontact) contributions to the hyperline shifts of heme and axial ligand protons for oxidized rat microsomal cytochrome at 313 K (heme numbering as in Fig. 5.7B) [127]... 

Santos JG, Silveira LB, Oliveira AC, Garg VK, Lacava BM, Tedesco AC, Morais PC (2007) Hyperline Interact 175 71... 

Different topological situations are possible for unavoided crossings between surfaces. One can have intersections between states of different spin multiplicity [an (n - l)-dimensional intersection space in this case, since the interstate coupling vector vanishes by symmetry], or between two singlet surfaces or two triplets [and one has an n - 2)-dimensional conical intersection hyperline in this case]. We have encountered situations in which both types of... 

In many cases, such structural or static information is not sufficient. The excited state may not decay at the point where the excited state path (MEP) intersects the n - 2 hyperline. Alternatively, the momentum developed on the excited state branch of the reaction coordinate may be sufficient to drive the ground state reactive trajectory along paths that are far from the ground state valleys. In such cases, a dynamics treatment of the excited state/ground state motion is required for mechanistic investigations. Furthermore a dynamics treatment is required to gain information of the time scales and quantum yields of the reaction. 

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